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MAASFundamentals Of Mathematics I
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Target : JEE (Main + Advanced) Fundamentals of Mathematics-I
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CONTENT
FUNDAMENTALS OF MATHEMATICS - I
Topic Page No.
Theory 01 – 12
Exercise # 1 13 – 20
Part - I : Subjective Question
Part - II : Only one option correct type
Part – III : Match the column
Exercise - 2 21 – 26
Part - I : Only one option correct type
Part - II : Numerical value questions
Part - III : One or More than one options correct type
Part - IV : Comprehension
Exercise - 3 26 – 27
Part - I : JEE(Advanced) / IIT-JEE Problems (Previous Years)
Part - II : JEE(Main) / AIEEE Problems (Previous Years)
Answer Key : 28 – 30
Logarithm Table : 31 – 32
Antilogarithm Table : 33 – 34
JEE (ADVANCED) SYLLABUS
Rational inequality, properties of Log, Exponential and Log equations and inequations.
JEE (MAIN) SYLLABUS
Sets : Sets and their representation; Union, intersection and complement of sets and their algebraic properties;
Power set; Rational inequality, properties of Log, Exponential and Log equations and inequations.
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SETS
A set is a collection of well defined objects which are distinct from each other. Sets are generally
denoted by capital letters A, B, C, ........ etc. and the elements of the set by small letters a, b, c
....... etc.
If a is an element of a set A, then we write a A and say a belongs to A.
If a does not belong to A then we write a A,
e.g. the collection of first five prime natural numbers is a set containing the elements 2, 3, 5, 7, 11.
METHODS TO WRITE A SET :
(i) Roster Method or Tabular Method : In this method a set is described by listing elements,
separated by commas and enclose then by curly brackets. Note that while writing the set in
roster form, an element is not generally repeated e.g. the set of letters of word SCHOOL may
be written as {S, C, H, O, L}.
(ii) Set builder form (Property Method) : In this we write down a property or rule which gives us
all the element of the set.
A = {x : P(x)} where P(x) is the property by which x A and colon ( : ) stands for ‘such that’
Example # 1 : Express set A = {x : x N and x = 2n for n N} in roster form
Solution : A = {2, 4, 8, 16, .........}
Example # 2 : Express set B = {x3 : x < 5, x W} in roster form
Solution : B = {0, 1, 8, 27, 64}
Example # 3 : Express set A = {0, 7, 26, 63, 124} in set builder form
Solution : A = {x : x = n3 – 1, nN, 1 n 5}
TYPES OF SETS
Null set or empty set : A set having no element in it is called an empty set or a null set or void set, it is
denoted by or { }. A set consisting of at least one element is called a non-empty set or a non-void set.
Singleton set : A set consisting of a single element is called a singleton set.
Finite set : A set which has only finite number of elements is called a finite set.
Order of a finite set : The number of distinct elements in a finite set A is called the order of this set
and denoted by O(A) or n(A). It is also called cardinal number of the set.
e.g. A = {a, b, c, d} n(A) = 4
Infinite set : A set which has an infinite number of elements is called an infinite set.
Equal sets : Two sets A and B are said to be equal if every element of A is member of B, and every
element of B is a member of A. If sets A and B are equal, we write A = B and if A and B are not equal
then
A B
Equivalent sets : Two finite sets A and B are equivalent if their cardinal number is same
i.e. n(A) = n(B)
e.g. A = {1, 3, 5, 7}, B = {a, b, c, d} n(A) = 4 and n(B) = 4
A and B are equivalent sets
Note - Equal sets are always equivalent but equivalent sets may not be equal
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Example # 4 : Identify the type of set :
(i) A = {x W : 3 x < 10} (ii) A = {}
(iii) A = {1, 0, –1, –2, –3,.......} (iv) A = {1, 8, –2, 6, 5} and B = {1, 8, –2, 6, 5}
(v) A = {x : x is number of students in a class room}
Solution : (i) finite set (ii) finite set
(iii) infinite set (iv) equal sets
(v) singleton set
Self Practice Problem :
(1) Write the set of all integers 'x' such that –2 < x – 4 < 5.
(2) Write the set {1, 2, 5, 10} in set builder form.
(3) If A = {x : x2 < 9, x Z} and B = {–2, –1, 1, 2} then find whether sets A and B are equal or not.
Answers (1) {3, 4, 5, 6, 7, 8}
(2) {x : x is a natural number and a divisor of 10}
(3) Not equal sets
SUBSET AND SUPERSET :
Let A and B be two sets. If every element of A is an element of B then A is called a subset of B and B is
called superset of A. We write it as A B.
e.g. A = {1, 2, 3, 4} and B = {1, 2, 3, 4, 5, 6, 7} A B
If A is not a subset of B then we write A B
PROPER SUBSET :
If A is a subset of B but A B then A is a proper subset of B. Set A is not proper subset of A so this is
improper subset of A
Note : (i) Every set is a subset of itself
(ii) Empty set is a subset of every set
(iii) A B and B A A = B
(iv) The total number of subsets of a finite set containing n elements is 2n.
(v) Number of proper subsets of a set having n elements is 2n – 1.
(vi) Empty set is proper subset of every set except itself.
POWER SET :
Let A be any set. The set of all subsets of A is called power set of A and is denoted by P(A)
Example # 5 : Examine whether the following statements are true or false :
(i) {a} {b, c, a}
(ii) {x, p} {x : x is a consonant in the English alphabet}
(iii) {} {}
(iv) {a, b} {a, {a}, b, c}
Solution : (i) False as {a} is subset of {b, c, a}
(ii) False as x, p are consonant
(iii) False as element , is not in the set {}
(iv) False as a, b {a, {a}, b, c} and {a, b} {a, {a}, b, c}
Example # 6 : Find power set of set A = {1, 2, 3}
Solution : P(A) = {, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
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Example # 7 : If denotes null set then find
(a) P() (b) P(P())
(c) n(P(P(P()))) (d) n(P(P(P(P()))))
Solution : (a) P() = {} (b) P(P()) = {,{}}
(c) n(P(P(P()))) = 22 = 4 (d) n(P(P(P(P())))) = 24 = 16
Self Practice Problem :
(4) State true/false : A = {p, q, r, s}, B = {p, q, r, p, t} then A B.
(5) State true/false : A = {p, q, r, s}, B = {s, r, q, p} then A B.
(6) State true/false : [4, 15) [–15, 15]
Answers (4) False (5) True (6) True
UNIVERSAL SET :
A set consisting of all possible elements which occur in the discussion is called a universal set and is
denoted by U.
e.g. if A = {1, 2, 3}, B = {2, 4, 5, 6}, C = {1, 3, 5, 7} then U = {1, 2, 3, 4, 5, 6, 7} can be taken as the
universal set.
SOME OPERATION ON SETS :
(i) Union of two sets : A B = {x : x A or x B}
e.g. A = {1, 2, 3}, B = {2, 3, 4} then A B = {1, 2, 3, 4}
(ii) Intersection of two sets : A B = {x : x A and x B}
e.g. A = {1, 2, 3}, B = {2, 3, 4} then A B = {2, 3}
(iii) Difference of two sets : A – B = {x : x A and x B}. It is also written as A B'.
Similarly B – A = B A' e.g. A = {1, 2, 3}, B = {2, 3, 4} ; A – B = {1}
(iv) Symmetric difference of sets : It is denoted by A B and A B = (A – B) (B – A)
(v) Complement of a set : A' = {x : x A but x U} = U – A
e.g. U = {1, 2,........, 10}, A = {1, 2, 3, 4, 5} then A' = {6, 7, 8, 9, 10}
(vi) Disjoint sets : If A B = , then A, B are disjoint sets.
e.g. If A = {1, 2, 3}, B = {7, 8, 9} then A B =
VENN DIAGRAM :
Most of the relationships between sets can be represented by means of diagrams which are known as
venn diagrams. These diagrams consist of a rectangle for universal set and circles in the rectangle for
subsets of universal set. The elements of the sets are written in respective circles.
For example If A = {1, 2, 3}, B = {3, 4, 5}, U = {1, 2, 3, 4, 5, 6, 7, 8} then their venn diagram is
A B A B A – B B – A
A' (A B) = (A – B) (B – A) Disjoint
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LAWS OF ALGEBRA OF SETS (PROPERTIES OF SETS):
(i) Commutative law : (A B) = B A ; A B = B A
(ii) Associative law : (A B) C = A (B C) ; (A B) C = A (B C)
(iii) Distributive law : A (B C) = (A B) (A C) ; A (B C) = (A B) (A C)
(iv) De-morgan law : (A B)' = A' B' ; (A B)' = A' B'
(v) Identity law : A U = A ; A = A
(vi) Complement law : A A' = U, A A' = , (A')' = A
(vii) Idempotent law : A A = A, A A = A
NOTE :
(i) A – (B C) = (A – B) (A – C) ; A – (B C) = (A – B) (A – C)
(ii) A = , A U = U
Example # 8 : Let A = {1, 2, 3, 4, 5, 6} and B = {4, 5, 6, 7, 8, 9} then find A B
Solution : A B = {1, 2, 3, 4, 5, 6, 7, 8, 9}
Example # 9 : Let A = {1, 2, 3, 4, 5, 6}, B = {4, 5, 6, 7, 8, 9}. Find A – B and B – A.
Solution : A – B = {x : x A and x B} = {1, 2, 3}
similarly B – A = {7, 8, 9}
Example # 10 : State true or false :
(i) A A = A (ii) U A = A
Solution : (i) false because A A' = U (ii) true as U A = A
Example # 11 : Use Venn diagram to prove that A – B = A B.
Solution :
From venn diagram we can conclude that A – B = A B.
Self Practice Problem :
(7) Find A B if A = {x : x = 2n + 1, n 5, n N} and B = {x : x = 3n – 2, n 4, n N}.
(8) Find A – (A – B) if A = {5, 9, 13, 17, 21} and B = {3, 6, 9, 12, 15, 18, 21, 24}
Answers (7) {1, 3, 4, 5, 7, 9, 10, 11} (8) {9, 21}
SOME IMPORTANT RESULTS ON NUMBER OF ELEMENTS IN SETS :
If A, B, C are finite sets and U be the finite universal set then
(i) n(A B) = n(A) + n(B) – n(A B)
(ii) n(A – B) = n(A) – n(A B)
(iii) n(A B C) = n(A) + n(B) + n(C) – n(A B) – n(B C) – n(A C) + n(A B C)
(iv) Number of elements in exactly two of the sets A, B, C
= n(A B) + n(B C) + n(C A) – 3n(A B C)
(v) Number of elements in exactly one of the sets A, B, C
= n(A) + n(B) + n(C) – 2n(A B) – 2n(B C) – 2n(A C) + 3n(A B C)
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Example # 12 : In a group of 60 students, 36 read English newspaper, 22 read Hindi newspaper and 12 read
neither of the two. How many read both English & Hindi news papers ?
Solution : n(U) = 60, n(E) = 36, n(H) = 22
n(E H) = 12 n(E H) = 12
n(U) – n(E H) = 12
n(E H) = 48
n(E) + n(H) – n(E H) = 48
n(E H) = 58 – 48 = 10
Example#13 : In a group of 50 persons, 14 drink tea but not coffee and 30 drink tea. Find
(i) How many drink tea and coffee both ? (ii) How many drink coffee but not tea ?
Solution : T : people drinking tea
C : people drinking coffee
(i) n(T) = n(T – C) + n(T C) 30 = 14 + n(T C) n(T C) = 16
(ii) n(C – T) = n(T C) – n(T) = 50 – 30 = 20
Self Practice Problem :
(9) Let A and B be two finite sets such that n(A – B) = 15, n(A B) = 90, n(A B) = 30. Find n(B)
(10) A market research group conducted a survey of 1000 consumers and reported that 720
consumers liked product A and 450 consumers liked product B. What is the least number that
must have liked both products ?
Answers (9) 75 (10) 170
Intervals :
Intervals are basically subsets of R and are commonly used in solving inequalities or in finding
domains. If there are two numbers a, b R such that a < b, we can define four types of intervals as
follows :
Name Representation Discription
Open Interval (a, b) {x : a < x < b} i.e. end points are not included.
{x : a x b} i.e. end points are also included. This is possible only when
Close Interval [a, b] both a and b are finite.
Open - Closed Interval (a, b] {x : a < x b} i.e. a is excluded and b is included.
Close - Open Interval [a, b) {x : ax < b} i.e. a is included and b is excluded.
Note : (1) The infinite intervals are defined as follows :
(i) (a, ) = {x : x > a} (ii) [a, ) = {x : x a}
(iii) (– , b) = {x : x < b} (iv) (–, b] = {x : x b}
(v) (– ) = {x : x R}
(2) x {1, 2} denotes some particular values of x, i.e. x = 1, 2
(3) If there is no value of x, then we say x (null set)
General Method to solve Inequalities :
(Method of intervals (Wavy curve method)
1 2 n (x – b1 )k (x – b 2 )k (x – bn )k Let g(x) = ... (i) r1 r2 rn (x – a1 ) (x – a 2 ) (x – an )
Where k1, k2 .....kn and r1, r2 ........rn N and b1, b2.....bn and a1, a2 ......an are real numbers.
Then to solve the inequality following steps are taken.
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Steps : -
Points where numerator becomes zero are called zeros or roots of the function and where denominator
becomes zero are called poles of the function.
(i) First we find the zeros and poles of the function.
(ii) Then we mark all the zeros and poles on the real line and put a vertical bar there
dividing the real line in many intervals.
(iii) Determine sign of the function in any of the interval and then alternates the sign
in the neghbouring interval if the poles or zeros dividing the two interval has
appeared odd number of times otherwise retain the sign.
(iv) Thus we consider all the intervals. The solution of the g(x) > 0 is the union of the
intervals in which we have put the plus sign and the solution of g(x) < 0 is the
union of all intervals in whichwe have put the minus sign.
5
10 3 1 2
(x – 2) (x 1) x – (x 8)
2 Example# 14 : Solve the inequality if f(x) = is > 0 or < 0.
x 24 (x – 3)3 (x 2)5
5
10 3 1 2 x – (x 8) (x – 2) (x 1) 1 2 Solution. Let f(x) = the poles and zeros are 0, 3,– 2,– 1, ,– 8, 2
x 24 (x – 3)3 (x 2)5 2
1
If f(x) > 0, then x (– , – 8) (– 8, – 2) (– 1, 0) 0, (3, )
2
(2, 3) Ans. and if f(x) < 0, then x (– 2, – 1) 1, 2
2
Exponential Function
A function f(x) = ax = ex In a (a > 0, a 1, x R) is called an exponential function. Graph of
exponential function can be as follows :
Case - Case -
For a > 1 For 0 < a 1
Logarithm of A Number :
The logarithm of the number N to the base ' a ' is the exponent indicating the power to which the base ' a
' must be raised to obtain the number N. This number is designated as loga N. Hence:
logaN = x ax = N , a > 0, a 1 & N > 0
If a = 10, then we write log b rather than log10 b.
If a = e, we write ln b rather than loge b. Here ' e ' is called as Napier's base & has numerical value equal
to 2.7182.
Remember
log102 ~ 0.3010 ; log103 ~ 0.4771
n 2 ~ 0.693 ; n 10 ~ 2.303
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Domain of Definition :
The existence and uniqueness of the number loga N can be determined with the help of set of
conditions, a > 0 & a 1 & N > 0.
The base of the logarithm ' a ' must not equal unity otherwise numbers not equal to unity will not have a
logarithm and any number will be the logarithm of unity.
Graph of Logarithmic function :
f(x) = logax is called logarithmic function where a > 0 and a 1 and x > 0. Its graph can be as follows:
Case- I Case- II
For a > 1 For 0 < a < 1
Fundamental Logarithmic Identity :
a N = N, a > 0, a 1 & N > 0 alog
The Principal Properties of Logarithm:
Let M & N are arbitrary positive numbers, a > 0, a 1, b > 0, b 1 and are any real numbers,
then :
(i) loga (M.N) = loga M + loga N ; in general loga (x1 x2 ......xn)
= logax1 + loga x2 + ........+ loga xn
(ii) loga (M/N) = loga M loga N
(iii) loga M = loga M
1
(iv) log M = logaM a
loga M
(v) logb M = (base changing theorem)
loga b
NOTE :
loga1 = 0 loga a = 1
1
log1/a a = 1 logba =
loga b
c a ax = e x n a alog c b blog
Note : (i) If the number and the base are on the same side of the unity, then the
logarithm is positive.
(ii) If the number and the base are on the opposite sides of unity, then the logarithm is
negative.
Example#15: Find the value of the followings :
32 9
(i) log272 + log2 + log2 Ans. 2
81 64
1
25 49 Ans. 5 (ii) 7 log
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Solution. (i) log272 + ..........
3 2 25 3 2
= log2 2 .3 . 4 . 6 = log24 = 2
3 2
1 2
log7 5 log7 7 log25 49 log49 25 2 (ii) 7 = 7 = 7 = 5 = 5
Self practice problem :
(11) Find the value of the followings :
(i) log49343 (ii) 4log27243
(iii) log(1/100)1000 (iv) log(7 4 3) (7 4 3)
(v) log125625
(12) log89.log910 ......... log6364
(13) Find the value of log cot1° + log cot2° + log cot3° + .... + log cot89°
Ans. (11) (i) 3/2 (ii) 20/3 (iii) – 3/2 (iv) –1 (v) 4/3
(12) 2 (13) 0
Logarithmic Equation :
The equality loga x = loga y is possible if and only if x = y i.e.
loga x = loga y x = y
Always check validity of given equation, (x > 0, y > 0, a > 0, a 1)
Example#16 : logx(4x – 3) = 2 Ans. x = 3
Solution. Domain : x > 0, 4x – 3 > 0, x 1
Hence 4x – 3 = x2 x2 – 4x + 3 = 0
x = 3 or x = 1 (rejected as not in domain)
Exmaple#17 : log2 (log3{log5 (x2 + 4)} = 0 Ans. x = ±11
Solution. log3(log5(x2 + 4)} = 2° = 1
log5(x2 + 4) = 31 = 3
(x2 + 4) = 53 = 125 x2 = 121 x = ±11
Example#18 : log2(x2) + log2 (x + 2) = 4 Ans. x = 2
Solution. log2 (x2(x + 2) = 4 x3 + 2x2 – 16 = 0 (x – 2) (x 2 4x 8) 0
D 0
x = 2
Self practice problem
(14) 33log 3 x 27 (15) (log10x)2 – (log10x) – 6 = 0
2 (x 2) 8(x 2)2 (16) 3(log7x + logx7) = 10 (17) (x 2)log
1 3
Ans. (14) x = 3 (15) x = 103, (16) x = 343, 7 (17) x = 6 or –3/2 2
10
Logarithmic Inequality :
Let 'a' is a real number such that
(i) If a > 1, then logax > loga y x > y
(ii) If a > 1, then logax < 0 < x < a
(iii) If a > 1, then logax > x > a
(iv) If 0 < a < 1, then logax > logay 0 < x < y
(v) If 0 < a < 1, then loga x < x > a
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Form - I : f(x) > 0, g(x) > 0, g(x) 1
Form Collection of system
f(x) 1 , g(x) 1
(a) logg(x) f(x) 0 0 f(x) 1 , 0 g(x) 1
f(x) 1 , 0 g(x) 1
(b) logg(x) f(x) 0 0 f(x) 1 , g(x) 1
f(x) (g(x))a , g(x) 1
(c) logg(x) f(x) a a
0 f(x) (g(x)) , 0 g(x) 1
0 f(x) (g(x))a , g(x) 1
(d) logg(x) f(x) a a
f(x) (g(x)) , 0 g(x) 1
From - II : When the inequality of the form
Form Collection of system
f(x) g(x), (x) 1,
(a) log(x) f(x) log(x) g(x) 0 f(x) g(x);0 (x) 1
0 f(x) g(x), (x) 1,
(b) log(x) f(x) log(x) g(x)
f(x) g(x) 0, 0 (x) 1
Example # 19 : Solve the logarithmic inequality log1/5 (2x2 + 7x + 7) 0.
