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Let $$X=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$$ and $$A=\left[\begin{array}{ccc}-1 & 2 & 3 \\ 0 & 1 & 6 \\...
JEE Main 2022 (Online) 29th July Evening Shift
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Let p and p + 2 be prime numbers and let $$ \Delta=\left|\begin{array}{ccc} \mathrm{p} ! & (\mathrm{p}+1) ! & (\mathrm{p...
JEE Main 2022 (Online) 29th July Morning Shift
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The number of matrices of order $$3 \times 3$$, whose entries are either 0 or 1 and the sum of all the entries is a prim...
JEE Main 2022 (Online) 29th July Morning Shift
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Let $$A=\left[\begin{array}{cc}1 & -1 \\ 2 & \alpha\end{array}\right]$$ and $$B=\left[\begin{array}{cc}\beta & 1 \\ 1 & ...
JEE Main 2022 (Online) 28th July Morning Shift
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Consider a matrix $$A=\left[\begin{array}{ccc}\alpha & \beta & \gamma \\ \alpha^{2} & \beta^{2} & \gamma^{2} \\ \beta+\g...
JEE Main 2022 (Online) 27th July Evening Shift
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Let $$S$$ be the set containing all $$3 \times 3$$ matrices with entries from $$\{-1,0,1\}$$. The total number of matric...
JEE Main 2022 (Online) 27th July Morning Shift
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The number of matrices $$A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$$, where $$a, b, c, d \in\{-1,0,1,2,3...
JEE Main 2022 (Online) 26th July Evening Shift
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Let $$A=\left[\begin{array}{lll} 1 & a & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{array}\right], a, b \in \mathbb{R}$$. If for s...
JEE Main 2022 (Online) 25th July Evening Shift
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Let $$A=\left(\begin{array}{rrr}2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0\end{array}\right)$$ and $$B=A-I$$. If $$\omega=\...
JEE Main 2022 (Online) 25th July Morning Shift
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Let $$S = \left\{ {\left( {\matrix{ { - 1} & a \cr 0 & b \cr } } \right);a,b \in \{ 1,2,3,....100\} } \right...
JEE Main 2022 (Online) 24th June Evening Shift
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Let A be a 3 $$\times$$ 3 matrix having entries from the set {$$-$$1, 0, 1}. The number of all such matrices A having su...
JEE Main 2022 (Online) 25th June Morning Shift
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Let $$A = \left( {\matrix{ 2 & { - 2} \cr 1 & { - 1} \cr } } \right)$$ and $$B = \left( {\matrix{ { - 1} ...
JEE Main 2022 (Online) 25th June Evening Shift
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Let $$X = \left[ {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr 0 & 0 & 0 \cr } } \right],\,Y = \alpha I + \beta...
JEE Main 2022 (Online) 26th June Evening Shift
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The positive value of the determinant of the matrix A, whose Adj(Adj(A)) = $$\left( {\matrix{ {14} & {28} & { - 14} ...
JEE Main 2022 (Online) 27th June Morning Shift
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Let A be a matrix of order 2 $$\times$$ 2, whose entries are from the set {0, 1, 2, 3, 4, 5}. If the sum of all the entr...
JEE Main 2022 (Online) 27th June Evening Shift
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Let $$A = \left( {\matrix{ {1 + i} & 1 \cr { - i} & 0 \cr } } \right)$$ where $$i = \sqrt { - 1} $$. Then, t...
JEE Main 2022 (Online) 28th June Evening Shift
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If the system of linear equations $$2x - 3y = \gamma + 5$$, $$\alpha x + 5y = \beta + 1$$, where $$\alpha$$, $$\beta$$...
JEE Main 2022 (Online) 28th June Evening Shift
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Let $$M = \left[ {\matrix{ 0 & { - \alpha } \cr \alpha & 0 \cr } } \right]$$, where $$\alpha$$ is a non-zer...
JEE Main 2022 (Online) 29th June Evening Shift
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The number of elements in the set $$\left\{ {A = \left( {\matrix{ a & b \cr 0 & d \cr } } \right):a,...
JEE Main 2021 (Online) 31st August Evening Shift
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If the system of linear equations2x + y $$-$$ z = 3x $$-$$ y $$-$$ z = $$\alpha$$3x + 3y + $$\beta$$z = 3has infinitely ...
JEE Main 2021 (Online) 27th August Morning Shift
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Let A be a 3 $$\times$$ 3 real matrix. If det(2Adj(2 Adj(Adj(2A)))) = 241, then the value of det(A2) equal __________....
JEE Main 2021 (Online) 26th August Evening Shift
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Let $$f(x) = \left| {\matrix{ {{{\sin }^2}x} & { - 2 + {{\cos }^2}x} & {\cos 2x} \cr {2 + {{\sin }^2}x} ...
JEE Main 2021 (Online) 27th July Morning Shift
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For real numbers $$\alpha$$ and $$\beta$$, consider the following system of linear equations :x + y $$-$$ z = 2, x + 2y ...
JEE Main 2021 (Online) 27th July Morning Shift
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If $$A = \left[ {\matrix{ 1 & 1 & 1 \cr 0 & 1 & 1 \cr 0 & 0 & 1 \cr } } \right]...
JEE Main 2021 (Online) 27th July Evening Shift
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Let $$M = \left\{ {A = \left( {\matrix{ a & b \cr c & d \cr } } \right):a,b,c,d \in \{ \pm 3, \pm 2...
JEE Main 2021 (Online) 25th July Morning Shift
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Let $$A = \left[ {\matrix{ 0 & 1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1 \cr } } \right...
JEE Main 2021 (Online) 22th July Evening Shift
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Let $$A = \{ {a_{ij}}\} $$ be a 3 $$\times$$ 3 matrix, where $${a_{ij}} = \left\{ {\matrix{ {{{( - 1)}^{j - i}}} &...
JEE Main 2021 (Online) 20th July Evening Shift
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Let a, b, c, d in arithmetic progression with common difference $$\lambda$$. If $$\left| {\matrix{ {x + a - c} & ...
