LFM Pure

jee-main 2019 Q79 Determinant of Parametric or Structured Matrix View

Let $d \in R$, and $A = \left[ \begin{array} { c c c } - 2 & 4 + d & ( \sin \theta ) - 2 \\ 1 & ( \sin \theta ) + 2 & d \\ 5 & ( 2 \sin \theta ) - d & ( - \sin \theta ) + 2 + 2 d \end{array} \right] , \theta \in [ 0,2 \pi ]$. If the minimum value of $\operatorname { det } ( A )$ is 8, then a value of $d$ is:
(1) $2 ( \sqrt { 2 } + 2 )$
(2) $2 ( \sqrt { 2 } + 1 )$
(3) $- 5$
(4) $- 7$

jee-main 2020 Q59 Matrix Power Computation and Application View

Let $\alpha$ be a root of the equation $x ^ { 2 } + x + 1 = 0$ and the matrix $A = \frac { 1 } { \sqrt { 3 } } \left[ \begin{array} { c c c } 1 & 1 & 1 \\ 1 & \alpha & \alpha ^ { 2 } \\ 1 & \alpha ^ { 2 } & \alpha ^ { 4 } \end{array} \right]$, then the matrix $A ^ { 31 }$ is equal to
(1) $A ^ { 3 }$
(2) $I _ { 3 }$
(3) $A ^ { 2 }$
(4) $A$

jee-main 2020 Q60 Determinant of Parametric or Structured Matrix View

If $A = \left[ \begin{array} { c c c } 1 & 1 & 2 \\ 1 & 3 & 4 \\ 1 & - 1 & 3 \end{array} \right] , B = \mathrm{adj}\, A$ and $C = 3 A$, then $\frac { | \mathrm{adj}\, B | } { | C | }$ is equal to
(1) 8
(2) 16
(3) 72
(4) 2

jee-main 2020 Q61 Determinant and Rank Computation View

Let $m$ and $M$ be respectively the minimum and maximum values of $\left| \begin{array} { c c c } \cos ^ { 2 } x & 1 + \sin ^ { 2 } x & \sin 2 x \\ 1 + \cos ^ { 2 } x & \sin ^ { 2 } x & \sin 2 x \\ \cos ^ { 2 } x & \sin ^ { 2 } x & 1 + \sin 2 x \end{array} \right|$. Then the ordered pair $( \mathrm { m } , \mathrm { M } )$ is equal to:
(1) $( 3,3 )$
(2) $( - 3 , - 1 )$
(3) $( 4,1 )$
(4) $( 1,3 )$

jee-main 2020 Q62 Determinant and Rank Computation View

If the minimum and the maximum values of the function $f : \left[ \frac { \pi } { 4 } , \frac { \pi } { 2 } \right] \rightarrow R$, defined by $f ( \theta ) = \left| \begin{array} { c c c } - \sin ^ { 2 } \theta & - 1 - \sin ^ { 2 } \theta & 1 \\ - \cos ^ { 2 } \theta & - 1 - \cos ^ { 2 } \theta & 1 \\ 12 & 10 & - 2 \end{array} \right|$ are $m$ and $M$ respectively, then the ordered pair $( \mathrm { m } , \mathrm { M } )$ is equal to :
(1) $( 0,2 \sqrt { 2 } )$
(2) $( - 4,0 )$
(3) $( - 4,4 )$
(4) $( 0,4 )$

jee-main 2020 Q62 Linear System and Inverse Existence View

If the system of linear equations $$x + y + 3z = 0$$ $$x + 3y + k^2z = 0$$ $$3x + y + 3z = 0$$ has a non-zero solution $(x, y, z)$ for some $k \in \mathrm{R}$, then $x + \left(\frac{y}{z}\right)$ is equal to:
(1) $-3$
(2) $9$
(3) $3$
(4) $-9$

jee-main 2020 Q63 Linear System Existence and Uniqueness via Determinant View

Let $\lambda \in \mathrm { R }$. The system of linear equations $2 x _ { 1 } - 4 x _ { 2 } + \lambda x _ { 3 } = 1$ $x _ { 1 } - 6 x _ { 2 } + x _ { 3 } = 2$ $\lambda x _ { 1 } - 10 x _ { 2 } + 4 x _ { 3 } = 3$ is inconsistent for :
(1) exactly one positive value of $\lambda$
(2) exactly one negative value of $\lambda$
(3) every value of $\lambda$
(4) exactly two values of $\lambda$

