LFM Pure

jee-main 2022 Q68 Determinant and Rank Computation View

Let $A$ be a matrix of order $3 \times 3$ and $\operatorname{det}(A) = 2$. Then $\operatorname{det}\left(\operatorname{det}(A)\operatorname{adj}\left(5\operatorname{adj}\left(A^3\right)\right)\right)$ is equal to
(1) $256 \times 10^6$
(2) $1024 \times 10^6$
(3) $512 \times 10^6$
(4) $256 \times 10^{11}$

jee-main 2022 Q69 Determinant and Rank Computation View

Let $S = \{ \sqrt { n } : 1 \leqslant n \leqslant 50$ and $n$ is odd $\}$. Let $a \in S$ and $A = \left[ \begin{array} { r r r } 1 & 0 & a \\ - 1 & 1 & 0 \\ - a & 0 & 1 \end{array} \right]$. If $\Sigma _ { a \in S } \operatorname { det } ( \operatorname { adj } A ) = 100 \lambda$, then $\lambda$ is equal to
(1) 218
(2) 221
(3) 663
(4) 1717

jee-main 2022 Q69 Matrix Power Computation and Application View

Let $A = \begin{pmatrix} 0 & -2 \\ 2 & 0 \end{pmatrix}$. If $M$ and $N$ are two matrices given by $M = \sum _ { k = 1 } ^ { 10 } A ^ { 2k }$ and $N = \sum _ { k = 1 } ^ { 10 } A ^ { 2k - 1 }$ then $MN^{2}$ is
(1) a non-identity symmetric matrix
(2) a skew-symmetric matrix
(3) neither symmetric nor skew-symmetric matrix
(4) an identity matrix

jee-main 2022 Q69 Linear System and Inverse Existence View

If the system of equations $\alpha x + y + z = 5 , x + 2 y + 3 z = 4 , x + 3 y + 5 z = \beta$ has infinitely many solutions, then the ordered pair $( \alpha , \beta )$ is equal to
(1) $( 1 , - 3 )$
(2) $( - 1,3 )$
(3) $( 1,3 )$
(4) $( - 1 , - 3 )$

jee-main 2022 Q69 Linear System and Inverse Existence View

If the system of linear equations $2x + 3y - z = -2$ $x + y + z = 4$ $x - y + |\lambda|z = 4\lambda - 4$ where $\lambda \in \mathbb{R}$, has no solution, then
(1) $\lambda = 7$
(2) $\lambda = -7$
(3) $\lambda = 8$
(4) $\lambda^2 = 1$

jee-main 2022 Q69 True/False or Multiple-Select Conceptual Reasoning View

Which of the following matrices can NOT be obtained from the matrix $\begin{pmatrix} -1 & 2 \\ 1 & -1 \end{pmatrix}$ by a single elementary row operation?
(1) $\begin{pmatrix} 0 & 1 \\ 1 & -1 \end{pmatrix}$
(2) $\begin{pmatrix} 1 & -1 \\ -1 & 2 \end{pmatrix}$
(3) $\begin{pmatrix} -1 & 2 \\ -2 & 7 \end{pmatrix}$
(4) $\begin{pmatrix} -1 & 2 \\ -1 & 3 \end{pmatrix}$

jee-main 2022 Q70 Linear System and Inverse Existence View

The number of values of $\alpha$ for which the system of equations $x + y + z = \alpha$ $\alpha x + 2 \alpha y + 3 z = - 1$ $x + 3 \alpha y + 5 z = 4$ is inconsistent, is
(1) 0
(2) 1
(3) 2
(4) 3

jee-main 2022 Q70 Linear System and Inverse Existence View

Let $A$ be a $3 \times 3$ real matrix such that $A \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$; $A \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$ and $A \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}$. If $X = \begin{pmatrix} x _ { 1 } \\ x _ { 2 } \\ x _ { 3 } \end{pmatrix}$ and $I$ is an identity matrix of order 3, then the system $( A - 2I ) X = \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}$ has
(1) no solution
(2) infinitely many solutions
(3) unique solution
(4) exactly two solutions