JEE Main 2021 (Online) 20th July Morning Shift
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Let $$A = \left( {\matrix{ 1 & { - 1} & 0 \cr 0 & 1 & { - 1} \cr 0 & 0 & 1 \cr ...
JEE Main 2021 (Online) 20th July Morning Shift
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Let I be an identity matrix of order 2 $$\times$$ 2 and P = $$\left[ {\matrix{ 2 & { - 1} \cr 5 & { - 3}...
JEE Main 2021 (Online) 18th March Evening Shift
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If 1, log10(4x $$-$$ 2) and log10$$\left( {{4^x} + {{18} \over 5}} \right)$$ are in arithmetic progression for a real nu...
JEE Main 2021 (Online) 17th March Evening Shift
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Let $$A = \left[ {\matrix{ a & b \cr c & d \cr } } \right]$$ and $$B = \left[ {\matrix{ \alpha ...
JEE Main 2021 (Online) 17th March Evening Shift
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If $$A = \left[ {\matrix{ 2 & 3 \cr 0 & { - 1} \cr } } \right]$$, then the value of det(A4) + det(A1...
JEE Main 2021 (Online) 17th March Morning Shift
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Let $$A = \left[ {\matrix{ {{a_1}} \cr {{a_2}} \cr } } \right]$$ and $$B = \left[ {\matrix{ {{b_1}} \cr ...
JEE Main 2021 (Online) 16th March Evening Shift
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The total number of 3 $$\times$$ 3 matrices A having entries from the set {0, 1, 2, 3} such that the sum of all the diag...
JEE Main 2021 (Online) 16th March Morning Shift
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Let $$P = \left[ {\matrix{ { - 30} & {20} & {56} \cr {90} & {140} & {112} \cr {120} & {...
JEE Main 2021 (Online) 16th March Morning Shift
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If the matrix $$A = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 2 & 0 \cr 3 & 0 & { - 1} \c...
JEE Main 2021 (Online) 26th February Evening Shift
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If the system of equationskx + y + 2z = 13x $$-$$ y $$-$$ 2z = 2$$-$$2x $$-$$2y $$-$$4z = 3has infinitely many solutions...
JEE Main 2021 (Online) 25th February Morning Slot
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Let $$A = \left[ {\matrix{ x & y & z \cr y & z & x \cr z & x & y \cr } } \right...
JEE Main 2021 (Online) 25th February Morning Slot
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If $$A = \left[ {\matrix{ 0 & { - \tan \left( {{\theta \over 2}} \right)} \cr {\tan \left( {{\theta \over ...
JEE Main 2021 (Online) 25th February Morning Slot
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Let M be any 3 $$ \times $$ 3 matrix with entries from the set {0, 1, 2}. The maximum number of such matrices, for which...
JEE Main 2021 (Online) 24th February Morning Slot
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Let P = $$\left[ {\matrix{ 3 & { - 1} & { - 2} \cr 2 & 0 & \alpha \cr 3 & { - 5} &...
JEE Main 2021 (Online) 24th February Morning Slot
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The sum of distinct values of $$\lambda $$ for which the system of equations$$\left( {\lambda - 1} \right)x + \left( {3...
JEE Main 2020 (Online) 6th September Evening Slot
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If the system of equations x - 2y + 3z = 9 2x + y + z = b x - 7y + az = 24, has infinitely many solutions, then a - b is...
JEE Main 2020 (Online) 4th September Morning Slot
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Let S be the set of all integer solutions, (x, y, z), of the system of equations x – 2y + 5z = 0 –2x + 4y + z = 0 –7x + ...
JEE Main 2020 (Online) 3rd September Evening Slot
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Let A = $$\left[ {\matrix{ x & 1 \cr 1 & 0 \cr } } \right]$$, x $$ \in $$ R and A4 = [aij]. If a11 =...
JEE Main 2020 (Online) 3rd September Morning Slot
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The number of all 3 × 3 matrices A, with enteries from the set {–1, 0, 1} such that the sum of the diagonal elements of ...
JEE Main 2020 (Online) 8th January Morning Slot
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MCQ (Single Correct Answer)
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If the system of equations $$ \begin{aligned} &x+y+z=6 \\ &2 x+5 y+\alpha z=\beta \\ &x+2 y+3 z=14 \end{aligned} $$ has ...
JEE Main 2022 (Online) 29th July Evening Shift
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Which of the following matrices can NOT be obtained from the matrix $$\left[\begin{array}{cc}-1 & 2 \\ 1 & -1\end{array}...
JEE Main 2022 (Online) 29th July Evening Shift
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Let A and B be two $$3 \times 3$$ non-zero real matrices such that AB is a zero matrix. Then
JEE Main 2022 (Online) 29th July Morning Shift
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Let $$\mathrm{A}$$ and $$\mathrm{B}$$ be any two $$3 \times 3$$ symmetric and skew symmetric matrices respectively. Then...
JEE Main 2022 (Online) 28th July Evening Shift
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Let the matrix $$A=\left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{array}\right]$$ and the matrix $$B_{0...
JEE Main 2022 (Online) 28th July Morning Shift
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Let $$A=\left(\begin{array}{rr}4 & -2 \\ \alpha & \beta\end{array}\right)$$. If $$\mathrm{A}^{2}+\gamma \mathrm{A}+18 \m...
JEE Main 2022 (Online) 27th July Evening Shift
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Let $$A=\left(\begin{array}{cc}1 & 2 \\ -2 & -5\end{array}\right)$$. Let $$\alpha, \beta \in \mathbb{R}$$ be such that $...
JEE Main 2022 (Online) 27th July Morning Shift
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$$ \text { Let } A=\left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right] \text { and } B=\left[\begin{array}{ccc} 9^{2} ...
JEE Main 2022 (Online) 26th July Evening Shift
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Let A be a 2 $$\times$$ 2 matrix with det (A) = $$-$$ 1 and det ((A + I) (Adj (A) + I)) = 4. Then the sum of the diagona...
