Mathematics: All important previous year questions of jee mains exam to get 99 percentile in mathematics

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AIR JEE Main (2026)
Determinants
Most Important JEE PYQs

Single Correct Type Questions 2 1 0 
Let A 1 2 −1 If |adj (adj (adj 2A)| = (16)n then n
5.=
1. Let p and p + 2 be prime numbers and let  
 0 −1 2 
p! ( p + 1)! ( p + 2)!
= ( p + 1)! ( p + 2)! ( p + 3)!
∆ is equal to [8 April, 2023 (Shift-I)]
( p + 2)! ( p + 3)! ( p + 4)! (a) 8 (b) 10 (c) 9 (d) 12

then the sum of the maximum values of α and β such that 6. Let the determinant of a square matrix A of order m be
pα and (p + 2)b divide ∆ is _____.[29 July, 2022 (Shift-I)] m – n, where m and n satisfy 4m + n = 22 and 17m + 4n =
(a) 0 (b) 1 93. If det (n adj(adj(mA))) = 3a5b6c. then a + b + c is equal
(c) 2 (d) 4 to: [15 April, 2023 (Shift-I)]
2. The maximum value of (a) 96 (b) 101 (c) 109 (d) 84
2 2
sin x 1 + cos x cos 2 x
1 2 3 
f ( x) =
1 + sin 2 x cos 2 x cos 2 x , x ∈ R is
7. Let for A =  . If |2adj (2adj (2A))| = 32n,
2
sin x 2
cos x sin 2 x
= a 3 1  , A 2
1 1 2 
[16 March, 2021 (Shift-II)]
(a) 3 (b) 5 then 3n + α is equal to [13 April, 2023 (Shift-II)]
4 (a) 10 (b) 9 (c) 12 (d) 11
(c) 5 (d) 7 8. Let A be a 3 × 3 matrix such that |adj (adj(adj A))| = 124.
3. Let α and β be the roots of the equation x2 + x + 1 = 0. Then |A–1adj A| is equal to [24 Jan, 2023 (Shift-II)]
y +1 α β
(a) 2 3 (b) 6
Then for y ≠ 0 in R, α y +β 1 is equal to
(c) 12 (d) 1
β 1 y+α
9. Let A be a 2 × 2 matrix with det (A) = –1 and det ((A + I)
[9 April, 2019 (Shift-I)]
(Adj (A) + I)) = 4. Then the sum of the diagonal elements
(a) y3 (b) y3 – 1
of A can be : [26 July, 2022 (Shift-I)]
(c) y(y2 – 1) (d) y(y2 – 3)
(a) –1 (b) 2 (c) 1 (d) – 2
1 1 1
4. Let the number 2, b, c be in an A.P. and A =  2 b c  . 0 1 0 
 4 b 2 c 2   
10. Let the matrix A = 0 0 1  and the matrix B0 = A49 +
If det (A) ∈ [2,16], then c lies in the interval 1 0 0 
[8 April, 2019 (Shift-II)] 2A . If Bn = Adj(Bn–1) for all n ≥ 1, then det(B4) is equal
98

(a) [2, 3) (b) (2 + 23/4, 4) to [28 July, 2022 (Shift-I)]
(d) [4, 6] (a) 3 28
(b) 3 30
(c) 332 (d) 336

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2 3 17. Let A = [aij] and B = [bij] be two 3 × 3 real matrices such
Let A  a 0  , a ∈ R be written as P + Q where P is
11.= that bij = (3)(i+j–2) aji, where i, j = 1, 2, 3. If the determinant
 