jee-main 2022 Q70 Matrix Algebra and Product Properties View

Let $A = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}$ and $B = \begin{pmatrix} 9 ^ { 2 } & - 10 ^ { 2 } & 11 ^ { 2 } \\ 12 ^ { 2 } & 13 ^ { 2 } & - 14 ^ { 2 } \\ - 15 ^ { 2 } & 16 ^ { 2 } & 17 ^ { 2 } \end{pmatrix}$, then the value of $A ^ { \prime } B A$ is:
(1) 1224
(2) 1042
(3) 540
(4) 539

jee-main 2022 Q70 Matrix Power Computation and Application View

Let the matrix $A = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ and the matrix $B_0 = A^{49} + 2A^{98}$. If $B_n = \text{Adj}(B_{n-1})$ for all $n \geq 1$, then $\det(B_4)$ is equal to
(1) $3^{28}$
(2) $3^{30}$
(3) $3^{32}$
(4) $3^{36}$

jee-main 2022 Q70 Linear Transformation and Endomorphism Properties View

Let $A$ and $B$ be two $3 \times 3$ non-zero real matrices such that $A B$ is a zero matrix. Then
(1) The system of linear equations $A X = 0$ has a unique solution
(2) The system of linear equations $A X = 0$ has infinitely many solutions
(3) $B$ is an invertible matrix
(4) $\operatorname { adj } ( A )$ is an invertible matrix

jee-main 2022 Q70 Linear System and Inverse Existence View

If the system of equations $x + y + z = 6$ $2x + 5y + \alpha z = \beta$ $x + 2y + 3z = 14$ has infinitely many solutions, then $\alpha + \beta$ is equal to
(1) 8
(2) 36
(3) 44
(4) 48

jee-main 2022 Q70 Determinant and Rank Computation View

Let $A = \left( \begin{array} { c c } 4 & - 2 \\ \alpha & \beta \end{array} \right)$. If $A ^ { 2 } + \gamma A + 18 I = O$, then $\operatorname { det } ( A )$ is equal to
(1) - 18
(2) 18
(3) - 50
(4) 50

jee-main 2022 Q71 Linear System and Inverse Existence View

The system of equations $-kx + 3y - 14z = 25$ $-15x + 4y - kz = 3$ $-4x + y + 3z = 4$ is consistent for all $k$ in the set
(1) $R$
(2) $R - \{-11, 13\}$
(3) $R - \{-13\}$
(4) $R - \{-11, 11\}$

jee-main 2022 Q71 Matrix Power Computation and Application View

Let $A = \left[ a _ { i j } \right]$ be a square matrix of order 3 such that $a _ { i j } = 2 ^ { j - i }$, for all $i , j = 1,2,3$. Then, the matrix $A ^ { 2 } + A ^ { 3 } + \ldots + A ^ { 10 }$ is equal to
(1) $\left( \frac { 3 ^ { 10 } - 1 } { 2 } \right) A$
(2) $\left( \frac { 3 ^ { 10 } + 1 } { 2 } \right) A$
(3) $\left( \frac { 3 ^ { 10 } + 3 } { 2 } \right) A$
(4) $\left( \frac { 3 ^ { 10 } - 3 } { 2 } \right) A$

jee-main 2022 Q71 Matrix Algebra and Product Properties View

Let $A = \left( \begin{array} { c c } 1 & 2 \\ - 2 & - 5 \end{array} \right)$. Let $\alpha , \beta \in \mathbb { R }$ be such that $\alpha A ^ { 2 } + \beta A = 2 I$. Then $\alpha + \beta$ is equal to
(1) - 10
(2) - 6
(3) 6
(4) 10

jee-main 2022 Q72 Linear System and Inverse Existence View

If the system of linear equations $2 x + y - z = 7$ $x - 3 y + 2 z = 1$ $x + 4 y + \delta z = k$, where $\delta , k \in R$ has infinitely many solutions, then $\delta + k$ is equal to
(1) $- 3$
(2) 3
(3) 6
(4) 9

jee-main 2022 Q72 Linear System and Inverse Existence View

The number of real values of $\lambda$, such that the system of linear equations $2x - 3y + 5z = 9$ $x + 3y - z = -18$ $3x - y + (\lambda^2 - |\lambda|)z = 16$ has no solutions, is
(1) 0
(2) 1
(3) 2
(4) 4

