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JEE Main

Matrices and Determinants

Mathematics

Previous Years Questions

Numerical

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Consider a matrix $$A=\left[\begin{array}{ccc}\alpha & \beta & \gamma \\ \alpha^{2} & \beta^{2} & \gamma^{2} \\ \beta+\g...
JEE Main 2022 (Online) 27th July Evening Shift
Let $$S$$ be the set containing all $$3 \times 3$$ matrices with entries from $$\{-1,0,1\}$$. The total number of matric...
JEE Main 2022 (Online) 27th July Morning Shift
The number of matrices $$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
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
Let $$A=\left(\begin{array}{rrr}2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0\end{array}\right)$$ and $$B=A-I$$. If $$\omega=\...
JEE Main 2022 (Online) 25th July Morning Shift
Let $$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
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
Let $$A = \left( {\matrix{ 2 & { - 2} \cr 1 & { - 1} \cr } } \right)$$ and $$B = \left( {\matrix{ { - 1} ...
JEE Main 2022 (Online) 25th June Evening Shift
Let $$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
The positive value of the determinant of the matrix A, whose Adj(Adj(A)) = $$\left( {\matrix{ {14} & {28} & { - 14} ...
JEE Main 2022 (Online) 27th June Morning Shift
Let A be a matrix of order 2 $$\times$$ 2, whose entries are from the set {0, 1, 2, 3, 4, 5}. If the sum of all the entr...
JEE Main 2022 (Online) 27th June Evening Shift
Let $$A = \left( {\matrix{ {1 + i} & 1 \cr { - i} & 0 \cr } } \right)$$ where $$i = \sqrt { - 1} $$. Then, t...
JEE Main 2022 (Online) 28th June Evening Shift
If the system of linear equations $$2x - 3y = \gamma + 5$$, $$\alpha x + 5y = \beta + 1$$, where $$\alpha$$, $$\beta$$...
JEE Main 2022 (Online) 28th June Evening Shift
Let $$M = \left[ {\matrix{ 0 & { - \alpha } \cr \alpha & 0 \cr } } \right]$$, where $$\alpha$$ is a non-zer...
JEE Main 2022 (Online) 29th June Evening Shift
The number of elements in the set $$\left\{ {A = \left( {\matrix{ a & b \cr 0 & d \cr } } \right):a,...
JEE Main 2021 (Online) 31st August Evening Shift
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
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
Let $$f(x) = \left| {\matrix{ {{{\sin }^2}x} & { - 2 + {{\cos }^2}x} & {\cos 2x} \cr {2 + {{\sin }^2}x} ...
JEE Main 2021 (Online) 27th July Morning Shift
For real numbers $$\alpha$$ and $$\beta$$, consider the following system of linear equations :x + y $$-$$ z = 2, x + 2y ...
JEE Main 2021 (Online) 27th July Morning Shift
If $$A = \left[ {\matrix{ 1 & 1 & 1 \cr 0 & 1 & 1 \cr 0 & 0 & 1 \cr } } \right]...
JEE Main 2021 (Online) 27th July Evening Shift
Let $$M = \left\{ {A = \left( {\matrix{ a & b \cr c & d \cr } } \right):a,b,c,d \in \{ \pm 3, \pm 2...
JEE Main 2021 (Online) 25th July Morning Shift
Let $$A = \left[ {\matrix{ 0 & 1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1 \cr } } \right...
JEE Main 2021 (Online) 22th July Evening Shift
Let $$A = \{ {a_{ij}}\} $$ be a 3 $$\times$$ 3 matrix, where $${a_{ij}} = \left\{ {\matrix{ {{{( - 1)}^{j - i}}} &amp...
JEE Main 2021 (Online) 20th July Evening Shift
Let a, b, c, d in arithmetic progression with common difference $$\lambda$$. If $$\left| {\matrix{ {x + a - c} & ...
