IIT JEE MAINS & ADVANCED MATHEMATICS BY OM SIR

MASTER Sheet: Determinant

EXERCISE # 1
Q.1 If ABC is a scalene triangle, then the value of Q.7 If A, B and C are the angles of a triangle, then
sin A sin B sin C sin 2A sin C sin B
cos A cos B cos C is sin C sin 2B sin A =
1 1 1 sin B sin A sin 2C
(A) = 0 (B)  0 (A) 0 (B) 1 (C) 2 (D) 3
(C) can not say (D) None of these
Q.8 The value of the determinant
Q.2 Co-factors of element of the second row of the
1 cos( − ) cos( − )
1 2 3
cos( − ) 1 cos( − )
determinant − 4 3 6 are-
cos( −  ) cos( −  ) 1
2 −7 9
is equal to
(A) 39, 3, 11 (B) –39, 3, 11
(C) 39, –3, 11 (D) 39, 3, –11 (A) cos  + cos  + cos 
(B) cos  cos  + cos  cos  + cos  cos 
Q.3 Without expanding value of the (C) –1
0 p−q p−r (D) 0
determinant q − p 0 q − r is- Q.9 If the following equations
r−p r−q 0 x+y–3=0
(A) (p – r) (q – r) (B) (q – r) (p – q) (1 + ) x + (2 + ) y – 8 = 0
(C) 0 (D) None of these x – (1 + ) y + (2 + ) = 0
are consistent then the value of  is
Q.4 If a, b,c are positive and are the pth, qth and rth (A) 1 (B) –1 (C) 0 (D) 2
terms respectively of a G.P., then the value of
log a p 1 Q.10 If a + b + c  0 and the system of equations
log b q 1 is ax + by + cz = 0, bx + cy + az = 0,
log c r 1 cx + ay + bz = 0 has a non-trivial solution, then
(A) 0 (B) p (C) q (D) r the roots of the equation at2 + bt + c = 0, are
(A) imaginary (B) real and distinct
b+c c b (C) real and of opposite sign (D) real and equal
Q.5 c c+a a =
b a a+b
(A) a + b + c (B) 2a + b + c ➢ True or false type questions
(C) ab + bc + ca (D) 4abc.
Q.11 If a, b, c are sides of a scalene triangle, then
Q.6 If ax4 + bx3 + cx2 + dx + e = a b c
2x x −1 x +1 value of b c a is negative
x + 1 x 2 − x x − 1 , then the value of e, is c a b
x −1 x +1 3x
(A) 0 (B) –2 (C) 3 (D) –1

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EXERCISE # 2
Only single correct answer type Q.7 If    are the roots of x3 – 3x + 2 = 0, then
Part-A
questions   
the value of the determinant    is
Q.1 Solution of the system of equations
  
a2x + ay + z = – a3, b2x + by + z = – b3,
c2x + cy + z = – c3 is- equal to
(A) x = – (a + b + c), y = ab + bc + ca, z = – abc (A) –3 (B) 2
(B) x = (a + b + c) , y = ab + bc + ca, z = – abc (C) 1 (D) none
(C) x = –(a – b – c), y = ab + bc + ca, z = – abc
Q.8 The value of the determinant
(D) None of these
(2 x − 2− x ) 2 (2 x + 2− x ) 2 1
Q.2 The value of the determinant (3x − 3− x )2 (3x + 3− x )2 1 is equal to
n −1 n −1 n −1
C r −1 Cr C r +1 (4 x − 4− x ) 2 (4 x + 4− x ) 2 1
n −1 n −1 n −1
Cr C r +1 C r+2 is -
n n n
(A) 0 (B) –9
Cr C r +1 C r+2 (C) 24 (D) 1
(A) 0 (B) 1 (C) –1 (D) None
One or more than one correct
Part-B
1+ x x x 2
answer type questions
Q.3 If x 1 + x x2
2r x n (n + 1)
x2 x 1+ x
Q.9 If r = 6r 2 − 1 y n 2 (2n + 3) , then the
= ax5 + bx4 + cx3 + dx2 + x +  be an identity
4r 3 − 2nr z n 3 (n + 1)
in x, where a, b, c, d, ,  are independent of x.
n
Then the value of  is
(A) 3 (B) 2 (C) 4 (D) None
value of 
r =1
r is independent of -