jee-main 2022 Q84 Determinant and Rank Computation View

Consider a matrix $\mathrm { A } = \left[ \begin{array} { c c c } \alpha & \beta & \gamma \\ \alpha ^ { 2 } & \beta ^ { 2 } & \gamma ^ { 2 } \\ \beta + \gamma & \gamma + \alpha & \alpha + \beta \end{array} \right]$, where $\alpha , \beta , \gamma$ are three distinct natural numbers. If $\frac { \operatorname { det } ( \operatorname { adj } ( \operatorname { adj } ( \operatorname { adj } ( \operatorname { adj } A ) ) ) } { ( \alpha - \beta ) ^ { 16 } ( \beta - \gamma ) ^ { 16 } ( \gamma - \alpha ) ^ { 16 } } = 2 ^ { 32 } \times 3 ^ { 16 }$, then the number of such 3-tuples $( \alpha , \beta , \gamma )$ is $\_\_\_\_$ .

jee-main 2022 Q86 Determinant and Rank Computation View

Let $S$ be the set containing all $3 \times 3$ matrices with entries from $\{ - 1,0,1 \}$. The total number of matrices $A \in S$ such that the sum of all the diagonal elements of $A ^ { T } A$ is 6 is $\_\_\_\_$ .

jee-main 2022 Q87 Matrix Power Computation and Application View

Let $A = \begin{pmatrix} 2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0 \end{pmatrix}$ and $B = A - I$. If $\omega = \frac { \sqrt { 3 }\, i - 1 } { 2 }$, then the number of elements in the set $\left\{ n \in \{1,2,\ldots,100\} : A ^ { n } + \omega B ^ { n } = A + B \right\}$ is equal to $\_\_\_\_$.

jee-main 2023 Q68 Matrix Algebra and Product Properties View

If $A$ and $B$ are two non-zero $n \times n$ matrices such that $A ^ { 2 } + B = A ^ { 2 } B$, then
(1) $A B = I$
(2) $A ^ { 2 } B = I$
(3) $A ^ { 2 } = I$ or $B = I$
(4) $A ^ { 2 } B = B A ^ { 2 }$

jee-main 2023 Q68 Matrix Power Computation and Application View

Let $A = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$. If $M = \sum_{k=1}^{20} (A^k + B^k)$, then $\det(M)$ is equal to
(1) 100
(2) 200
(3) 0
(4) 400

jee-main 2023 Q68 Matrix Power Computation and Application View

Let $A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 12 & -3 \end{pmatrix}$. Then the sum of the diagonal elements of the matrix $(A+I)^{11}$ is equal to:
(1) 6144
(2) 4094
(3) 4097
(4) 2050

jee-main 2023 Q68 Determinant and Rank Computation View

If $A$ is a $3 \times 3$ matrix and $|A| = 2$, then $|3\, \text{adj}(3A)| \cdot |A^2|$ is equal to
(1) $3^{12} \cdot 6^{11}$
(2) $3^{12} \cdot 6^{10}$
(3) $3^{10} \cdot 6^{11}$
(4) $3^{11} \cdot 6^{10}$

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63

Load questions...

LFM Pure

Straight Lines & Coordinate Geometry 247

Circles 443

Simultaneous equations 79

Proof 711

Proof by induction 25

Sine and Cosine Rules 195

Radians, Arc Length and Sector Area 35

Trig Graphs & Exact Values 75

Standard trigonometric equations 106

Trig Proofs 53

Quadratic trigonometric equations 36

Trigonometric equations in context 18

Modulus function 49

Matrices 1553

Linear transformations 26

Invariant lines and eigenvalues and vectors 188

Reciprocal Trig & Identities 44

Addition & Double Angle Formulae 98

Harmonic Form 12

Small angle approximation 20

Chain Rule 113

Product & Quotient Rules 18

Differentiating Transcendental Functions 183

Connected Rates of Change 56

Standard Integrals and Reverse Chain Rule 43

Integration by Substitution 116

Integration by Parts 74

Implicit equations and differentiation 43

Differential equations 352

3x3 Matrices 144