JEE Main 2021 (Online) 20th July Morning Shift
Let $$A = \left( {\matrix{ 1 & { - 1} & 0 \cr 0 & 1 & { - 1} \cr 0 & 0 & 1 \cr ...
JEE Main 2021 (Online) 20th July Morning Shift
Let I be an identity matrix of order 2 $$\times$$ 2 and P = $$\left[ {\matrix{ 2 & { - 1} \cr 5 & { - 3}...
JEE Main 2021 (Online) 18th March Evening Shift
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
Let $$A = \left[ {\matrix{ a & b \cr c & d \cr } } \right]$$ and $$B = \left[ {\matrix{ \alpha ...
JEE Main 2021 (Online) 17th March Evening Shift
If $$A = \left[ {\matrix{ 2 & 3 \cr 0 & { - 1} \cr } } \right]$$, then the value of det(A4) + det(A1...
JEE Main 2021 (Online) 17th March Morning Shift
Let $$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
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
Let $$P = \left[ {\matrix{ { - 30} & {20} & {56} \cr {90} & {140} & {112} \cr {120} & {...
JEE Main 2021 (Online) 16th March Morning Shift
If the matrix $$A = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 2 & 0 \cr 3 & 0 & { - 1} \c...
JEE Main 2021 (Online) 26th February Evening Shift
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
Let $$A = \left[ {\matrix{ x & y & z \cr y & z & x \cr z & x & y \cr } } \right...
JEE Main 2021 (Online) 25th February Morning Slot
If $$A = \left[ {\matrix{ 0 & { - \tan \left( {{\theta \over 2}} \right)} \cr {\tan \left( {{\theta \over ...
JEE Main 2021 (Online) 25th February Morning Slot
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
Let P = $$\left[ {\matrix{ 3 & { - 1} & { - 2} \cr 2 & 0 & \alpha \cr 3 & { - 5} &...
JEE Main 2021 (Online) 24th February Morning Slot
The sum of distinct values of $$\lambda $$ for which the system of equations$$\left( {\lambda - 1} \right)x + \left( {3...
JEE Main 2020 (Online) 6th September Evening Slot
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
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
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
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

MCQ (Single Correct Answer)

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Let $$A=\left(\begin{array}{rr}4 & -2 \\ \alpha & \beta\end{array}\right)$$. If $$\mathrm{A}^{2}+\gamma \mathrm{A}+18 \m...
JEE Main 2022 (Online) 27th July Evening Shift
Let $$A=\left(\begin{array}{cc}1 & 2 \\ -2 & -5\end{array}\right)$$. Let $$\alpha, \beta \in \mathbb{R}$$ be such that $...
JEE Main 2022 (Online) 27th July Morning Shift
$$ \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
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
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
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
The number of $$\theta \in(0,4 \pi)$$ for which the system of linear equations $$ \begin{aligned} &3(\sin 3 \theta) x-y...
JEE Main 2022 (Online) 25th July Morning Shift
Let A and B be two square matrices of order 2. If $$det\,(A) = 2$$, $$det\,(B) = 3$$ and $$\det \left( {(\det \,5(det\,A...
JEE Main 2022 (Online) 30th June Morning Shift
Let $$A = \left[ {\matrix{ 1 & { - 2} & \alpha \cr \alpha & 2 & { - 1} \cr } } \right]$$ and $$B = \left[ ...
JEE Main 2022 (Online) 30th June Morning Shift
Let S = {$$\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
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
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
Let $$A = \left[ {\matrix{ 0 & { - 2} \cr 2 & 0 \cr } } \right]$$. If M and N are two matrices given by $$M ...
JEE Main 2022 (Online) 25th June Morning Shift
Let A be a 3 $$\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
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
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
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
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
Let the system of linear equations $$x + 2y + z = 2$$, $$\alpha x + 3y - z = \alpha $$, $$ - \alpha x + y + 2z = - \alp...