Q.4 The system of equations (A) x (B) y

2x – y + z = 0 (C) z (D) n
x – 2y + z = 0
x +a x +b x +a −c
x – y + 2z = 0
Q.10 Let (x) = x + b x + c x −1 and
has infinite number of nontrivial solutions for -
x+c x+d x−b+d
(A)  = 1 (B)  = 5
(C)  = – 5 (D) no real value of  2

Q.5 If a  b  c such that

 (x)dx = – 16, where a, b, c, d are in A.P.,
0

a −1 b −1 c −1
3 3 3 then the common difference of the AP is -
(A) 1 (B) 2
a b c = 0 then (C) –2 (D) None
2 2 2
a b c
Q.11 Let x  – 1 and let a, b, c nonzero real numbers.
(A) ab + bc + ca = 0 (B) a + b + c = 0 Then the determinant
(C) abc = 1 (D) a + b + c = 1 a (1 + x ) b c
1 + a 2 − b2 2ab − 2b a b(1 + x ) c is divisible by -
Q.6 2ab 1− a + b
2 2
2a = a b c(1 + x )
2b − 2a 1− a − b
2 2
(A) abcx
(B) (1 + x)2
(A) (1 – a2 – b2)3 (B) (1 + a2 + b2)3
(C) (1 + x)3
(C) (1 + a2 – b2)3 (D) None of these
(D) x(1 + x)2

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Q.12 Let {1, 2, 3, ......, k} be the set of third- Q.15 Assertion:
order determinants that can be made with the cos( + ) cos( + ) cos( +  )
distinct nonzero real numbers a1, a2, a3, .... a9.
sin(  + ) sin(  + ) sin(  +  )
then -
k
sin(  −  ) sin(  − ) sin(  − )
(A) k = 9 ! (B) 
i =1
i =0
is independent of 
(C) at least one i = 0 (D) None of these Reason: If f() = c, then f() is independent of .

Q.13 System of equation x + 3y + 2z = 6 Part-D Column Matching type questions

x + y + 2z = 7 Q.16 Column-I Column-II
x + 3y + 2z =  has
(A) Let |A| = |aij|3×3  0. (P) 0
(A) unique solution if  = 2,   6
(B) infinitely many solution if  = 4,  = 6 Each element aij is multiplied
(C) no solution if  = 5,  = 7 by ki–j. Let |B| the resulting
(D) no solution if  = 3,  = 5 Determinant, where
k1|A| + k2|B| = 0. Then
Part-C Assertion-Reason type questions
k1 + k 2 =
The following questions consist of two (B) The maximum value of a third (Q) 4
statements each, printed as Assertion-1 and
order determinant each of its
Reason-2. While answering these questions
you are to choose any one of the following entries are ± 1 equals
four responses. 1 cos  cos 
(A) If both Assertion -1 and Reason-2 are (C) If cos  1 cos  (R) 1
true and the Reason-2 is correct cos  cos  1
explanation of the Assertion -1.
(B) If both Assertion -1 and Reason-2 are 0 cos  cos 
true but Reason -2 is not correct = cos  0 cos 
explanation of the Assertion -1. cos  cos  0
(C) If Assertion-1 is true but the Reason-2 is
false. then cos2 + cos2 + cos2 =
(D) If Assertion -1 is false but Reason-2 is true. x2 + x x +1 x − 2
Q.14 Assertion: The system of equations possess a (D) 2x + 3x − 1 3x
2
3x − 3 (S) 2
non trivial solution for the equations x + 2x + 3 2x − 1 2x − 1
2

x + xy + 3z = 0, 3x + xy – 2z = 0
= Ax + B where A and B are
& 2x + 3y – 4z = 0
29 determinants of order 3. Then
then value of k is
2 A + 2B =
Reason: for non trivial solution  = 0

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EXERCISE # 3
JEE MAINS PYQ
1 1 1
1. If D = 1 1 + x 1 for x  0, y  0 the D is - [AIEEE - 2007]
1 1 1+ y
(1) Divisible by both x and y (2) Divisible by x but not y
(3) Divisible by y but not x (4) Divisible by neither x nor y

