LFM Pure

jee-main 2023 Q70 Determinant and Rank Computation View

Let for $A = \begin{pmatrix} 1 & 2 & 3 \\ \alpha & 3 & 1 \\ 1 & 1 & 2 \end{pmatrix}$, $|A| = 2$. If $| 2 \operatorname { adj } ( 2 \operatorname { adj } ( 2 A ) ) | = 32 ^ { n }$, then $3 n + \alpha$ is equal to
(1) 9
(2) 11
(3) 12
(4) 10

jee-main 2023 Q70 Determinant and Rank Computation View

Let the determinant of a square matrix $A$ of order $m$ be $m - n$, where m and $n$ satisfy $4 m + n = 22$ and $17 m + 4 n = 93$. If $\operatorname { det } ( n \operatorname { adj } ( \operatorname { adj } ( m A ) ) ) = 3 ^ { a } 5 ^ { b } 6 ^ { c }$, then $a + b + c$ is equal to
(1) 84
(2) 96
(3) 101
(4) 109

jee-main 2023 Q71 Linear System and Inverse Existence View

Let $S$ denote the set of all real values of $\lambda$ such that the system of equations $$\lambda x + y + z = 1$$ $$x + \lambda y + z = 1$$ $$x + y + \lambda z = 1$$ is inconsistent, then $\sum_{\lambda \in S} (\lambda^2 + \lambda)$ is equal to
(1) 2
(2) 12
(3) 4
(4) 6

jee-main 2023 Q72 Determinant and Rank Computation View

If $A = \frac { 1 } { 5! \cdot 6! \cdot 7! } \begin{vmatrix} 5! & 6! & 7! \\ 6! & 7! & 8! \\ 7! & 8! & 9! \end{vmatrix}$, then $|\text{adj}(\text{adj}(2A))|$ is equal to
(1) $2 ^ { 20 }$
(2) $2 ^ { 8 }$
(3) $2 ^ { 12 }$
(4) $2 ^ { 16 }$

jee-main 2023 Q74 Determinant and Rank Computation View

Let $f(x) = \begin{vmatrix} 1 + \sin^2 x & \cos^2 x & \sin 2x \\ \sin^2 x & 1 + \cos^2 x & \sin 2x \\ \sin^2 x & \cos^2 x & 1 + \sin 2x \end{vmatrix}$, $x \in \left[\frac{\pi}{6}, \frac{\pi}{3}\right]$. If $\alpha$ and $\beta$ respectively are the maximum and the minimum values of $f$, then
(1) $\beta^2 - 2\sqrt{\alpha} = \frac{19}{4}$
(2) $\beta^2 + 2\sqrt{\alpha} = \frac{19}{4}$
(3) $\alpha^2 - \beta^2 = 4\sqrt{3}$
(4) $\alpha^2 + \beta^2 = \frac{9}{2}$

jee-main 2023 Q74 Determinant and Rank Computation View

Let $x , y , z > 1$ and $A = \left[ \begin{array} { l l l } 1 & \log _ { x } y & \log _ { x } z \\ \log _ { y } x & 2 & \log _ { y } z \\ \log _ { z } x & \log _ { z } y & 3 \end{array} \right]$. Then $\left| \operatorname { adj } \left( \operatorname { adj } \mathrm { A } ^ { 2 } \right) \right|$ is equal to
(1) $6 ^ { 4 }$
(2) $2 ^ { 8 }$
(3) $4 ^ { 8 }$
(4) $2 ^ { 4 }$

jee-main 2023 Q74 Determinant and Rank Computation View

If $P$ is a $3 \times 3$ real matrix such that $P^{T} = aP + (a-1)I$, where $a > 1$, then
(1) $P$ is a singular matrix
(2) $|\operatorname{Adj} P| > 1$
(3) $|\operatorname{Adj} P| = \frac{1}{2}$
(4) $|\operatorname{Adj} P| = 1$

jee-main 2023 Q77 Determinant of Parametric or Structured Matrix View

Let $A$ be a $3 \times 3$ matrix and $\det(A) = 2$. If $n = \det(\text{adj}(\text{adj}(\cdots(\text{adj}(A))\cdots)))$ where adj is applied 6 times, then the remainder when $n$ is divided by 9 is $\_\_\_\_$.