a symmetric matrix and Q is skew symmetric matrix. If of B is 81, then the determinant of A is
det(Q) = 9, then the modulus of the sum of all possible
values of determinant of P is equal to: [7 Jan, 2020 (Shift-II)]
[20 July, 2021 (Shift-I)] (a) 1/9 (b) 1/81
(a) 18 (b) 36
(c) 3 (d) 1/3
(c) 24 (d) 45
1 1 2  x − 2 2 x − 3 3x − 4
1 3 4  , B adjA 18. If D = x − 3 3 x − 4 4 x − 5 = Ax3 + Bx2 + Cx + D, then
2
matrices A =
12. If the=  and C = 3A,
1 −1 3  3 x − 5 5 x − 8 10 x − 17
B + C is equal to [3 Sep, 2020 (Shift I)]
adjB
then is equal to: [9 Jan, 2020 (Shift-I)] (a) 9 (b) –1
C
(c) 1 (d) –3
(a) 72 (b) 8
19. If the minimum and the maximum values of the function
(c) 16 (d) 2 π π
f :  ,  → R, defined b  [5 Sep, 2020 (Shift-I)]
13. Let A and B be two invertible matrices of order 3×3. 4 2
If det (ABAT) = 8 and det (AB–1) = 8, then det (BA–1 BT) is
equal to: [11 Jan, 2019 (Shift-II)] − sin 2 θ −1 − sin 2 θ 1
2 2
1 f (θ) = − cos θ −1 − cos θ 1
(a) (b) 1
4 12 10 −2

(c) 1 (d) 16 are m and M respectively, then the ordered pair (m, M) is
16 equal to:
x +1 x x (a) (0, 2 2 ) (b) (0, 4)
9
14. If x x+λ = x (103x + 81) , then λ, λ are (c) (– 4, 4) (d) (– 4, 0)
8 3
x x x + λ2 20. The sum of the real roots of the equation
the roots of the equation [11 April, 2023 (Shift-II)] x −6 −1
(a) 4x2 + 24x – 27 = 0 (b) 4x2 – 24x + 27 = 0 2 −3 x x − 3 = 0, is equal to
(c) 4x2 + 24x + 27 = 0 (d) 4x2 – 24x – 27 = 0 −3 2 x x + 2

 4 −2  [10 April, 2019 (Shift-II)]
15. Let A =  
α β  (a) 6 (b) 1
If A2 + γA + 18I = O, then det(A) is equal to (c) 0 (d) –4
[27 July, 2022 (Shift-II)] x sin θ cos θ
(a) –18 (b) 18 (c) –50 (d) 50 21. If ∆1 = − sin θ − x 1 and
16. Let S = { n :1 ≤ n ≤ 50 and n is odd} cos θ 1 x
x sin 2θ cos 2θ
 1 0 a ∆ 2 = − sin 2θ , x ≠ 0; then for all
−x 1
Let a ∈ S and A=  −1 1 0 
  cos 2θ 1 x
 −a 0 1 
 π
θ ∈  0,   [10 April, 2019 (Shift-I)]
If ∑ det(adj =
A) 100 λ then λ is equal to  2
a∈S
[24 June, 2022 (Shift-I)] (a) ∆1 – ∆2 = x (cos 2θ – cos 4θ)
(b) ∆1 + ∆2 = – 2x3
(a) 218 (b) 221
(c) ∆1 – ∆2 = – 2x3
(c) 663 (d) 1717
(d) ∆1 + ∆2 = – 2(x3 + x – 1)

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 0 2q r  27. If a + x = b + y = c + z + 1, where a, b, c, x, y, z are non-
  zero distinct real numbers, then
22. Let A =  p q −r  . If AAT = I3, then |p| is:
 p −q r  x a+ y x+a
 
y b+ y y + b is equal to. [5 Sep, 2020 (Shift-II)]
[11 Jan, 2019 (Shift-I)]
z c+ y z+c
1 1
(a) (b)
5 3 (a) y(b – a)
1 1 (b) y(a – b)
(c) (d)
2 6 (c) y(a – c)
23. Let A and B be two 3 × 3 matrices such that AB = I (d) 0

1 Integer Type Questions
and |A| = then |adj (B adj(2A))| is equal to
8
1 2k 2k − 1 n
[27 June, 2022 (Shift-II)] 28. L e t =
Dk n n + n + 22
n2 . If ∑D k = 96,
(a) 16 (b) 32 (c) 64 (d) 128 n n2 + n n2 + n + 2
k =1


