Matrices and Determinants jee mains questions | JEE Main Maths Matrices and Determinants Previous Year Question

Q 11 :
Let $A = [\begin{matrix} α & - 1 \\ 6 & β \end{matrix}], α > 0$ , such that det(A) = 0 and $α + β = 1 .$ If $I$ denotes $2 \times 2$ identity matrix, then the matrix ${(I + A)}^{8}$ is: [2025]

$[\begin{matrix} 257 & - 64 \\ 514 & - 127 \end{matrix}]$
$[\begin{matrix} 1025 & - 511 \\ 2024 & - 1024 \end{matrix}]$
$[\begin{matrix} 4 & - 1 \\ 6 & - 1 \end{matrix}]$
$[\begin{matrix} 766 & - 255 \\ 1530 & - 509 \end{matrix}]$

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(4)

We have, $A = [\begin{matrix} α & - 1 \\ 6 & β \end{matrix}], | A | = 0$

$\Rightarrow α β + 6 = 0 \Rightarrow α β = - 6$

Also, $α + β = 1$ [Given]

$\Rightarrow α = 3 and β = - 2$ ( $∵ α > 0$ )

Now, $A = [\begin{matrix} 3 & - 1 \\ 6 & - 2 \end{matrix}] \Rightarrow A^{2} = [\begin{matrix} 3 & - 1 \\ 6 & - 2 \end{matrix}] [\begin{matrix} 3 & - 1 \\ 6 & - 2 \end{matrix}] = [\begin{matrix} 3 & - 1 \\ 6 & - 2 \end{matrix}]$

$\Rightarrow A^{2} = A$

So, $A = A^{2} = A^{3} = A^{4} = A^{5} = A^{6} = A^{7}$

$∴ {(I + A)}^{8} = I + {}^{8}C_{1} A^{7} + {}^{8}C_{2} A^{6} + . . . + {}^{8}C_{8} A^{8}$

$= I + A ({}^{8}C_{1} + {}^{8}C_{2} + . . . + {}^{8}C_{8}) = I + A (2^{8} - 1)$

$= [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] + [\begin{matrix} 765 & - 255 \\ 1530 & - 510 \end{matrix}] = [\begin{matrix} 766 & - 255 \\ 1530 & - 509 \end{matrix}]$ .

Q 12 :
For some, a, b, let $f (x) =$ $| \begin{matrix} a + \frac{\sin x}{x} & 1 & b \\ a & 1 + \frac{\sin x}{x} & b \\ a & 1 & b + \frac{\sin x}{x} \end{matrix} |$ $, x \neq 0, \lim_{x \to 0} f (x) = λ + μ a + ν b$ . Then ${(λ + μ + ν)}^{2}$ is equal to : [2025]

25
9
16
36

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(3)

$f (x) =$ $| \begin{matrix} a + \frac{\sin x}{x} & 1 & b \\ a & 1 + \frac{\sin x}{x} & b \\ a & 1 & b + \frac{\sin x}{x} \end{matrix} |$ $, x \neq 0$

Since, $\lim_{x \to 0} f (x) = λ + μ a + ν b$

$\Rightarrow λ + μ a + ν b = \lim_{x \to 0}$ $| \begin{matrix} a + \frac{\sin x}{x} & 1 & b \\ a & 1 + \frac{\sin x}{x} & b \\ a & 1 & b + \frac{\sin x}{x} \end{matrix} |$

$=$ $| \begin{matrix} a + 1 & 1 & b \\ a & 1 + 1 & b \\ a & 1 & b + 1 \end{matrix} |$ $=$ $| \begin{matrix} a + 1 & 1 & b \\ a & 2 & b \\ a & 1 & b + 1 \end{matrix} |$

$= (a + 1) [2 (b + 1) - b] - 1 [a (b + 1) - a b] + b \times (a - 2 a)$

$= (a + 1) (b + 2) - a - a b = a + b + 2$

On comparing, we get

$λ = 2, μ = 1 and ν = 1$

Hence, ${(λ + μ + ν)}^{2} = {(2 + 1 + 1)}^{2} = 16$ .