JEE Main 2022 (Online) 26th July Morning Shift
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If the system of linear equations. $$8x + y + 4z = - 2$$ $$x + y + z = 0$$ $$\lambda x - 3y = \mu $$ has infinitely man...
JEE Main 2022 (Online) 26th July Morning Shift
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The number of real values of $$\lambda$$, such that the system of linear equations 2x $$-$$ 3y + 5z = 9 x + 3y $$-$$ z =...
JEE Main 2022 (Online) 25th July Evening Shift
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The number of $$\theta \in(0,4 \pi)$$ for which the system of linear equations $$ \begin{aligned} &3(\sin 3 \theta) x-y...
JEE Main 2022 (Online) 25th July Morning Shift
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Let A and B be two square matrices of order 2. If $$det\,(A) = 2$$, $$det\,(B) = 3$$ and $$\det \left( {(\det \,5(det\,A...
JEE Main 2022 (Online) 30th June Morning Shift
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Let $$A = \left[ {\matrix{ 1 & { - 2} & \alpha \cr \alpha & 2 & { - 1} \cr } } \right]$$ and $$B = \left[ ...
JEE Main 2022 (Online) 30th June Morning Shift
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Let S = {$$\sqrt{n}$$ : 1 $$\le$$ n $$\le$$ 50 and n is odd}. Let a $$\in$$ S and $$A = \left[ {\matrix{ 1 & 0 & a \...
JEE Main 2022 (Online) 24th June Morning Shift
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The number of values of $$\alpha$$ for which the system of equations : x + y + z = $$\alpha$$ $$\alpha$$x + 2$$\alpha$$y...
JEE Main 2022 (Online) 24th June Morning Shift
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Let the system of linear equations x + y + $$\alpha$$z = 2 3x + y + z = 4 x + 2z = 1 have a unique solution (x$$^ * $$, ...
JEE Main 2022 (Online) 24th June Evening Shift
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Let $$A = \left[ {\matrix{ 0 & { - 2} \cr 2 & 0 \cr } } \right]$$. If M and N are two matrices given by $$M ...
JEE Main 2022 (Online) 25th June Morning Shift
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Let A be a 3 $$\times$$ 3 real matrix such that $$A\left( {\matrix{ 1 \cr 1 \cr 0 \cr } } \right) = \le...
JEE Main 2022 (Online) 25th June Morning Shift
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The system of equations $$ - kx + 3y - 14z = 25$$ $$ - 15x + 4y - kz = 3$$ $$ - 4x + y + 3z = 4$$ is consistent for all ...
JEE Main 2022 (Online) 25th June Evening Shift
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If the system of equations $$\alpha$$x + y + z = 5, x + 2y + 3z = 4, x + 3y + 5z = $$\beta$$ has infinitely many solutio...
JEE Main 2022 (Online) 26th June Evening Shift
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The ordered pair (a, b), for which the system of linear equations 3x $$-$$ 2y + z = b 5x $$-$$ 8y + 9z = 3 2x + y + az =...
JEE Main 2022 (Online) 26th June Morning Shift
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Let A be a 3 $$\times$$ 3 invertible matrix. If |adj (24A)| = |adj (3 adj (2A))|, then |A|2 is equal to :...
JEE Main 2022 (Online) 26th June Morning Shift
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Let the system of linear equations $$x + 2y + z = 2$$, $$\alpha x + 3y - z = \alpha $$, $$ - \alpha x + y + 2z = - \alp...
JEE Main 2022 (Online) 27th June Morning Shift
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Let A and B be two 3 $$\times$$ 3 matrices such that $$AB = I$$ and $$|A| = {1 \over 8}$$. Then $$|adj\,(B\,adj(2A))|$$ ...
JEE Main 2022 (Online) 27th June Evening Shift
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Let $$f(x) = \left| {\matrix{ a & { - 1} & 0 \cr {ax} & a & { - 1} \cr {a{x^2}} & {ax} & a \cr } } \rig...
JEE Main 2022 (Online) 27th June Evening Shift
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Let A be a matrix of order 3 $$\times$$ 3 and det (A) = 2. Then det (det (A) adj (5 adj (A3))) is equal to _____________...
JEE Main 2022 (Online) 28th June Morning Shift
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If the system of linear equations $$2x + 3y - z = - 2$$ $$x + y + z = 4$$ $$x - y + |\lambda |z = 4\lambda - 4$$ where...
JEE Main 2022 (Online) 28th June Morning Shift
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Let $$A = \left( {\matrix{ 2 & { - 1} \cr 0 & 2 \cr } } \right)$$. If $$B = I - {}^5{C_1}(adj\,A) + {}^5{C_2...
JEE Main 2022 (Online) 29th June Evening Shift
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Let $$A = [{a_{ij}}]$$ be a square matrix of order 3 such that $${a_{ij}} = {2^{j - i}}$$, for all i, j = 1, 2, 3. Then,...
JEE Main 2022 (Online) 29th June Morning Shift
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If the system of linear equations 2x + y $$-$$ z = 7 x $$-$$ 3y + 2z = 1 x + 4y + $$\delta$$z = k, where $$\delta$$, k $...
JEE Main 2022 (Online) 29th June Morning Shift
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Let $${J_{n,m}} = \int\limits_0^{{1 \over 2}} {{{{x^n}} \over {{x^m} - 1}}dx} $$, $$\forall$$ n > m and n, m $$\in$$ ...
JEE Main 2021 (Online) 1st September Evening Shift
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Consider the system of linear equations$$-$$x + y + 2z = 03x $$-$$ ay + 5z = 12x $$-$$ 2y $$-$$ az = 7Let S1 be the set ...
JEE Main 2021 (Online) 1st September Evening Shift
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If $$\alpha$$ + $$\beta$$ + $$\gamma$$ = 2$$\pi$$, then the system of equations x + (cos $$\gamma$$)y + (cos $$\beta$$)z...