JEE Main 2022 (Online) 27th June Morning Shift
Let A and B be two 3 $$\times$$ 3 matrices such that $$AB = I$$ and $$|A| = {1 \over 8}$$. Then $$|adj\,(B\,adj(2A))|$$ ...
JEE Main 2022 (Online) 27th June Evening Shift
Let $$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
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
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
Let $$A = \left( {\matrix{ 2 & { - 1} \cr 0 & 2 \cr } } \right)$$. If $$B = I - {}^5{C_1}(adj\,A) + {}^5{C_2...
JEE Main 2022 (Online) 29th June Evening Shift
Let $$A = [{a_{ij}}]$$ be a square matrix of order 3 such that $${a_{ij}} = {2^{j - i}}$$, for all i, j = 1, 2, 3. Then,...
JEE Main 2022 (Online) 29th June Morning Shift
If the system of linear equations 2x + y $$-$$ z = 7 x $$-$$ 3y + 2z = 1 x + 4y + $$\delta$$z = k, where $$\delta$$, k $...
JEE Main 2022 (Online) 29th June Morning Shift
Let $${J_{n,m}} = \int\limits_0^{{1 \over 2}} {{{{x^n}} \over {{x^m} - 1}}dx} $$, $$\forall$$ n > m and n, m $$\in$$ ...
JEE Main 2021 (Online) 1st September Evening Shift
Consider the system of linear equations$$-$$x + y + 2z = 03x $$-$$ ay + 5z = 12x $$-$$ 2y $$-$$ az = 7Let S1 be the set ...
JEE Main 2021 (Online) 1st September Evening Shift
If $$\alpha$$ + $$\beta$$ + $$\gamma$$ = 2$$\pi$$, then the system of equations x + (cos $$\gamma$$)y + (cos $$\beta$$)z...
JEE Main 2021 (Online) 31st August Evening Shift
If $${a_r} = \cos {{2r\pi } \over 9} + i\sin {{2r\pi } \over 9}$$, r = 1, 2, 3, ....., i = $$\sqrt { - 1} $$, then the d...
JEE Main 2021 (Online) 31st August Morning Shift
If the following system of linear equations2x + y + z = 5x $$-$$ y + z = 3x + y + az = bhas no solution, then :...
JEE Main 2021 (Online) 31st August Morning Shift
Let [$$\lambda$$] be the greatest integer less than or equal to $$\lambda$$. The set of all values of $$\lambda$$ for wh...
JEE Main 2021 (Online) 27th August Evening Shift
Let A(a, 0), B(b, 2b + 1) and C(0, b), b $$\ne$$ 0, |b| $$\ne$$ 1, be points such that the area of triangle ABC is 1 sq....
JEE Main 2021 (Online) 27th August Evening Shift
Let $$A = \left( {\matrix{ {[x + 1]} & {[x + 2]} & {[x + 3]} \cr {[x]} & {[x + 3]} & {[x + 3]} ...
JEE Main 2021 (Online) 27th August Evening Shift
If the matrix $$A = \left( {\matrix{ 0 & 2 \cr K & { - 1} \cr } } \right)$$ satisfies $$A({A^3} + 3I...
JEE Main 2021 (Online) 27th August Morning Shift
Let $$A = \left( {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 1 \cr 1 & 0 & 0 \cr } } \right...
JEE Main 2021 (Online) 26th August Evening Shift
If $$A = \left( {\matrix{ {{1 \over {\sqrt 5 }}} & {{2 \over {\sqrt 5 }}} \cr {{{ - 2} \over {\sqrt 5 }}} &a...
JEE Main 2021 (Online) 26th August Morning Shift
Let $$\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
Let $$A = \left[ {\matrix{ 1 & 2 \cr { - 1} & 4 \cr } } \right]$$. If A$$-$$1 = $$\alpha$$I + $$\bet...