Solution. Since log1/5 1 = 0, the given inequality can be written as.
log1/5 (2x2 + 7x + 7) log1/5 1
when the domain of the function is taken into account the inequality is equivalent to the system
2x 2 7x 7 0 of inequalities
2x 2 7x 7 1
–3 Solving the inequalities by using method of intervals x –2, 2
Example # 20 : Solve the inequality log1/3 (5x – 1) > 0.
Solution. by using the basic property of logarithm.
2 5x 2 x 5 5x – 1 1
5x – 1 0 1 5x 1 x
5
1 2
The solution of the inequality is given by , Ans.
5 5
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Example # 21 : Solve the inequality log(2x + 3) x2 < log(2x + 3) (2x + 3).
Solution. The given inequality is equivalent to the collection of the systems
0 2x 3 1 (i)
2
x 2x 3
2x 3 1 (ii) 2
0 x 2x 3
Solving system (i) we obtain
–3
2 x –1
(x – 3)(x 1) 0 (iii)
System (iii) is equivalent to the collection of two systems
–3 x –1, x 3; (iv)
2
–3 x –1, x –1 (v) 2
system (iv) has no solution. The solution of system (v) is x –3, – 1 ,
2
solving system (ii) we obtain.
x –1 x –1
or x (– 1, 3)
(x – 3)(x 1) 0 –1 x 3
x–3,–1 (– 1, 3)
2
2x
> 0. log2 Example # 22 : Solve the in equation log x 2 –12x 30 5 10
Solution. This in equation is equivalent to the collection of following systems.
x 2 – 12x 30 x 2 – 12x 30 1, 0 1,
10 10 and
2x 2x 1, 1 0 log2 log2 5 5
Solving the first system we have.
x 2 – 12x 20 0
2x (x –10)(x – 2) 0 x 2 or x 10 2 x 5 x 5 5
Therefore the system has solution x > 10
Solving the second system we have.
0 x 2 – 12x 30 10 x 2 –12x 30 0 and x 2 –12x 20 0
2x 5 1 2 x 5 5 2
x 6 – 6 or x 6 6 and 2 x 10
5
2 x 5
5
The system has solutions < x < 6 – 6 combining both systems, then solution of
2
the original inequation is.
5
x ( , 6 – 6 (10, ) Ans.
2
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Self practice problems :
(18) Solve the following inequalities
(i) log3x + 5 (9x2 + 8x + 8) > 2
(ii) log0.2 (x2 – x – 2) > log0.2 (– x2 + 2x + 3)
(iii) logx (x3 – x2 – 2x) < 3
4 17 5
Answers : (18) (i) – ,– (ii) 2, (iii) (2, )
3 22 2
Characteristic & Mantissa
[loga N] is called characteristic of log of N with base ‘a’. It is always an integer.
{loga N} is called mantissa of log of N with base ‘a’. Mantissa [0, 1)
Characteristic of log of 1 with base 10 = 0
characteristic of log of 10 with base 10 = 1
characteristic of log of 100 with base 10 = 2
characteristic of log of 1000 with base 10 = 3
characteristic of log of 83.5609 with base 10 = 1
characteristic of log of 613.0965 with base 10 = 2
Interval, Cha.(Base 10) number of digits No. of integers in the interval
in no
[1, 10) 0 1 9 = 9 × 10º
[10, 100) 1 2 90 = 9 × 101
[100, 1000) 2 3 900 = 9 × 102
[100, 10000) 3 4 9000 = 9 × 103
|
|
|
n (n + 1) 9 × 10n
Note :
If characteristic of a number (base of log is 10) is found to be n, then there would be (n + 1) digits in that
number.
1
* Characteristic of log of = 0.1 with base 10 = – 1
10
1
Characteristic of log of = 0.01 with base 10 = – 2
100
1
Characteristic of log of = 0.001 with base 10 = – 3
1000
3
Characteristic of log of with base 10 = – 2
100
3
Characteristic of log of with base 10 = – 3
1000
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Interval Characteristic (base 10) No. of zeros immedi- No.ofinteger resiprocal
ately after decimal of which lies in interval.
[1/10, 1) –1 0 9 = 9×101–1
[1/100, 1/10) –2 1 90 = 9×102–1
[0.01, 0.1)
[1/103, 1/102) [0.0001, 0.01) –3 2 900 = 9×103–1
[0.0001, 0.001) –4 3 9000 = 9×104–1
|
|
|
|
– n (n – 1) = 9 × 10n–1
Note :
If characteristic of a number (base of log is 10) is found to be –n, then there would be (n –1) zeros
immediately after decimal before first significant digit.
Example # 23 Find the total number of digits in the number 1850.
(Given log102 = 0.3010 ; log103 = 0.4771) Ans. 63
Solution. N = 1850
log10N = 50 log1018 = 50 (0.3010 + 0.9542) = 50(1.2552) = 62.76
Characterstic = [log10N] = 62
No. of digits = 62 + 1 = 63
Self practice problem
(19) Find the total number of zeros immediately after the decimal in 6– 200 .
Ans. (19) 155
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Marked questions are recommended for Revision.
PART - I : SUBJECTIVE QUESTIONS
Section (A) : Representation of sets, Types of sets, subset and power set
A-1. State wether the following collections is a set or not ?
(i) The collection of natural numbers between 2 and 20
(ii) The collection of numbers which satisfy the equation x2 –5x + 6 = 0
(iii) The collection of prime numbers between 1 and 100.
(iv) The collection of all intelligent women in Jalandhar.
A-2. Write the following set in tabular form
(i) A = {x : x is a positive prime < 10}
(ii) B = {x : x = 3, x , 1 3}
A-3. Write the following set in builder form
(i) set of all rational number
(ii) {2, 5, 10, 17, 26, 37, .........}
A-4. Identifiy type of set in terms of empty/singleton/finite/infinite
(i) {x : x is a real number and x2 – 1 = 0}
(ii) {x : x is a real number and x2 + 1 = 0}
(iii) {x : x is positive real number and x2 – 9 = 0}
(iv) {x : x is a real number and x2 + 2x + 2 0}
A-5. Write power set of the set A = {, {}}.
Section (B) : Operations on sets, Law of Algebra of sets
B-1. Given the sets A = {1, 2, 3}, B = {3, 4}, C = {4, 5, 6}, then find the following
(i) A (B C) (ii) A – (B C) (iii) (B C) – A
B-2. Find the smallest set A such that A {1, 2} = {1, 2, 3, 5, 9}
B-3. If aN = {ax : x N} and bN cN = dN, where b, c N, b 2, c 2 are relatively prime, then write 'd' in
terms of b and c.
B-4. Sets A and B have 3 and 6 elements respectively. What can be the minimum and maximum number of
elements in
(i) A B (ii) A B
Section (C) : Cardinal number Problems
C-1. Let n(U) = 700, n(A) = 200, n(B) = 300 and n(A B) = 100, then find n(A' B')
C-2. In a college of 300 students, every student reads 5 newspapers and every newspaper is read by 60
students. Find the number of newspaper.
C-3. In a town of 10,000 families it was found that 40% families buy newspaper A, 20% families buy
newspaper B and 10% families buy newspaper C, 5% families buy A and B, 3 % buy B and C and 4%
buy A and C. If 2% families buy all the three news papers, then find number of families which buy
newspaper A only.
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C-4. In a survery, it was found that 21 persons liked product A, 26 liked product B and 29 liked product C. If
14 persons liked products A and B, 12 liked products C and A, 13 persons liked products B and C and
8 liked all the three products then
(i) Find the number of persons who liked atleast one product
(ii) The number of persons who like the products A and B but not C
Section (D) : Rational Inequalities
D-1. Solve the following rational in equalities
(x –1)(x 2) (1– x)3 (x 2)4
(i) < 0 (ii) 2 0 (x – 3)(x 3) (x 9) (x – 8)
x 2 4 x 4 ( 2 x 2 )( x 3 ) 3 0 (iii) 0 (iv) ( x 1)( x 2 3 x 4 ) 2 x 2 x 1
( x 2)( x 2 2 x 1)
(v) 2 0 4 3 x x
D-2. Solve the following Inequalities
7 x 5 14x 9x 30 (i) 4 (ii)
8 x 3 x 1 x 4
(x –1)(x – 2)(x – 3) x 2 2 (iii) 1 (iv) 2
(x 1)(x 2)(x 3) x 2 1
D-3. Solve the following rational in equalities
(x 2 – 3x 1)3 2 x 2 3 x 459 (i) 0 (ii) 1
(x 1)(x 2) x 2 1
x 2 5 x 12 x 4 x 2 1
(iii) 3 (iv) 2 0 2 x 4 x 5 x 4 x 5
D-4. Solve the following rational in equalities
(i) x4 – 5x2 + 4 0 (ii) x4 – 2x2 – 63 0
(iii) (x2+3x+1) (x2+3x–3) 5
x – 1D-5. If 1 < 7 then find the range of
x 2
1
(i) x (ii) x2 (iii)
x
(3 x – 5 x )(x – 2)
D-6. Find the number of positive integral value of x satisfying the inequality 2 0 (x 5x 2)
Section (E) : Logarithmic Properties
E-1. Find the value of
2 5 log 5 2 9log 3 7 8log (i) log105.log1020 + (log102)2 (ii) 5
1
3 log7 5 1 1
(iii) 5 ( log10 0.1) (iv) log0.75 log2 0.125
1 log7 2
1 –log1/ 5 7 log3 5 log5 7 log3 7 log5 3 (v) + 5 (vi) 7 3 5 7
49
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E-2. Which of the following numbers are positive/negative
(i) log 2 (ii) log1/7(2) (iii) log1/3(1/5) 3
(iv) log3(4) (v) log7(2.11) (vi) log3 ( 7 2)
2 1 2· 3 3
(ix) log10 (log10 9) (viii) log3 (vii) log4 3 2 1
E-3. Let log102 = a and log103 = b then determine the following logarithms in terms of a and b.
2
(i) log10 sin (ii) log1004 + 2 log10027
3
(iii) log29 + log38 (iv) log 45 144
4 3 2 1
E-4. (i) Let n = 75600, then find the value of
log2 n log3 n log5 n log7 n
(ii) If log2 log3 log4 (x) 0 and log3 log4 log2 (y) 0 and log4 log2 log3 (z) 0 then find
the sum of x, y and z is
1
(iii) Suppose n be an integer greater than 1. let an = . Suppose b = a2 + a3 + a4 + a5 and
logn 2002
c = a10 + a11 + a12 + a13 + a14. Then find the value of (b – c)
E-5. Show that the number log27 is an irrational number.
loga logb logc
E-6. If , show that aa. bb. cc = 1.
b c c a a b
Section (F) : Logarithmic Equation
F-1. Solve the following equations :
(i) logx(4x – 3) = 2 (ii) log2(log3(x2 – 1)) = 0
(iii) 4log 2 x 2x 3 0 (iv) log4 (log2x) + log2 (log4x) = 2.
1 x
(v) log3 log9 x 9 = 2x. (vi) 2log4 (4 – x) = 4 – log2 (– 2 – x).
2
(log
x 0.5log x (x 2 x) 9 4 . (vii) x 2 x) 4 (viii) x 3log
F-2. (i) Find the product of roots of the equation
(log3x)2 – 2(log3x) – 5 = 0
(ii) Find sum of roots of the equation 4x – 7.2x + 6 = 0
10 x 2 (iii) Solve for x : xlog 10 x 2 10log
log10 x 5
10 x (iv) Solve for x : x 3 = 105 log
Section (G) : Logarithmic inequalities
G-1. Solve the following inequalities
2 3 2 (i) log 5 2x x 1 (ii) log 1 ( x 5 x 6 ) 1
2 8 8
2 x 6 2
(iii) log7 0 (iv) log1/4(2 – x) > log1/4 2 x 1 x 1
(v) log1/3(2x+2 – 4x) – 2
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4x 6
G-2. Find the number of integers satisfying log1/5 0
x
G-3 Solve the inequalities
(i) (log.5x)2 + log.5x – 2 0 (ii) 15x – 25.3x – 9.5x + 225 0
3 x–2 2 x
(iii) 8. x x > 1 +
3 – 2 3
G-4. Solve the following inequalities :
1
2 (i) logx(4x – 3) 2 (ii) log(3x 21) < 2
(iii) logx² (2 + x) < 1
PART - II : ONLY ONE OPTION CORRECT TYPE
Section (A) : Representation of sets, Types of sets, subset and power set
A-1. The set of intelligent students in a class is-
(A) a null set (B) a singleton set
(C) a finite set (D) not a well defined collection
A-2. The set A = {x : x R, x2 = 16 and 2x = 6} is
(A) Null set (B) Singleton set
(C) Infinite set (D) not a well defined collection
A-3. If A = {x : –3 < x < 3, x Z} then the number of subsets of A is –
(A) 120 (B) 30 (C) 31 (D) 32
A-4. Which of the following are true ?
(A) [3, 7] (2, 10) (B) (0, ) (4, ) (C) (5, 7] [5, 7) (D) [2, 7] (2.9, 8)
A-5. The number of subsets of the power set of set A = {7, 10, 11} is
(A) 32 (B) 16 (C) 64 (D) 256
A-6. Which of the folowing sets is an infinite set ?
(A) Set of divisors of 24
(B) Set of all real number which lie between 1 and 2
(C) Set of all humman beings living in India.
(D) Set of all three digit natural numbers
Section (B) : Operations on sets, Law of Algebra of sets
B-1. Let A = {x : x R, –1 < x < 1} , B = {x : x R, x 0 or x 2} and A B = R – D, then the set D is
(A) {x : 1 < x 2} (B) {x : 1 x < 2} (C) {x : 1 x 2} (D) {x : 1 < x < 2}
B-2. If A = {2, 3, 4, 8, 10}, B = {3, 4, 5, 10, 12}, C = {4, 5, 6, 12, 14} then (A B) (A C) is equal to
(A) {3, 4, 10} (B) {2, 8, 10} (C) {4, 5, 6} (D) {3, 5, 14}
B-3. The shaded region in the given figure is
(A) A (B C) (B) A (B C) (C) A (B – C) (D) A – (B C)
B-4. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 5}, B = {6, 7}, then A Bis
(A) B (B) A (C) A (D) B
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B-5. If A = {x : x = 4n + 1, n 5, n N} and B = {3n : n 8, n N}, then A – (A – B) is :
(A) {9, 21} (B) {9, 12} (C) {6, 12} (D) {6, 21}
B-6. A B = A B iff :
(A) A B (B) A = B (C) A B (D) A B
B-7. Consider the following statements :
1. N (B Z) = (N B) Z for any subset B of R, where N is the set of positive integers, Z is
the set of integers, R is the set of real numbers.
2. Let A = {n N : 1 n 24, n is a multiple of 3}. There exists no subset B of N such that the
number of elemets in A is equal to the number of elements in B.
Which of the above statements is/are correct ?
(A) 1 only (B) 2 only (C) Both 1 and 2 (D) Neither 1 nor 2
B-8.Which of the following venn-diagrams best represents the sets of females, mothers and doctors ?
(A) (B) (C) (D)
Section (C) : Cardinal number Problems
C-1. Let A and B be two sets. Then
(A) n(A B) n(A B) (B) n(A B) n(A B)
(C) n(A B) = n(A B) (D) can't be say
C-2. In a city 20 percent of the population travels by car, 50 percent travels by bus and 10 percent travels by
both car and bus. Then persons travelling by car or bus is
(A) 80 percent (B) 40 percent (C) 60 percent (D) 70 percent
C-3. A class has 175 students. The following data shows the number of students obtaining one or more
subjects : Mathematics 100, Physics 70, Chemistry 40, Mathematics and Physics 30, Mathematics and
Chemistry 28, Physics and Chemistry 23, Mathematics & Physics & Chemistry 18. How many students
have offered Mathematics alone ?
(A) 35 (B) 48 (C) 60 (D) 22
C-4. 31 candidates appeared for an examination, 15 candidates passed in English, 15 candidates passed in
Hindi, 20 candidates passed in Sanskrit. 3 candidates passed only in English. 4. candidates passed
only in Hindi, 7 candidates passed only in Sanskrit. 2 candidates passed in all the three subjects How
many candidates passed only in two subjects ?
(A) 17 (B) 15 (C) 22 (D) 14
Section (D) : Rational Inequalities
x 4 3x 3 2x 2
D-1. The complete solution set of the inequality 0 is:
x 2 x 30
(A) (, 5) (1, 2) (6, ) {0} (B) (, 5) [1, 2] (6, ) {0}
(C) (, 5] [1, 2] [6, ) {0} (D) (, 5] [1, 2] [6, )
D-2. Number of positive integral values of x satisfying the inequality
(x 4)2017 . (x 8)2016 (x 1)
0 is
x 2016 (x 2)3 . (x 3)5 . (x 6) (x 9)2018
(A) 0 (B) 1 (C) 2 (D) 3
x 2 1
D-3. The number of prime numbers satisfying the inequality < 3 is
2x 5
(A) 1 (B) 2 (C) 3 (D) 4
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x 2 1
D-4. The complete solution of 0 & x2 – 5x + 2 0 is :
x 3
17 5 17 5 17 (A) x 5 , (B) x 1,
2 2 2
(C) x (–3, –1] (D) x (–3, –1] [1, )
D-5 The solution of the inequality 2x – 1 x2 + 3 x – 1 is
(A) x R (B) [–2, 2] (C) (–2, 2) (D) x
D-6. The number of the integral solutions of x2 + 9 < (x + 3)2 < 8x + 25 is :
(A) 1 (B) 3 (C) 4 (D) 5
2 1 2x 1
D-7. Number of non-negative integral values of x satisfying the inequality 0 is
x 2 x 1 x 1 x 3 1
(A) 0 (B) 1 (C) 2 (D) 3
Section (E) : Logarithmic Properties
E-1. If a4 · b5 = 1 then the value of loga(a5b4) equals
(A) 9/5 (B) 4 (C) 5 (D) 8/5
1 1 1
E-2. + + has the value equal to
1 logb a logb c 1 logc a logc b 1 loga b loga c
1
(A) abc (B) (C) 0 (D) 1
abc
1 1 1
E-3. + + has the value equal to :
log abc log abc log abc
bc c a ab
(A) 1/2 (B) 1 (C) 2 (D) 4
E-4. (log210) . (log280) – (log25) . (log2160) is equal to :
(A) log25 (B) log220 (C) log210 (D) log216
3
log a log27 a 2 1 21/ 4 2 3 2a
E-5. The ratio simplifies to :
7 4log49 a a 1
(A) a2 a 1 (B) a2 + a 1 (C) a2 a + 1 (D) a2 + a + 1
E-6. If loga(ab) = x, then logb(ab) is equal to
1 x x x
(A) (B) (C) (D)
x 1 x 1 x x 1
logp logq (logr x) E-7. 10 = 1 and logq logr (logp x) = 0 then 'p' equals
(A) rq/r (B) rq (C) 1 (D)rr/q
E-8. Which one of the following is the smallest?
