LFM Pure

jee-main 2019 Q75 Matrix Algebraic Properties and Abstract Reasoning View

Let $A = \begin{pmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{pmatrix}$, $\alpha \in R$ such that $A^{32} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$. Then, a value of $\alpha$ is:
(1) 0
(2) $\frac{\pi}{16}$
(3) $\frac{\pi}{64}$
(4) $\frac{\pi}{32}$

jee-main 2019 Q75 Determinant of Parametric or Structured Matrix View

Let $\alpha$ and $\beta$ be the roots of the equation $x ^ { 2 } + x + 1 = 0$. Then for $y \neq 0$ in $R , \left| \begin{array} { c c c } y + 1 & \alpha & \beta \\ \alpha & y + \beta & 1 \\ \beta & 1 & y + \alpha \end{array} \right|$ is equal to
(1) $y ^ { 3 }$
(2) $y \left( y ^ { 2 } - 1 \right)$
(3) $y ^ { 3 } - 1$
(4) $y \left( y ^ { 2 } - 3 \right)$

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 Q60 Linear System Existence and Uniqueness via Determinant View

If the system of linear equations $$\begin{aligned} & 2x + 2ay + az = 0 \\ & 2x + 3by + bz = 0 \\ & 2x + 4cy + cz = 0 \end{aligned}$$ where $a, b, c \in R$ are non-zero and distinct; has a non-zero solution, then
(1) $\frac { 1 } { a } , \frac { 1 } { b } , \frac { 1 } { c }$ are in $A.P$.
(2) $a, b, c$ are in $G.P$.
(3) $a + b + c = 0$
(4) $a, b, c$ are in $A.P$.

jee-main 2020 Q60 Linear System Existence and Uniqueness via Determinant View

The system of linear equations $\lambda x + 2y + 2z = 5$ $2\lambda x + 3y + 5z = 8$ $4x + \lambda y + 6z = 10$ has
(1) no solution when $\lambda = 8$
(2) a unique solution when $\lambda = -8$
(3) no solution when $\lambda = 2$
(4) infinitely many solutions when $\lambda = 2$

jee-main 2020 Q60 Linear System Existence and Uniqueness via Determinant View

The following system of linear equations $7 x + 6 y - 2 z = 0$ $3 x + 4 y + 2 z = 0$ $x - 2 y - 6 z = 0$, has
(1) infinitely many solutions, ( $x , y , z$ ) satisfying $y = 2z$
(2) no solution
(3) infinitely many solutions, $( x , y , z )$ satisfying $x = 2z$
(4) only the trivial solution

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 Q63 Matrix Algebraic Properties and Abstract Reasoning View

Let $a , b , c \in R$ be all non-zero and satisfies $a ^ { 3 } + b ^ { 3 } + c ^ { 3 } = 2$. If the matrix $A = \left[ \begin{array} { c c c } a & b & c \\ b & c & a \\ c & a & b \end{array} \right]$ satisfies $A ^ { T } A = I$, then a value of $a b c$ can be
(1) $- \frac { 1 } { 3 }$
(2) $\frac { 1 } { 3 }$
(3) 3
(4) $\frac { 2 } { 3 }$

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 2020 Q63 Direct Determinant Computation View

If $a + x = b + y = c + z + 1$, where $a, b, c, x, y, z$ are non-zero distinct real numbers, then $\left|\begin{array}{lll} x & a+y & x+a \\ y & b+y & y+b \\ z & c+y & z+c \end{array}\right|$ is equal to:
(1) $y(b-a)$
(2) $y(a-b)$
(3) $0$
(4) $y(a-c)$

jee-main 2021 Q69 Direct Determinant Computation View

Let $A$ be a $3 \times 3$ matrix with $\operatorname { det } ( A ) = 4$. Let $R _ { i }$ denote the $i ^ { \text {th} }$ row of $A$. If a matrix $B$ is obtained by performing the operation $R _ { 2 } \rightarrow 2 R _ { 2 } + 5 R _ { 3 }$ on $2 A$, then $\operatorname { det } ( B )$ is equal to:
(1) 64
(2) 16
(3) 128
(4) 80

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 Q85 Matrix Algebraic Properties and Abstract Reasoning View

Let $P = \left[ \begin{array} { c c c } - 30 & 20 & 56 \\ 90 & 140 & 112 \\ 120 & 60 & 14 \end{array} \right]$ and $A = \left[ \begin{array} { c c c } 2 & 7 & \omega ^ { 2 } \\ - 1 & - \omega & 1 \\ 0 & - \omega & - \omega + 1 \end{array} \right]$ where $\omega = \frac { - 1 + i \sqrt { 3 } } { 2 }$, and $I _ { 3 }$ be the identity matrix of order 3 . If the determinant of the matrix $\left( P ^ { - 1 } A P - I _ { 3 } \right) ^ { 2 }$ is $\alpha \omega ^ { 2 }$, then the value of $\alpha$ is equal to $\_\_\_\_$.

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 Q69 Determinant of Parametric or Structured Matrix View

Let $A$ be a $3 \times 3$ invertible matrix. If $| \operatorname { adj } ( 24 A ) | = | \operatorname { adj } ( 3 \operatorname { adj } ( 2A ) ) |$, then $| A | ^ { 2 }$ is equal to
(1) $2 ^ { 6 }$
(2) $2 ^ { 12 }$
(3) 512
(4) $6 ^ { 6 }$

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 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 2023 Q74 Matrix Algebraic Properties and Abstract Reasoning View

Let $\alpha$ and $\beta$ be real numbers. Consider a $3 \times 3$ matrix $A$ such that $A ^ { 2 } = 3 A + \alpha I$. If $A ^ { 4 } = 21 A + \beta I$, then
(1) $\alpha = 1$
(2) $\alpha = 4$
(3) $\beta = 8$
(4) $\beta = - 8$

jee-main 2023 Q76 Determinant of Parametric or Structured Matrix View

Let $A$ be a $n \times n$ matrix such that $|A| = 2$. If the determinant of the matrix $\operatorname{Adj}\left(2 \cdot \operatorname{Adj}\left(2A^{-1}\right)\right)$ is $2^{84}$, then $n$ is equal to $\_\_\_\_$.

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 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

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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