Q 13 :
Let $A = [a_{i j}] = [\begin{matrix} \log_{5} 128 & \log_{4} 5 \\ \log_{5} 8 & \log_{4} 25 \end{matrix}]$ . If $A_{i j}$ is the cofactor of $a_{i j}, C_{i j} = \sum_{k = 1}^{2} a_{i k} A_{j k}, 1 \leq i, j \leq 2$ , and $C = [C_{i j}]$ , then $8$ $| C |$ is equal to : [2025]

242
288
262
222

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(1)

From the given matrix A, $| A |$ $= \frac{11}{2}$

Here, $C_{i j} = \sum_{k = 1}^{2} a_{i k} A_{j k}, 1 \leq i, j \leq 2$

$C_{11} = \sum_{k = 1}^{2} a_{1 k} A_{1 k} = a_{11} A_{11} + a_{12} A_{12} = | A | = \frac{11}{2}$

$C_{12} = \sum_{k = 1}^{2} a_{1 k} A_{2 k} = 0$

$C_{21} = \sum_{k = 1}^{2} a_{2 k} A_{1 k} = 0$

$C_{22} = \sum_{k = 1}^{2} a_{2 k} A_{2 k} = | A | = \frac{11}{2}$

Now, $C = [\begin{matrix} 11 / 2 & 0 \\ 0 & 11 / 2 \end{matrix}] \Rightarrow$ $| C |$ $= \frac{121}{4}$

Hence, $8$ $| C |$ $= 242$ .

Q 14 :
Let M and m respectively be the maximum and the minimum values of $f (x) =$ $| \begin{matrix} 1 + \sin^{2} x & \cos^{2} x & 4 \sin 4 x \\ \sin^{2} x & 1 + \cos^{2} x & 4 \sin 4 x \\ \sin^{2} x & \cos^{2} x & 1 + 4 \sin 4 x \end{matrix} |$ $, x \in R$ . Then $M^{4} - m^{4}$ is equal to : [2025]

1295
1040
1215
1280

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4

(4)

$f (x) =$ $| \begin{matrix} 1 + \sin^{2} x & \cos^{2} x & 4 \sin 4 x \\ \sin^{2} x & 1 + \cos^{2} x & 4 \sin 4 x \\ \sin^{2} x & \cos^{2} x & 1 + 4 \sin 4 x \end{matrix} |$

$=$ $| \begin{matrix} 2 & \cos^{2} x & 4 \sin 4 x \\ 2 & 1 + \cos^{2} x & 4 \sin 4 x \\ 1 & \cos^{2} x & 1 + 4 \sin 4 x \end{matrix} |$ $(Applying C_{1} \to C_{1} + C_{2})$

$=$ $| \begin{matrix} 2 & \cos^{2} x & 4 \sin 4 x \\ 0 & 1 & 0 \\ 1 & \cos^{2} x & 1 + 4 \sin 4 x \end{matrix} |$ $(Applying R_{2} \to R_{2} - R_{1})$

Expanding along $R_{2}$ , we get f(x) = 2(1 + 4 sin 4x) – 4 sin 4x

$\Rightarrow f (x) = 2 + 4 \sin 4 x$

Maximum value of f(x), M = 6

Minimum value of f(x), m = –2

$∴ M^{4} - m^{4} = 1280$ .

Q 15 :
Let $I$ be the identity matrix of order $3 \times 3$ and for the matrix $A = [\begin{matrix} λ & 2 & 3 \\ 4 & 5 & 6 \\ 7 & - 1 & 2 \end{matrix}], | A | = - 1$ . Let B be the inverse of the matrix adj $(A a d j (A^{2}))$ . Then $| (λ B + I) |$ is equal to __________. [2025]

0%

(38)

Given, $A = [\begin{matrix} λ & 2 & 3 \\ 4 & 5 & 6 \\ 7 & - 1 & 2 \end{matrix}]$

$\Rightarrow | A | = [\begin{matrix} λ & 2 & 3 \\ 4 & 5 & 6 \\ 7 & - 1 & 2 \end{matrix}] = - 1$