2 1 3 1 (A) log10 (B) log10 (C) (D) log10 log10
E-9. log10(log23) + log10(log34) + log10(log45) + ........ + log10(log10231024) simplifies to
(A) a composite (B) a prime number
(C) rational which is not an integer (D) an integer
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Section (F) : Logarithmic Equation
F-1. The sum of all the solutions to the equation 2 log10 x – log10(2x – 75) = 2
(A) 30 (B) 350 (C) 75 (D) 200
F-2. If the solution of the equation logx(125x) . log252 x = 1 are and ( < ). Then the value of 1/is :
(A) 5 (B) 25 (C) 125 (D) 625
9 2
F-3. The positive integral solution of the equation logx 5 + logx5x = + log x 5 is : 4
(A) Composite number (B) Prime number
(C) Even number (D) Divisible by 3
F-4. The expression logp , where p 2, p N ; n N when simplified is
(A) independent of p (B) independent of p and of n
(C) dependent on both p and n (D) positive
1
= . Then the value of 1000 x is equal to 2 8F-5. If logx log18 3
(A) 8 (B) 1/8 (C) 1/125 (D) 125
2 is :F-6. Number of real solutions of the equation log10 x = log10 x
(A) zero (B) exactly 1 (C) exactly 2 (D) 4
x 2
x isF-7. The correct graph of y = x log
(A) (B) (C) (D)
Section (G) : Logarithmic inequalities
G-1. The solution set of the inequality log (x2 3x 2) 2 is
sin
3
5 1 5 1, 2 (A) 1,2 (B) 1, (C) , 1 2, (D) 2 2 2 2
G-2. If log0.3 (x 1) < log0.09 (x 1), then x lies in the interval
3
(A) (2, ) (B) (1, 2) (C) ( 2, 1) (D) 1,
2
G-3. Solution set of the inequality 2 log2 (x2 + 3x) 0 is :
(A) [ 4, 1] (B) [ 4, 3) (0, 1]
(C) ( 3) (1, ) (D) ( 4) [1, )
G-4. If log0.5 log5 (x2 – 4) > log0.51, then ‘x’ lies in the interval
(A) (– 3, – 5 ) ( 5 , 3) (B) (– 3, – 5 ) ( 5 , 2)
(C) ( 5 , 3 5 ) (D)
x 2 2x
G-5. The set of all solutions of the inequality (1/2) < 1/4 contains the set
(A) (– , 0) (B) (– , 1) (C) (1, ) (D) (3, )
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G-6. The number of positive integers not satisfying the inequality log2(4x – 2.2x + 17) > 5.
(A) 2 (B) 3 (C) 4 (D) 1
G-7. The set of all the solutions of the inequality log1 – x (x – 2) – 1 is
(A) (– , 0) (B) (2, ) (C) (– , 1) (D)
PART - III : MATCH THE COLUMN
1. Match the set P in column one with its super set Q in column II
Column – (set P) Column– (set Q)
(A) [32n – 8n – 1 : n N} (p) {49 (n – 1) : n N}
(B) {23n – 1 : n N} (q) {64 (n –1) : n N}
(C) {32n –1 : n N} (r) {7n : n N}
(D) {23n – 7n – 1 : n N} (s) {8n : n N}
2. Column-I Column-II
(A) If a = 3 8 2 7 8 2 7 , b = (42)(30) 36 (p) – 1
then the value of logab is equal to
(B) If a = 4 2 3 – 4 2 3 , b = 11 6 2 – 11 6 2 , (q) 1
then the value of logab is equal to
(C) If a = 3 2 2 , b = 3 2 2 (r) 2
then the value of logab is equal to
(D) If a = 7 7 2 1 , b = 7 7 2 1 , (s) 3
2
then the value of logab is equal to
(E) The number of zeroes at the end of the product of first 20 (t) None
prime numbers, is
(F) The number of solutions of 22x – 32y = 55, in which x and y
are integers, is
3. Column-I Column-II
(A) When the repeating decimal 0.363636............. is written as a rational (p) 4
fraction in the simplest form, the sum of the numerator
and denominator is
(B) Given positive integer p, q and r with p = 3q · 2r and 100 < p < 1000. (q) 0
The difference between maximum and minimum values of (q + r), is
(C) If log8a + log8b = (log8a)(log8b) and logab = 3, then the value of 'a' is (r) 15
(D) If P = 3 log3 2 – 2 log2 3 then value of P is (s) 16
4. Column-I Column-II
(A) Anti logarithm 0.6 to the base 27 has the value equal to (p) 5 of
(B) Characteristic of the logarithm of 2008 to the base 2 is
(C) The value of b satisfying the equation, (q) 7
loge2 · logb625 = log1016 · loge10 is
(D) Number of naughts after decimal before a significant figure (r) 9
5 100
comes in the number , is (s) 10
6
(Given log102 = 0.3010 and log103 = 0.4771)
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Marked questions are recommended for Revision.
* Marked Questions may have more than one correct option.
PART - I : ONLY ONE OPTION CORRECT TYPE
1. Let A1, A2 and A3 be subsets of a set X. Which one of the following is correct ?
(A) A1 A2 A3 is the smallest subset of X containing elements of each of A1, A2 and A3
(B) A1 A2 A3 is the smallest subset of X containing either A1 or A2 A3 but not both
(C) The smallest subset of X containing A1 A2 and A3 equals the smallest subset of X containing
both A1 and A2 A3 only if A2 = A3
(D) None of these
2. Let A, B, C be distinct subsets of a universal set U. For a subset X of U, let X ' denote the complement
of X in U.
Consider the following sets :
1. (((A B) C) B) = B C
2. (A B )(A B C) = (A (B C))
Which of the above statements is/are correct ?
(A) 1 only (B) 2 only (C) Both 1 and 2 (D) Neither 1 nor 2
3. In an examination of a certain class, at least 70% of the students failed in Physics, at least 72% failed in
Chemistry, at least 80% failed in Mathematics and at least 85% failed in English. How many at least
must have failed in all the four subjects ?
(A) 9% (B) 7% (C) 15%
(D) Cannot be determined due to insufficient data
4. Let X and Y be two sets.
Statement-1 X (Y X)' =
Statement-2 If X Y has m elements and X Y has n elements then symmetric difference X Y
has m – n elements.
(A) Both the statements are true. (B) Statement- is true, but Statement- is false.
(C) Statement- is false, but Statement- is true. (D) Both the statements are false.
6x 2 5x 3
5. If 4, then the least and the highest values of 4x2 are:
x 2 2x 6
(A) 36 & 81 (B) 9 & 81 (C) 0 & 81 (D) 9 & 36
(x 2 2)( x 2 16)
6. Sum of all the real solutions of the inequality 0 is
(x 4 2)(x 2 9)
(A) 5 (B) 4 (C) 8 (D) 0
7. If logab = 2; logbc = 2 and log3c = 3 + log3a then (a + b + c) equals
(A) 90 (B) 93 (C) 102 (D) 243
3log289 11 log28 17 log1331 784 y and y = 25 , then value of is8. Let x log1 3 5 log125 343 log49 729 x
5 5 4 3
(A) (B) (C) (D)
3 3 5 7
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Fundamentals of Mathematics-I
x 2 3x 2 x(x 3 1)
3x log2 8 (x 1)(x 2 x 1) x 2 9. The expression: reduces to
(x 1)(log2 3)(log3 4)(log4 5)(log5 2)
x 1 x 2 3x 2 3x
(A) (B) (C) (D) x
x 1 (log2 5)x 1 x 1
10. If a, b, c are positive real numbers such that alog 11 25 = 11 . The value 3 7 = 27 ; blog 7 11 = 49 and c log
11 25)2 equals of a(log 3 7)2 b(log 7 11)2 c (log
(A) 489 (B) 469 (C) 464 (D) 400
11. Consider the statement : x ( – x) < y ( – y) for all x, y with 0 < x < y < 1. The statement is true
(A) if and only if 2 (B) if and only if 2
(C) if and only if –1 (D) for no values of
(x 8) (2 x)
12. The set of values of x satisfying simultaneously the inequalities 0 and
10
log0.3 (log2 5 1)
7
2x 3 31 > 0 is :
(A) a unit set (B) an empty set
(C) an infinite set (D) a set consisting of exactly two elements.
(3 x 4 x ) · n(x 2)
13. The solution set of the inequality 0 is
x 2 3x 4
(A) (–, 0] (4, ) (B) (–2, 0] (4, )
(C) (–1, 0] (4, ) (D) (–2, –1) (– 1, 0] (4, )
14. If log4 {log3 {log2 (x 2 2x a)}} is defined xR, then the set of values of 'a' is
(A) [9, ) (B) [10, ) (C) [15, ) (D) [2, )
15. If log(2x + 3) (6x2 + 23x + 21) = 4 – log(3x + 7) (4x2 + 12x + 9) then value of x is equal to
3 3 1 3
(A) (B) – (C) – (D)
4 4 4 2
PART-II: NUMERICAL VALUE QUESTIONS
INSTRUCTION :
The answer to each question is NUMERICAL VALUE with two digit integer and decimal upto two digit.
If the numerical value has more than two decimal places truncate/round-off the value to TWO decimal
placed.
1. Let U be set with number of elements in it is 2009 and A, B are subsets of U with n (A B) = 280. If
n(A B) = x 13 + x 32 = y 13 + y 32 for some positive integers x1 < y1 < y2 < x2, then find value of
x 2 y 2
x1 y 1
2. Let U be set with number of elements in it is 2009. A is a subset of U with n (A) = 1681 and out of these
1681 elements, exactly 1075 elements belong to a subset B of U. If n (A – B) = m2 + p1 p2 p3 for some
p1 p3
positive integer m and distinct primes p1 < p2 < p3 then for least m find
p2
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3. Let A = {(x, y) : x R, y R, x3 + y3 = 1], B = {(x, y) : x R, y R, x – y = 1}
and C = {(x, y) : x R, y R, x + y = 1}. If A B contains 'p' elements and A C contains 'q' elements
then find (q – p).
4. In a class of 42 students, the number of students studying different subjects are 23 in Mathematics, 24
in Physics, 19 in Chemistry, 12 in Mathematics and Physics 9 in Mathematics and Chemistry, 7 in
Physics and Chemistry and 4 in all the three subjects. Then find number of students who have taken
exactly one subject.
log(a c) log(a 2b c)
5. If c(a – b) = a(b – c) then find the value of (Assume all terms are defined)
log(a c)
6. If logb a. logc a + loga b. logc b + loga c. logb c = 3 ( where a, b, c are different positive real numbers 1),
then find the value of a b c.
7. If 4A + 9B = 10C, where A = log164, B = log39 & C = logx83, then find x.
3 5
8. Let a, b, c, d are positive integers such that logab = and logcd = . If (a – c) = 9, find
2 4
b d
the value of .
a c
9. Find the positive number, x , which satisfies the equation log10 (2x² 21 x + 50) = 2
10. Find the value of x satisfying the equation log 1 (x 1) log 1 (x 1) log 1 (7 x) 1
2 2 2
11. Find sum of roots of equation log 102 x + log10x2 = log 102 2 1
2009 x logx (2010)12. If the product of all solutions of the equation 2009 can be expressed in the lowest
2010
m
form as then the value of (m – n) is
n
13. If the complete solution set of the inequality (log10 x)2 log10 x + 2 is (0, a] [100, ) then find the value
of a.
1 1
14. The complete solution set of the inequality 4 ( x 3 ) , is (–a, ), then determine ‘a’. log4 xx21 log
15. If complete solution set of inequality log1/2 (x + 5)2 > log1/2 (3x – 1)2 is (–,p) (q,r) (s,) then find
p 2 q 2 r 2
s 2
PART - III : ONE OR MORE THAN ONE OPTIONS CORRECT TYPE
1. Let a > 2, a N be a constant. If there are just 18 positive integers satisfying the inequality
(x – a)(x – 2a)(x – a2) < 0 then which of the option(s) is/are correct?
(A) 'a' is composite (B) 'a' is odd
(C) 'a' is greater than 8 (D) 'a' lies in the interval (3, 11)
log3 135 log3 52. Let N = – . Then N is :
log15 3 log405 3
(A) a natural number (B) a prime number (C) a rational number (D) an integer
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2 5 3. Values of x satisfying the equation log5 x + log5x = 1 are
x
1
(A) 1 (B) 5 (C) (D) 3
25
4. The equation log 16 + log2x64 = 3 has : x 2
(A) one irrational solution (B) no prime solution
(C) two real solutions (D) one integral solution
2 9
(log3 x) log3 x 5
5. The equation x 2 = 3 3 has
(A) exactly three real solution (B) at least one real solution
(C) exactly one irrational solution (D) complex roots.
2
6. The solution set of the system of equations log3x + log3y = 2 + log32 and log27(x + y) = is :
3
(A) {6, 3} (B) {3, 6} (C) {6, 12} (D) {12, 6}
7. Consider the quadratic equation, (log108)x2 – (log105)x = 2(log210)–1 – x. Which of the following
quantities are irrational.
(A) sum of the roots (B) product of the roots
(C) sum of the coefficients (D) discriminant
8. If loga x = b for permissible values of a and x then identify the statement(s) which can be correct?
(A) If a and b are two irrational numbers then x can be rational.
(B) If a rational and b irrational then x can be rational.
(C) If a irrational and b rational then x can be rational.
(D) If a rational and b rational then x can be rational.
9. Which of the following statements are true
(A) log2 3 < log12 10 (B) log6 5 < log7 8
(C) log326 < log29 (D) log1615 > log1011 > log76
1
10. If log0.1 x 2, then
2
1 1 1
(A) maximum value of x is (B) x lies between and
10 100 10
1 1
(C) minimum value of x is (D) minimum value of x is
10 100
PART - IV : COMPREHENSION
Comprehension # 1 (1 to 3)
In a group of 1000 people, there are 750 people, who can speak Hindi and 400 people, who can speak
Bengali.
1. Number of people who can speak Hindi only is
(A) 300 (B) 400 (C) 500 (D) 600
2. Number of people who can speak Bengali only is
(A) 150 (B) 250 (C) 50 (D) 100
3. Number of people who can speak both Hindi and Bengali is
(A) 50 (B) 100 (C) 150 (D) 200
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Fundamentals of Mathematics-I
Comprehension # 2 (4 to 6)
1 4
Let A denotes the sum of the roots of the equation + = 3.
5 4log4 x 1 log 4 x
B denotes the value of the product of m and n, if 2m = 3 and 3n = 4.
3
C denotes the sum of the integral roots of the equation log3x + (log3x)2 = 1.
x
4. The value of A + B equals
(A) 10 (B) 6 (C) 8 (D) 4
5. The value of B + C equals
(A) 6 (B) 2 (C) 4 (D) 8
6. The value of A + C B equals
(A) 5 (B) 8 (C) 7 (D) 4
Comprehension # 3 (Q.7- to Q.9)
A function f(x) = ax(a > 0, a 1, x R) is called an exponential function. Graph of exponential function
can be as follows :
Case - Case -
For a > 1 For 0 < a < 1
7*. Which of the following is correct :
(A) (B)
Y Y
y=3–x y=2–x
y=3–x y=2–x
(C) X (D) X
8. Number of solutions of 3x + x – 2 = 0 is/are :
(A) 1 (B) 2 (C) 3 (D) 4
9. The number of positive solutions of log1/2x = 7x is/are :
(A) 0 (B) 1 (C) 2 (D) 3
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Fundamentals of Mathematics-I
Marked questions are recommended for Revision.
* Marked Questions may have more than one correct option.
PART - I : JEE (ADVANCED) / IIT-JEE PROBLEMS (PREVIOUS YEARS)
1. Indicate all correct alternatives, where base of the log is 2. [ JEE ’89 ]
The equation x(3/4) (logx)² + logx(5/4) = 2 has :
(A) at least one real solution (B) exactly three real solutions
(C) exactly one irrational solution (D) complex roots
2. The number log27 is :
(A) an integer (B) a rational number
(C) an irrational number (D) a prime number [ JEE ’90 ]
3. Find all real numbers x which satisfy the equation
2 log2 log2 x + log1/2 log2 (2 2 x)= 1. [REE – 1999, 6 ]
4. Solve the equation log3/4 log8 (x2 + 7) + log1/2 log1/4 (x2 + 7) 1 = 2. [REE– 2000, 5]
5. The number of solution(s) of log4(x – 1) = log2(x – 3) is/are [IIT-JEE-2002, Scr., (1, 0)/35]
(A) 3 (B) 1 (C) 2 (D) 0
x 2 6x 5
6. Let f(x) = [IIT-JEE 2007, Paper-2, (6, 0), 81]
x 2 5x 6
Column – Column –
(A) If – 1 < x < 1, then f(x) satisfies (p) 0 < f(x) < 1
(B) If 1 < x < 2, then f(x) satisfies (q) f(x) < 0
(C) If 3 < x < 5, then f(x) satisfies (r) f(x) > 0
(D) If x > 5, then f(x) satisfies (s) f(x) < 1
7. Let (x0, y0) be the solution of the following equations
(2x)n2 = (3y)n3
3nx = 2ny .
Then x0 is [IIT-JEE 2011, Paper-1, (3, –1), 80]
1 1 1
(A) (B) (C) (D) 6
6 3 2
1 1 1 1 8. The value of 6 log 3 4 – 4 – 4 – .... is 2 3 2 3 2 3 2 3 2
[IIT-JEE 2012, Paper-1, (4, 0), 70]
9*. If 3x = 4x – 1 , then x = [JEE (Advanced) 2013, Paper-2, (3, –1)/60]
2log3 2 2 1 2log2 3 (A) (B) (C) (D)
2log3 2 – 1 2 – log2 3 1– log4 3 2log2 3 – 1
1 1
2 (log2 9) ( 7) log 4 7 is ___________.10. The value of ((log 2 9)2 ) log
[JEE(Advanced) 2018, Paper-1,(4, –2)/60]
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PART - II : PREVIOUS YEARS PROBLEMS OF MAINS LEVEL
1. If logp x = and logq x = , then the value of logp/q x is [KCET-1997]
– –
(1) (2) (3) (4)
– –
2. If logxa, ax/2 and logbx are in G.P. Then x is equal to [KCET-1998]
(1) loga (logba) (2) loga(logea)+ loga logbb
(3) – loga(logab) (4) none of these
3. If logx 256 = 8/5, then x is equal to [KCET-2000]
(1) 64 (2) 16 (3) 32 (4) 8
4. If log 2, log(2x – 1) and log (2x + 3) are in A.P., then x is equal to [KCET-2000]
(1) 5/2 (2) log25 (3) log23 (4) log32
5. The number log27 is [DCE-2000]
(1) an integer (2) a rational (3) an irrational (4) a prime number
6. The roots of the equation log2(x2 – 4x + 5) = (x – 2) are [KCET-2001]
(1) 4, 5 (2) 2, – 3 (3) 2, 3 (4) 3, 5
1 1 1
7. If x = 198 !, then value of the expression .... equals
log2 x log3 x log198 x
[DCE-2005]
(1) –1 (2) 0 (3) 1 (4) 198
8. If log0.3 (x – 1) < log0.09 (x – 1), then x lies in the intervel [DCE-2006]
(1) (2, ) (2) (1, 2) (3) (–2, –1) (4) none of these
9. If A, B and C are three sets such that A B = A C and A B = A C, then
[AIEEE-2009, (4, – 1), 144]
(1) A = C (2) B = C (3) A B = (4) A = B
10. Let X = {1, 2, 3, 4, 5}. The number of different ordered pairs (Y, Z) that can formed such that Y X, Z
X and Y Z is empty, is : [AIEEE-2012, (4, – 1), 120]
(1) 52 (2) 35 (3) 25 (4) 53
11. If X = {4n – 3n – 1 : n N} and Y = {9(n – 1) : n N}, where N is the set of natural numbers, then X Y
is equal to [JEE(Main) 2014, (4, – 1), 120]
(1) X (2) Y (3) N (4) Y – X
12. The sum of all real values of x satisfying the equation ( x 2 – 5 x 5) x 2 4 x – 60 = 1 is
[JEE(Main) 2016, (4, – 1), 120]
(1) – 4 (2) 6 (3) 5 (4) 3
13. In a class 140 students numbered 1 to 140, all even numbered students opted Mathematics course,
those whose number is divisible by 3 opted Physics course and those whose number is divisible 5
opted Chemistry course. Then the number of student who did not opt for any of the three courses is :
[JEE(Main) 2019, Online (10-01-19),P-1 (4, – 1), 120]
(1) 38 (2) 42 (3) 102 (4) 1
14. Let X = {n N: 1 n 50}. If A = {n X: n is a multiple of 2}; B = {n X: n is a multiple of 7}, then the
number of elements in the smallest subset of X containing both A and B is__________.
[JEE(Main) 2020, Online (07-01-20),P-2 (4, 0), 120]
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Fundamentals of Mathematics-I
EXERCISE - 1
PART -I
Section (A) :
A-1. (i) Yes (ii) Yes (iii) Yes (iv) No
A-2. (i) {2,3,5,7} (ii) {3, 4, 5, 6, 7, 8, 9}
p
A-3. (i) {x : x = , p , q N} (ii) {x : x = + 1, N}
q
A-4. (i) Finite (ii) Finite and empty
(iii) Singleton & finite (iv) Infinite
A-5. {, {}, {{}}, A}
Section (B) :
B-1. (i) {1, 2, 3, 4} (ii) {1, 2, 3} (iii) {4, 5, 6}
B-2. {3, 5, 9}
B-3. d = bc
B-4. (i) minimum n(A B) = 0, maximum n(A B) = 3
(ii) minimum n(A B) = 6, maximum n(A B) = 9
Section (C) :
C-1. 300 C-2. 25 C-3. 3300 C-4. (i) 45 (ii) 6
Section (D) :
D-1. (i) (–3,–2) (1,3) (ii) {–2} [1, 8)
(iii) (–, –2)(–2, –1/2)(1, ) (iv) [ 2, 1) 1, 2 3, 4
(v) (– , – 2] (–1, 4)
D-2. (i) (–17/25, –3/8) (ii) x (–6, –1) (1, 4)
(iii) (–3,–2) (–1,) (iv) x (–1,0) (0,1)
3 – 5 3 5
1, –2,D-3. (i) 2 (ii) , 20 23, 2
(iii) 1, 3 , 1 5, 2 (iv)
D-4. (i) x [–2, –1] [1, 2] (ii) x [–3, 3]
(iii) x(–, –4][–2, –1) [1,)
5 2 D-5. (i) – ,– (ii) 25, (iii) – ,0
2 4 5
D-6. 2
Section (E) :
1
E-1. (i) 1 (ii)–72 (iii) 2 (iv) 1 (v) 7 + (vi) 0
196
E-2. (i) +ve (ii) – ve (iii) +ve (iv) +ve
(v) +ve (vi) – ve (vii) +ve (viii) – ve (ix) – ve
2b 2 3a 2 4(2a b)
E-3. (i) b – 2a (ii) a + 3b (iii) (iv)
ab 1 a 2b
E-4. (i) 1 (ii) 89 (iii) –1
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Fundamentals of Mathematics-I
Section (F) :
F-1. (i) 3 (ii) ± 2 (iii) 3 (iv) 16
(v) {1/3} (vi) {– 4} (vii) no root (viii) (2)
1 5
F-2. (i) 9 (ii) log26 (iii) 10 or (iv) {10 , 103} 100
Section (G) :
1 1 3 G-1. (i) , ,1 (ii) (1, 2) (3, 4) 2 4 4
1
(iii) , (iv) (–1, 0) (1, 2)
2
(v) (–, 2)
G-2. 1
1,4 G-3 (i) (ii) R (iii) (0, log 3 3 )
2 2
3
G-4. (i) , 1 1, 3
4
(ii) (, 1) (1, )
(iii) x (– 2, –1) (– 1, 0) (0, 1) (2, )
PART - II
Section (A) :
A-1. (D) A-2. (A) A-3. (D) A-4. (A) A-5. (D) A-6. (B)
Section (B) :
B-1. (B) B-2. (A) B-3. (D) B-4. (B) B-5. (A) B-6. (B) B-7. (A)
B-8. (D)
Section (C) :
C-1. (B) C-2. (C) C-3. (C) C-4. (B)
Section (D) :
D-1. (B) D-2. (D) D-3. (D) D-4. (B) D-5 (D) D-6. (D) D-7. (D)
Section (E) :
E-1. (A) E-2. (D) E-3. (B) E-4. (D) E-5. (D) E-6. (D) E-7. (A)
E-8. (A) E-9. (D)
Section (F) :
F-1. (D) F-2. (C) F-3. (B) F-4. (A) F-5. (D) F-6. (C) F-7. (B)
Section (G) :
G-1. (C) G-2. (A) G-3. (B) G-4. (A) G-5. (D) G-6. (A) G-7. (D)
PART - II
1. (A) (q), (B) (r), (C) (s), (D) (p)
2. (A) r, (B) s, (C) p, (D) p, (E) q, (F) q
3. (A) r, (B) p, (C) s, (D) q
4. (A) r, (B) s, (C) p, (D) q
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Fundamentals of Mathematics-I
EXERCISE - 2
PART - I
1. (A) 2. (B) 3. (B) 4. (A) 5. (C) 6. (D) 7. (B)
8. (B) 9. (A) 10. (B) 11. (A) 12. (A) 13. (D) 14. (A)
15. (C)
PART - II
1. 02.20 2. 12.28 or 12.29 3. 01.00 4. 22.00 5. 02.00 6. 01.00
7. 10.00 8. 03.82 or 03.83 9. 12.50 10. 03.00 11. 00.25 12. 01.00
13. 00.01 14. 01.00 15. 05.66 or 05.67
PART - III
1. (BD) 2. (ABCD) 3. (ABC) 4. (ABCD) 5. (ABCD) 6. (AB)
7. (CD) 8. (ABCD) 9. (BC) 10. (ABD)
PART - IV
1. (D) 2. (B) 3. (C) 4. (C) 5. (A) 6. (B) 7. (BC)
8. (A) 9. (B)
EXERCISE - 3
PART - I
1. (ABCD) 2. (C) 3. x = 8 4. x = 3 or – 3 5. (B)
6. (A) (p), (r), (s) ; (B) (q), (s) ; (C) (q), (s) ; (D) (p), (r), (s)
7. (C) 8. (4) 9. (ABC) 10. (8)
PART - II
1. (4) 2. (1) 3. (3) 4. (2) 5. (3) 6. (3) 7. (3)
8. (1) 9. (2) 10. (2) 11. (2) 12. (4) 13. (1) 14. 29
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Fundamentals of Mathematics-I
LOGARITHM TABLE
0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
0000 0043 0086 0128 0170 5 9 13 17 21 26 30 34 38
10
0212 0253 0294 0334 0374 4 8 12 16 20 24 28 32 36
0414 0453 0492 0531 0569 4 8 12 16 20 23 27 31 35
11
0607 0645 0682 0719 0755 4 7 11 15 18 22 26 29 33
0792 0828 0864 0899 0934 3 7 11 14 18 21 25 28 32
12
0969 1004 1038 1072 1106 3 7 10 14 17 20 24 27 31
1139 1173 1206 1239 1271 36 10 13 16 19 23 26 29
13
1303 1335 1367 1399 1430 37 10 13 16 19 22 25 29
1461 1492 1523 1553 1584 36 9 12 15 19 22 25 28
14
1614 1644 1673 1703 1732 36 9 12 14 17 22 25 26
1761 1790 1818 1847 1875 3 6 9 11 14 17 20 23 26
15
1903 1931 1959 1987 2014 3 6 8 11 14 17 19 22 25
2041 2068 2095 2122 2148 3 6 8 11 14 16 19 22 24
16
2175 2201 2227 2253 2279 3 5 8 10 13 16 18 21 23
2304 2330 2355 2380 2405 3 5 8 10 13 15 18 20 23
17
2430 2455 2480 2504 2529 3 5 8 10 12 15 17 20 22
2553 2577 2601 2625 2648 2 5 7 9 12 14 17 19 21
18
2672 2695 2718 2742 2765 2 4 7 9 11 14 16 18 21
2788 2810 2833 2856 2878 2 4 7 9 11 13 16 18 20
19
2900 2923 2945 2967 2989 2 4 6 8 11 13 15 17 19
20 3010 3032 3054 3075 3096 3118 3139 3160 3181 3201 2 4 6 8 11 13 15 17 19
21 3222 3243 3263 3284 3304 3324 3345 3365 3385 3404 2 4 6 8 10 12 14 16 18
22 3424 3444 3464 3483 3502 3522 3541 3560 3579 3598 2 4 6 8 10 12 14 15 17
23 3617 3636 3655 3674 3692 3711 3729 3747 3766 3784 2 4 6 7 9 11 13 15 17
24 3802 3820 3838 3856 3874 3892 3909 3927 3945 3962 2 4 5 7 9 11 12 14 16
25 3979 3997 4014 4031 4048 4065 4082 4099 4116 4133 2 3 5 7 9 10 12 14 15
26 4150 4166 4183 4200 4216 4232 4249 4265 4281 4298 2 3 5 7 8 10 11 13 15
27 4314 4330 4346 4362 4378 4393 4409 4425 4440 4456 2 3 5 6 8 9 11 13 14
28 4472 4487 4502 4518 4533 4548 4564 4579 4594 4609 2 3 5 6 8 9 11 12 14
29 4624 4639 4654 4669 4683 4698 4713 4728 4742 4757 1 3 4 6 7 9 10 12 13
30 4771 4786 4800 4814 4829 4843 4857 4871 4886 4900 1 3 4 6 7 9 10 11 13
31 4914 4928 4942 4955 4969 4983 4997 5011 5024 5038 1 3 4 6 7 8 10 11 12
32 5051 5065 5079 5092 5105 5119 5132 5145 5159 5172 1 3 4 5 7 8 9 11 12
33 5185 5198 5211 5224 5237 5250 5263 5276 5289 5302 1 3 4 5 6 8 9 10 12
34 5315 5328 5340 5353 5366 5378 5391 5403 5416 5428 1 3 4 5 6 8 9 10 11
35 5441 5453 5465 5478 5490 5502 5514 5527 5539 5551 1 2 4 5 6 7 9 10 11
36 5563 5575 5587 5599 5611 5623 5635 5647 5658 5670 1 2 4 5 6 7 8 10 11
37 5682 5694 5705 5717 5729 5740 5752 5763 5775 5786 1 2 3 5 6 7 8 9 10
38 5798 5809 5821 5832 5843 5855 5866 5877 5888 5899 1 2 3 5 6 7 8 9 10
39 5911 5922 5933 5944 5955 5966 5977 5988 5999 6010 1 2 3 4 5 7 8 9 10
40 6021 6031 6042 6053 6064 6075 6085 6096 6107 6117 1 2 3 4 5 6 8 9 10
41 6128 6138 6149 6160 6170 6180 6191 6201 6212 6222 1 2 3 4 5 6 7 8 9
42 6232 6243 6253 6263 6274 6284 6294 6304 6314 6325 1 2 3 4 5 6 7 8 9
43 6335 6345 6355 6365 6375 6385 6395 6405 6415 6425 1 2 3 4 5 6 7 8 9
44 6435 6444 6454 6464 6474 6484 6493 6503 6513 6522 1 2 3 4 5 6 7 8 9
45 6532 6542 6551 6561 6571 6580 6590 6599 6609 6618 1 2 3 4 5 6 7 8 9
46 6628 6637 6646 6656 6665 6675 6684 6693 6702 6712 1 2 3 4 5 6 7 7 8
47 6721 6730 6739 6749 6758 6767 6776 6785 6794 6803 1 2 3 4 5 5 6 7 8
48 6812 6821 6830 6839 6848 6857 6866 6875 6884 6893 1 2 3 4 4 5 6 7 8
49 6902 6911 6920 6928 6937 6946 6955 6964 6972 6981 1 2 3 4 4 5 6 7 8
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Fundamentals of Mathematics-I
LOGARITHM TABLE
0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
50 6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 1 2 3 3 4 5 6 7 8
51 7076 7084 7093 7101 7110 7118 7126 7135 7143 7152 1 2 3 3 4 5 6 7 8
52 7160 7168 7177 7185 7193 7202 7210 7218 7226 7235 1 2 2 3 4 5 6 7 7
53 7243 7251 7259 7267 7275 7284 7292 7300 7308 7316 1 2 2 3 4 5 6 6 7
54 7324 7332 7340 7348 7356 7364 7372 7380 7388 7396 1 2 2 3 4 5 6 6 7
55 7404 7412 7419 7427 7435 7443 7451 7459 7466 7474 1 2 2 3 4 5 5 6 7
56 7482 7490 7497 7505 7513 7520 7528 7536 7543 7551 1 2 2 3 4 5 5 6 7
57 7559 7566 7574 7582 7589 7597 7604 7612 7619 7627 1 2 2 3 4 5 5 6 7
58 7634 7642 7649 7657 7664 7672 7679 7686 7694 7701 1 1 2 3 4 4 5 6 7
59 7709 7716 7723 7731 7738 7745 7752 7760 7767 7774 1 1 2 3 4 4 5 6 7
60 7782 7789 7796 7803 7810 7818 7825 7832 7839 7846 1 1 2 3 4 4 5 6 6
61 7853 7860 7868 7875 7882 7889 7896 7903 7910 7917 1 1 2 3 4 4 5 6 6
62 7924 7931 7938 7945 7952 7959 9766 7973 7980 7987 1 1 2 3 3 4 5 6 6
63 7993 8000 8007 8014 8021 8028 8035 8041 8048 8055 1 1 2 3 3 4 5 5 6
64 8062 8069 8075 8082 8089 8096 8102 8109 8116 8122 1 1 2 3 3 4 5 5 6
65 8129 8136 8142 8149 8156 8162 8169 8176 8182 8189 1 1 2 3 3 4 5 5 6
66 8195 8202 8209 8215 8222 8228 8235 8241 8248 8254 1 1 2 3 3 4 5 5 6
67 8261 8267 8274 8280 8287 8293 8299 8306 8312 8319 1 1 2 3 3 4 5 5 6
68 8325 8331 8338 8344 8351 8357 8363 8370 8376 8382 1 1 2 3 3 4 4 5 6
69 8388 8395 8401 8407 8414 8420 8426 8432 8439 8445 1 1 2 2 3 4 4 5 6
70 8451 8457 8463 8470 8476 8482 8488 8494 8500 8506 1 1 2 2 3 4 4 5 6
71 8513 8519 8525 8531 8537 8543 8549 8555 8561 8567 1 1 2 2 3 4 4 5 5
72 8573 8579 8585 8591 8597 8603 8609 8615 8621 8627 1 1 2 2 3 4 4 5 5
73 8633 8639 8645 8651 8657 8663 8669 8675 8681 8686 1 1 2 2 3 4 4 5 5
74 8692 8698 8704 8710 8716 8722 8727 8733 8739 8745 1 1 2 2 3 4 4 5 5
75 8751 8756 8762 8768 8774 8779 8785 8791 8797 8802 1 1 2 2 3 3 4 5 5
76 8808 8814 8820 8825 8831 8837 8842 8848 8854 8859 1 1 2 2 3 3 4 5 5
77 8865 8871 8876 8882 8887 8893 8899 8904 8910 8915 1 1 2 2 3 3 4 4 5
78 8921 8927 8932 8938 8943 8949 8954 8960 8965 8971 1 1 2 2 3 3 4 4 5
79 8976 9882 8987 8993 8998 9004 9009 9015 9020 9025 1 1 2 2 3 3 4 4 5
80 9031 9036 9042 9047 9053 9058 9063 9069 9074 9079 1 1 2 2 3 3 4 4 5
81 9085 9090 9096 9101 9106 9112 9117 9122 9128 9133 1 1 2 2 3 3 4 4 5
82 9138 9143 9149 9154 9159 9165 9170 9175 9180 9186 1 1 2 2 3 3 4 4 5
83 9191 9196 9201 9206 9212 9217 9222 9227 9232 9238 1 1 2 2 3 3 4 4 5
84 9243 9248 9253 9258 9263 9269 9274 9279 9284 9289 1 1 2 2 3 3 4 4 5
85 9294 9299 9304 9309 9315 9320 9325 9330 9335 9340 1 1 2 2 3 3 4 4 5
86 9345 9350 9355 9360 9365 9370 9375 9380 9385 9390 1 1 2 2 3 3 4 4 5
87 9395 9400 9405 9410 9415 9420 9425 9430 9435 9440 0 1 1 2 2 3 3 4 4
88 9445 9450 9455 9460 9465 9469 9474 9479 9484 9489 0 1 1 2 2 3 3 4 4
89 9494 9499 9504 9509 9513 9518 9523 9528 9533 9538 0 1 1 2 2 3 3 4 4
90 9542 9547 9552 9557 9562 9566 9571 9576 9581 9586 0 1 1 2 2 3 3 4 4
91 9590 9595 9600 9605 9609 9614 9619 9624 9628 9633 0 1 1 2 2 3 3 4 4
92 9638 9643 9647 9652 9657 9661 9666 9671 9675 9680 0 1 1 2 2 3 3 4 4
93 9685 9689 9694 8699 9703 9708 9713 9717 9722 9727 0 1 1 2 2 3 3 4 4
94 9731 9736 9741 9745 9750 9754 9759 9763 9768 9773 0 1 1 2 2 3 3 4 4
95 9777 9782 9786 9791 9795 9800 9805 9809 9814 9818 0 1 1 2 2 3 3 4 4
96 9823 9827 9832 9836 9841 9845 9850 9854 9859 9863 0 1 1 2 2 3 3 4 4
97 9868 9872 9877 9881 9886 9890 9894 9899 9903 9908 0 1 1 2 2 3 3 4 4
98 9912 9917 9921 9926 9930 9934 9939 9943 9948 9952 0 1 1 2 2 3 3 4 4
99 9956 9961 9965 9969 9974 9978 9983 9987 9991 9996 0 1 1 2 2 3 3 4 4
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Fundamentals of Mathematics-I
ANTILOGARITHM TABLE
0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
.00 1000 1002 1005 1007 1009 1012 1014 1016 1019 1021 0 0 1 1 1 1 2 2 2
.01 1023 1026 1028 1030 1033 1035 1038 1040 1042 1045 0 0 1 1 1 1 2 2 2
.02 1047 1050 1052 1054 1057 1059 1062 1064 1067 1069 0 0 1 1 1 1 2 2 2
.03 1072 1074 1076 1079 1081 1084 1086 1089 1091 1094 0 0 1 1 1 1 2 2 2
.04 1096 1099 1102 1104 1107 1109 1112 1114 1117 1119 0 1 1 1 1 2 2 2 2
.05 1122 1125 1127 1130 1132 1135 1138 1140 1143 1146 0 1 1 1 1 2 2 2 2
.06 1148 1151 1153 1156 1159 1161 1164 1167 1169 1172 0 1 1 1 1 2 2 2 2
.07 1175 1178 1180 1183 1186 1189 1191 1194 1197 1199 0 1 1 1 1 2 2 2 2
.08 1202 1205 1208 1211 1213 1216 1219 1222 1225 1227 0 1 1 1 1 2 2 2 3
.09 1230 1233 1236 1239 1242 1245 1247 1250 1253 1256 0 1 1 1 1 2 2 2 3
.10 1259 1262 1265 1268 1271 1274 1276 1279 1282 1285 0 1 1 1 1 2 2 2 3
.11 1288 1291 1294 1297 1300 1303 1306 1309 1312 1315 0 1 1 1 2 2 2 2 3