jee-main 2023 Q78 Matrix Power Computation and Application View

Let $P = \left[ \begin{array} { c c } \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \\ - \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \end{array} \right] , A = \left[ \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right]$ and $Q = P A P ^ { T }$. If $P ^ { T } Q ^ { 2007 } P = \left[ \begin{array} { l l } a & b \\ c & d \end{array} \right]$ then $2 a + b - 3 c - 4 d$ is equal to
(1) 2004
(2) 2005
(3) 2007
(4) 2006

jee-main 2024 Q68 Matrix Algebraic Properties and Abstract Reasoning View

Let $R = \begin{pmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{pmatrix}$ be a non-zero $3 \times 3$ matrix, where $x\sin\theta = y\sin\left(\theta + \frac{2\pi}{3}\right) = z\sin\left(\theta + \frac{4\pi}{3}\right) \neq 0$, $\theta \in (0, 2\pi)$. For a square matrix $M$, let Trace $M$ denote the sum of all the diagonal entries of $M$. Then, among the statements: I. Trace$(R) = 0$ II. If Trace$(\operatorname{adj}(\operatorname{adj}(R))) = 0$, then $R$ has exactly one non-zero entry.
(1) Both (I) and (II) are true
(2) Only (II) is true
(3) Neither (I) nor (II) is true
(4) Only (I) is true

jee-main 2024 Q68 Matrix Algebraic Properties and Abstract Reasoning View

Let $A = \left[ \begin{array} { l l l } 2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b \end{array} \right]$. If $A ^ { 3 } = 4 A ^ { 2 } - A - 21 I$, where $I$ is the identity matrix of order $3 \times 3$, then $2 a + 3 b$ is equal to
(1) $- 9$
(2) $- 13$
(3) $- 10$
(4) $- 12$

jee-main 2024 Q69 Determinant and Rank Computation View

Let $A = \left[ \begin{array} { l l l } 1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha \end{array} \right]$ and $| 2 A | ^ { 3 } = 2 ^ { 21 }$ where $\alpha , \beta \in Z$, Then a value of $\alpha$ is
(1) 3
(2) 5
(3) 17
(4) 9

jee-main 2024 Q69 Determinant with Cofactor or Expansion Relationship View

Let $\alpha \beta \neq 0$ and $A = \left[ \begin{array} { r r r } \beta & \alpha & 3 \\ \alpha & \alpha & \beta \\ - \beta & \alpha & 2 \alpha \end{array} \right]$. If $B = \left[ \begin{array} { r r r } 3 \alpha & - 9 & 3 \alpha \\ - \alpha & 7 & - 2 \alpha \\ - 2 \alpha & 5 & - 2 \beta \end{array} \right]$ is the matrix of cofactors of the elements of $A$, then $\operatorname { det } ( A B )$ is equal to :
(1) 64
(2) 216
(3) 343
(4) 125

jee-main 2024 Q70 Determinant of Parametric or Structured Matrix View

Let $\mathrm { A } = \left[ \begin{array} { c c c } 2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2 \end{array} \right]$ and $\mathrm { P } = \left[ \begin{array} { l l l } 1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5 \end{array} \right]$. The sum of the prime factors of $\left| \mathrm { P } ^ { - 1 } \mathrm { AP } - 2 \mathrm { I } \right|$ is equal to
(1) 26
(2) 27
(3) 66
(4) 23

jee-main 2024 Q70 Direct Determinant Computation View

$f ( x ) = \left| \begin{array} { c c c } 2 \cos ^ { 4 } x & 2 \sin ^ { 4 } x & 3 + \sin ^ { 2 } 2 x \\ 3 + 2 \cos ^ { 4 } x & 2 \sin ^ { 4 } x & \sin ^ { 2 } 2 x \\ 2 \cos ^ { 4 } x & 3 + 2 \sin ^ { 4 } x & \sin ^ { 2 } 2 x \end{array} \right|$ then $\frac { 1 } { 5 } f ^ { \prime } ( 0 )$ is equal to:
(1) 0
(2) 1
(3) 2
(4) 6