JEE Main 2021 (Online) 31st August Evening Shift
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If $${a_r} = \cos {{2r\pi } \over 9} + i\sin {{2r\pi } \over 9}$$, r = 1, 2, 3, ....., i = $$\sqrt { - 1} $$, then the d...
JEE Main 2021 (Online) 31st August Morning Shift
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If the following system of linear equations2x + y + z = 5x $$-$$ y + z = 3x + y + az = bhas no solution, then :...
JEE Main 2021 (Online) 31st August Morning Shift
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Let [$$\lambda$$] be the greatest integer less than or equal to $$\lambda$$. The set of all values of $$\lambda$$ for wh...
JEE Main 2021 (Online) 27th August Evening Shift
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Let A(a, 0), B(b, 2b + 1) and C(0, b), b $$\ne$$ 0, |b| $$\ne$$ 1, be points such that the area of triangle ABC is 1 sq....
JEE Main 2021 (Online) 27th August Evening Shift
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Let $$A = \left( {\matrix{ {[x + 1]} & {[x + 2]} & {[x + 3]} \cr {[x]} & {[x + 3]} & {[x + 3]} ...
JEE Main 2021 (Online) 27th August Evening Shift
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If the matrix $$A = \left( {\matrix{ 0 & 2 \cr K & { - 1} \cr } } \right)$$ satisfies $$A({A^3} + 3I...
JEE Main 2021 (Online) 27th August Morning Shift
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Let $$A = \left( {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 1 \cr 1 & 0 & 0 \cr } } \right...
JEE Main 2021 (Online) 26th August Evening Shift
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If $$A = \left( {\matrix{ {{1 \over {\sqrt 5 }}} & {{2 \over {\sqrt 5 }}} \cr {{{ - 2} \over {\sqrt 5 }}} &a...
JEE Main 2021 (Online) 26th August Morning Shift
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Let $$\theta \in \left( {0,{\pi \over 2}} \right)$$. If the system of linear equations$$(1 + {\cos ^2}\theta )x + {\si...
JEE Main 2021 (Online) 26th August Morning Shift
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Let $$A = \left[ {\matrix{ 1 & 2 \cr { - 1} & 4 \cr } } \right]$$. If A$$-$$1 = $$\alpha$$I + $$\bet...
JEE Main 2021 (Online) 27th July Morning Shift
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If $$P = \left[ {\matrix{ 1 & 0 \cr {{1 \over 2}} & 1 \cr } } \right]$$, then P50 is :
JEE Main 2021 (Online) 25th July Evening Shift
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The number of distinct real roots of $$\left| {\matrix{ {\sin x} & {\cos x} & {\cos x} \cr {\cos x} &...
JEE Main 2021 (Online) 25th July Evening Shift
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Let A and B be two 3 $$\times$$ 3 real matrices such that (A2 $$-$$ B2) is invertible matrix. If A5 = B5 and A3B2 = A2B3...
JEE Main 2021 (Online) 27th July Evening Shift
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The values of a and b, for which the system of equations 2x + 3y + 6z = 8x + 2y + az = 53x + 5y + 9z = bhas no solution,...
JEE Main 2021 (Online) 25th July Morning Shift
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Let A = [aij] be a real matrix of order 3 $$\times$$ 3, such that ai1 + ai2 + ai3 = 1, for i = 1, 2, 3. Then, the sum of...
JEE Main 2021 (Online) 22th July Evening Shift
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The values of $$\lambda$$ and $$\mu$$ such that the system of equations $$x + y + z = 6$$, $$3x + 5y + 5z = 26$$, $$x + ...
JEE Main 2021 (Online) 22th July Evening Shift
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The value of k $$\in$$R, for which the following system of linear equations3x $$-$$ y + 4z = 3,x + 2y $$-$$ 3z = $$-$$26...
JEE Main 2021 (Online) 20th July Evening Shift
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Let $$A = \left[ {\matrix{ 2 & 3 \cr a & 0 \cr } } \right]$$, a$$\in$$R be written as P + Q where P ...
JEE Main 2021 (Online) 20th July Morning Shift
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Define a relation R over a class of n $$\times$$ n real matrices A and B as "ARB iff there exists a non-singular matrix ...
JEE Main 2021 (Online) 18th March Evening Shift
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Let the system of linear equations 4x + $$\lambda$$y + 2z = 02x $$-$$ y + z = 0$$\mu$$x + 2y + 3z = 0, $$\lambda$$, $$\m...
JEE Main 2021 (Online) 18th March Evening Shift
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Let $$A + 2B = \left[ {\matrix{ 1 & 2 & 0 \cr 6 & { - 3} & 3 \cr { - 5} & 3 & 1 \c...
JEE Main 2021 (Online) 18th March Morning Shift
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The solutions of the equation $$\left| {\matrix{ {1 + {{\sin }^2}x} & {{{\sin }^2}x} & {{{\sin }^2}x} \cr ...
JEE Main 2021 (Online) 18th March Morning Shift
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If x, y, z are in arithmetic progression with common difference d, x $$\ne$$ 3d, and the determinant of the matrix $$\le...
JEE Main 2021 (Online) 17th March Evening Shift
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If $$A = \left( {\matrix{ 0 & {\sin \alpha } \cr {\sin \alpha } & 0 \cr } } \right)$$ and $$\det \le...
JEE Main 2021 (Online) 17th March Morning Shift
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The system of equations kx + y + z = 1, x + ky + z = k and x + y + zk = k2 has no solution if k is equal to :
JEE Main 2021 (Online) 17th March Morning Shift
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Let $$A = \left[ {\matrix{ i & { - i} \cr { - i} & i \cr } } \right],i = \sqrt { - 1} $$. Then, the ...
JEE Main 2021 (Online) 16th March Morning Shift
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Consider the following system of equations :x + 2y $$-$$ 3z = a2x + 6y $$-$$ 11z = bx $$-$$ 2y + 7z = c,where a, b and c...