$\Rightarrow λ (16) - 2 (- 34) + 3 (- 39) = - 1$

$\Rightarrow 16 λ = 48 \Rightarrow λ = 3$

Given, $B^{- 1} = adj (A \cdot adj (A^{2}))$

Let $C = A \cdot adj (A^{2})$

$A C = A^{2} adj (A^{2}) = | A |^{2} \cdot I = I$

$\Rightarrow C = A^{- 1}$ [ $∵$ |A| = – 1]

Now, $B^{- 1} = adj (A^{- 1}) \Rightarrow B = adj (A)$

Now, $λ B + I = 3 B + I$

Let P = 3B + $I$

$\Rightarrow$ P = 3 adj(A) + $I$

$\Rightarrow$ AP = A 3 adj(A)+ (A)

$\Rightarrow$ AP = 3 $| A |$ $\cdot$ $I$ + A $\Rightarrow$ AP = A – 3 $I$

$\Rightarrow$ $| A P |$ $=$ $| A - 3 I |$

$| A |$ $\cdot$ $| P |$ $=$ $[\begin{matrix} 0 & 2 & 3 \\ 4 & 2 & 6 \\ 7 & - 1 & - 1 \end{matrix}]$

= 0 – 2(–46) + 3(–18)

= 92 – 54 = 38

$\Rightarrow$ $| P |$ = 38.

Q 16 :
Let integers $a, b \in [- 3, 3]$ be such that $a + b \neq 0$ . Then the number of all possible ordered pairs (a, b) for which $| \frac{z - a}{z + b} |$ $= 1$ and $| \begin{matrix} z + 1 & ω & ω^{2} \\ ω & z + ω^{2} & 1 \\ ω^{2} & 1 & z + ω \end{matrix} |$ $= 1, z \in C$ , where $ω$ and $ω^{2}$ are the roots of $x^{2} + x + 1 = 0$ , is equal to __________. [2025]

0%

(10)

We have, $a, b \in z, - 3 \leq a, b \leq 3$ and $a + b \neq 0$ .

Also, $| \frac{z - a}{z + b} |$ $= 1$ and $| \begin{matrix} z + 1 & ω & ω^{2} \\ ω & z + ω^{2} & 1 \\ ω^{2} & 1 & z + ω \end{matrix} |$ $= 1$

Applying $C_{1} \to C_{1} + C_{2} + C_{3}$

$z$ $| \begin{matrix} 1 & ω & ω^{2} \\ 1 & z + ω^{2} & 1 \\ 1 & 1 & z + ω \end{matrix} |$ $= 1$ [ $∵ 1 + ω + ω^{2} = 0$ ]

On expanding, we get

$z [(z^{2} + (ω^{2} + ω) z + ω^{3} - 1) - ω z - ω^{2} - ω^{2} + ω - ω^{2} z - ω] = 1$

$\Rightarrow z (z^{2}) = 1 \Rightarrow z^{3} = 1 \Rightarrow z = 1, ω, ω^{2}$

Case 1 : $z = ω$ , then $a + b \neq 0$ and a – b = –1

a = –3, b = –2; a = –2; b = –1;

a = –1, b = 0; a = 0, b = 1

a = 1, b = 2; a = 2, b = 3

Case 2 : z = 1; then a – b = 2 and $a + b \neq 0$

a = –1, b = –3; a = 0, b = –2; a = 2, b = 0; a = 3, b = 1

Total pairs = 10.

Topic Question Set

Q 11 :
Let $A = [\begin{matrix} α & - 1 \\ 6 & β \end{matrix}], α > 0$ , such that det(A) = 0 and $α + β = 1 .$ If $I$ denotes $2 \times 2$ identity matrix, then the matrix ${(I + A)}^{8}$ is: [2025]

Q 12 :
For some, a, b, let $f (x) =$ $| \begin{matrix} a + \frac{\sin x}{x} & 1 & b \\ a & 1 + \frac{\sin x}{x} & b \\ a & 1 & b + \frac{\sin x}{x} \end{matrix} |$ $, x \neq 0, \lim_{x \to 0} f (x) = λ + μ a + ν b$ . Then ${(λ + μ + ν)}^{2}$ is equal to : [2025]

