Mathematics BASIC MATHS MODULE

EXERCISE-I (JEE Mains Pattern)

1. Let A1, A2 and A3 be subsets of a set X. Which one of the following is correct?
(a) A1  A 2  A 3 is the smallest subset of X containing elements of each of A1, A2
and A3 (b) A1  A 2  A 3 is the smallest subset of X containing either A1 or A 2  A 3
but not both
(c) The smallest subset of X containing A1  A 2 and A3 equals the smallest subset
of X containing
(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

x 1
3. Number of integral values of ‘x’ satisfying the equation 3 – 2.3x  2. 3x – 1  1 are
(a) 1 (b) 2 (c) 3 (d) 4

4. x 2  6x  p  x 2  6x  p  x  R where p is a prime number then least possible

value p is
(a) 7 (b) 11 (c) 5 (d) 13

2
5. If  log10 x  – 4 log10 x  3  0 , the product of roots of the equation is:
(a) 3 (b) 104 (c) 108 (d) 1

6x 2 – 5x – 3
6. 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

7. Sum of all the real solutions of the inequality

 x  2  x
2 2
– 16   0 is
 x  2 x
4 2
– 9
(a) 5 (b) 4 (c) 8 (d) 0

1–a –b
a b 21–b 
8. If 60  3 and 60  5 then the value of 12 equals
(a) 2 (b) 3 (c) 3 (d) 12

 2x – 2007 
9. Number of integral values of x the inequality log10    0 holds true, is
 x 1 
(a) 1004 (b) 1005 (c) 2007 (d) 2008

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 Mathematics BASIC MATHS MODULE

x 2 – 4x  3
10. Number of prime numbers satisfying the inequality log 3  0 is equal to
x2  x – 5
(a) 1 (b) 2 (c) 3 (d) 4

11. If |x + 2| + y = 5 and x – |y| = 1 then the value of (x + y) is

(a) 1 (b) 2 (c) 3 (d) 4

12. In an examination of creation 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

13. Let X and Y be two sets.

Statement-I : X   Y  X  '  
Statement-II : 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-I is true, but statement-II is false.
(c) Statement-I is false, but Statement-II is true.
(d) Both the statements are false.

x
14. If p,q  N satisfy the equation x x
  x then p & q are
(a) relatively prime
(b) twin prime
(c) coprime
(d) if logqp is defined then logpq is not and vice versa.

15. Let x  3 – 5 and y  3  5 . If the value of expression

x – y + 2x2y + 2xy2 – x4y + xy4 can be expressed in the form p  q where p,q  N ,
then find the value of (p + q).

16. If log a b  2; log b c  2 and log 3 c  3  log 3 a then (a + b + c) equals

(a) 90 (b) 93 (c) 102 (d) 243

17. Let x   log1/3 5   log125 343   log 49 729  and y  253 log 289 11log 28 17 log1331 784
, then value of
y
is
x
5 5 4 3
(a) (b) – (c) – (d)
3 3 5 7

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 Mathematics BASIC MATHS MODULE

18. The sum of all possible integral solutions of equation

x 2 – 6x  5 – 2x 2 – 3x  1  3 x 2 – 3x  2 is
(a) 10 (b) 12 (c) 13 (d) 15

19. The set of values of x satisfying simultaneously the inequalities

 x – 8  2 – x 
 0 and 2x –3 – 31  0 is:
 10
log 0.3   log 2 5 – 1 
 7 
(a) a unit set
(b) an empty set
(c) an infinite set
(d) a set consisting of exactly two elements.

20. The solution set of the inequality

 3x – 4x  .n  x  2
 0 is
x 2 – 3x – 4
(a)  – ,0    4,   (b)  –2,0    4,  
(c)  –1,0    4,   (d)  –2, –1   –1,0    4,  

21. Column-I Column-II

(A) If a  3  8  2 7 – 8 – 2 7 ,b   42 30   36 (P) 0
then the value of logab is equal to
(B) Number of real solutions of the equation (Q) 1
3
x –1  x – 3  is
2
(C) If a  6  2 5 – 6 – 2 5,b  3 17 5  38 – 3 17 5 – 38 (R) 2
then the value of logab is equal to
(S) 3

COMPREHENSION
A function f  x   a x  a  0,a  1, x  R  is called an exponential function. Graph of
exponential function can be as follows:

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 Mathematics BASIC MATHS MODULE

22. Which of the following is correct:

(a) (b)

(c) (d)

23. Number of solutions of 3x  x – 2  0 is/are:

(a) 1 (b) 2 (c) 3 (d) 4

24. The number of positive solutions of log1/2 x  7 x is/are:

(a) 0 (b) 1 (c) 2 (d) 3

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 Mathematics BASIC MATHS MODULE

EXERCISE-II (JEE AdvancedPattern)

1. The product of all the integers which do not belong to the solution set of the
3 x –2
inequality  2 is
x –1
(a) – 1 (b) – 4 (c) 4 (d) 0

