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Determinants · Question 5

JEE Main 2026 (Online) 5th April Morning ShiftMCQHard📈 Topic in 42% papers+4−1
1

Let A be a 3 3 matrix such that AT bmatrix1\ 0\ 1 bmatrix= bmatrix5\ 2\ 2 bmatrix, AT bmatrix0\ 0\ 1 bmatrix= bmatrix3\ 1\ 1 bmatrix, A bmatrix1\ 0\ 1 bmatrix= bmatrix3\ 4\ 4 bmatrix and A bmatrix0\ 0\ 1 bmatrix= bmatrix1\ 3\ 1 bmatrix. If (A)=1, then (adj (A2+A ) ) is equal to:

  1. 16
  2. 25
  3. 49
  4. 64

Answer

(4) 64

Solution

Let e1= bmatrix1\\0\\0 bmatrix, e3= bmatrix0\\0\\1 bmatrix. Then ATek = k-th row of A and Aek = k-th column of A.

ATe3=(3, 1, 1) R3=(3, 1, 1)

AT(e1+e3)=R1+R3=(5, 2, 2) R1=(2, 1, 1)

Ae3=C3=(1, 3, 1); A(e1+e3)=C1+C3=(3, 4, 4) C1=(2, 1, 3)

A= bmatrix2 & 1 & 1\ 1 & k & 3\ 3 & 1 & 1 bmatrix

A=2(k-3)-1(1-9)+1(1-3k)=3-k=1 k=2

A2+A=A(A+I), A+I= bmatrix3 & 1 & 1\ 1 & 3 & 3\ 3 & 1 & 2 bmatrix

(A+I)=3(6-3)-1(2-9)+1(1-9)=9+7-8=8

(A2+A )= A (A+I)=1 8=8

|adj M|=|M|n-1 with n=3

(adj (A2+A ) )=82=64

Answer: (4)

JEE Main 2026 (Online) 5th April Morning Shift

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2

Consider the system of linear equations in x, y, z : x+2y+tz=0 6x+y+5tz=0 3x+t2y+f(t)z=0 where f : R R is a differentiable function. If this system has infinitely many solutions for all t R, then f

  1. is a constant function
  2. is strictly increasing on R
  3. is strictly decreasing on R
  4. has two critical points
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3

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4

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  1. 3
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5

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  2. 18
  3. 19
  4. 21
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  1. -4
  2. 4
  3. 8
  4. -8
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  1. y-x=3
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  3. x+y=11
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9

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  1. 2
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11

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  1. both are true
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