Q 13 :
Let $A = [a_{i j}] = [\begin{matrix} \log_{5} 128 & \log_{4} 5 \\ \log_{5} 8 & \log_{4} 25 \end{matrix}]$ . If $A_{i j}$ is the cofactor of $a_{i j}, C_{i j} = \sum_{k = 1}^{2} a_{i k} A_{j k}, 1 \leq i, j \leq 2$ , and $C = [C_{i j}]$ , then $8$ $| C |$ is equal to : [2025]

Q 15 :
Let $I$ be the identity matrix of order $3 \times 3$ and for the matrix $A = [\begin{matrix} λ & 2 & 3 \\ 4 & 5 & 6 \\ 7 & - 1 & 2 \end{matrix}], | A | = - 1$ . Let B be the inverse of the matrix adj $(A a d j (A^{2}))$ . Then $| (λ B + I) |$ is equal to __________. [2025]

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Topic Question Set

Q 11 : Let A=[α–16β], α>0, such that det(A) = 0 and α+β=1. If I denotes 2×2 identity matrix, then the matrix (I+A)8 is: [2025]

Q 12 : For some, a, b, let f(x)=|a+sin xx1ba1+sin xxba1b+sin xx|, x≠0, limx→0f(x)=λ+μa+νb. Then (λ+μ+ν)2 is equal to : [2025]

Q 13 : Let A=[aij]=[log5128log45log58log425]. If Aij is the cofactor of aij,Cij=∑k=12aik Ajk, 1≤i, j≤2, and C=[Cij], then 8|C| is equal to : [2025]

Q 14 : Let M and m respectively be the maximum and the minimum values of f(x)=|1+sin2xcos2x4sin4xsin2x1+cos2x4sin4xsin2xcos2x1+4sin4x|, x∈R. Then M4–m4 is equal to : [2025]

Q 15 : Let I be the identity matrix of order 3×3 and for the matrix A=[λ234567–12], |A|=–1. Let B be the inverse of the matrix adj(A adj (A2)). Then |(λB+I)| is equal to __________. [2025]

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Q 16 : Let integers a, b∈[–3,3] be such that a+b≠0. Then the number of all possible ordered pairs (a, b) for which |z–az+b|=1 and |z+1ωω2ωz+ω21ω21z+ω|=1, z∈C, where ω and ω2 are the roots of x2+x+1=0, is equal to __________. [2025]

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Q 11 :
Let $A = [\begin{matrix} α & - 1 \\ 6 & β \end{matrix}], α > 0$ , such that det(A) = 0 and $α + β = 1 .$ If $I$ denotes $2 \times 2$ identity matrix, then the matrix ${(I + A)}^{8}$ is: [2025]

Q 12 :
For some, a, b, let $f (x) =$ $| \begin{matrix} a + \frac{\sin x}{x} & 1 & b \\ a & 1 + \frac{\sin x}{x} & b \\ a & 1 & b + \frac{\sin x}{x} \end{matrix} |$ $, x \neq 0, \lim_{x \to 0} f (x) = λ + μ a + ν b$ . Then ${(λ + μ + ν)}^{2}$ is equal to : [2025]

Q 13 :
Let $A = [a_{i j}] = [\begin{matrix} \log_{5} 128 & \log_{4} 5 \\ \log_{5} 8 & \log_{4} 25 \end{matrix}]$ . If $A_{i j}$ is the cofactor of $a_{i j}, C_{i j} = \sum_{k = 1}^{2} a_{i k} A_{j k}, 1 \leq i, j \leq 2$ , and $C = [C_{i j}]$ , then $8$ $| C |$ is equal to : [2025]

Q 15 :
Let $I$ be the identity matrix of order $3 \times 3$ and for the matrix $A = [\begin{matrix} λ & 2 & 3 \\ 4 & 5 & 6 \\ 7 & - 1 & 2 \end{matrix}], | A | = - 1$ . Let B be the inverse of the matrix adj $(A a d j (A^{2}))$ . Then $| (λ B + I) |$ is equal to __________. [2025]