2. Let a, b, c be any real numbers, Suppose that there are real numbers x, y, z not all zero such that x = cy + bz,
y = az + cx and z = bx + ay, then a2 + b2 + c2 + 2abc is equal to - [AIEEE - 2008]
(1) 2 (2) –1 (3) 0 (4) 1

a a +1 a –1 a +1 b +1 c –1
3. Let a, b, c be such that b(a + c)  0. If –b b + 1 b – 1 + a –1 b –1 c + 1 = 0. [AIEEE - 2009]
n+ 2 n +1
c c –1 c +1 (–1) a ( −1) b (–1)n c
Then the value of n is
(1) Any odd integer (2)Any integer (3) Zero (4) Any even integer

4. Consider the system of linear equation x1 + 2x2 + x3 = 3, 2x1 + 3x2 + x3 = 3, 3x1 + 5x2 + 2x3 = 1 The system has
[AIEEE - 2010]
(1) Infinite number of solutions (2) Exactly 3 solutions
(3) A unique solution (4) No solution

5. The number of values of k for which the linear equations 4x + ky + 2z = 0,

kx + 4y + z = 0, 2x + 2y + z = 0 possess a non-zero solutions is - [AIEEE - 2011]
(1) 1 (2) zero (3) 3 (4) 2

6. The number of values of k, for which the system of equations

( k + 1)x + 8y = 4k, kx + (k + 3)y = 3k –1 has no solution, is - [JEE MAIN - 2013]
(1) infinite (2) 1 (3) 2 (4) 3

3 1 + f(1) 1 + f(2)
7. If    0, and f(n) =n + n and 1 + f(1) 1 + f(2) 1 + f(3) = K (1–)2 (1 – )2 ( – )2 , then K is equal to :
1 + f(2) 1 + f(3) 1 + f(4)
[JEE MAIN - 2014]
1
(1)  (2) (3) 1 (4) –1

8. The set of all values of  for which the system of linear equations :
2x1 –2x2 + x3 = x1, 2x1 – 3x2 + 2x3 = x2, –x1 + 2x2 = x3 has a non-trivial solution [JEE MAIN - 2015]
(1) contains two elements (2) contains more then two elements
(3) is an empty set (4) is a singleton

9. The system of linear equations x + y –z = 0, x – y – z = 0, x + y – z = 0 has non- trivial solution

[JEE MAIN - 2016]
(1) exactly three values of  (2) infinitely many values of 
(3) exactly one value of . (4) exactly two values of .

10. If S is the set of distinct values of ‘b’ for which the following system of linear equations
x + y + z = 1, x + ay + z = 1,ax + by + z = 0 has no solution, then S is [JEE MAIN - 2017]
(1) a singleton (2) an empty set
(3) an infinite set (4) a finite set containing two or more elements

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x – 4 2x 2x
11. If 2x x – 4 2x = (A + Bx)(x–A)2. Then ordered pair (A, B) is equal to : [JEE MAIN - 2018]
2x 2x x – 4
(1) (–4, 3) (2) (–4, 5) (3) (4, 5) (4) (–4,–5)

12. If the system of linear equations x + ky + 3z = 0, 3x + ky – 2z = 0,2x + 4y – 3z = 0 has a non-zero solution

xz
(x, y, z), then is equal to [JEE MAIN - 2018]
y2
(1)10 (2) –30 (3) 30 (4) –10

13. The system of linear equations x + y + z = 2, 2x + 3y + 2z = 5, 2x + 3y + (a2 – 1)z = a + 1 [JEE MAIN 2019]
(1) has infinitely many solutions for a = 4 (2) has a unique solution for |a| = 3
(3) is inconsistent when |a| = 3 (4) is inconsistent when a = 4

14. If the system of linear equation x – 4y + 7z = g, 3y – 5z = h, –2x + 5y – 9z = k is consistent, then : [JEE MAIN 2019]
(1) g + 2h + k = 0 (2) g + h + 2k = 0 (3) g + h + k = 0 (4) 2g + h + k = 0

15. If the system of equations x + y + z = 5, x + 2y + 3z = 9, x + 3y + z =  has infinitely many solutions, then  –  equals -
[JEE MAIN 2019]
(1) 8 (2) 21 (3) 18 (4) 5

16. Let a1, a2, a3, …, a10 be in G.P. with ai > 0 for i = 1, 2, …, 10 and S be the set of pairs (r, k), r, k  N (the set of natural
log e a1r a 2k log e a 2r a 3k log e a 3r a 4k
numbers) for which log e a 4r a 5k log e a 5r a 6r log e a 6r a 7k = 0. Then the number of elements in S, is: [JEE MAIN 2019]
log e a 7r a 8k log e a 8r a 9k log e a 9r a10
k