JEE Main 2021 (Online) 26th February Evening Shift
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The value of $$\left| {\matrix{ {(a + 1)(a + 2)} & {a + 2} & 1 \cr {(a + 2)(a + 3)} & {a + 3} & ...
JEE Main 2021 (Online) 26th February Morning Shift
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Let A be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of A2 is 1, then the po...
JEE Main 2021 (Online) 26th February Morning Shift
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The following system of linear equations2x + 3y + 2z = 93x + 2y + 2z = 9x $$-$$ y + 4z = 8
JEE Main 2021 (Online) 25th February Evening Shift
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If for the matrix, $$A = \left[ {\matrix{ 1 & { - \alpha } \cr \alpha & \beta \cr } } \right]$$, $...
JEE Main 2021 (Online) 25th February Evening Shift
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Let A be a 3 $$\times$$ 3 matrix with det(A) = 4. Let Ri denote the ith row of A. If a matrix B is obtained by performin...
JEE Main 2021 (Online) 25th February Evening Shift
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For the system of linear equations:$$x - 2y = 1,x - y + kz = - 2,ky + 4z = 6,k \in R$$,consider the following statement...
JEE Main 2021 (Online) 24th February Evening Slot
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Let A and B be 3 $$\times$$ 3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then the sys...
JEE Main 2021 (Online) 24th February Evening Slot
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The system of linear equations 3x - 2y - kz = 10 2x - 4y - 2z = 6 x+2y - z = 5m is inconsistent if :
JEE Main 2021 (Online) 24th February Morning Slot
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Let $$\theta = {\pi \over 5}$$ and $$A = \left[ {\matrix{ {\cos \theta } & {\sin \theta } \cr { - \sin \th...
JEE Main 2020 (Online) 6th September Evening Slot
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The values of $$\lambda $$ and $$\mu $$ for which the system of linear equations x + y + z = 2 x + 2y + 3z = 5 x + 3y + ...
JEE Main 2020 (Online) 6th September Morning Slot
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Let m and M be respectively the minimum and maximum values of $$\left| {\matrix{ {{{\cos }^2}x} & {1 + {{\sin }^2...
JEE Main 2020 (Online) 6th September Morning Slot
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If the system of linear equations x + y + 3z = 0 x + 3y + k2z = 0 3x + y + 3z = 0 has a non-zero solution (x, y, z) for ...
JEE Main 2020 (Online) 5th September Evening Slot
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If a + x = b + y = c + z + 1, where a, b, c, x, y, z are non-zero distinct real numbers, then $$\left| {\matrix{ x &a...
JEE Main 2020 (Online) 5th September Evening Slot
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Let $$\lambda \in $$ R . The system of linear equations 2x1 - 4x2 + $$\lambda $$x3 = 1 x1 - 6x2 + x3 = 2 $$\lambda $$x1...
JEE Main 2020 (Online) 5th September Morning Slot
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If the system of equations x+y+z=2 2x+4y–z=6 3x+2y+$$\lambda $$z=$$\mu $$ has infinitely many solutions, then...
JEE Main 2020 (Online) 4th September Evening Slot
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Suppose the vectors x1, x2 and x3 are the solutions of the system of linear equations, Ax = b when the vector b on the r...
JEE Main 2020 (Online) 4th September Evening Slot
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If $$A = \left[ {\matrix{ {\cos \theta } & {i\sin \theta } \cr {i\sin \theta } & {\cos \theta } \cr ...
JEE Main 2020 (Online) 4th September Morning Slot
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Let A be a 3 $$ \times $$ 3 matrix such that adj A = $$\left[ {\matrix{ 2 & { - 1} & 1 \cr { - 1} & ...
JEE Main 2020 (Online) 3rd September Evening Slot
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If $$\Delta $$ = $$\left| {\matrix{ {x - 2} & {2x - 3} & {3x - 4} \cr {2x - 3} & {3x - 4} & {4x ...
JEE Main 2020 (Online) 3rd September Morning Slot
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Let A = {X = (x, y, z)T: PX = 0 and x2 + y2 + z2 = 1} where $$P = \left[ {\matrix{ 1 & 2 & 1 \cr { - 2} ...
JEE Main 2020 (Online) 2nd September Evening Slot
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Let a, b, c $$ \in $$ R be all non-zero and satisfy a3 + b3 + c3 = 2. If the matrix A = $$\left( {\matrix{ a & b ...
JEE Main 2020 (Online) 2nd September Evening Slot
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Let A be a 2 $$ \times $$ 2 real matrix with entries from {0, 1} and |A| $$ \ne $$ 0. Consider the following two stateme...
JEE Main 2020 (Online) 2nd September Morning Slot
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Let S be the set of all $$\lambda $$ $$ \in $$ R for which the system of linear equations 2x – y + 2z = 2 x – 2y + $$\la...
JEE Main 2020 (Online) 2nd September Morning Slot
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The following system of linear equations 7x + 6y – 2z = 0 3x + 4y + 2z = 0 x – 2y – 6z = 0, has
JEE Main 2020 (Online) 9th January Evening Slot
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If for some $$\alpha $$ and $$\beta $$ in R, the intersection of the following three places x + 4y – 2z = 1 x + 7y – 5z ...
JEE Main 2020 (Online) 9th January Morning Slot
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If the matrices A = $$\left[ {\matrix{ 1 & 1 & 2 \cr 1 & 3 & 4 \cr 1 & { - 1} & 3 ...
JEE Main 2020 (Online) 9th January Morning Slot
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If $$A = \left( {\matrix{ 2 & 2 \cr 9 & 4 \cr } } \right)$$ and $$I = \left( {\matrix{ 1 & 0 ...
JEE Main 2020 (Online) 8th January Evening Slot
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The system of linear equations $$\lambda $$x + 2y + 2z = 5 2$$\lambda $$x + 3y + 5z = 8 4x + $$\lambda $$y + 6z = 10 has...
JEE Main 2020 (Online) 8th January Evening Slot
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For which of the following ordered pairs ($$\mu $$, $$\delta $$), the system of linear equations x + 2y + 3z = 1 3x + 4y...
JEE Main 2020 (Online) 8th January Morning Slot
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Let A = [aij] and B = [bij] be two 3 × 3 real matrices such that bij = (3)(i+j-2)aji, where i, j = 1, 2, 3. If the deter...
JEE Main 2020 (Online) 7th January Evening Slot
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If the system of linear equations 2x + 2ay + az = 0 2x + 3by + bz = 0 2x + 4cy + cz = 0, where a, b, c $$ \in $$ R are n...
JEE Main 2020 (Online) 7th January Morning Slot
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Let $$\alpha $$ be a root of the equation x2 + x + 1 = 0 and the matrix A = $${1 \over {\sqrt 3 }}\left[ {\matrix{ 1 ...
JEE Main 2020 (Online) 7th January Morning Slot
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A value of $$\theta \in \left( {0,{\pi \over 3}} \right)$$, for which $$\left| {\matrix{ {1 + {{\cos }^2}\theta } &...
JEE Main 2019 (Online) 12th April Evening Slot
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If $$B = \left[ {\matrix{ 5 & {2\alpha } & 1 \cr 0 & 2 & 1 \cr \alpha & 3 & { - 1}...
JEE Main 2019 (Online) 12th April Morning Slot
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If A is a symmetric matrix and B is a skew-symmetric matrix such that A + B = $$\left[ {\matrix{ 2 & 3 \cr 5...
JEE Main 2019 (Online) 12th April Morning Slot
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The sum of the real roots of the equation $$\left| {\matrix{ x & { - 6} & { - 1} \cr 2 & { - 3x} &am...
JEE Main 2019 (Online) 10th April Evening Slot
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Let $$\lambda $$ be a real number for which the system of linear equations x + y + z = 6, 4x + $$\lambda $$y – $$\lambda...
JEE Main 2019 (Online) 10th April Evening Slot
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If the system of linear equations x + y + z = 5 x + 2y + 2z = 6 x + 3y + $$\lambda $$z = $$\mu $$, ($$\lambda $$, $$\mu ...
JEE Main 2019 (Online) 10th April Morning Slot
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If $${\Delta _1} = \left| {\matrix{ x & {\sin \theta } & {\cos \theta } \cr { - \sin \theta } & { - ...
JEE Main 2019 (Online) 10th April Morning Slot
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The total number of matrices $$A = \left( {\matrix{ 0 & {2y} & 1 \cr {2x} & y & { - 1} \cr ...
JEE Main 2019 (Online) 9th April Evening Slot
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If the system of equations 2x + 3y – z = 0, x + ky – 2z = 0 and 2x – y + z = 0 has a non-trival solution (x, y, z), then...
JEE Main 2019 (Online) 9th April Evening Slot
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If $$\left[ {\matrix{ 1 & 1 \cr 0 & 1 \cr } } \right]\left[ {\matrix{ 1 & 2 \cr 0 & ...
JEE Main 2019 (Online) 9th April Morning Slot
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Let $$\alpha $$ and $$\beta $$ be the roots of the equation x2 + x + 1 = 0. Then for y $$ \ne $$ 0 in R, $$$\left| {\mat...
JEE Main 2019 (Online) 9th April Morning Slot
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Let the number 2,b,c be in an A.P. and A = $$\left[ {\matrix{ 1 & 1 & 1 \cr 2 & b & c \cr 4...
JEE Main 2019 (Online) 8th April Evening Slot
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The greatest value of c $$ \in $$ R for which the system of linear equations x – cy – cz = 0 cx – y + cz = 0 cx + cy – z...
JEE Main 2019 (Online) 8th April Morning Slot
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Let $$A = \left( {\matrix{ {\cos \alpha } & { - \sin \alpha } \cr {\sin \alpha } & {\cos \alpha } \cr ...
JEE Main 2019 (Online) 8th April Morning Slot
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The set of all values of $$\lambda $$ for which the system of linear equations x – 2y – 2z = $$\lambda $$x x + 2y + z = ...
JEE Main 2019 (Online) 12th January Evening Slot
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If A = $$\left[ {\matrix{ 1 & {\sin \theta } & 1 \cr { - \sin \theta } & 1 & {\sin \...
JEE Main 2019 (Online) 12th January Evening Slot
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Let P = $$\left[ {\matrix{ 1 & 0 & 0 \cr 3 & 1 & 0 \cr 9 & 3 & 1 \cr } } \right...
JEE Main 2019 (Online) 12th January Morning Slot
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An ordered pair ($$\alpha $$, $$\beta $$) for which the system of linear equations (1 + $$\alpha $$) x + $$\beta $$y + ...
JEE Main 2019 (Online) 12th January Morning Slot
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Let A and B be two invertible matrices of order 3 $$ \times $$ 3. If det(ABAT) = 8 and det(AB–1) = 8, then det (BA–1 BT...
JEE Main 2019 (Online) 11th January Evening Slot
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If $$\left| {\matrix{ {a - b - c} & {2a} & {2a} \cr {2b} & {b - c - a} & {2b} \cr ...
JEE Main 2019 (Online) 11th January Evening Slot
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Let A = $$\left( {\matrix{ 0 & {2q} & r \cr p & q & { - r} \cr p & { - q} & r \cr ...
JEE Main 2019 (Online) 11th January Morning Slot
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If the system of linear equations 2x + 2y + 3z = a 3x – y + 5z = b x – 3y + 2z = c where a, b, c are non zero real numbe...
JEE Main 2019 (Online) 11th January Morning Slot
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The number of values of $$\theta $$ $$ \in $$ (0, $$\pi $$) for which the system of linear equations x + 3y + 7z = 0 $$-...
JEE Main 2019 (Online) 10th January Evening Slot
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Let A = $$\left[ {\matrix{ 2 & b & 1 \cr b & {{b^2} + 1} & b \cr 1 & b & 2 \cr ...
JEE Main 2019 (Online) 10th January Evening Slot
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Let d $$ \in $$ R, and $$A = \left[ {\matrix{ { - 2} & {4 + d} & {\left( {\sin \theta } \rig...
JEE Main 2019 (Online) 10th January Morning Slot
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If the system of equations x + y + z = 5 x + 2y + 3z = 9 x + 3y + az = $$\beta $$ has infinitely many solutions, then $$...
JEE Main 2019 (Online) 10th January Morning Slot
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If $$A = \left[ {\matrix{ {{e^t}} & {{e^{ - t}}\cos t} & {{e^{ - t}}\sin t} \cr {{e^t}}...
JEE Main 2019 (Online) 9th January Evening Slot
GO TO QUESTION
If the system of linear equations x $$-$$ 4y + 7z = g 3y $$-$$ 5z = h $$-$$2x...
JEE Main 2019 (Online) 9th January Evening Slot
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The system of linear equations x + y + z = 2 2x + 3y + 2z = 5 2x + 3y + (a2 – 1) z = a + 1 then ...
JEE Main 2019 (Online) 9th January Morning Slot
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If $$A = \left[ {\matrix{ {\cos \theta } & { - \sin \theta } \cr {\sin \theta } & {\cos \theta } \cr ...
JEE Main 2019 (Online) 9th January Morning Slot
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The number of values of k for which the system of linear equations, (k + 2)x + 10y = k kx + (k +3)y = k -1 has no solu...
JEE Main 2018 (Online) 16th April Morning Slot
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Let A = $$\left[ {\matrix{ 1 & 0 & 0 \cr 1 & 1 & 0 \cr 1 & 1 & 1 \cr } } \right...
JEE Main 2018 (Online) 16th April Morning Slot
GO TO QUESTION
If the system of linear equations x + ay + z = 3 x + 2y + 2z = 6 x + 5y + 3z = b has no solution, then :...
JEE Main 2018 (Online) 15th April Evening Slot
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Suppose A is any 3$$ \times $$ 3 nonsingular matrx and ( A $$-$$ 3I) (A $$-$$ 5I) = O where I = I3 and O = O3. If $$\alp...
JEE Main 2018 (Online) 15th April Evening Slot
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Let S be the set of all real values of k for which the systemof linear equations x + y + z = 2 2x + y $$-$$ z = 3 3x + 2...
JEE Main 2018 (Online) 15th April Morning Slot
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Let $$A$$ be a matrix such that $$A.\left[ {\matrix{ 1 & 2 \cr 0 & 3 \cr } } \right]$$ is a scalar m...
JEE Main 2018 (Online) 15th April Morning Slot
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If the system of linear equations x + ky + 3z = 0 3x + ky - 2z = 0 2x + 4y - 3z = 0 has a non-zero solution (x, y, z), t...
JEE Main 2018 (Offline)
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If $$\left| {\matrix{ {x - 4} & {2x} & {2x} \cr {2x} & {x - 4} & {2x} \cr {2x} & {2x} &...
JEE Main 2018 (Offline)
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For two 3 × 3 matrices A and B, let A + B = 2BT and 3A + 2B = I3, where BT is the transpose of B and I3 is 3 × 3 identi...
JEE Main 2017 (Online) 9th April Morning Slot
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If $$S = \left\{ {x \in \left[ {0,2\pi } \right]:\left| {\matrix{ 0 & {\cos x} & { - \sin x} \cr {\sin ...
JEE Main 2017 (Online) 8th April Morning Slot
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The number of real values of $$\lambda $$ for which the system of linear equations 2x + 4y $$-$$ $$\lambda $$z = 0 4x + ...
JEE Main 2017 (Online) 8th April Morning Slot
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Let A be any 3 $$ \times $$ 3 invertible matrix. Then which one of the following is not always true ?
JEE Main 2017 (Online) 8th April Morning Slot
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If S is the set of distinct values of 'b' for which the following system of linear equations x + y + z = 1 x + ay + z =...
JEE Main 2017 (Offline)
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If $$A = \left[ {\matrix{ 2 & { - 3} \cr { - 4} & 1 \cr } } \right]$$, then adj(3A2 + 12A) is equal ...
JEE Main 2017 (Offline)
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If A = $$\left[ {\matrix{ { - 4} & { - 1} \cr 3 & 1 \cr } } \right]$$, then the determ...
JEE Main 2016 (Online) 10th April Morning Slot
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Let A be a 3 $$ \times $$ 3 matrix such that A2 $$-$$ 5A + 7I = 0 Statement - I : A$$-$$1 = $${1 \over 7}$$...
JEE Main 2016 (Online) 10th April Morning Slot
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The number of distinct real roots of the equation, $$\left| {\matrix{ {\cos x} & {\sin x} & {\sin x} \cr ...
JEE Main 2016 (Online) 9th April Morning Slot
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If P = $$\left[ {\matrix{ {{{\sqrt 3 } \over 2}} & {{1 \over 2}} \cr { - {1 \over 2}} & {{{\sqrt 3 } \ov...
JEE Main 2016 (Online) 9th April Morning Slot
GO TO QUESTION
The system of linear equations $$\matrix{ {x + \lambda y - z = 0} \cr {\lambda x - y - z = 0} \cr {x + y -...
JEE Main 2016 (Offline)
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If $$A = \left[ {\matrix{ {5a} & { - b} \cr 3 & 2 \cr } } \right]$$ and $$A$$ adj $$A=A$$ $${A^T},$...
JEE Main 2016 (Offline)
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If $$A = \left[ {\matrix{ 1 & 2 & 2 \cr 2 & 1 & { - 2} \cr a & 2 & b \cr } } \r...
JEE Main 2015 (Offline)
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The set of all values of $$\lambda $$ for which the system of linear equations: $$\matrix{ {2{x_1} - 2{x_2} + {x_3} =...
JEE Main 2015 (Offline)
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If $$A$$ is a $$3 \times 3$$ non-singular matrix such that $$AA'=A'A$$ and $$B = {A^{ - 1}}A',$$ then $$BB'$$ equals:...
JEE Main 2014 (Offline)
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If $$\alpha ,\beta \ne 0,$$ and $$f\left( n \right) = {\alpha ^n} + {\beta ^n}$$ and $$$\left| {\matrix{ 3 & {1...
JEE Main 2014 (Offline)
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If $$P = \left[ {\matrix{ 1 & \alpha & 3 \cr 1 & 3 & 3 \cr 2 & 4 & 4 \cr } } \...
JEE Main 2013 (Offline)
GO TO QUESTION
Let $$A = \left( {\matrix{ 1 & 0 & 0 \cr 2 & 1 & 0 \cr 3 & 2 & 1 \cr } } \right...
AIEEE 2012
GO TO QUESTION
Let $$P$$ and $$Q$$ be $$3 \times 3$$ matrices $$P \ne Q.$$ If $${P^3} = {Q^3}$$ and $${P^2}Q = {Q^2}P$$ then determina...
AIEEE 2012
GO TO QUESTION
The number of values of $$k$$ for which the linear equations $$4x + ky + 2z = 0,kx + 4y + z = 0$$ and $$2x+2y+z=0$$ pos...
AIEEE 2011
GO TO QUESTION
Let $$A$$ and $$B$$ be two symmetric matrices of order $$3$$. Statement - 1: $$A(BA)$$ and $$(AB)$$$$A$$ are symmetric...
AIEEE 2011
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The number of $$3 \times 3$$ non-singular matrices, with four entries as $$1$$ and all other entries as $$0$$, is
AIEEE 2010
GO TO QUESTION
Consider the system of linear equations; $$$\matrix{ {{x_1} + 2{x_2} + {x_3} = 3} \cr {2{x_1} + 3{x_2} + {x_3} ...
AIEEE 2010
GO TO QUESTION
Let $$A$$ be a $$\,2 \times 2$$ matrix with non-zero entries and let $${A^2} = I,$$ where $$I$$ is $$2 \times 2$$ ident...
AIEEE 2010
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Let $$a, b, c$$ be such that $$b\left( {a + c} \right) \ne 0$$ if $$\left| {\matrix{ a & {a + 1} & {a - 1} \...
AIEEE 2009
GO TO QUESTION
Let $$A$$ be a $$\,2 \times 2$$ matrix Statement - 1 : $$adj\left( {adj\,A} \right) = A$$ Statement - 2 :$$\left| {adj\,...
AIEEE 2009
GO TO QUESTION
Let $$A$$ be a square matrix all of whose entries are integers. Then which one of the following is true?
AIEEE 2008
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Let $$a, b, c$$ be any real numbers. Suppose that there are real numbers $$x, y, z$$ not all zero such that $$x=cy+bz,$$...
AIEEE 2008
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Let $$A$$ be $$a\,2 \times 2$$ matrix with real entries. Let $$I$$ be the $$2 \times 2$$ identity matrix. Denote by tr$$...
AIEEE 2008
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If$$D = \left| {\matrix{ 1 & 1 & 1 \cr 1 & {1 + x} & 1 \cr 1 & 1 & {1 + y} \cr ...
AIEEE 2007
GO TO QUESTION
Let $$A = \left| {\matrix{ 5 & {5\alpha } & \alpha \cr 0 & \alpha & {5\alpha } \cr 0 &...
AIEEE 2007
GO TO QUESTION
Let $$A = \left( {\matrix{ 1 & 2 \cr 3 & 4 \cr } } \right)$$ and $$B = \left( {\matrix{ a & 0...
AIEEE 2006
GO TO QUESTION
If $$A$$ and $$B$$ are square matrices of size $$n\, \times \,n$$ such that $${A^2} - {B^2} = \left( {A - B} \right)\le...
AIEEE 2006
GO TO QUESTION
If $${a^2} + {b^2} + {c^2} = - 2$$ and f$$\left( x \right) = \left| {\matrix{ {1 + {a^2}x} & {\left( {1 + {b^2}...
AIEEE 2005
GO TO QUESTION
If $${a_1},{a_2},{a_3},........,{a_n},.....$$ are in G.P., then the determinant $$$\Delta \left| {\matrix{ {\log {a_...
AIEEE 2005
GO TO QUESTION
The system of equations $$\matrix{ {\alpha \,x + y + z = \alpha - 1} \cr {x + \alpha y + z = \alpha - 1} \cr...
AIEEE 2005
GO TO QUESTION
If $${A^2} - A + 1 = 0$$, then the inverse of $$A$$ is
AIEEE 2005
GO TO QUESTION
If $${a_1},{a_2},{a_3},.........,{a_n},......$$ are in G.P., then the value of the determinant $$\left| {\matrix{ {\...
AIEEE 2004
GO TO QUESTION
Let $$A = \left( {\matrix{ 1 & { - 1} & 1 \cr 2 & 1 & { - 3} \cr 1 & 1 & 1 \cr ...
AIEEE 2004
GO TO QUESTION
Let $$A = \left( {\matrix{ 0 & 0 & { - 1} \cr 0 & { - 1} & 0 \cr { - 1} & 0 & 0 \c...
AIEEE 2004
GO TO QUESTION
If the system of linear equations $$x + 2ay + az = 0;$$ $$x + 3by + bz = 0;\,\,x + 4cy + cz = 0;$$ has a non - zero so...
AIEEE 2003
GO TO QUESTION
If $$1,$$ $$\omega ,{\omega ^2}$$ are the cube roots of unity, then $$\Delta = \left| {\matrix{ 1 & {{\omega ^...
AIEEE 2003
GO TO QUESTION
If $$A = \left[ {\matrix{ a & b \cr b & a \cr } } \right]$$ and $${A^2} = \left[ {\matrix{ \alpha...
AIEEE 2003
GO TO QUESTION
If $$a>0$$ and discriminant of $$\,a{x^2} + 2bx + c$$ is $$-ve$$, then $$\left| {\matrix{ a & b & {ax + b}...
AIEEE 2002
GO TO QUESTION
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