.12 1318 1321 1324 1327 1330 1334 1337 1340 1343 1346 0 1 1 1 2 2 2 2 3
.13 1349 1352 1355 1358 1361 1365 1368 1371 1374 1377 0 1 1 1 2 2 2 3 3
.14 1380 1384 1387 1390 1393 1396 1400 1403 1406 1409 0 1 1 1 2 2 2 3 3
.15 1413 1416 1419 1422 1426 1429 1432 1435 1439 1442 0 1 1 1 2 2 2 3 3
.16 1445 1449 1452 1455 1459 1462 1466 1469 1472 1476 0 1 1 1 2 2 2 3 3
.17 1479 1483 1486 1489 1493 1496 1500 1503 1507 1510 0 1 1 1 2 2 2 3 3
.18 1514 1517 1521 1524 1528 1531 1535 1538 1542 1545 0 1 1 1 2 2 2 3 3
.19 1549 1552 1556 1560 1563 1567 1570 1574 1578 1581 0 1 1 1 2 2 3 3 3
.20 1585 1589 1592 1596 1600 1603 1607 1611 1614 1618 0 1 1 1 2 2 3 3 3
.21 1622 1626 1629 1633 1637 1641 1644 1648 1652 1656 0 1 1 2 2 2 3 3 3
.22 1660 1663 1667 1671 1675 1679 1683 1687 1690 1694 0 1 1 2 2 2 3 3 3
.23 1698 1702 1706 1710 1714 1718 1722 1726 1730 1734 0 1 1 2 2 2 3 3 4
.24 1738 1742 1746 1750 1754 1758 1762 1766 1770 1774 0 1 1 2 2 2 3 3 4
.25 1778 1782 1786 1791 1795 1799 1803 1807 1811 1816 0 1 1 2 2 2 3 3 4
.26 1820 1824 1828 1832 1837 1841 1845 1849 1854 1858 0 1 1 2 2 3 3 3 4
.27 1862 1866 1871 1875 1879 1884 1888 1892 1897 1901 0 1 1 2 2 3 3 3 4
.28 1905 1910 1914 1919 1923 1928 1932 1936 1841 1845 0 1 1 2 2 3 3 4 4
.29 1950 1954 1959 1963 1968 1972 1977 1982 1986 1991 0 1 1 2 2 3 3 4 4
.30 1995 2000 2004 2009 2014 2018 2023 2028 2032 2037 0 1 1 2 2 3 3 4 4
.31 2042 2046 2051 2056 2061 2065 2070 2075 2080 2084 0 1 1 2 2 3 3 4 4
.32 2089 2094 2099 2104 2109 2113 2118 2123 2128 2133 0 1 1 2 2 3 3 4 4
.33 2138 2143 2148 2153 2158 2163 2168 2173 2178 2183 0 1 1 2 2 3 3 4 4
.34 2188 2193 2198 2203 2208 2213 2218 2223 2328 2234 1 1 2 2 3 3 4 4 5
.35 2239 2244 2249 2254 2259 2265 2270 2275 2280 2286 1 1 2 2 3 3 4 4 5
.36 2291 2296 2301 2307 2312 2317 2323 2328 2333 2339 1 1 2 2 3 3 4 4 5
.37 2344 2350 2355 2360 2366 2371 2377 2382 2388 2393 1 1 2 2 3 3 4 4 5
.38 2399 2404 2410 2415 2421 2432 2427 2432 2443 2449 1 1 2 2 3 3 4 4 5
.39 2455 2460 2466 2472 2477 2483 2489 2495 2500 2506 1 1 2 2 3 3 4 5 5
.40 2512 2518 2523 2529 2535 2541 2547 2553 2559 2564 1 1 2 2 3 4 4 5 5
.41 2570 2576 2582 2588 2594 2600 2606 2612 2618 2624 1 1 2 2 3 4 4 5 5
.42 2630 2636 2642 2649 2655 2661 2667 2673 2679 2685 1 1 2 2 3 4 4 5 6
.43 2692 2698 2704 2710 2716 2723 2729 2735 2742 2748 1 1 2 3 3 4 4 5 6
.44 2754 2761 2767 2773 2780 2786 2793 2799 2805 2812 1 1 2 3 3 4 4 5 6
.45 2818 2825 2831 2838 2844 2851 2858 2864 2871 2877 1 1 2 3 3 4 5 5 6
.46 2884 2891 2897 2904 2911 2917 2924 2931 2938 2944 1 1 2 3 3 4 5 5 6
.47 2951 2958 2965 2972 2979 2985 2992 2999 3006 3013 1 1 2 3 3 4 5 5 6
.48 3020 3027 3034 3041 3048 3055 3062 3069 3076 3083 1 1 2 3 4 4 5 6 6
.49 3090 3097 3105 3112 3119 3126 3133 3141 3148 3155 1 1 2 3 4 4 5 6 6
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Fundamentals of Mathematics-I
ANTILOGARITHM TABLE
0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
.50 3162 3170 3177 3184 3192 3199 3206 3214 3221 3228 1 1 2 3 4 4 5 6 7
.51 3236 3243 3251 3258 3266 3273 3281 3289 3296 3304 1 2 2 3 4 5 5 6 7
.52 3311 3319 3327 3334 3342 3350 3357 3365 3373 3381 1 2 2 3 4 5 5 6 7
.53 3388 3396 3404 3412 3420 3428 3436 3443 3451 3459 1 2 2 3 4 5 6 6 7
.54 3467 3475 3483 3491 3499 3508 3516 3524 3532 3540 1 2 2 3 4 5 6 6 7
.55 3548 3556 3565 3573 3581 3589 3597 3606 3614 3622 1 2 2 3 4 5 6 6 7
.56 3631 3639 3648 3656 3664 3673 3681 3690 3698 3707 1 2 3 3 4 5 6 7 8
.57 3715 3724 3733 3741 3750 3758 3767 3776 3784 3793 1 2 3 3 4 5 6 7 8
.58 3802 3811 3819 3828 3837 3846 3855 3864 3873 3882 1 2 3 4 4 5 6 7 8
.59 3890 3899 3908 3917 3926 3936 3945 3954 3963 3972 1 2 3 4 5 5 6 7 8
.60 3981 3990 3999 4009 4018 4027 4036 4046 4055 4064 1 2 3 4 5 6 6 7 8
.61 4074 4083 4093 4102 4111 4121 4130 4140 4150 4156 1 2 3 4 5 6 7 8 9
.62 4169 4178 4188 4198 4207 4217 4227 4236 4246 4256 1 2 3 4 5 6 7 8 9
.63 4266 4276 4285 4295 4305 4315 4325 4335 4345 4355 1 2 3 4 5 6 7 8 9
.64 4365 4375 4385 4395 4406 4416 4426 4436 4446 4457 1 2 3 4 5 6 7 8 9
.65 4467 4477 4487 4498 4508 4519 4529 4539 4550 4560 1 2 3 4 5 6 7 8 9
.66 4571 4581 4592 4603 4613 4624 4634 4645 4656 4667 1 2 3 4 5 6 7 9 10
.67 4677 4688 4699 4710 4721 4732 4742 4753 4764 4775 1 2 3 4 5 7 8 9 10
.68 4786 4797 4808 4819 4831 4842 4853 4864 4875 4887 1 2 3 4 6 7 8 9 10
.69 4898 4909 4920 4932 4943 4955 4966 4977 4989 5000 1 2 3 5 6 7 8 9 10
.70 5012 5023 5035 5047 5058 5070 5082 5093 5105 5117 1 2 4 5 6 7 8 9 11
.71 5129 5140 5152 5164 5176 5188 5200 5212 5224 5236 1 2 4 5 6 7 8 10 11
.72 5248 5260 5272 5284 5297 5309 5321 5333 5346 5358 1 2 4 5 6 7 9 10 11
.73 5370 5383 5395 5408 5420 5433 5445 5458 5470 5483 1 3 4 5 6 8 9 10 11
.74 5495 5508 5521 5534 5546 5559 5572 5585 5598 5610 1 3 4 5 6 8 9 10 12
.75 5623 5636 5649 5662 5675 5689 5702 5715 5728 5741 1 3 4 5 7 8 9 10 12
.76 5754 5768 5781 5794 5808 5821 5834 5848 5861 5875 1 3 4 5 7 8 9 11 12
.77 5888 5902 5916 5929 5943 5957 5970 5984 5998 6012 1 3 4 5 7 8 10 11 12
.78 6026 6039 6053 6067 6081 6095 6109 6124 6138 6152 1 3 4 6 7 8 10 11 13
.79 6166 6180 6194 6209 6223 6237 6252 6266 6281 6295 1 3 4 6 7 9 10 11 13
.80 6310 6324 6339 6353 6368 6383 6397 6412 6427 6442 1 3 4 6 7 9 10 12 13
.81 6457 6471 6486 6501 6516 6531 6546 6561 6577 6592 2 3 5 6 8 9 11 12 14
.82 6607 6622 6637 6653 6668 6683 6699 6714 6730 6745 2 3 5 6 8 9 11 12 14
.83 6761 6776 6792 6808 6823 6839 6855 6871 6887 6902 2 3 5 6 8 9 11 13 14
.84 6918 6934 6950 6966 6982 6998 7015 7031 7047 7063 2 3 5 6 8 10 11 13 15
.85 7079 7096 7112 7129 7145 7161 7178 7194 7211 7228 2 3 5 7 8 10 12 13 15
.86 7244 7261 7278 7295 7311 7328 7345 7362 7379 7396 2 3 5 7 8 10 12 13 15
.87 7413 7430 7447 7464 7482 7499 7516 7534 7551 7568 2 3 5 7 9 10 12 14 16
.88 7586 7603 7621 7638 7656 7674 7691 7709 7727 7745 2 4 5 7 9 11 12 14 16
.89 7762 7780 7798 7816 7834 7852 7870 7889 7907 7925 2 4 5 7 9 11 13 14 16
.90 7943 7962 7980 7998 8017 8035 8054 8072 8091 8110 2 4 6 7 9 11 13 15 17
.91 8128 8147 8166 8185 8204 8222 8241 8260 8279 8299 2 4 6 8 9 11 13 15 17
.92 8318 8337 8356 8375 8395 8414 8433 8453 8472 8492 2 4 6 8 10 12 14 15 17
.93 8511 8531 8551 8570 8590 8610 8630 8650 8670 8690 2 4 6 8 10 12 14 16 18
.94 8710 8730 8750 8770 8790 8810 8831 8851 8872 8892 2 4 6 8 10 12 14 16 18
.95 8913 8933 8954 8974 8995 9016 9036 9057 9078 9099 2 4 6 8 10 12 15 17 19
.96 9120 9141 9162 9183 9204 9226 9247 9268 9290 9311 2 4 6 8 11 13 15 17 19
.97 9333 9354 9376 9397 9419 9441 9462 9484 9506 9528 2 4 7 9 11 13 15 17 20
.98 9550 9572 9594 9616 9638 9661 9683 9705 9727 9750 2 4 7 9 11 13 16 18 20
.99 9772 9795 9817 9849 9863 9886 9908 9931 9954 9977 2 5 7 9 11 14 16 18 20
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Fundamentals of Mathematics-I
Marked questions are recommended for Revision.
PART - I : SUBJECTIVE QUESTIONS
Section (A) : Representation of sets, Types of sets, subset and power set
A-1. State wether the following collections is a set or not ?
(i) The collection of natural numbers between 2 and 20
(ii) The collection of numbers which satisfy the equation x2 –5x + 6 = 0
(iii) The collection of prime numbers between 1 and 100.
(iv) The collection of all intelligent women in Jalandhar.
A-2. Write the following set in tabular form
(i) A = {x : x is a positive prime < 10}
(ii) B = {x : x = 3, x , 1 3}
A-3. Write the following set in builder form
(i) set of all rational number
(ii) {2, 5, 10, 17, 26, 37, .........}
A-4. Identifiy type of set in terms of empty/singleton/finite/infinite
(i) {x : x is a real number and x2 – 1 = 0}
(ii) {x : x is a real number and x2 + 1 = 0}
(iii) {x : x is positive real number and x2 – 9 = 0}
(iv) {x : x is a real number and x2 + 2x + 2 0}
A-5. Write power set of the set A = {, {}}.
Section (B) : Operations on sets, Law of Algebra of sets
B-1. Given the sets A = {1, 2, 3}, B = {3, 4}, C = {4, 5, 6}, then find the following
(i) A (B C) (ii) A – (B C) (iii) (B C) – A
B-2. Find the smallest set A such that A {1, 2} = {1, 2, 3, 5, 9}
B-3. If aN = {ax : x N} and bN cN = dN, where b, c N, b 2, c 2 are relatively prime, then write 'd' in
terms of b and c.
B-4. Sets A and B have 3 and 6 elements respectively. What can be the minimum and maximum number of
elements in
(i) A B (ii) A B
Section (C) : Cardinal number Problems
C-1. Let n(U) = 700, n(A) = 200, n(B) = 300 and n(A B) = 100, then find n(A' B')
C-2. In a college of 300 students, every student reads 5 newspapers and every newspaper is read by 60
students. Find the number of newspaper.
C-3. In a town of 10,000 families it was found that 40% families buy newspaper A, 20% families buy
newspaper B and 10% families buy newspaper C, 5% families buy A and B, 3 % buy B and C and 4%
buy A and C. If 2% families buy all the three news papers, then find number of families which buy
newspaper A only.
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Fundamentals of Mathematics-I
C-4. In a survery, it was found that 21 persons liked product A, 26 liked product B and 29 liked product C. If
14 persons liked products A and B, 12 liked products C and A, 13 persons liked products B and C and
8 liked all the three products then
(i) Find the number of persons who liked atleast one product
(ii) The number of persons who like the products A and B but not C
Section (D) : Rational Inequalities
D-1. Solve the following rational in equalities
(x –1)(x 2) (1– x)3 (x 2)4
(i) < 0 (ii) 2 0 (x – 3)(x 3) (x 9) (x – 8)
2 2 3 x 4 x 4 ( 2 x )( x 3 ) 0 (iii) 0 (iv) ( x 1)( x 2 3 x 4 ) 2 x 2 x 1
2 ( x 2)( x 2 x 1)
(v) 2 0 4 3 x x
D-2. Solve the following Inequalities
7 x 5 14x 9x 30 (i) 4 (ii)
8 x 3 x 1 x 4
2 (x –1)(x – 2)(x – 3) x 2
(iii) 1 (iv) 2
(x 1)(x 2)(x 3) x 2 1
D-3. Solve the following rational in equalities
(x 2 – 3x 1)3 2 x 2 3 x 459
(i) 0 (ii) 1
(x 1)(x 2) x 2 1
x 2 5 x 12 x 4 x 2 1
(iii) 3 (iv) 2 0 2 x 4 x 5 x 4 x 5
D-4. Solve the following rational in equalities
(i) x4 – 5x2 + 4 0 (ii) x4 – 2x2 – 63 0
(iii) (x2+3x+1) (x2+3x–3) 5
x – 1
D-5. If 1 < 7 then find the range of
x 2
1
(i) x (ii) x2 (iii)
x
(3 x – 5 x )(x – 2)
D-6. Find the number of positive integral value of x satisfying the inequality 2 0 (x 5x 2)
Section (E) : Logarithmic Properties
E-1. Find the value of
log 5 2 log3 7 log2 5 (i) log105.log1020 + (log102)2 (ii) 5 9 8
1
3 log7 5 1 1
(iii) 5 ( log10 0.1) (iv) log0.75 log2 0.125
1 log7 2
1 –log1/ 5 7 log3 5 log5 7 log3 7 log5 3 (v) + 5 (vi) 7 3 5 7
49
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Fundamentals of Mathematics-I
E-2. Which of the following numbers are positive/negative
(i) log 2 (ii) log1/7(2) (iii) log1/3(1/5) 3
(iv) log3(4) (v) log7(2.11) (vi) log3 ( 7 2)
2 1 2· 3 3
(vii) log4 (viii) log3 (ix) log10 (log10 9) 2 1 3
E-3. Let log102 = a and log103 = b then determine the following logarithms in terms of a and b.
2
(i) log10 sin (ii) log1004 + 2 log10027
3
(iii) log29 + log38 (iv) log 45 144
4 3 2 1
E-4. (i) Let n = 75600, then find the value of
log2 n log3 n log5 n log7 n
(ii) If log2 log3 log4 (x) 0 and log3 log4 log2 (y) 0 and log4 log2 log3 (z) 0 then find
the sum of x, y and z is
1
(iii) Suppose n be an integer greater than 1. let an = . Suppose b = a2 + a3 + a4 + a5 and
logn 2002
c = a10 + a11 + a12 + a13 + a14. Then find the value of (b – c)
E-5. Show that the number log27 is an irrational number.
loga logb logc
E-6. If , show that aa. bb. cc = 1.
b c c a a b
Section (F) : Logarithmic Equation
F-1. Solve the following equations :
(i) logx(4x – 3) = 2 (ii) log2(log3(x2 – 1)) = 0
(iii) 4log 2 x 2x 3 0 (iv) log4 (log2x) + log2 (log4x) = 2.