(1) 4 (2) infinitely many (3) 2 (4) 10

17. The number of values of   (0, ) for which the system of linear equations
x + 3y + 7z = 0, – x + 4y + 7z = 0, (sin 3)x + (cos 2)y + 2z = 0 has a non-trivial solution, is - [JEE MAIN 2019]
(1) two (2) one (3) four (4) three

18. If the system of linear equations 2x + 2y + 3z = a, 3x – y + 5z = b, x – 3y + 2z = c where a, b, c are non zero real numbers,
has more than one solution, then : [JEE MAIN 2019]
(1) b – c – a = 0 (2) a + b + c = 0 (3) b – c + a = 0 (4) b + c – a = 0

a –b–c 2a 2a
19. If 2b b–c–a 2b = (a + b + c) (x + a + b + c)2, x  0 and a + b + c  0, then x is equal to :
2c 2c c–a –b
[JEE MAIN 2019]
(1) – 2(a + b + c) (2) 2(a + b + c) (3) abc (4) – (a + b + c)

20. An ordered pair (, ) for which the system of linear equations
(1 + ) x + y + z = 2, x + (1 + )y + z = 3, x + y + 2z = 2 has a unique solution, is : [JEE MAIN 2019]
(1) (– 3, 1) (2) (1, – 3) (3) (– 4, 2) (4) (2, 4)

21. The set of all values of  for which the system of linear equations
x – 2y – 2z = x, x + 2y + z = y, – x – y = z has a non-trivial solutions : [JEE MAIN 2019]
(1) is an empty set (2) contains more than two elements
(3) is a singleton (4) contains exactly two elements

22. The greatest value of c  R for which the system of linear equations
x – cy – cz = 0, cx – y + cz = 0, cx + cy – z = 0 has a non-trivial solution, is - [JEE MAIN 2019]
(1)1/2 (2) 0 (3) 2 (4) – 1

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y +1  
23. Let  and  be the roots of the equation x2 + x + 1 = 0. Then for y  0 in R,  y+ 1 is equal to
 1 y+
[JEE MAIN 2019]
(1) y(y2 – 3) (2) y3 (3) y(y2 – 1) (4) y3 – 1

24. If the system of equations 2x + 3y – z = 0, x + ky – 2z = 0 and 2x – y + z = 0 has a non-trivial solution (x, y, z), then
x y z
+ + + k is equal to : [JEE MAIN 2019]
y z x
1 3 1
(1) (2) (3) – (4) –4
2 4 4

x sin  cos  x sin 2 cos 2

 
25. If 1 = – sin  –x 1 and 2 = – sin 2 –x 1 , x  0; then for all    0, 2  : [JEE MAIN 2019]
 
cos  1 x cos 2 1 x
(1) 1 – 2 = x(cos 2 – cos4) (2) 1 – 2 = – 2x3
(3) 1 + 2 = – 2(x3 + x – 1) (4) 1 + 2 = – 2x3

26. Let  be a real number for which the system of linear equations x + y + z = 6, 4x + y – z =  – 2, 3x + 2y – 4z = – 5 has
infinitely many solutions. Then  is a root of the quadratic equation [JEE MAIN 2019]
(1) 2 +  – 6 = 0 (2) 2 –  – 6 = 0 (3) 2 – 3 – 4 = 0 (4) 2 + 3 – 4 = 0

x –6 –1
27. The sum of the real roots of the equation 2 –3x x – 3 = 0, is equal to : [JEE MAIN 2019]
–3 2x x + 2
(1) – 4 (2) 0 (3) 1 (4) 6

1 + cos 2  sin 2  4cos 6

 
28. A value of    0,  , for which cos 
2
1 + sin 
2
4cos 6 = 0, is [JEE MAIN 2019]
 3
cos2  sin 2  1 + 4cos 6

  7 7
(1) (2) (3) (4)
18 9 24 36

29. The sum of distinct values of  for which the system of equations
(–1)x + (3 + 1)y + 2z = 0,(–1)x + (4 – 2)y + ( + 3)z = 0, 2x + (3 + 1)y + 3(–1)z = 0, has non-zero solutions, is
_________.
[JEE MAIN 2020]