jee-main 2024 Q70 Linear System with Parameter — Infinite Solutions View

If the system of linear equations $$\begin{aligned} & x - 2 y + z = - 4 \\ & 2 x + \alpha y + 3 z = 5 \\ & 3 x - y + \beta z = 3 \end{aligned}$$ has infinitely many solutions, then $12 \alpha + 13 \beta$ is equal to
(1) 60
(2) 64
(3) 54
(4) 58

jee-main 2024 Q71 Solving a 3×3 Linear System Explicitly View

Let $A$ be a $3 \times 3$ real matrix such that $A\begin{pmatrix}0\\1\\0\end{pmatrix} = \begin{pmatrix}2\\0\\0\end{pmatrix}$, $A\begin{pmatrix}0\\0\\1\end{pmatrix} = \begin{pmatrix}4\\0\\0\end{pmatrix}$, $A\begin{pmatrix}1\\1\\1\end{pmatrix} = \begin{pmatrix}2\\1\\1\end{pmatrix}$. Then, the system $(A - 3I)\begin{pmatrix}x\\y\\z\end{pmatrix} = \begin{pmatrix}1\\2\\3\end{pmatrix}$ has
(1) unique solution
(2) exactly two solutions
(3) no solution
(4) infinitely many solutions

jee-main 2024 Q71 Direct Determinant Computation View

For $\alpha , \beta \in \mathbb { R }$ and a natural number $n$, let $A _ { r } = \left| \begin{array} { c c c } r & 1 & \frac { n ^ { 2 } } { 2 } + \alpha \\ 2 r & 2 & n ^ { 2 } - \beta \\ 3 r - 2 & 3 & \frac { n ( 3 n - 1 ) } { 2 } \end{array} \right|$. Then $\sum_{r=1}^{n} A_r$ is
(1) 0
(2) $4 \alpha + 2 \beta$
(3) $2 \alpha + 4 \beta$
(4) $2 n$

jee-main 2024 Q84 Linear System and Inverse Existence View

Let $\mathrm { A } = \left[ \begin{array} { l l l } 2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{array} \right] , \mathrm { B } = \left[ \begin{array} { l l l } \mathrm { B } _ { 1 } & \mathrm {~B} _ { 2 } & \mathrm {~B} _ { 3 } \end{array} \right]$, where $\mathrm { B } _ { 1 } , \mathrm {~B} _ { 2 } , \mathrm {~B} _ { 3 }$ are column matrices, and $\mathrm { AB } _ { 1 } = \left[ \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right]$, $\mathrm { AB } _ { 2 } = \left[ \begin{array} { l } 2 \\ 3 \\ 0 \end{array} \right] , \mathrm { AB } _ { 3 } = \left[ \begin{array} { l } 3 \\ 2 \\ 1 \end{array} \right]$ If $\alpha = | \mathrm { B } |$ and $\beta$ is the sum of all the diagonal elements of B, then $\alpha ^ { 3 } + \beta ^ { 3 }$ is equal to

jee-main 2025 Q1 Determinant of Parametric or Structured Matrix View

For a $3 \times 3$ matrix $M$, let trace ( $M$ ) denote the sum of all the diagonal elements of $M$. Let $A$ be a $3 \times 3$ matrix such that $| A | = \frac { 1 } { 2 }$ and trace $( A ) = 3$. If $B = \operatorname { adj } ( \operatorname { adj } ( 2 A ) )$, then the value of $| B | +$ trace $( \mathrm { B } )$ equals :
(1) 56
(2) 132
(3) 174
(4) 280