2. Let  be set with number of elements in the is 2009 and A, B are subsets of 
with n  A  B   280 . If n  A ' B '  x13  x 23  y13  y 23 for some positive integers
x2  y2
x1  y1  y 2  x 2 , then find value of .
x1  y1

3. Let  be set with number of elements in it is 2009. A is a subset of  with

n (A) = 1681 and out of these 1681 elements, exactly 1075 elements belong to a
subset B of  . If n  A – B   m2  p1p2p3 for some positive integer m and distinct
p1p3
primes then for least m find
p2

4. If f  x   x  1 – 2 x – 1 then
(a) maximum value of f(x) is 2.
(b) there are two solutions of f (x) = 1.
(c) there is one solution of f(x) = 2.
(d) there are two solutions of f(x) = 3.

5. Which of the following is a null set?

(a) A  x : x  1 and x  1 (b) B  x : x  3  3
(c) C   (d) D  x : x  1 and x  1

6. 6  6  6  6  ... 
(a) 3 (b) 2 (c) 1 (d) 3

1 2 
7. If x  8 – 60 , then  x 
2 x 
(a) 5 (b) 3 (c) 2 5 (d) 2 3

1 – 21 – 4x – x 2
8. Find the complete solution set of the inequality 0
x 1
(a) 2 6 – 2,3 (b)  –2 – 2 6, –1 
(c)  –2 – 2 6, –1  2 6 – 2,3  
(d)  –2 – 2 6, –1  2 6 – 2,3 

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 Mathematics BASIC MATHS MODULE

x2 – x
9. The solution set of the inequality  0 is
4 – x3
(a)  –1, 3 4  (b) 1, 3 4  
(c)  –1, 3 2 (d) 0, 3 4 
log  a  c   log  a – 2b  c 
10. If c(a – b) = a(b – c) then find the value of (Assume all
log  a – c 
terms are defined)

11. If log b a.log c a  log a b.log c b  log b c  3 (where a, b, c are different positive real
number  1 ). Then find the value of a b c.

12. If 4A  9B  10C , where A  log16 4,B  log 3 9 & C  log x 83 , then find x.

13. If graph of y = f (x) in (– 3, 1), is as shown in the following figure

and g  x   n  f  x   , then the graph of y  g  – x  is

(a) (b)

(c) (d)

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 Mathematics BASIC MATHS MODULE

2 9 5

14. If
 x  3   x – 1  x  1  0 , then number of possible integral values of x is
4 5
 x – 3  x – 5   x – 6
(a) 6 (b) 3 (c) 4 (d) 5

x x
15. If   
2 1  2 –1  – 2 2  0 , then sum of all possible values of x is
(a) 0 (b) 1 (c) 2 (d) 3

16. If P(x) is a cubic polynomial such that P(1) = 1; P(2) = 2; P(3) = 3 with leading
coefficient 3 then find the value of P(4).

1 1 1
17. If x + y + z = 12 & x2 + y2 + z2 = 96 and    36 . Find the value of
x y z
x3  y3  z3
.
4

2
18. If x  4 – 2 3 and y  9 – 4 5 then the value of  5x – 3y  is equal to

a – b c  where
19. a, b, c are coprime numbers then a + b + c is equal to (where ‘c’ is an odd integer)
If x, y, z  R and 121x2 + 4y2 + 9z2 – 22x + 4y + 6z + 3 = 0 then value
of x–1 – y–1 – z –1 is equal to

20. Find the positive number, x which satisfies the equation log10 2x 2 – 21x  50  2  
21. Find the value of x satisfying the equation log 1  x – 1  log 1  x  1 – log 1 7 – x   1 .
2 2 2

22. The equation log x2 16  log 2x 64  3 has:

(a) one irrational solution (b) no prime solution
(c) two real solutions (d) one integral solution

 2 9 
 log 3 x  – 2 log 3 x 5
23. The equation x  3 3 has
(a) exactly three real solution (b) at least one real solution
(c) exactly one irrational solution (d) complex roots

24. The solution set of the system of equations log 3 x  log 3 y  2  log 3 2 and
2
log 27  x  y   is:
3
(a) {6, 3} (b) {3, 6} (c) {6, 12} (d) {12, 6}

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25. Consider the equation x 2 – 4 x  3  p

(a) for p = 2 the equation has four solutions
(b) for p = 2 the equation has eight solutions
(c) there exists only one real value of p for which the equation has odd number of
solutions
(d) sum of roots of the equation is zero irrespective of value of p

26. The set of values of x satisfying simultaneously the inequalities

 x – 8  2 – x 
 0 and 2x –3 – 31  0 is:
 10
log 0.3   log 2 5 – 1 
 7 
(a) a unit set
(b) an empty set
(c) an infinite set
(d) a set consisting of exactly two elements