30. If the system of equations x – 2y + 3z = 9, 2x + y + z = b, x – 7y + az = 24 has infinitely many solutions, then a – b is equal
to ______. [JEE MAIN 2020]

31. Let S be the set of all integer solutions (x, y, z) of the system of equations
x – 2y + 5z = 0; – 2x + 4y + z = 0, – 7x + 14y + 9z = 0 such that 15  x2 + y2 + z2  150. Then, the number of elements in the
set S is equal to ______. [JEE MAIN 2020]

32. If the system of linear equations x + y + z = 6, x + 2y + 3z = 10, 3x + 2y + z = 

has more than two solutions, then  – 2 is equal to ________. [JEE MAIN 2020]

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33. If the system of linear equations 2x + 2ay + az = 0, 2x + 3by + bz = 0, 2x + 4cy + cz = 0 where a, b, c  R are non-zero and
distinct; has a non-zero solution, then : [JEE MAIN 2020]
1 1 1
(1) a + b + c = 0 (2) a, b, c are in G.P. (3) , , are in A.P. (4) a, b, c are in A.P.
a b c

34. For which of the following ordered pairs (, ), the system of linear equations
x + 2y + 3z = 1; 3x + 4y + 5z = 4x + 4y + 4z =  is inconsistent ? [JEE MAIN 2020]
(1) (1, 0) (2) (4, 6) (3) (3, 4) (4) (4, 3)

35. The system of linear equations x + 2y + 2z = 5; 2x + 3y + 5z = 8; 4x + y + 6z = 10 has : [JEE MAIN 2020]
(1) no solution when  = 2 (2) infinitely many solutions when  = 2
(3) a unique solution when  = – 8 (4) no solution when  = 8

36. If for some  and  in R, the intersection of the following three planes
x + 4y – 2z = 1; x + 7y – 5z = ; x + 5y + z = 5 is a line in R3, then  +  is equal to : [JEE MAIN 2020]
(1) 0 (2) 2 (3) 10 (4) –10

x + a x + 2 x +1
37. Let a – 2b + c = 1, If f(x) = x + b x + 3 x + 2 , then : [JEE MAIN 2020]
x +c x +4 x +3
(1) f(– 50) = 501 (2) f(50) = 1 (3) f(– 50) = – 1 (4) f(50) = – 501

38. The following system of linear equations 7x + 6y – 2z = 0; 3x + 4y + 2z = 0; x – 2y – 6z = 0, has [JEE MAIN 2020]
(1) infinitely many solutions, (x, y, z) satisfying y = 2z.
(2) infinitely many solutions, (x, y, z) satisfying x = 2z.
(3) no solution.
(4) only the trivial solution.

39. Let S be the set of all   R for which the system of linear equations
2x – y + 2z = 2; x – 2y + z = – 4; x + y + z = 4 has no solution. Then the set S [JEE MAIN 2020]
(1) contains more than two elements (2) is a singleton
(3) contains exactly two elements (4) is an empty set

x – 2 2x – 3 3x – 4
40. If  = 2x – 3 3x – 4 4x – 5 = Ax3 + Bx2 + Cx + D, then B + C is equal to : [JEE MAIN 2020]
3x – 5 5x – 8 10x –17
(1) – 1 (2) 1 (3) – 3 (4) 9

41. If the system of equations x + y + z = 2, 2x + 4y – z = 6, 3x + 2y + z =  has infinitely many solutions, then

[JEE MAIN 2020]
(1)  – 2 = – 5 (2) 2 –  = 5 (3) 2 +  = 14 (4)  + 2 = 14

42. If the system of linear equations x + y + 3z = 0, x + 3y + k 2z = 0, 3x + y + 3z = 0 has a non-zero solution (x, y, z) for some
y
k  R, then x +   is equal to : [JEE MAIN 2020]
z
(1) 9 (2) –3 (3) –9 (4) 3

x a+y x+a
43. If a + x = b + y = c + z + 1, where a, b, c, x, y, z are non-zero distinct real numbers, then y b + y y + b is equal to :
z c+y z+c
[JEE MAIN 2020]
(1) 0 (2) y(a – b) (3) y(b – a) (4) y(a – c)