JEE Main 2021 (Online) 27th July Morning Shift
If $$P = \left[ {\matrix{ 1 & 0 \cr {{1 \over 2}} & 1 \cr } } \right]$$, then P50 is :
JEE Main 2021 (Online) 25th July Evening Shift
The number of distinct real roots of $$\left| {\matrix{ {\sin x} & {\cos x} & {\cos x} \cr {\cos x} &amp...
JEE Main 2021 (Online) 25th July Evening Shift
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
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
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
The values of $$\lambda$$ and $$\mu$$ such that the system of equations $$x + y + z = 6$$, $$3x + 5y + 5z = 26$$, $$x + ...
JEE Main 2021 (Online) 22th July Evening Shift
The value of k $$\in$$R, for which the following system of linear equations3x $$-$$ y + 4z = 3,x + 2y $$-$$ 3z = $$-$$26...
JEE Main 2021 (Online) 20th July Evening Shift
Let $$A = \left[ {\matrix{ 2 & 3 \cr a & 0 \cr } } \right]$$, a$$\in$$R be written as P + Q where P ...
JEE Main 2021 (Online) 20th July Morning Shift
Define a relation R over a class of n $$\times$$ n real matrices A and B as "ARB iff there exists a non-singular matrix ...
JEE Main 2021 (Online) 18th March Evening Shift
Let the system of linear equations 4x + $$\lambda$$y + 2z = 02x $$-$$ y + z = 0$$\mu$$x + 2y + 3z = 0, $$\lambda$$, $$\m...
JEE Main 2021 (Online) 18th March Evening Shift
Let $$A + 2B = \left[ {\matrix{ 1 & 2 & 0 \cr 6 & { - 3} & 3 \cr { - 5} & 3 & 1 \c...
JEE Main 2021 (Online) 18th March Morning Shift
The solutions of the equation $$\left| {\matrix{ {1 + {{\sin }^2}x} & {{{\sin }^2}x} & {{{\sin }^2}x} \cr ...
JEE Main 2021 (Online) 18th March Morning Shift
If x, y, z are in arithmetic progression with common difference d, x $$\ne$$ 3d, and the determinant of the matrix $$\le...
JEE Main 2021 (Online) 17th March Evening Shift
If $$A = \left( {\matrix{ 0 & {\sin \alpha } \cr {\sin \alpha } & 0 \cr } } \right)$$ and $$\det \le...
JEE Main 2021 (Online) 17th March Morning Shift
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
Let $$A = \left[ {\matrix{ i & { - i} \cr { - i} & i \cr } } \right],i = \sqrt { - 1} $$. Then, the ...
JEE Main 2021 (Online) 16th March Morning Shift
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
The value of $$\left| {\matrix{ {(a + 1)(a + 2)} & {a + 2} & 1 \cr {(a + 2)(a + 3)} & {a + 3} & ...
JEE Main 2021 (Online) 26th February Morning Shift
Let A be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of A2 is 1, then the po...
JEE Main 2021 (Online) 26th February Morning Shift
The following system of linear equations2x + 3y + 2z = 93x + 2y + 2z = 9x $$-$$ y + 4z = 8
JEE Main 2021 (Online) 25th February Evening Shift
If for the matrix, $$A = \left[ {\matrix{ 1 & { - \alpha } \cr \alpha & \beta \cr } } \right]$$, $...
JEE Main 2021 (Online) 25th February Evening Shift
Let A be a 3 $$\times$$ 3 matrix with det(A) = 4. Let Ri denote the ith row of A. If a matrix B is obtained by performin...
JEE Main 2021 (Online) 25th February Evening Shift
For the system of linear equations:$$x - 2y = 1,x - y + kz = - 2,ky + 4z = 6,k \in R$$,consider the following statement...
JEE Main 2021 (Online) 24th February Evening Slot
Let A and B be 3 $$\times$$ 3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then the sys...