27. If log 0.3  x – 1  log 0.09  x – 1 , then x lies in the interval

(a)  2,   (b) (1, 2) (c) 1,   (d) None of these

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 Mathematics BASIC MATHS MODULE

EXERCISE-III (JEE Mains Pyq’s)

1. Number of solutions of log 4  x – 1  log 2  x – 3  is JEE-M 2021
(a) 3 (b) 1 (c) 2 (d) 0

2. Let (x0, y0) be solution of the following equations

ln 2 ln 3
 2x    3y  Then x0 is: JEE-M 2024
3ln x  2ln y
1 1 1
(a) (b) (c) (d) 6
6 3 2

 
 1 1 1 1
3. The value of 6  log 3 4– 4– 4– ...  is JEE-M 2012
2
 3 2 3 2 3 2 3 2 
 

1 1

4. The value of  log 2 9 

2 log  log 9
2 2
  7 log 4 7
is JEE-M 2018

5. In a class of 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 by 5 opted chemistry course. Then the
number of students who did not opt for any of the three courses is:
(a) 102 (b) 42 (c) 1 (d) 38

6. Two newspapers A and B are published in a city. It is known that 25% of the city
populations reads A and 20% reads B while 8% reads both A and B. Further, 30%
of those who read A but not B look into advertisements and 40% of those who read
B but not A also look into advertisements, while 50% of those who read both A and
B look into advertisements. Then the percentage of the populations who look into
advertisement is
(a) 12.8 (b) 13.5 (c) 13.9 (d) 13

7. Let Z be the set of integers. If A  x  Z : 2   x  2 x 2 –5x  6


 1 and

B  x  Z : –3  2x – 1  9 , then the number of subsets of set A  B , is:

(a) 218 (b) 210 (c) 215 (d) 212

8. If 3x  4x –1 , then x = [JEE-Advanced 2013]

2 log 3 2 2 1 2 log 2 3
(a) (b) (c) (d)
2 log 3 2 – 1 2 – log 2 3 1 – log 4 3 2 log 2 3 – 1

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 Mathematics BASIC MATHS MODULE

x 1 x 1
9. If x  0, y  0, x  y   and  x  y   – , then x = ….. and y = ……
y 2 y 2

2 2
10. The equation x – 1– has
x –1 x –1
(a) no root b) one root
(c) two equal roots (d) infinitely many roots

11. 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 :
(a) 52 (b) 35 (c) 25 (d) 53

 
12. 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
(a) X (b) Y (c) N (d) Y – X

x 2  4x –60
13. The sum of all real values of x satisfying the equation  x 2 – 5x  5   1 is
(a) – 4 (b) 6 (c) 5 (d) 3

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 number of elements in the smallest subset of X containing
both A and B is __________.

15. If x satisfies x – 1  x – 2  x – 3  6 , then

(a) 0  x  4 (b) x  –2 or x  4 (c) x  0 or x  4 (d) None of these

16. Solve x 2  4x  3  2x  5  0 .

17. If p, q, r are positive and are in A.P., then roots of the quadratic equation
px2 + qx + r= 0 are real for
r r
(a) –7 4 3 (b) –7 4 3 (c) all p and r (d) no p and r
p p

y
18. Find the set of all solutions of the equation 2 – 2y –1 – 1  2y –1  1

2
19. The sum of all the real roots of the equation x – 2  x – 2 – 2  0 is ________.

20. The equation 3x 2  x  5  x – 3 , where x is real, has:

(a) exactly four solutions (b) exactly one solutions
(c) exactly two solutions (d) no solution

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 1
21. If x is a solution of the equation, 2x  1 – 2x – 1  1,  x   , then 4x 2 – 1 is
 2
equal to
3 1
(a) 2 (b) (c) 2 2 (d)
4 2

22. Let and  be the roots of equation px 2  qx  r  0, p  0 . If p, q, r are in the A.P.

1 1
and   4 , then the value of  –  is :
 
3 3 1 5
(a) (b) (c) (d)
4 2 4 4

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 Mathematics BASIC MATHS MODULE

ANSWER KEY
EXERCISE-I
1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
A B B B D C D A B A
11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
C B A ACD 610 B B D A D
21. 22. 23. 24.
A-R; B-P; C-R; D-Q BC A B

EXERCISE-II
1. 2. 3. 4. 5. 6. 7. 8.
A 2.20 12.28 to 12.29 ABC A A A D
9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
A 2 1 10 D D A 22 216.50 36
19. 20. 21. 22. 23. 24. 25. 26. 27.
16 12.5 3 ABCD ABCD AB ACD A A

EXERCISE-III
1. 2. 3. 4. 5. 6. 7. 8. 9.
B C 4 8 D C C ABC –1 –1
x ,y 
4 4
10. 11. 12. 13. 14. 15. 16 17.
A B B D 29 C x  –1 – 3 or – 4 A

18. 19. 20. 21. 22.

–1  1,   4 D B B

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