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 
44. If the minimum and the maximum values of the function f :  ,  → R. defined by :
4 2
– sin 2  –1 – sin 2  1
f() = – cos 2  –1 – cos 2  1 are m and M respectively, then the ordered pair (m, M) is equal to : [JEE MAIN 2020]
12 10 –2
(1) (0, 4) (2) (–4, 4) (3) (0, 2 2 ) (4) (–4, 0)

45. Let   R. The system of linear equations

2x1 – 4x2 + x3 = 1;x1 – 6x2 + x3 = 2;x1 – 10x2 + 4x3 = 3 is inconsistent for : [JEE MAIN 2020]
(1) exactly one negative value of . (2) exactly one positive value of .
(3) every value of . (4) exactly two values of .

46. The values of  and  for which the system of linear equations
x + y + z = 2, x + 2y + 3z = 5, x + 3y + z =  has infinitely many solutions are, respectively [JEE MAIN 2020]
(1) 5 and 7 (2) 6 and 8 (3) 4 and 9 (4) 5 and 8

47. Let m and M be respectively the minimum and maximum values of

cos 2 x 1 + sin 2 x sin 2x
1 + cos x
2 2
sin x sin 2x then the ordered pair (m, M) is equal to [JEE MAIN 2020]
2
cos x sin x 1 + sin 2x
2

(1) (–3, –1) (2) (–4, –1) (3) (1, 3) (4) (–3, 3)

PYQ JEE ADVANCED

1. Consider the system of equations x – 2y + 3z = –1, –x + y – 2z = k, x – 3y + 4z = 1

Statement-1 : The system of equations has no solution for k  3.
1 3 −1
Statement-2 : The determinant − 1 − 2 k  0, for k  3. [IIT 2008]
1 4 1
(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(B) Statement-1 is True, Statement-2 is True; Statement-2 is not a correct explanation for Statement-1
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True.
2. The number of all possible values of , where 0 <  < , for which the system of equations
2 cos 3 2 sin 3
(y + z) cos 3 = (xyz) sin 3 x sin 3 = + ,(xyz) sin3 = (y + 2z) cos 3 + y sin 3)
y z
have a solution (x0, y0, z0) with y0z0  0, is [IIT- 2010]
(1 + ) (1 + 2) (1 + 3)
2 2 2

3. Which of the following values of  satisfy the equation (2 + )2 (2 + 2)2 (2 + 3)2 = –648 ?
(3 + )2 (3 + 2)2 (3 + 3)2
[JEE (Advanced)-2015]
(A) –4 (B) 9 (C) –9 (D) 4
x x 1+ x 2 3

4. The total number of distinct x  R for which 2x 4x 2

1 + 8x 3 = 10 is [JEE (Advanced)-2016]
3x 9x 2 1 + 27x 3

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 IIT JEE Mains & Advanced Mathematics by OM Sir
5. Let    R. Consider the system of linear equation αx + 2y = ; 3x –2y = 
Which of the following statement(s) is(are) correct? [JEE (Advanced)-2016]
(A)If  = –3, then the system has infinitely many solutions for all values of  and 
(B) If  ≠ –3, then the system has a unique solutions for all values of  and 
(C)If  +  = 0, then the system has infinitely many solutions for  = –3
(D) If  +  ≠ 0, then the system has no solutions for  = –3

ANSWER KEY

EXERCISE # 1

Q.No. 1 2 3 4 5 6 7 8 9 10
Ans. B B C A D A A D A A
11. True

EXERCISE # 2

Q.No. 1 2 3 4 5 6 7 8
Ans. A A A B C B D A

Q.No. 9 10 11 12 13
Ans. A,B,C,D B,C A A,B B,C,D
14. D 15. B 16. A → P; B → Q; C → R; D → P

EXERCISE # 3
JEE- MAIN

1 2 3 4 5 6 7 8 9 10
1 4 1 4 4 2 3 1 1 1
11 12 13 14 15 16 17 18 19 20
2 1 3 4 1 2 1 1 1 4
21 22 23 24 25 26 27 28 29 30
3 1 2 1 4 2 2 2 3.00 5
31 32 33 34 35 36 37 38 39 40
8 13.00 3 4 1 3 2 2 3 3
41 42 43 44 45 46 47
3 2 2 4 1 4 1

JEE Advanced

1.(A) 2.(3) 3(BC) 4(2) 5(BCD)

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