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The values of \( m, n \) for which the system of equations
\( x+y+z=4 \),
\( 2x+5y+5z=17 \),
\( x+2y+mz=n \)
has infinitely many solutions, satisfy the equation:

If the shortest distance between the lines \( \frac{x-\lambda}{3}=\frac{y-2}{-1}=\frac{z-1}{1} \) and \( \frac{x+2}{-3}=\frac{y+5}{2}=\frac{z-4}{4} \) is \( \frac{44}{\sqrt{30}} \), then the largest possible value of \( |\lambda| \) is equal to ______.

If the system of equations \( x+4y-z=\lambda, \ 7x+9y+\mu z=-3, \ 5x+y+2z=-1 \) has infinitely many solutions, then \( (2\mu+3\lambda )\) is equal to:

If the system of linear equations
\( x-2y+z=-4 \)
\( 2x+\alpha y+3z=5 \)
\( 3x-y+\beta z=3 \)
has infinitely many solutions, then \( 12\alpha+13\beta \) is equal to:

for all \( x\in \mathbb{R} \), then \( 2f(0)+f'(0) \) is equal to:

Given below are two statements:
Statement I: \( f(-x) \) is the inverse of the matrix \( f(x) \).
Statement II: \( f(x)f(y)=f(x+y) \).
In the light of the above statements, choose the correct answer from the options given below:

Consider the system of linear equations \( x+y+z=5 \), \( x+2y+\lambda^{2}z=9 \), \( x+3y+\lambda z=\mu \), where \( \lambda,\mu \in \mathbb{R} \). Then, which of the following statement is \(\text {NOT}\) correct?

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

Let \( A \) be a \( 2 \times 2 \) real matrix and \( 1 \) be the identity matrix of order \( 2 \). If the roots of the equation \( |A - xI| = 0 \) be \( -1 \) and \( 3 \), then the sum of the diagonal elements of the matrix \( A^2 \) is __________.

. The sum of the prime factors of \( \left|P^{-1}AP-2I\right| \) is equal to:

Let \( A \) be a \( 2 \times 2 \) symmetric matrix such that and the determinant of \( A \) be \( 1 \).
If \( A^{-1} = \alpha A + \beta I, \) where \( I \) is an identity matrix of order \( 2 \times 2 \), then \( \alpha + \beta \) equals _______________ .

If \( A \) is a square matrix of order \( 3 \) such that \( \det(A)=3 \) and \( \det(\text{adj}(-4 \, \text{adj}(-3 \, \text{adj}(3 \, \text{adj}((2A)^{-1}))))) = 2^m 3^n\), then \( m+2n \) is equal to:

. If \( A^3 = 4A^2 - A - 21I \), where \( I \) is the identity matrix of order \( 3 \times 3 \), then \( 2a + 3b \) is equal to:

If the system of equations
\(11x+y+\lambda z=-5\)
\(2x+3y+5z=3\)
\(8x-19y-39z=\mu\)
has infinitely many solutions, then \( \lambda^{4}-\mu \) is equal to:

Let \( A \) be a \( 3\times 3 \) real matrix such that
.Then, the system has.

and \( A \) be a \( 2 \times 2 \) matrix such that \( AB^{-1} = A^{-1} \). If \( BCB^{-1} = A \) and \( C^{4} + \alpha C^{2} + \beta I = 0 \), then \( 2\beta - \alpha \) is equal to

If the system of equations
\( x+\big(\sqrt{2}\sin\alpha\big)y+\big(\sqrt{2}\cos\alpha\big)z=0 \)
\( x+(\cos\alpha)y+(\sin\alpha)z=0 \)
\( x+(\sin\alpha)y-(\cos\alpha)z=0 \)
has a non-trivial solution, then \( \alpha\in\left(0,\dfrac{\pi}{2}\right) \) is equal to:

Let \( A \) be a square matrix of order \( 2 \) such that \( |A| = 2 \) and the sum of its diagonal elements is \( -3 \). If the points \( (x, y) \) satisfying \( A^2 + xA + yI = 0 \) lie on a hyperbola, whose transverse axis is parallel to the \( x \)-axis, eccentricity is \( e \) and the length of the latus rectum is \( l \), then \( e^4 + l^4 \) is equal to \( \_\_\_\_\_\_\_\_\_\_ \).

and \( |2A|^{3}=2^{21} \) where \( \alpha,\beta \in \mathbb{Z} \), then a value of \( \alpha \) is:

If \( \alpha \ne a,\ \beta \ne b,\ \gamma \ne c \) and then \( \dfrac{a}{\alpha-a}+\dfrac{b}{\beta-b}+\dfrac{\gamma}{\gamma-c} \) is equal to:

. If the sum of the diagonal elements of \( A^{13} \) is \( 3^n \), then \( n \) is equal to ______.

, \( C=ABA^{T} \) and \( X=A^{T}C^{2}A \), then \( \det X \) is equal to:

If the system of equations
\( 2x+7y+\lambda z=3 \)
\( 3x+2y+5z=4 \)
\( x+\mu y+3z=-1 \)
has infinitely many solutions, then \( (\lambda-\mu) \) is equal to ______.