π  cos θ sin θ 
24. Let θ = and A =   . If B = A + A ,
4
then n is equal to [12 April, 2023 (Shift-I)]
5  − sin θ cos θ 
then det (B) [6 Sep, 2020 (Shift-II)] 29. Let A = {aij} be a 3×3 matrix, where
(a) lies in (2, 3) (b) is zero (−1) j −i if i < j
(c) is one (d) lies in (1, 2) 
= aij = 2 if i j
25. Let a – 2b + c = 1. (−1)i + j if i > j
x + a x + 2 x +1 
If f ( x) =+x b x + 3 x + 2 , then then det (3 Adj (2A–1)) is equal to......
x+c x+4 x+3 [20 July, 2021 (Shift-II)]
[9 Jan, 2020 (Shift-II)] 30. If x, y, z are in arithmetic progression with common
(a) f(–50) = –1 (b) f(50) = 1
difference d, x ≠ 3d, and the determinant of the matrix
(c) f(50) = –501 (d) f(–50) = 501
3 4 2 x
26. Let d ∈ R, and  
4 5 2 y  is zero, then the value of k2 is:
 −2 4+d (sin θ) − 2   
 5 k z
= A  1 (sin θ) + 2 d ,
 q ∈ [0, 2p].

[17 March, 2021 (Shift-II)]
 5 (2sin θ) − d (− sin θ) + 2 + 2d 
If the minimum value of det. (A) is 8, then a value of d is
[10 Jan, 2019 (Shift-I)]
(a) –5 (b) –7
(c) 2( 2 + 1) (d) 2( 2 + 2)




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AIR JEE Main (2026)
Quadratic Equations
Most Important JEE PYQs

Single Correct Type Questions 6. Let α and β be the roots of the equation x2 + (2i – 1) = 0.
Then, the value of |α8 + β8| is equal to :
1. Let a, b be the roots of the quadratic equation
α 23 + β23 + α14 + β14 [29 June, 2022 (Shift-I)]
x2 + 6x + 3 =0. Then 15 is equal to
(a) 50 (b) 250
α + β15 + α10 + β10
(c) 1250 (d) 1500
[12 April, 2023 (Shift-I)]
(a) 729 (b) 72 7. Let α and β be the roots of x2 – 6x – 2 = 0. If an= an – βn
(c) 81 (d) 9 a – 2a8
for n ≥ 1, then the value of 10 is:
3a9
2. Let a, b be the roots of the equation x 2 − 2 x + 2 =.
0
Then a14 + b14 is equal to [13 April, 2023 (Shift-II)] [25 Feb, 2021 (Shift-II)]
(a) −64 2 (b) −128 2 (a) 2 (b) 1 (c) 4 (d) 3
(c) –64 (d) –128 8. The number of real roots of the equation
3. The sum of all the roots of the equation |x – 8x +15| – 2x 2 e6 x - e 4 x - 2e3 x - 12e 2 x + e x + 1 =0 is:
+ 7 = 0 is: [6 April, 2023 (Shift-I)] [25 July, 2021 (Shift-I)]
(a) 9 + 3 (b) 11 + 3 (a) 1 (b) 4 (c) 2 (d) 6
(c) 9 − 3 (d) 11 − 3 9. If a and b are the distinct roots of the equation x2 + (3)1/4x
4. Let f(x) be a quadratic polynomial such that f(–2) + f(3) = 0. + 31/2 = 0, then the value of a96(a12 – 1) + b96(b12 – 1) is
If one of the roots of f(x) = 0 is –1 , then the sum of the roots equal to: [20 July, 2021 (Shift-I)]
of f(x) = 0 is equal to: [28 June, 2022 (Shift-II)] (a) 56×325 (b) 28×325 (c) 52×324 (d) 56×324
(a) 11 (b) 7 10. The value of 4+ 1 [17 March, 2021 (Shift-I)]
1
3 3 5+
4+ 1
1
5+
(c) 13 (d) 14 4+

3 3 2 4
(a) 2 + 30 (b) 2 + 30
5. If a, b are the roots of the equation 5 5

) x + 3(3 )
1 2
4 2
(
x2 − 5 + 3
log3 5
−5
log5 3 (log3 5) 3
− 5(log5 3) −1 =
3
0 (c) 4 +
5
30 (d) 5 +
5
30

1 1 11. cosec 18° is a root of the equation:
then the equation, whose roots are
and

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[31 Aug, 2021 (Shift-I)]
[27 July, 2022 (Shift-II)] (a) x2 + 2x – 4 = 0
(a) 3x – 20x – 12 = 0
2
(b) x2 – 2x + 4 = 0
(b) 3x2 – 10x – 4 = 0 (c) 4x2 + 2x – 1 = 0
(c) 3x2 – 10x + 2 = 0 (d) x2 – 2x – 4 = 0
(d) 3x2 – 20x + 16 = 0