1 x
(v) log3 log9 x 9 = 2x. (vi) 2log4 (4 – x) = 4 – log2 (– 2 – x). 2
(log
x 0.5log x (x 2 x) (vii) x 9 4 . 2 x) 4 (viii) x 3log
F-2. (i) Find the product of roots of the equation
(log3x)2 – 2(log3x) – 5 = 0
(ii) Find sum of roots of the equation 4x – 7.2x + 6 = 0
10 x 2 (iii) Solve for x : xlog 10 x 2 10log
log10 x 5
10 x (iv) Solve for x : x 3 = 105 log
Section (G) : Logarithmic inequalities
G-1. Solve the following inequalities
2 3 2 (i) log 5 2x x 1 (ii) log 1 ( x 5 x 6 ) 1
2 8 8
2 x 6 2
(iii) log7 0 (iv) log1/4(2 – x) > log1/4 2 x 1 x 1
(v) log1/3(2x+2 – 4x) – 2
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4x 6
G-2. Find the number of integers satisfying log1/5 0
x
G-3 Solve the inequalities
(i) (log.5x)2 + log.5x – 2 0 (ii) 15x – 25.3x – 9.5x + 225 0
3 x–2 2 x
(iii) 8. x x > 1 +
3 – 2 3
G-4. Solve the following inequalities :
1
2 (i) logx(4x – 3) 2 (ii) log(3x 21) < 2
(iii) logx² (2 + x) < 1
PART - II : ONLY ONE OPTION CORRECT TYPE
Section (A) : Representation of sets, Types of sets, subset and power set
A-1. The set of intelligent students in a class is-
(A) a null set (B) a singleton set
(C) a finite set (D) not a well defined collection
A-2. The set A = {x : x R, x2 = 16 and 2x = 6} is
(A) Null set (B) Singleton set
(C) Infinite set (D) not a well defined collection
A-3. If A = {x : –3 < x < 3, x Z} then the number of subsets of A is –
(A) 120 (B) 30 (C) 31 (D) 32
A-4. Which of the following are true ?
(A) [3, 7] (2, 10) (B) (0, ) (4, ) (C) (5, 7] [5, 7) (D) [2, 7] (2.9, 8)
A-5. The number of subsets of the power set of set A = {7, 10, 11} is
(A) 32 (B) 16 (C) 64 (D) 256
A-6. Which of the folowing sets is an infinite set ?
(A) Set of divisors of 24
(B) Set of all real number which lie between 1 and 2
(C) Set of all humman beings living in India.
(D) Set of all three digit natural numbers
Section (B) : Operations on sets, Law of Algebra of sets
B-1. Let A = {x : x R, –1 < x < 1} , B = {x : x R, x 0 or x 2} and A B = R – D, then the set D is
(A) {x : 1 < x 2} (B) {x : 1 x < 2} (C) {x : 1 x 2} (D) {x : 1 < x < 2}
B-2. If A = {2, 3, 4, 8, 10}, B = {3, 4, 5, 10, 12}, C = {4, 5, 6, 12, 14} then (A B) (A C) is equal to
(A) {3, 4, 10} (B) {2, 8, 10} (C) {4, 5, 6} (D) {3, 5, 14}
B-3. The shaded region in the given figure is
(A) A (B C) (B) A (B C) (C) A (B – C) (D) A – (B C)
B-4. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 5}, B = {6, 7}, then A Bis
(A) B (B) A (C) A (D) B
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B-5. If A = {x : x = 4n + 1, n 5, n N} and B = {3n : n 8, n N}, then A – (A – B) is :
(A) {9, 21} (B) {9, 12} (C) {6, 12} (D) {6, 21}
B-6. A B = A B iff :
(A) A B (B) A = B (C) A B (D) A B
B-7. Consider the following statements :
1. N (B Z) = (N B) Z for any subset B of R, where N is the set of positive integers, Z is
the set of integers, R is the set of real numbers.
2. Let A = {n N : 1 n 24, n is a multiple of 3}. There exists no subset B of N such that the
number of elemets in A is equal to the number of elements in B.
Which of the above statements is/are correct ?
(A) 1 only (B) 2 only (C) Both 1 and 2 (D) Neither 1 nor 2
B-8.Which of the following venn-diagrams best represents the sets of females, mothers and doctors ?
(A) (B) (C) (D)
Section (C) : Cardinal number Problems
C-1. Let A and B be two sets. Then
(A) n(A B) n(A B) (B) n(A B) n(A B)
(C) n(A B) = n(A B) (D) can't be say
C-2. In a city 20 percent of the population travels by car, 50 percent travels by bus and 10 percent travels by
both car and bus. Then persons travelling by car or bus is
(A) 80 percent (B) 40 percent (C) 60 percent (D) 70 percent
C-3. A class has 175 students. The following data shows the number of students obtaining one or more
subjects : Mathematics 100, Physics 70, Chemistry 40, Mathematics and Physics 30, Mathematics and
Chemistry 28, Physics and Chemistry 23, Mathematics & Physics & Chemistry 18. How many students
have offered Mathematics alone ?
(A) 35 (B) 48 (C) 60 (D) 22
C-4. 31 candidates appeared for an examination, 15 candidates passed in English, 15 candidates passed in
Hindi, 20 candidates passed in Sanskrit. 3 candidates passed only in English. 4. candidates passed
only in Hindi, 7 candidates passed only in Sanskrit. 2 candidates passed in all the three subjects How
many candidates passed only in two subjects ?
(A) 17 (B) 15 (C) 22 (D) 14
Section (D) : Rational Inequalities
x 4 3x 3 2x 2
D-1. The complete solution set of the inequality 0 is:
x 2 x 30
(A) (, 5) (1, 2) (6, ) {0} (B) (, 5) [1, 2] (6, ) {0}
(C) (, 5] [1, 2] [6, ) {0} (D) (, 5] [1, 2] [6, )
D-2. Number of positive integral values of x satisfying the inequality
(x 4)2017 . (x 8)2016 (x 1)
0 is
x 2016 (x 2)3 . (x 3)5 . (x 6) (x 9)2018
(A) 0 (B) 1 (C) 2 (D) 3
x 2 1
D-3. The number of prime numbers satisfying the inequality < 3 is
2x 5
(A) 1 (B) 2 (C) 3 (D) 4
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x 2 1
D-4. The complete solution of 0 & x2 – 5x + 2 0 is :
x 3
17 5 17 5 17 (A) x 5 , (B) x 1,
2 2 2
(C) x (–3, –1] (D) x (–3, –1] [1, )
D-5 The solution of the inequality 2x – 1 x2 + 3 x – 1 is
(A) x R (B) [–2, 2] (C) (–2, 2) (D) x
D-6. The number of the integral solutions of x2 + 9 < (x + 3)2 < 8x + 25 is :
(A) 1 (B) 3 (C) 4 (D) 5
2 1 2x 1
D-7. Number of non-negative integral values of x satisfying the inequality 0 is
x 2 x 1 x 1 x 3 1
(A) 0 (B) 1 (C) 2 (D) 3
Section (E) : Logarithmic Properties
E-1. If a4 · b5 = 1 then the value of loga(a5b4) equals
(A) 9/5 (B) 4 (C) 5 (D) 8/5
1 1 1
E-2. + + has the value equal to
1 logb a logb c 1 logc a logc b 1 loga b loga c
1
(A) abc (B) (C) 0 (D) 1
abc
1 1 1
E-3. + + has the value equal to :
log abc log abc log abc
bc c a ab
(A) 1/2 (B) 1 (C) 2 (D) 4
E-4. (log210) . (log280) – (log25) . (log2160) is equal to :
(A) log25 (B) log220 (C) log210 (D) log216
3
a2 1 log a log27 21/ 4 2 3 2a
E-5. The ratio simplifies to :
4log49 a 7 a 1
(A) a2 a 1 (B) a2 + a 1 (C) a2 a + 1 (D) a2 + a + 1
E-6. If loga(ab) = x, then logb(ab) is equal to
1 x x x
(A) (B) (C) (D)
x 1 x 1 x x 1
logp logq (logr x) = 0 then 'p' equalsE-7. 10 = 1 and logq logr (logp x)
(A) rq/r (B) rq (C) 1 (D)rr/q
E-8. Which one of the following is the smallest?
2 1 3 1 (A) log10 (B) log10 (C) (D) log10 log10
E-9. log10(log23) + log10(log34) + log10(log45) + ........ + log10(log10231024) simplifies to
(A) a composite (B) a prime number
(C) rational which is not an integer (D) an integer
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Section (F) : Logarithmic Equation
F-1. The sum of all the solutions to the equation 2 log10 x – log10(2x – 75) = 2
(A) 30 (B) 350 (C) 75 (D) 200
F-2. If the solution of the equation logx(125x) . log252 x = 1 are and ( < ). Then the value of 1/is :
(A) 5 (B) 25 (C) 125 (D) 625
9 2
F-3. The positive integral solution of the equation logx 5 + logx5x = + log x 5 is : 4
(A) Composite number (B) Prime number
(C) Even number (D) Divisible by 3
F-4. The expression logp , where p 2, p N ; n N when simplified is
(A) independent of p (B) independent of p and of n
(C) dependent on both p and n (D) positive
1
= . Then the value of 1000 x is equal to 2 8F-5. If logx log18 3
(A) 8 (B) 1/8 (C) 1/125 (D) 125
F-6. Number of real solutions of the equation log10 x = log10 x 2 is :
(A) zero (B) exactly 1 (C) exactly 2 (D) 4
x 2
x isF-7. The correct graph of y = x log
(A) (B) (C) (D)
Section (G) : Logarithmic inequalities
G-1. The solution set of the inequality log (x2 3x 2) 2 is
sin
3
5 1 5 1,2 1, 2 (A) (B) 1, (C) , 1 2, (D) 2 2 2 2
G-2. If log0.3 (x 1) < log0.09 (x 1), then x lies in the interval
3
(A) (2, ) (B) (1, 2) (C) ( 2, 1) (D) 1, 2
G-3. Solution set of the inequality 2 log2 (x2 + 3x) 0 is :
(A) [ 4, 1] (B) [ 4, 3) (0, 1]
(C) ( 3) (1, ) (D) ( 4) [1, )
G-4. If log0.5 log5 (x2 – 4) > log0.51, then ‘x’ lies in the interval
(A) (– 3, – 5 ) ( 5 , 3) (B) (– 3, – 5 ) ( 5 , 2)
(C) ( 5 , 3 5 ) (D)
x 2 2xG-5. The set of all solutions of the inequality (1/2) < 1/4 contains the set
(A) (– , 0) (B) (– , 1) (C) (1, ) (D) (3, )
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G-6. The number of positive integers not satisfying the inequality log2(4x – 2.2x + 17) > 5.
(A) 2 (B) 3 (C) 4 (D) 1
G-7. The set of all the solutions of the inequality log1 – x (x – 2) – 1 is
(A) (– , 0) (B) (2, ) (C) (– , 1) (D)
PART - III : MATCH THE COLUMN
1. Match the set P in column one with its super set Q in column II
Column – (set P) Column– (set Q)
(A) [32n – 8n – 1 : n N} (p) {49 (n – 1) : n N}
(B) {23n – 1 : n N} (q) {64 (n –1) : n N}
(C) {32n –1 : n N} (r) {7n : n N}
(D) {23n – 7n – 1 : n N} (s) {8n : n N}
2. Column-I Column-II
(A) If a = 3 8 2 7 8 2 7 , b = (42)(30) 36 (p) – 1
then the value of logab is equal to
(B) If a = 4 2 3 – 4 2 3 , b = 11 6 2 – 11 6 2 , (q) 1
then the value of logab is equal to
(C) If a = 3 2 2 , b = 3 2 2 (r) 2
then the value of logab is equal to
2 2 3 (D) If a = 7 7 1 , b = 7 7 1 , (s)
2
then the value of logab is equal to
(E) The number of zeroes at the end of the product of first 20 (t) None
prime numbers, is
(F) The number of solutions of 22x – 32y = 55, in which x and y
are integers, is
3. Column-I Column-II
(A) When the repeating decimal 0.363636............. is written as a rational (p) 4
fraction in the simplest form, the sum of the numerator
and denominator is
(B) Given positive integer p, q and r with p = 3q · 2r and 100 < p < 1000. (q) 0
The difference between maximum and minimum values of (q + r), is
(C) If log8a + log8b = (log8a)(log8b) and logab = 3, then the value of 'a' is (r) 15
(D) If P = 3 log3 2 – 2 log2 3 then value of P is (s) 16
4. Column-I Column-II
(A) Anti logarithm 0.6 to the base 27 has the value equal to (p) 5 of
(B) Characteristic of the logarithm of 2008 to the base 2 is
(C) The value of b satisfying the equation, (q) 7
loge2 · logb625 = log1016 · loge10 is
(D) Number of naughts after decimal before a significant figure (r) 9
5 100
comes in the number , is (s) 10
6
(Given log102 = 0.3010 and log103 = 0.4771)
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Marked questions are recommended for Revision.
* Marked Questions may have more than one correct option.
PART - I : ONLY ONE OPTION CORRECT TYPE
1. Let A1, A2 and A3 be subsets of a set X. Which one of the following is correct ?
(A) A1 A2 A3 is the smallest subset of X containing elements of each of A1, A2 and A3
(B) A1 A2 A3 is the smallest subset of X containing either A1 or A2 A3 but not both
(C) The smallest subset of X containing A1 A2 and A3 equals the smallest subset of X containing
both A1 and A2 A3 only if A2 = A3
(D) None of these
2. Let A, B, C be distinct subsets of a universal set U. For a subset X of U, let X ' denote the complement
of X in U.
Consider the following sets :
1. (((A B) C) B) = B C
2. (A B )(A B C) = (A (B C))
Which of the above statements is/are correct ?
(A) 1 only (B) 2 only (C) Both 1 and 2 (D) Neither 1 nor 2
3. In an examination of a certain class, at least 70% of the students failed in Physics, at least 72% failed in
Chemistry, at least 80% failed in Mathematics and at least 85% failed in English. How many at least
must have failed in all the four subjects ?
(A) 9% (B) 7% (C) 15%
(D) Cannot be determined due to insufficient data
4. Let X and Y be two sets.
Statement-1 X (Y X)' =
Statement-2 If X Y has m elements and X Y has n elements then symmetric difference X Y
has m – n elements.
(A) Both the statements are true. (B) Statement- is true, but Statement- is false.
(C) Statement- is false, but Statement- is true. (D) Both the statements are false.
6x 2 5x 3
5. If 4, then the least and the highest values of 4x2 are:
x 2 2x 6
(A) 36 & 81 (B) 9 & 81 (C) 0 & 81 (D) 9 & 36
(x 2 2)( x 2 16)
6. Sum of all the real solutions of the inequality 0 is
(x 4 2)(x 2 9)
(A) 5 (B) 4 (C) 8 (D) 0
7. If logab = 2; logbc = 2 and log3c = 3 + log3a then (a + b + c) equals
(A) 90 (B) 93 (C) 102 (D) 243
3log289 11 log28 17 log1331 784 y8. Let x log125 343 log49 729 and y = 25 , then value of is log1 3 5 x
5 5 4 3
(A) (B) (C) (D)
3 3 5 7
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x 2 3x 2 x(x 3 1)
3x log2 8 (x 1)(x 2 x 1) x 2 9. The expression: reduces to
(x 1)(log2 3)(log3 4)(log4 5)(log5 2)
x 1 x 2 3x 2 3x
(A) (B) (C) (D) x
x 1 (log2 5)x 1 x 1
10. If a, b, c are positive real numbers such that alog 3 7 = 27 ; blog 11 25 = 11 . The value 7 11 = 49 and c log
(log3 7)2 (log7 11)2 (log11 25)2 of a b c equals
(A) 489 (B) 469 (C) 464 (D) 400
11. Consider the statement : x ( – x) < y ( – y) for all x, y with 0 < x < y < 1. The statement is true
(A) if and only if 2 (B) if and only if 2
(C) if and only if –1 (D) for no values of
(x 8) (2 x)
12. The set of values of x satisfying simultaneously the inequalities 0 and
10
log0.3 (log2 5 1) 7
2x 3 31 > 0 is :
(A) a unit set (B) an empty set
(C) an infinite set (D) a set consisting of exactly two elements.
(3 x 4 x ) · n(x 2)
13. The solution set of the inequality 0 is
x 2 3x 4
(A) (–, 0] (4, ) (B) (–2, 0] (4, )
(C) (–1, 0] (4, ) (D) (–2, –1) (– 1, 0] (4, )
14. If log4 {log3 {log2 (x 2 2x a)}} is defined xR, then the set of values of 'a' is
(A) [9, ) (B) [10, ) (C) [15, ) (D) [2, )
15. If log(2x + 3) (6x2 + 23x + 21) = 4 – log(3x + 7) (4x2 + 12x + 9) then value of x is equal to
3 3 1 3
(A) (B) – (C) – (D)
4 4 4 2
PART-II: NUMERICAL VALUE QUESTIONS
INSTRUCTION :
The answer to each question is NUMERICAL VALUE with two digit integer and decimal upto two digit.
If the numerical value has more than two decimal places truncate/round-off the value to TWO decimal
placed.
1. Let U be set with number of elements in it is 2009 and A, B are subsets of U with n (A B) = 280. If
n(A B) = x 13 + x 32 = y 13 + y 32 for some positive integers x1 < y1 < y2 < x2, then find value of
x 2 y 2
x1 y 1
2. Let U be set with number of elements in it is 2009. A is a subset of U with n (A) = 1681 and out of these
1681 elements, exactly 1075 elements belong to a subset B of U. If n (A – B) = m2 + p1 p2 p3 for some
p1 p3 positive integer m and distinct primes p1 < p2 < p3 then for least m find
p2
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3. Let A = {(x, y) : x R, y R, x3 + y3 = 1], B = {(x, y) : x R, y R, x – y = 1}
and C = {(x, y) : x R, y R, x + y = 1}. If A B contains 'p' elements and A C contains 'q' elements
then find (q – p).
4. In a class of 42 students, the number of students studying different subjects are 23 in Mathematics, 24
in Physics, 19 in Chemistry, 12 in Mathematics and Physics 9 in Mathematics and Chemistry, 7 in
Physics and Chemistry and 4 in all the three subjects. Then find number of students who have taken
exactly one subject.
log(a c) log(a 2b c)
5. If c(a – b) = a(b – c) then find the value of (Assume all terms are defined)
log(a c)
6. If logb a. logc a + loga b. logc b + loga c. logb c = 3 ( where a, b, c are different positive real numbers 1),
then find the value of a b c.
7. If 4A + 9B = 10C, where A = log164, B = log39 & C = logx83, then find x.
3 5
8. Let a, b, c, d are positive integers such that logab = and logcd = . If (a – c) = 9, find 2 4
b d
the value of .
a c
9. Find the positive number, x , which satisfies the equation log10 (2x² 21 x + 50) = 2
10. Find the value of x satisfying the equation log 1 (x 1) log 1 (x 1) log 1 (7 x) 1
2 2 2
11. Find sum of roots of equation log 102 x + log10x2 = log 102 2 1
2009 x logx (2010)12. If the product of all solutions of the equation 2009 can be expressed in the lowest
2010
m
form as then the value of (m – n) is
n
13. If the complete solution set of the inequality (log10 x)2 log10 x + 2 is (0, a] [100, ) then find the value
of a.
1 1
14. The complete solution set of the inequality 4 ( x 3 ) , is (–a, ), then determine ‘a’. log4 xx21 log
15. If complete solution set of inequality log1/2 (x + 5)2 > log1/2 (3x – 1)2 is (–,p) (q,r) (s,) then find
p 2 q 2 r 2
s 2
PART - III : ONE OR MORE THAN ONE OPTIONS CORRECT TYPE
1. Let a > 2, a N be a constant. If there are just 18 positive integers satisfying the inequality
(x – a)(x – 2a)(x – a2) < 0 then which of the option(s) is/are correct?