jee-main 2025 Q18 Determinant of Parametric or Structured Matrix View

For some $\mathrm{a}, \mathrm{b}$, let $f(x) = \left| \begin{array}{ccc} \mathrm{a} + \frac{\sin x}{x} & 1 & \mathrm{b} \\ \mathrm{a} & 1 + \frac{\sin x}{x} & \mathrm{b} \\ \mathrm{a} & 1 & \mathrm{b} + \frac{\sin x}{x} \end{array} \right|$, $x \neq 0$, $\lim_{x \rightarrow 0} f(x) = \lambda + \mu\mathrm{a} + \nu\mathrm{b}$. Then $(\lambda + \mu + \nu)^{2}$ is equal to:
(1) 16
(2) 25
(3) 9
(4) 36

jee-main 2025 Q21 Determinant and Rank Computation View

Let $A$ be a square matrix of order 3 such that $\operatorname { det } ( A ) = - 2$ and $\operatorname { det } ( 3 \operatorname { adj } ( - 6 \operatorname { adj } ( 3 A ) ) ) = 2 ^ { \mathrm { m } + \mathrm { n } } \cdot 3 ^ { \mathrm { mn } } , \mathrm { m } > \mathrm { n }$. Then $4 \mathrm {~m} + 2 \mathrm { n }$ is equal to $\_\_\_\_$

jee-main 2025 Q68 Determinant of Parametric or Structured Matrix View

Q68. Let $A$ and $B$ be two square matrices of order 3 such that $| A | = 3$ and $| B | = 2$. Then $\left| \mathrm { A } ^ { \mathrm { T } } \mathrm { A } ( \operatorname { adj } ( 2 \mathrm {~A} ) ) ^ { - 1 } ( \operatorname { adj } ( 4 \mathrm {~B} ) ) ( \operatorname { adj } ( \mathrm { AB } ) ) ^ { - 1 } \mathrm { AA } ^ { \mathrm { T } } \right|$ is equal to :
(1) 108
(2) 32
(3) 81
(4) 64

jee-main 2025 Q68 Determinant of Parametric or Structured Matrix View

Q68. Let $A = \left[ \begin{array} { l l l } 2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b \end{array} \right]$. If $A ^ { 3 } = 4 A ^ { 2 } - A - 21 I$, where $I$ is the identity matrix of order $3 \times 3$, then $2 a + 3 b$ is equal to
(1) - 9
(2) - 13
(3) - 10
(4) - 12

jee-main 2025 Q69 Determinant of Parametric or Structured Matrix View

Q69. Let $\alpha \in ( 0 , \infty )$ and $A = \left[ \begin{array} { c c c } 1 & 2 & \alpha \\ 1 & 0 & 1 \\ 0 & 1 & 2 \end{array} \right]$. If $\operatorname { det } \left( \operatorname { adj } \left( 2 A - A ^ { T } \right) \cdot \operatorname { adj } \left( A - 2 A ^ { T } \right) \right) = 2 ^ { 8 }$, then $( \operatorname { det } ( A ) ) ^ { 2 }$ is equal to:
(1) 36
(2) 16
(3) 1
(4) 49

1
2
3
4
5
6
7

LFM Pure

Straight Lines & Coordinate Geometry 242

Circles 702

Simultaneous equations 134

Proof 868

Proof by induction 36

Sine and Cosine Rules 185

Radians, Arc Length and Sector Area 22

Trig Graphs & Exact Values 95

Standard trigonometric equations 142

Trig Proofs 92

Quadratic trigonometric equations 8

Trigonometric equations in context 11

Modulus function 37

Matrices 1085

Linear transformations 72

Invariant lines and eigenvalues and vectors 339

Reciprocal Trig & Identities 63

Addition & Double Angle Formulae 88

Harmonic Form 14

Small angle approximation 14

Chain Rule 281

Product & Quotient Rules 13

Differentiating Transcendental Functions 160

Connected Rates of Change 48

Standard Integrals and Reverse Chain Rule 82

Integration by Substitution 213

Integration by Parts 44

Implicit equations and differentiation 27

Differential equations 308

3x3 Matrices 164