jee-main 2021 Q70 Determinant of Parametric or Structured Matrix View

Let $A = \left[ \begin{array} { c c c } { [ x + 1 ] } & { [ x + 2 ] } & { [ x + 3 ] } \\ { [ x ] } & { [ x + 3 ] } & { [ x + 3 ] } \\ { [ x ] } & { [ x + 2 ] } & { [ x + 4 ] } \end{array} \right]$, where $[ x ]$ denotes the greatest integer less than or equal to $x$. If $\operatorname { det } ( A ) = 192$, then the set of values of $x$ is in the interval: (1) $[ 62,63 )$ (2) $[ 65,66 )$ (3) $[ 60,61 )$ (4) $[ 68,69 )$

jee-main 2021 Q71 Determinant of Parametric or Structured Matrix View

If $x , y , z$ are in arithmetic progression with common difference $d , x \neq 3 d$, and the determinant of the matrix $\left[ \begin{array} { c c c } 3 & 4 \sqrt { 2 } & x \\ 4 & 5 \sqrt { 2 } & y \\ 5 & k & z \end{array} \right]$ is zero, then the value of $k ^ { 2 }$ is
(1) 72
(2) 12
(3) 36
(4) 6

jee-main 2021 Q71 Determinant of Parametric or Structured Matrix View

If $\mathrm { a } _ { \mathrm { r } } = \cos \frac { 2 \mathrm { r } \pi } { 9 } + i \sin \frac { 2 \mathrm { r } \pi } { 9 } , \mathrm { r } = 1,2,3 , \ldots , i = \sqrt { - 1 }$, then the determinant $\left| \begin{array} { l l l } a _ { 1 } & a _ { 2 } & a _ { 3 } \\ a _ { 4 } & a _ { 5 } & a _ { 6 } \\ a _ { 7 } & a _ { 8 } & a _ { 9 } \end{array} \right|$ is equal to $:$
(1) $\mathrm { a } _ { 9 }$
(2) $a _ { 1 } a _ { 9 } - a _ { 3 } a _ { 7 }$
(3) $a _ { 5 }$
(4) $a _ { 2 } a _ { 6 } - a _ { 4 } a _ { 8 }$

jee-main 2021 Q81 Direct Determinant Computation View

If $1 , \log _ { 10 } \left( 4 ^ { x } - 2 \right)$ and $\log _ { 10 } \left( 4 ^ { x } + \frac { 18 } { 5 } \right)$ are in arithmetic progression for a real number $x$ then the value of the determinant $\left| \begin{array} { c c c } 2 \left( x - \frac { 1 } { 2 } \right) & x - 1 & x ^ { 2 } \\ 1 & 0 & x \\ x & 1 & 0 \end{array} \right|$ is equal to:

jee-main 2021 Q85 Matrix Algebraic Properties and Abstract Reasoning View

Let $A = \left[ \begin{array} { l l l } x & y & z \\ y & z & x \\ z & x & y \end{array} \right]$, where $x , y$ and $z$ are real numbers such that $x + y + z > 0$ and $x y z = 2$. If $A ^ { 2 } = I _ { 3 }$, then the value of $x ^ { 3 } + y ^ { 3 } + z ^ { 3 }$ is

jee-main 2021 Q87 Determinant of Parametric or Structured Matrix View

Let $A = \left\{ a _ { i j } \right\}$ be a $3 \times 3$ matrix, where $a _ { i j } = \left\{ \begin{array} { l l } ( - 1 ) ^ { j - i } & \text { if } i < j \\ 2 & \text { if } i = j \\ ( - 1 ) ^ { i + j } & \text { if } i > j \end{array} \right.$ then $\det \left( 3 \operatorname{Adj} \left( 2 A ^ { - 1 } \right) \right)$ is equal to $\underline{\hspace{1cm}}$.

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 Linear System Existence and Uniqueness via Determinant View

The number of $\theta \in (0,4\pi)$ for which the system of linear equations $3(\sin 3\theta) x - y + z = 2$ $3(\cos 2\theta) x + 4y + 3z = 3$ $6x + 7y + 7z = 9$ has no solution is
(1) 6
(2) 7
(3) 8
(4) 9

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 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 Determinant of Parametric or Structured Matrix View

Let $A$ and $B$ be two $3 \times 3$ matrices such that $A B = I$ and $| A | = \frac { 1 } { 8 }$ then $| \operatorname{adj} ( B \operatorname{adj} ( 2 A ) ) |$ is equal to
(1) 128
(2) 32
(3) 64
(4) 102

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 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 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) $3^{11} \cdot 6^{10}$

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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