(A) 'a' is composite (B) 'a' is odd
(C) 'a' is greater than 8 (D) 'a' lies in the interval (3, 11)
log3 135 log3 52. Let N = – . Then N is :
log15 3 log405 3
(A) a natural number (B) a prime number (C) a rational number (D) an integer
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2 5 3. Values of x satisfying the equation log5 x + log5x = 1 are x
1
(A) 1 (B) 5 (C) (D) 3
25
4. The equation log 16 + log2x64 = 3 has : x 2
(A) one irrational solution (B) no prime solution
(C) two real solutions (D) one integral solution
2 9
(log3 x) log3 x 5
5. The equation x 2 = 3 3 has
(A) exactly three real solution (B) at least one real solution
(C) exactly one irrational solution (D) complex roots.
2
6. The solution set of the system of equations log3x + log3y = 2 + log32 and log27(x + y) = is :
3
(A) {6, 3} (B) {3, 6} (C) {6, 12} (D) {12, 6}
7. Consider the quadratic equation, (log108)x2 – (log105)x = 2(log210)–1 – x. Which of the following
quantities are irrational.
(A) sum of the roots (B) product of the roots
(C) sum of the coefficients (D) discriminant
8. If loga x = b for permissible values of a and x then identify the statement(s) which can be correct?
(A) If a and b are two irrational numbers then x can be rational.
(B) If a rational and b irrational then x can be rational.
(C) If a irrational and b rational then x can be rational.
(D) If a rational and b rational then x can be rational.
9. Which of the following statements are true
(A) log2 3 < log12 10 (B) log6 5 < log7 8
(C) log326 < log29 (D) log1615 > log1011 > log76
1
10. If log0.1 x 2, then
2
1 1 1
(A) maximum value of x is (B) x lies between and
10 100 10
1 1
(C) minimum value of x is (D) minimum value of x is
10 100
PART - IV : COMPREHENSION
Comprehension # 1 (1 to 3)
In a group of 1000 people, there are 750 people, who can speak Hindi and 400 people, who can speak
Bengali.
1. Number of people who can speak Hindi only is
(A) 300 (B) 400 (C) 500 (D) 600
2. Number of people who can speak Bengali only is
(A) 150 (B) 250 (C) 50 (D) 100
3. Number of people who can speak both Hindi and Bengali is
(A) 50 (B) 100 (C) 150 (D) 200
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Comprehension # 2 (4 to 6)
1 4
Let A denotes the sum of the roots of the equation + = 3.
5 4log4 x 1 log 4 x
B denotes the value of the product of m and n, if 2m = 3 and 3n = 4.
3
C denotes the sum of the integral roots of the equation log3x + (log3x)2 = 1. x
4. The value of A + B equals
(A) 10 (B) 6 (C) 8 (D) 4
5. The value of B + C equals
(A) 6 (B) 2 (C) 4 (D) 8
6. The value of A + C B equals
(A) 5 (B) 8 (C) 7 (D) 4
Comprehension # 3 (Q.7- to Q.9)
A function f(x) = ax(a > 0, a 1, x R) is called an exponential function. Graph of exponential function
can be as follows :
Case - Case -
For a > 1 For 0 < a < 1
7*. Which of the following is correct :
(A) (B)
Y Y
y=3–x y=2–x
y=3–x y=2–x
(C) X (D) X
8. Number of solutions of 3x + x – 2 = 0 is/are :
(A) 1 (B) 2 (C) 3 (D) 4
9. The number of positive solutions of log1/2x = 7x is/are :
(A) 0 (B) 1 (C) 2 (D) 3
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Marked questions are recommended for Revision.
* Marked Questions may have more than one correct option.
PART - I : JEE (ADVANCED) / IIT-JEE PROBLEMS (PREVIOUS YEARS)
1. Indicate all correct alternatives, where base of the log is 2. [ JEE ’89 ]
The equation x(3/4) (logx)² + logx(5/4) = 2 has :
(A) at least one real solution (B) exactly three real solutions
(C) exactly one irrational solution (D) complex roots
2. The number log27 is :
(A) an integer (B) a rational number
(C) an irrational number (D) a prime number [ JEE ’90 ]
3. Find all real numbers x which satisfy the equation
2 log2 log2 x + log1/2 log2 (2 2 x)= 1. [REE – 1999, 6 ]
4. Solve the equation log3/4 log8 (x2 + 7) + log1/2 log1/4 (x2 + 7) 1 = 2. [REE– 2000, 5]
5. The number of solution(s) of log4(x – 1) = log2(x – 3) is/are [IIT-JEE-2002, Scr., (1, 0)/35]
(A) 3 (B) 1 (C) 2 (D) 0
x 2 6x 5
6. Let f(x) = [IIT-JEE 2007, Paper-2, (6, 0), 81]
x 2 5x 6
Column – Column –
(A) If – 1 < x < 1, then f(x) satisfies (p) 0 < f(x) < 1
(B) If 1 < x < 2, then f(x) satisfies (q) f(x) < 0
(C) If 3 < x < 5, then f(x) satisfies (r) f(x) > 0
(D) If x > 5, then f(x) satisfies (s) f(x) < 1
7. Let (x0, y0) be the solution of the following equations
(2x)n2 = (3y)n3
3nx = 2ny .
Then x0 is [IIT-JEE 2011, Paper-1, (3, –1), 80]
1 1 1
(A) (B) (C) (D) 6
6 3 2
1 1 1 1 8. The value of 6 log 3 4 – 4 – 4 – .... is 3 2 3 2 3 2 3 2 2
[IIT-JEE 2012, Paper-1, (4, 0), 70]
9*. If 3x = 4x – 1 , then x = [JEE (Advanced) 2013, Paper-2, (3, –1)/60]
2log3 2 2 1 2log2 3 (A) (B) (C) (D)
2log3 2 – 1 2 – log2 3 1– log4 3 2log2 3 – 1
1 1
2 (log2 9) ( 7) log 4 7 is ___________.10. The value of ((log 2 9)2 ) log
[JEE(Advanced) 2018, Paper-1,(4, –2)/60]
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Fundamentals of Mathematics-I
PART - II : PREVIOUS YEARS PROBLEMS OF MAINS LEVEL
1. If logp x = and logq x = , then the value of logp/q x is [KCET-1997]
– –
(1) (2) (3) (4)
– –
2. If logxa, ax/2 and logbx are in G.P. Then x is equal to [KCET-1998]
(1) loga (logba) (2) loga(logea)+ loga logbb
(3) – loga(logab) (4) none of these
3. If logx 256 = 8/5, then x is equal to [KCET-2000]
(1) 64 (2) 16 (3) 32 (4) 8
4. If log 2, log(2x – 1) and log (2x + 3) are in A.P., then x is equal to [KCET-2000]
(1) 5/2 (2) log25 (3) log23 (4) log32
5. The number log27 is [DCE-2000]
(1) an integer (2) a rational (3) an irrational (4) a prime number
6. The roots of the equation log2(x2 – 4x + 5) = (x – 2) are [KCET-2001]
(1) 4, 5 (2) 2, – 3 (3) 2, 3 (4) 3, 5
1 1 1
7. If x = 198 !, then value of the expression .... equals
log2 x log3 x log198 x
[DCE-2005]
(1) –1 (2) 0 (3) 1 (4) 198
8. If log0.3 (x – 1) < log0.09 (x – 1), then x lies in the intervel [DCE-2006]
(1) (2, ) (2) (1, 2) (3) (–2, –1) (4) none of these
9. If A, B and C are three sets such that A B = A C and A B = A C, then
[AIEEE-2009, (4, – 1), 144]
(1) A = C (2) B = C (3) A B = (4) A = B
10. Let X = {1, 2, 3, 4, 5}. The number of different ordered pairs (Y, Z) that can formed such that Y X, Z
X and Y Z is empty, is : [AIEEE-2012, (4, – 1), 120]
(1) 52 (2) 35 (3) 25 (4) 53
11. If X = {4n – 3n – 1 : n N} and Y = {9(n – 1) : n N}, where N is the set of natural numbers, then X Y
is equal to [JEE(Main) 2014, (4, – 1), 120]
(1) X (2) Y (3) N (4) Y – X
12. The sum of all real values of x satisfying the equation ( x 2 – 5 x 5) x 2 4 x – 60 = 1 is
[JEE(Main) 2016, (4, – 1), 120]
(1) – 4 (2) 6 (3) 5 (4) 3
13. In a class 140 students numbered 1 to 140, all even numbered students opted Mathematics course,
those whose number is divisible by 3 opted Physics course and those whose number is divisible 5
opted Chemistry course. Then the number of student who did not opt for any of the three courses is :
[JEE(Main) 2019, Online (10-01-19),P-1 (4, – 1), 120]
(1) 38 (2) 42 (3) 102 (4) 1
14. Let X = {n N: 1 n 50}. If A = {n X: n is a multiple of 2}; B = {n X: n is a multiple of 7}, then the
number of elements in the smallest subset of X containing both A and B is__________.
[JEE(Main) 2020, Online (07-01-20),P-2 (4, 0), 120]
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Fundamentals of Mathematics-I
EXERCISE - 1
PART -I
Section (A) :
A-1. (i) Yes (ii) Yes (iii) Yes (iv) No
A-2. (i) {2,3,5,7} (ii) {3, 4, 5, 6, 7, 8, 9}
p
A-3. (i) {x : x = , p , q N} (ii) {x : x = + 1, N}
q
A-4. (i) Finite (ii) Finite and empty
(iii) Singleton & finite (iv) Infinite
A-5. {, {}, {{}}, A}
Section (B) :
B-1. (i) {1, 2, 3, 4} (ii) {1, 2, 3} (iii) {4, 5, 6}
B-2. {3, 5, 9}
B-3. d = bc
B-4. (i) minimum n(A B) = 0, maximum n(A B) = 3
(ii) minimum n(A B) = 6, maximum n(A B) = 9
Section (C) :
C-1. 300 C-2. 25 C-3. 3300 C-4. (i) 45 (ii) 6
Section (D) :
D-1. (i) (–3,–2) (1,3) (ii) {–2} [1, 8)
(iii) (–, –2)(–2, –1/2)(1, ) (iv) [ 2, 1) 1, 2 3, 4
(v) (– , – 2] (–1, 4)
D-2. (i) (–17/25, –3/8) (ii) x (–6, –1) (1, 4)
(iii) (–3,–2) (–1,) (iv) x (–1,0) (0,1)
3 – 5 3 5
–2, 1,D-3. (i) 2 (ii) , 20 23, 2
(iii) 1, 3 , 1 5, 2 (iv)
D-4. (i) x [–2, –1] [1, 2] (ii) x [–3, 3]
(iii) x(–, –4][–2, –1) [1,)
5 2 D-5. (i) – ,– (ii) 25, (iii) – ,0
2 4 5
D-6. 2
Section (E) :
1
E-1. (i) 1 (ii)–72 (iii) 2 (iv) 1 (v) 7 + (vi) 0
196
E-2. (i) +ve (ii) – ve (iii) +ve (iv) +ve
(v) +ve (vi) – ve (vii) +ve (viii) – ve (ix) – ve
2b 2 3a 2 4(2a b)
E-3. (i) b – 2a (ii) a + 3b (iii) (iv)
ab 1 a 2b
E-4. (i) 1 (ii) 89 (iii) –1
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Fundamentals of Mathematics-I
Section (F) :
F-1. (i) 3 (ii) ± 2 (iii) 3 (iv) 16
(v) {1/3} (vi) {– 4} (vii) no root (viii) (2)
1 5F-2. (i) 9 (ii) log26 (iii) 10 or (iv) {10 , 103} 100
Section (G) :
1 1 G-1. (i) , 3 ,1 (ii) (1, 2) (3, 4)
2 4 4
1
(iii) , (iv) (–1, 0) (1, 2)
2
(v) (–, 2)
G-2. 1
1,4 G-3 (i) (ii) R (iii) (0, log 3 3 )
2 2
3
G-4. (i) , 1 1, 3 4
(ii) (, 1) (1, )
(iii) x (– 2, –1) (– 1, 0) (0, 1) (2, )
PART - II
Section (A) :
A-1. (D) A-2. (A) A-3. (D) A-4. (A) A-5. (D) A-6. (B)
Section (B) :
B-1. (B) B-2. (A) B-3. (D) B-4. (B) B-5. (A) B-6. (B) B-7. (A)
B-8. (D)
Section (C) :
C-1. (B) C-2. (C) C-3. (C) C-4. (B)
Section (D) :
D-1. (B) D-2. (D) D-3. (D) D-4. (B) D-5 (D) D-6. (D) D-7. (D)
Section (E) :
E-1. (A) E-2. (D) E-3. (B) E-4. (D) E-5. (D) E-6. (D) E-7. (A)
E-8. (A) E-9. (D)
Section (F) :
F-1. (D) F-2. (C) F-3. (B) F-4. (A) F-5. (D) F-6. (C) F-7. (B)
Section (G) :
G-1. (C) G-2. (A) G-3. (B) G-4. (A) G-5. (D) G-6. (A) G-7. (D)
PART - II
1. (A) (q), (B) (r), (C) (s), (D) (p)
2. (A) r, (B) s, (C) p, (D) p, (E) q, (F) q
3. (A) r, (B) p, (C) s, (D) q
4. (A) r, (B) s, (C) p, (D) q
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Fundamentals of Mathematics-I
EXERCISE - 2
PART - I
1. (A) 2. (B) 3. (B) 4. (A) 5. (C) 6. (D) 7. (B)
8. (B) 9. (A) 10. (B) 11. (A) 12. (A) 13. (D) 14. (A)
15. (C)
PART - II
1. 02.20 2. 12.28 or 12.29 3. 01.00 4. 22.00 5. 02.00 6. 01.00
7. 10.00 8. 03.82 or 03.83 9. 12.50 10. 03.00 11. 00.25 12. 01.00
13. 00.01 14. 01.00 15. 05.66 or 05.67
PART - III
1. (BD) 2. (ABCD) 3. (ABC) 4. (ABCD) 5. (ABCD) 6. (AB)
7. (CD) 8. (ABCD) 9. (BC) 10. (ABD)
PART - IV
1. (D) 2. (B) 3. (C) 4. (C) 5. (A) 6. (B) 7. (BC)
8. (A) 9. (B)
EXERCISE - 3
PART - I
1. (ABCD) 2. (C) 3. x = 8 4. x = 3 or – 3 5. (B)
6. (A) (p), (r), (s) ; (B) (q), (s) ; (C) (q), (s) ; (D) (p), (r), (s)
7. (C) 8. (4) 9. (ABC) 10. (8)
PART - II
1. (4) 2. (1) 3. (3) 4. (2) 5. (3) 6. (3) 7. (3)
8. (1) 9. (2) 10. (2) 11. (2) 12. (4) 13. (1) 14. 29
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Fundamentals of Mathematics-I
LOGARITHM TABLE
0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
0000 0043 0086 0128 0170 5 9 13 17 21 26 30 34 38
10
0212 0253 0294 0334 0374 4 8 12 16 20 24 28 32 36
0414 0453 0492 0531 0569 4 8 12 16 20 23 27 31 35
11
0607 0645 0682 0719 0755 4 7 11 15 18 22 26 29 33
0792 0828 0864 0899 0934 3 7 11 14 18 21 25 28 32
12
0969 1004 1038 1072 1106 3 7 10 14 17 20 24 27 31
1139 1173 1206 1239 1271 36 10 13 16 19 23 26 29
13
1303 1335 1367 1399 1430 37 10 13 16 19 22 25 29
1461 1492 1523 1553 1584 36 9 12 15 19 22 25 28
14
1614 1644 1673 1703 1732 36 9 12 14 17 22 25 26
1761 1790 1818 1847 1875 3 6 9 11 14 17 20 23 26
15
1903 1931 1959 1987 2014 3 6 8 11 14 17 19 22 25
2041 2068 2095 2122 2148 3 6 8 11 14 16 19 22 24
16
2175 2201 2227 2253 2279 3 5 8 10 13 16 18 21 23
2304 2330 2355 2380 2405 3 5 8 10 13 15 18 20 23
17
2430 2455 2480 2504 2529 3 5 8 10 12 15 17 20 22
2553 2577 2601 2625 2648 2 5 7 9 12 14 17 19 21
18
2672 2695 2718 2742 2765 2 4 7 9 11 14 16 18 21
2788 2810 2833 2856 2878 2 4 7 9 11 13 16 18 20
19
2900 2923 2945 2967 2989 2 4 6 8 11 13 15 17 19
20 3010 3032 3054 3075 3096 3118 3139 3160 3181 3201 2 4 6 8 11 13 15 17 19
21 3222 3243 3263 3284 3304 3324 3345 3365 3385 3404 2 4 6 8 10 12 14 16 18
22 3424 3444 3464 3483 3502 3522 3541 3560 3579 3598 2 4 6 8 10 12 14 15 17
23 3617 3636 3655 3674 3692 3711 3729 3747 3766 3784 2 4 6 7 9 11 13 15 17
24 3802 3820 3838 3856 3874 3892 3909 3927 3945 3962 2 4 5 7 9 11 12 14 16
25 3979 3997 4014 4031 4048 4065 4082 4099 4116 4133 2 3 5 7 9 10 12 14 15
26 4150 4166 4183 4200 4216 4232 4249 4265 4281 4298 2 3 5 7 8 10 11 13 15
27 4314 4330 4346 4362 4378 4393 4409 4425 4440 4456 2 3 5 6 8 9 11 13 14
28 4472 4487 4502 4518 4533 4548 4564 4579 4594 4609 2 3 5 6 8 9 11 12 14
29 4624 4639 4654 4669 4683 4698 4713 4728 4742 4757 1 3 4 6 7 9 10 12 13
30 4771 4786 4800 4814 4829 4843 4857 4871 4886 4900 1 3 4 6 7 9 10 11 13
31 4914 4928 4942 4955 4969 4983 4997 5011 5024 5038 1 3 4 6 7 8 10 11 12
32 5051 5065 5079 5092 5105 5119 5132 5145 5159 5172 1 3 4 5 7 8 9 11 12
33 5185 5198 5211 5224 5237 5250 5263 5276 5289 5302 1 3 4 5 6 8 9 10 12
34 5315 5328 5340 5353 5366 5378 5391 5403 5416 5428 1 3 4 5 6 8 9 10 11
35 5441 5453 5465 5478 5490 5502 5514 5527 5539 5551 1 2 4 5 6 7 9 10 11
36 5563 5575 5587 5599 5611 5623 5635 5647 5658 5670 1 2 4 5 6 7 8 10 11
37 5682 5694 5705 5717 5729 5740 5752 5763 5775 5786 1 2 3 5 6 7 8 9 10
38 5798 5809 5821 5832 5843 5855 5866 5877 5888 5899 1 2 3 5 6 7 8 9 10
39 5911 5922 5933 5944 5955 5966 5977 5988 5999 6010 1 2 3 4 5 7 8 9 10
40 6021 6031 6042 6053 6064 6075 6085 6096 6107 6117 1 2 3 4 5 6 8 9 10
41 6128 6138 6149 6160 6170 6180 6191 6201 6212 6222 1 2 3 4 5 6 7 8 9
42 6232 6243 6253 6263 6274 6284 6294 6304 6314 6325 1 2 3 4 5 6 7 8 9
43 6335 6345 6355 6365 6375 6385 6395 6405 6415 6425 1 2 3 4 5 6 7 8 9
44 6435 6444 6454 6464 6474 6484 6493 6503 6513 6522 1 2 3 4 5 6 7 8 9
45 6532 6542 6551 6561 6571 6580 6590 6599 6609 6618 1 2 3 4 5 6 7 8 9
46 6628 6637 6646 6656 6665 6675 6684 6693 6702 6712 1 2 3 4 5 6 7 7 8
47 6721 6730 6739 6749 6758 6767 6776 6785 6794 6803 1 2 3 4 5 5 6 7 8
48 6812 6821 6830 6839 6848 6857 6866 6875 6884 6893 1 2 3 4 4 5 6 7 8
49 6902 6911 6920 6928 6937 6946 6955 6964 6972 6981 1 2 3 4 4 5 6 7 8
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Fundamentals of Mathematics-I
LOGARITHM TABLE
0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
50 6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 1 2 3 3 4 5 6 7 8
51 7076 7084 7093 7101 7110 7118 7126 7135 7143 7152 1 2 3 3 4 5 6 7 8
52 7160 7168 7177 7185 7193 7202 7210 7218 7226 7235 1 2 2 3 4 5 6 7 7
53 7243 7251 7259 7267 7275 7284 7292 7300 7308 7316 1 2 2 3 4 5 6 6 7
54 7324 7332 7340 7348 7356 7364 7372 7380 7388 7396 1 2 2 3 4 5 6 6 7
55 7404 7412 7419 7427 7435 7443 7451 7459 7466 7474 1 2 2 3 4 5 5 6 7
56 7482 7490 7497 7505 7513 7520 7528 7536 7543 7551 1 2 2 3 4 5 5 6 7
57 7559 7566 7574 7582 7589 7597 7604 7612 7619 7627 1 2 2 3 4 5 5 6 7
58 7634 7642 7649 7657 7664 7672 7679 7686 7694 7701 1 1 2 3 4 4 5 6 7
59 7709 7716 7723 7731 7738 7745 7752 7760 7767 7774 1 1 2 3 4 4 5 6 7
60 7782 7789 7796 7803 7810 7818 7825 7832 7839 7846 1 1 2 3 4 4 5 6 6
61 7853 7860 7868 7875 7882 7889 7896 7903 7910 7917 1 1 2 3 4 4 5 6 6
62 7924 7931 7938 7945 7952 7959 9766 7973 7980 7987 1 1 2 3 3 4 5 6 6
63 7993 8000 8007 8014 8021 8028 8035 8041 8048 8055 1 1 2 3 3 4 5 5 6
64 8062 8069 8075 8082 8089 8096 8102 8109 8116 8122 1 1 2 3 3 4 5 5 6
65 8129 8136 8142 8149 8156 8162 8169 8176 8182 8189 1 1 2 3 3 4 5 5 6
66 8195 8202 8209 8215 8222 8228 8235 8241 8248 8254 1 1 2 3 3 4 5 5 6
67 8261 8267 8274 8280 8287 8293 8299 8306 8312 8319 1 1 2 3 3 4 5 5 6
68 8325 8331 8338 8344 8351 8357 8363 8370 8376 8382 1 1 2 3 3 4 4 5 6
69 8388 8395 8401 8407 8414 8420 8426 8432 8439 8445 1 1 2 2 3 4 4 5 6
70 8451 8457 8463 8470 8476 8482 8488 8494 8500 8506 1 1 2 2 3 4 4 5 6
71 8513 8519 8525 8531 8537 8543 8549 8555 8561 8567 1 1 2 2 3 4 4 5 5
72 8573 8579 8585 8591 8597 8603 8609 8615 8621 8627 1 1 2 2 3 4 4 5 5
73 8633 8639 8645 8651 8657 8663 8669 8675 8681 8686 1 1 2 2 3 4 4 5 5
74 8692 8698 8704 8710 8716 8722 8727 8733 8739 8745 1 1 2 2 3 4 4 5 5
75 8751 8756 8762 8768 8774 8779 8785 8791 8797 8802 1 1 2 2 3 3 4 5 5
76 8808 8814 8820 8825 8831 8837 8842 8848 8854 8859 1 1 2 2 3 3 4 5 5
77 8865 8871 8876 8882 8887 8893 8899 8904 8910 8915 1 1 2 2 3 3 4 4 5
78 8921 8927 8932 8938 8943 8949 8954 8960 8965 8971 1 1 2 2 3 3 4 4 5
79 8976 9882 8987 8993 8998 9004 9009 9015 9020 9025 1 1 2 2 3 3 4 4 5
80 9031 9036 9042 9047 9053 9058 9063 9069 9074 9079 1 1 2 2 3 3 4 4 5
81 9085 9090 9096 9101 9106 9112 9117 9122 9128 9133 1 1 2 2 3 3 4 4 5
82 9138 9143 9149 9154 9159 9165 9170 9175 9180 9186 1 1 2 2 3 3 4 4 5
83 9191 9196 9201 9206 9212 9217 9222 9227 9232 9238 1 1 2 2 3 3 4 4 5
84 9243 9248 9253 9258 9263 9269 9274 9279 9284 9289 1 1 2 2 3 3 4 4 5
85 9294 9299 9304 9309 9315 9320 9325 9330 9335 9340 1 1 2 2 3 3 4 4 5
86 9345 9350 9355 9360 9365 9370 9375 9380 9385 9390 1 1 2 2 3 3 4 4 5
87 9395 9400 9405 9410 9415 9420 9425 9430 9435 9440 0 1 1 2 2 3 3 4 4
88 9445 9450 9455 9460 9465 9469 9474 9479 9484 9489 0 1 1 2 2 3 3 4 4
89 9494 9499 9504 9509 9513 9518 9523 9528 9533 9538 0 1 1 2 2 3 3 4 4
90 9542 9547 9552 9557 9562 9566 9571 9576 9581 9586 0 1 1 2 2 3 3 4 4
91 9590 9595 9600 9605 9609 9614 9619 9624 9628 9633 0 1 1 2 2 3 3 4 4
92 9638 9643 9647 9652 9657 9661 9666 9671 9675 9680 0 1 1 2 2 3 3 4 4
93 9685 9689 9694 8699 9703 9708 9713 9717 9722 9727 0 1 1 2 2 3 3 4 4
94 9731 9736 9741 9745 9750 9754 9759 9763 9768 9773 0 1 1 2 2 3 3 4 4
95 9777 9782 9786 9791 9795 9800 9805 9809 9814 9818 0 1 1 2 2 3 3 4 4
96 9823 9827 9832 9836 9841 9845 9850 9854 9859 9863 0 1 1 2 2 3 3 4 4
97 9868 9872 9877 9881 9886 9890 9894 9899 9903 9908 0 1 1 2 2 3 3 4 4
98 9912 9917 9921 9926 9930 9934 9939 9943 9948 9952 0 1 1 2 2 3 3 4 4
99 9956 9961 9965 9969 9974 9978 9983 9987 9991 9996 0 1 1 2 2 3 3 4 4
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Fundamentals of Mathematics-I
ANTILOGARITHM TABLE
0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
.00 1000 1002 1005 1007 1009 1012 1014 1016 1019 1021 0 0 1 1 1 1 2 2 2
.01 1023 1026 1028 1030 1033 1035 1038 1040 1042 1045 0 0 1 1 1 1 2 2 2
.02 1047 1050 1052 1054 1057 1059 1062 1064 1067 1069 0 0 1 1 1 1 2 2 2
.03 1072 1074 1076 1079 1081 1084 1086 1089 1091 1094 0 0 1 1 1 1 2 2 2
.04 1096 1099 1102 1104 1107 1109 1112 1114 1117 1119 0 1 1 1 1 2 2 2 2
.05 1122 1125 1127 1130 1132 1135 1138 1140 1143 1146 0 1 1 1 1 2 2 2 2
.06 1148 1151 1153 1156 1159 1161 1164 1167 1169 1172 0 1 1 1 1 2 2 2 2
.07 1175 1178 1180 1183 1186 1189 1191 1194 1197 1199 0 1 1 1 1 2 2 2 2
.08 1202 1205 1208 1211 1213 1216 1219 1222 1225 1227 0 1 1 1 1 2 2 2 3
.09 1230 1233 1236 1239 1242 1245 1247 1250 1253 1256 0 1 1 1 1 2 2 2 3
.10 1259 1262 1265 1268 1271 1274 1276 1279 1282 1285 0 1 1 1 1 2 2 2 3
.11 1288 1291 1294 1297 1300 1303 1306 1309 1312 1315 0 1 1 1 2 2 2 2 3
.12 1318 1321 1324 1327 1330 1334 1337 1340 1343 1346 0 1 1 1 2 2 2 2 3
.13 1349 1352 1355 1358 1361 1365 1368 1371 1374 1377 0 1 1 1 2 2 2 3 3
.14 1380 1384 1387 1390 1393 1396 1400 1403 1406 1409 0 1 1 1 2 2 2 3 3
.15 1413 1416 1419 1422 1426 1429 1432 1435 1439 1442 0 1 1 1 2 2 2 3 3
.16 1445 1449 1452 1455 1459 1462 1466 1469 1472 1476 0 1 1 1 2 2 2 3 3
.17 1479 1483 1486 1489 1493 1496 1500 1503 1507 1510 0 1 1 1 2 2 2 3 3
.18 1514 1517 1521 1524 1528 1531 1535 1538 1542 1545 0 1 1 1 2 2 2 3 3
.19 1549 1552 1556 1560 1563 1567 1570 1574 1578 1581 0 1 1 1 2 2 3 3 3
.20 1585 1589 1592 1596 1600 1603 1607 1611 1614 1618 0 1 1 1 2 2 3 3 3
.21 1622 1626 1629 1633 1637 1641 1644 1648 1652 1656 0 1 1 2 2 2 3 3 3
.22 1660 1663 1667 1671 1675 1679 1683 1687 1690 1694 0 1 1 2 2 2 3 3 3
.23 1698 1702 1706 1710 1714 1718 1722 1726 1730 1734 0 1 1 2 2 2 3 3 4
.24 1738 1742 1746 1750 1754 1758 1762 1766 1770 1774 0 1 1 2 2 2 3 3 4
.25 1778 1782 1786 1791 1795 1799 1803 1807 1811 1816 0 1 1 2 2 2 3 3 4
.26 1820 1824 1828 1832 1837 1841 1845 1849 1854 1858 0 1 1 2 2 3 3 3 4
.27 1862 1866 1871 1875 1879 1884 1888 1892 1897 1901 0 1 1 2 2 3 3 3 4
.28 1905 1910 1914 1919 1923 1928 1932 1936 1841 1845 0 1 1 2 2 3 3 4 4
.29 1950 1954 1959 1963 1968 1972 1977 1982 1986 1991 0 1 1 2 2 3 3 4 4
.30 1995 2000 2004 2009 2014 2018 2023 2028 2032 2037 0 1 1 2 2 3 3 4 4
.31 2042 2046 2051 2056 2061 2065 2070 2075 2080 2084 0 1 1 2 2 3 3 4 4
.32 2089 2094 2099 2104 2109 2113 2118 2123 2128 2133 0 1 1 2 2 3 3 4 4
.33 2138 2143 2148 2153 2158 2163 2168 2173 2178 2183 0 1 1 2 2 3 3 4 4
.34 2188 2193 2198 2203 2208 2213 2218 2223 2328 2234 1 1 2 2 3 3 4 4 5
.35 2239 2244 2249 2254 2259 2265 2270 2275 2280 2286 1 1 2 2 3 3 4 4 5
.36 2291 2296 2301 2307 2312 2317 2323 2328 2333 2339 1 1 2 2 3 3 4 4 5
.37 2344 2350 2355 2360 2366 2371 2377 2382 2388 2393 1 1 2 2 3 3 4 4 5
.38 2399 2404 2410 2415 2421 2432 2427 2432 2443 2449 1 1 2 2 3 3 4 4 5
.39 2455 2460 2466 2472 2477 2483 2489 2495 2500 2506 1 1 2 2 3 3 4 5 5
.40 2512 2518 2523 2529 2535 2541 2547 2553 2559 2564 1 1 2 2 3 4 4 5 5
.41 2570 2576 2582 2588 2594 2600 2606 2612 2618 2624 1 1 2 2 3 4 4 5 5
.42 2630 2636 2642 2649 2655 2661 2667 2673 2679 2685 1 1 2 2 3 4 4 5 6
.43 2692 2698 2704 2710 2716 2723 2729 2735 2742 2748 1 1 2 3 3 4 4 5 6
.44 2754 2761 2767 2773 2780 2786 2793 2799 2805 2812 1 1 2 3 3 4 4 5 6
.45 2818 2825 2831 2838 2844 2851 2858 2864 2871 2877 1 1 2 3 3 4 5 5 6
.46 2884 2891 2897 2904 2911 2917 2924 2931 2938 2944 1 1 2 3 3 4 5 5 6
.47 2951 2958 2965 2972 2979 2985 2992 2999 3006 3013 1 1 2 3 3 4 5 5 6
.48 3020 3027 3034 3041 3048 3055 3062 3069 3076 3083 1 1 2 3 4 4 5 6 6
.49 3090 3097 3105 3112 3119 3126 3133 3141 3148 3155 1 1 2 3 4 4 5 6 6
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MAASFundamentals Of Mathematics I
© MAAS. All rights reserved.Page 58 / 58
Extracted text · page 58
Fundamentals of Mathematics-I
ANTILOGARITHM TABLE
0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
.50 3162 3170 3177 3184 3192 3199 3206 3214 3221 3228 1 1 2 3 4 4 5 6 7
.51 3236 3243 3251 3258 3266 3273 3281 3289 3296 3304 1 2 2 3 4 5 5 6 7
.52 3311 3319 3327 3334 3342 3350 3357 3365 3373 3381 1 2 2 3 4 5 5 6 7
.53 3388 3396 3404 3412 3420 3428 3436 3443 3451 3459 1 2 2 3 4 5 6 6 7
.54 3467 3475 3483 3491 3499 3508 3516 3524 3532 3540 1 2 2 3 4 5 6 6 7
.55 3548 3556 3565 3573 3581 3589 3597 3606 3614 3622 1 2 2 3 4 5 6 6 7
.56 3631 3639 3648 3656 3664 3673 3681 3690 3698 3707 1 2 3 3 4 5 6 7 8
.57 3715 3724 3733 3741 3750 3758 3767 3776 3784 3793 1 2 3 3 4 5 6 7 8
.58 3802 3811 3819 3828 3837 3846 3855 3864 3873 3882 1 2 3 4 4 5 6 7 8
.59 3890 3899 3908 3917 3926 3936 3945 3954 3963 3972 1 2 3 4 5 5 6 7 8
.60 3981 3990 3999 4009 4018 4027 4036 4046 4055 4064 1 2 3 4 5 6 6 7 8
.61 4074 4083 4093 4102 4111 4121 4130 4140 4150 4156 1 2 3 4 5 6 7 8 9
.62 4169 4178 4188 4198 4207 4217 4227 4236 4246 4256 1 2 3 4 5 6 7 8 9
.63 4266 4276 4285 4295 4305 4315 4325 4335 4345 4355 1 2 3 4 5 6 7 8 9
.64 4365 4375 4385 4395 4406 4416 4426 4436 4446 4457 1 2 3 4 5 6 7 8 9
.65 4467 4477 4487 4498 4508 4519 4529 4539 4550 4560 1 2 3 4 5 6 7 8 9
.66 4571 4581 4592 4603 4613 4624 4634 4645 4656 4667 1 2 3 4 5 6 7 9 10
.67 4677 4688 4699 4710 4721 4732 4742 4753 4764 4775 1 2 3 4 5 7 8 9 10
.68 4786 4797 4808 4819 4831 4842 4853 4864 4875 4887 1 2 3 4 6 7 8 9 10
.69 4898 4909 4920 4932 4943 4955 4966 4977 4989 5000 1 2 3 5 6 7 8 9 10
.70 5012 5023 5035 5047 5058 5070 5082 5093 5105 5117 1 2 4 5 6 7 8 9 11
.71 5129 5140 5152 5164 5176 5188 5200 5212 5224 5236 1 2 4 5 6 7 8 10 11
.72 5248 5260 5272 5284 5297 5309 5321 5333 5346 5358 1 2 4 5 6 7 9 10 11
.73 5370 5383 5395 5408 5420 5433 5445 5458 5470 5483 1 3 4 5 6 8 9 10 11
.74 5495 5508 5521 5534 5546 5559 5572 5585 5598 5610 1 3 4 5 6 8 9 10 12
.75 5623 5636 5649 5662 5675 5689 5702 5715 5728 5741 1 3 4 5 7 8 9 10 12
.76 5754 5768 5781 5794 5808 5821 5834 5848 5861 5875 1 3 4 5 7 8 9 11 12
.77 5888 5902 5916 5929 5943 5957 5970 5984 5998 6012 1 3 4 5 7 8 10 11 12
.78 6026 6039 6053 6067 6081 6095 6109 6124 6138 6152 1 3 4 6 7 8 10 11 13
.79 6166 6180 6194 6209 6223 6237 6252 6266 6281 6295 1 3 4 6 7 9 10 11 13
.80 6310 6324 6339 6353 6368 6383 6397 6412 6427 6442 1 3 4 6 7 9 10 12 13
.81 6457 6471 6486 6501 6516 6531 6546 6561 6577 6592 2 3 5 6 8 9 11 12 14
.82 6607 6622 6637 6653 6668 6683 6699 6714 6730 6745 2 3 5 6 8 9 11 12 14
.83 6761 6776 6792 6808 6823 6839 6855 6871 6887 6902 2 3 5 6 8 9 11 13 14
.84 6918 6934 6950 6966 6982 6998 7015 7031 7047 7063 2 3 5 6 8 10 11 13 15
.85 7079 7096 7112 7129 7145 7161 7178 7194 7211 7228 2 3 5 7 8 10 12 13 15
.86 7244 7261 7278 7295 7311 7328 7345 7362 7379 7396 2 3 5 7 8 10 12 13 15
.87 7413 7430 7447 7464 7482 7499 7516 7534 7551 7568 2 3 5 7 9 10 12 14 16
.88 7586 7603 7621 7638 7656 7674 7691 7709 7727 7745 2 4 5 7 9 11 12 14 16
.89 7762 7780 7798 7816 7834 7852 7870 7889 7907 7925 2 4 5 7 9 11 13 14 16
.90 7943 7962 7980 7998 8017 8035 8054 8072 8091 8110 2 4 6 7 9 11 13 15 17
.91 8128 8147 8166 8185 8204 8222 8241 8260 8279 8299 2 4 6 8 9 11 13 15 17
.92 8318 8337 8356 8375 8395 8414 8433 8453 8472 8492 2 4 6 8 10 12 14 15 17
.93 8511 8531 8551 8570 8590 8610 8630 8650 8670 8690 2 4 6 8 10 12 14 16 18
.94 8710 8730 8750 8770 8790 8810 8831 8851 8872 8892 2 4 6 8 10 12 14 16 18
.95 8913 8933 8954 8974 8995 9016 9036 9057 9078 9099 2 4 6 8 10 12 15 17 19
.96 9120 9141 9162 9183 9204 9226 9247 9268 9290 9311 2 4 6 8 11 13 15 17 19
.97 9333 9354 9376 9397 9419 9441 9462 9484 9506 9528 2 4 7 9 11 13 15 17 20
.98 9550 9572 9594 9616 9638 9661 9683 9705 9727 9750 2 4 7 9 11 13 16 18 20
.99 9772 9795 9817 9849 9863 9886 9908 9931 9954 9977 2 5 7 9 11 14 16 18 20
ADVFOMI-34