Consider the system of linear equations
\( x + y + z = 4\mu, \quad x + 2y + 2\lambda z = 10\mu, \quad x + 3y + 4\lambda^{2} z = \mu^{2} + 15 \), where, \( \lambda, \mu \in \mathbb{R} \). Which one of the following statements is NOT correct?

Let \( A \) be a square matrix such that \( AA^{T} = I \). Then \( \frac{1}{2} A \left[ (A + A^{T})^{2} + (A - A^{T})^{2} \right] \) is equal to:

For \( \alpha, \beta \in \) \(\mathbb{R}\) and a natural number \( n \), let . Then \( 2A_{10} - A_8 \) is:

Let \( A \) be \( 3 \times 3 \) matrix and \( \det(A) = 2 \). If \( n = \det\dfrac{(\operatorname{adj}(\operatorname{adj}(\ldots(\operatorname{adj} A))) )}{ 2024 \text{-times} }\) remainder when \( n \) is divided by \( 9 \) is equal to ______.

If the system of equations
\( 2x+3y-z=5 \)
\( x+\alpha y+3z=-4 \)
\( 3x-y+\beta z=7 \)
has infinitely many solutions, then \( 13\alpha\beta \) is equal to ______.

, \( B = [B_{1}, B_{2}, B_{3}] \) are column matrices, and If \( \alpha = |B| \) and \( \beta \) is the sum of all the diagonal element of \( B \), then \( \alpha^{3} + \beta^{3} \) is equal to ____________.

Let \( A = I_{2} - 2MM^{T} \) where \( M \) is a real matrix of order \( 2 \times 1 \) such that the relation \( M^{T}M = I_{1} \) holds. If \( \lambda \) is a real number such that the relation \( AX = \lambda X \) holds for some non-zero real matrix \( X \) of order \( 2 \times 1 \), then the sum of squares of all possible values of \( \lambda \) is equal to ________.

Let \( A \) be a non-singular matrix of order \( 3 \). If \( \det\!\big(3\,\operatorname{adj}(2\,\operatorname{adj}((\det A)\,A))\big) = 3^{-13}\cdot 2^{-10} \) and \( \det\!\big(3\,\operatorname{adj}(2A)\big) = 2^{m}\cdot 3^{n} \), then \( |3m + 2n| \) is equal to ________.

Let \( \alpha \in (0,\infty) \) and . If \( \det\!\big(\operatorname{adj}(2A-A^{T})\cdot \operatorname{adj}(A-2A^{T})\big)=2^{8} \), then \( (\det(A))^{2} \) is equal to:

Let \( A \) be a \( 3 \times 3 \) matrix of non-negative real elements such that . Then the maximum value of \( \det(A) \) is _________.

Let \( \alpha \beta \gamma = 45; \alpha, \beta, \gamma \in \; \)\(\mathbb{R}\). If \( x(\alpha, 1, 2) + y(1, \beta, 2) + z(2, 3, \gamma) = (0, 0, 0) \) for some \( x, y, z \in \; \)\(\mathbb{R}\), \( xyz \ne 0 \), then \( 6\alpha + 4\beta + \gamma \) is equal to __________.

Let for any three distinct consecutive terms \( a, b, c \) of an A.P. the lines \( ax + by + c = 0 \) be concurrent at the point \( P \) and \( Q \equiv (\alpha,\beta) \) be a point such that the system of equations
\( x + y + z = 6 \)
\( 2x + 5y + \alpha z = \beta \) and
\( x + 2y + 3z = 4 \), has infinitely many solutions. The \( (PQ)^2 \) is equal to ________.

Let and \( B = I + \operatorname{adj}(A) + (\operatorname{adj}A)^2 + \ldots + (\operatorname{adj}A)^{10} \). Then, the sum of all the elements of the matrix \( B \) is:

is the matrix of cofactors of the elements of \( A \), then \( \det(AB) \) is equal to:

The values of \( \alpha \), for which lie in the interval.

be a non-zero \( 3 \times 3 \) matrix, where \( x \sin \theta = y \sin \!\left(\theta + \dfrac{2\pi}{3}\right) = z \sin \!\left(\theta + \dfrac{4\pi}{3}\right) \neq 0, \\ \theta \in (0, 2\pi) \). For a square matrix \( M \), Let trace \( (M) \) denote the sum of all the diagonal entries of \( M \). Then, among the statements:
(I) \( \text{Trace} (R) = 0 \)
(II) If \( \text{trace}(\operatorname{adj}(\operatorname{adj}(R))) = 0 \), then \( R \) has exactly one non-zero entry.

Let \( \lambda,\mu \in \mathbb{R} \). If the system of equations
\( 3x+5y+\lambda z=3 \)
\( 7x+11y-9z=2 \)
\( 97x+155y-189z=\mu \)
has infinitely many solutions; then \( \mu+2\lambda \) is equal to:

Let the system of equations \( x + 2y + 3z = 5, \; 2x + 3y + z = 9, \; 4x + 3y + \lambda z = \mu \) have infinite number of solutions. Then \( \lambda + 2\mu \) is equal to:

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JEE Main 2024 27th Jan Morning Shift

Given below are two statements:
Statement I: \( f(-x) \) is the inverse of the matrix \( f(x) \).
Statement II: \( f(x)f(y)=f(x+y) \).
In the light of the above statements, choose the correct answer from the options given below: