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Determinants · Question 10

JEE Main 2026 (Online) 4th April Evening ShiftMCQMedium📈 Topic in 6% papers+4−1
1

Let A= bmatrix1 & 2 & 7\ 4 & -2 & 8\ 3 & 8 & -7 bmatrix and (A- I)=0, where is a real number. If the largest possible value of is p, then the circle (x-p)2+(y-2p)2=320, intersects the co-ordinate axes at

  1. 1 point
  2. 2 points
  3. 3 points
  4. 4 points

Answer

(3) 3 points

Solution

(A- I)=0 3-(tr A) 2+(sum of principal minors) - A=0

tr A=1-2-7=-8

Principal minors: (14-64)+(-7-21)+(-2-8)=-50-28-10=-88

A=1(14-64)-2(-28-24)+7(32+6)=-50+104+266=320

3+8 2-88 -320=0

=8 satisfies it: 512+512-704-320=0

( -8) ( 2+16 +40 )=0 =8, -8 2 6

p=8 and the circle is (x-8)2+(y-16)2=320, centre C(8, 16), radius 8 5

[figure]

x-axis (y=0): (x-8)2=320-256=64 x=0, 16

y-axis (x=0): (y-16)2=320-64=256 y=0, 32

Distinct points: (0, 0), (16, 0), (0, 32) (the origin is common to both axes)

the circle meets the co-ordinate axes at 3 points.

Answer: (3)

JEE Main 2026 (Online) 4th April Evening Shift

Keep practising — Determinants

2

Let , R be such that the system of linear equations x+2y+z=5 2x+y+ z=5 8x+4y+ z=18 has no solution. Then is equal to :

  1. -4
  2. 4
  3. 8
  4. -8
JEE Main 2026 (Online) 2nd April Morning Shift
3

If the system of equations x+5y+6z=4, 2x+3y+4z=7, x+6y+az=b has infinitely many solutions, then the point (a, b) lies on the line

  1. y-x=3
  2. x-y=3
  3. x+y=11
  4. x+y=12
JEE Main 2026 (Online) 2nd April Evening Shift
4

Let P = [pij] and Q = [qij] be two square matrices of order 3 such that qij = 2(i+j-1)pij and (Q) = 210. Then the value of ( adj( adj P)) is:

  1. 81
  2. 16
  3. 124
  4. 32
JEE Main 2026 (Online) 24th January Evening Shift
5

Let f(x) = 7x10 + 9x8(1 + x2 + 2x9)2 dx, x > 0, x 0 f(x) = 0 and f(1) = 14. If A = bmatrix 0 & 0 & 1 \ 14 & f'(1) & 1 \ 2 & 4 & 1 bmatrix and B = adj(adj A) be such that |B| = 81, then 2 is equal to

  1. 2
  2. 4
  3. 3
  4. 1
JEE Main 2026 (Online) 24th January Evening Shift
6

Among the statements : I: If vmatrix 1 & & \ & 1 & \ & & 1 vmatrix = vmatrix 0 & & \ & 0 & \ & & 0 vmatrix, then 2 + 2 + 2 = 32, and II: If vmatrix x2 + x & x + 1 & x - 2 \ 2x2 + 3x - 1 & 3x & 3x - 3 \ x2 + 2x + 3 & 2x - 1 & 2x - 1 vmatrix = px + q, then p2 = 196q2.

  1. both are true
  2. both are false
  3. only I is true
  4. only II is true
JEE Main 2026 (Online) 23rd January Morning Shift
7

Let |A| = 6, where A is a 3 3 matrix. If |adj (3 adj (A2 adj(2A) ) ) | = 2m 3n, m, n N, then m + n is equal to .

JEE Main 2026 (Online) 23rd January Morning Shift
8

The system of linear equations x + y + z = 6 2x + 5y + az = 36 x + 2y + 3z = b has :

  1. unique solution for a = 8 and b = 16
  2. infinitely many solutions for a = 8 and b = 14
  3. infinitely many solutions for a = 8 and b = 16
  4. unique solution for a = 8 and b = 14
JEE Main 2026 (Online) 23rd January Evening Shift
9

Let A be a 3 3 matrix such that A + AT = O. If A bmatrix 1 \ -1 \ 0 bmatrix = bmatrix 3 \ 3 \ 2 bmatrix, A2 bmatrix 1 \ -1 \ 0 bmatrix = bmatrix -3 \ 19 \ -24 bmatrix and (adj(2 adj(A + I))) = (2) (3) (11) , , , are non-negative integers, then + + is equal to

JEE Main 2026 (Online) 22nd January Morning Shift
10

Let n be the number obtained on rolling a fair die. If the probability that the system x - ny + z = 6 x + (n - 2)y + (n + 1)z = 8 (n - 1)y + z = 1 has a unique solution is k6, then the sum of k and all possible values of n is :

  1. 22
  2. 20
  3. 24
  4. 21
JEE Main 2026 (Online) 22nd January Evening Shift
11

If X = bmatrix x \ y \ z bmatrix is a solution of the system of equations AX = B, where adj A = bmatrix 4 & 2 & 2 \ -5 & 0 & 5 \ 1 & -2 & 3 bmatrix and B = bmatrix 4 \ 0 \ 2 bmatrix, then |x + y + z| is equal to :

  1. 3
  2. 2
  3. 32
  4. 1
JEE Main 2026 (Online) 22nd January Evening Shift

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