JEE Main 2021 (Online) 24th February Evening Slot
The system of linear equations 3x - 2y - kz = 10 2x - 4y - 2z = 6 x+2y - z = 5m is inconsistent if :
JEE Main 2021 (Online) 24th February Morning Slot
Let $$\theta = {\pi \over 5}$$ and $$A = \left[ {\matrix{ {\cos \theta } & {\sin \theta } \cr { - \sin \th...
JEE Main 2020 (Online) 6th September Evening Slot
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
Let m and M be respectively the minimum and maximum values of $$\left| {\matrix{ {{{\cos }^2}x} & {1 + {{\sin }^2...
JEE Main 2020 (Online) 6th September Morning Slot
If the system of linear equations x + y + 3z = 0 x + 3y + k2z = 0 3x + y + 3z = 0 has a non-zero solution (x, y, z) for ...
JEE Main 2020 (Online) 5th September Evening Slot
If a + x = b + y = c + z + 1, where a, b, c, x, y, z are non-zero distinct real numbers, then $$\left| {\matrix{ x &a...
JEE Main 2020 (Online) 5th September Evening Slot
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
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
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
If $$A = \left[ {\matrix{ {\cos \theta } & {i\sin \theta } \cr {i\sin \theta } & {\cos \theta } \cr ...
JEE Main 2020 (Online) 4th September Morning Slot
Let A be a 3 $$ \times $$ 3 matrix such that adj A = $$\left[ {\matrix{ 2 & { - 1} & 1 \cr { - 1} & ...
JEE Main 2020 (Online) 3rd September Evening Slot
If $$\Delta $$ = $$\left| {\matrix{ {x - 2} & {2x - 3} & {3x - 4} \cr {2x - 3} & {3x - 4} & {4x ...
JEE Main 2020 (Online) 3rd September Morning Slot
Let A = {X = (x, y, z)T: PX = 0 and x2 + y2 + z2 = 1} where $$P = \left[ {\matrix{ 1 & 2 & 1 \cr { - 2} ...
JEE Main 2020 (Online) 2nd September Evening Slot
Let a, b, c $$ \in $$ R be all non-zero and satisfy a3 + b3 + c3 = 2. If the matrix A = $$\left( {\matrix{ a & b ...
JEE Main 2020 (Online) 2nd September Evening Slot
Let A be a 2 $$ \times $$ 2 real matrix with entries from {0, 1} and |A| $$ \ne $$ 0. Consider the following two stateme...
JEE Main 2020 (Online) 2nd September Morning Slot
Let S be the set of all $$\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
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
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
If the matrices A = $$\left[ {\matrix{ 1 & 1 & 2 \cr 1 & 3 & 4 \cr 1 & { - 1} & 3 ...
JEE Main 2020 (Online) 9th January Morning Slot
If $$A = \left( {\matrix{ 2 & 2 \cr 9 & 4 \cr } } \right)$$ and $$I = \left( {\matrix{ 1 & 0 ...
JEE Main 2020 (Online) 8th January Evening Slot
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
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
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
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
Let $$\alpha $$ be a root of the equation x2 + x + 1 = 0 and the matrix A = $${1 \over {\sqrt 3 }}\left[ {\matrix{ 1 ...
JEE Main 2020 (Online) 7th January Morning Slot
A value of $$\theta \in \left( {0,{\pi \over 3}} \right)$$, for which $$\left| {\matrix{ {1 + {{\cos }^2}\theta } &...
JEE Main 2019 (Online) 12th April Evening Slot
If $$B = \left[ {\matrix{ 5 & {2\alpha } & 1 \cr 0 & 2 & 1 \cr \alpha & 3 & { - 1}...
JEE Main 2019 (Online) 12th April Morning Slot
If A is a symmetric matrix and B is a skew-symmetric matrix such that A + B = $$\left[ {\matrix{ 2 & 3 \cr 5...
JEE Main 2019 (Online) 12th April Morning Slot
The sum of the real roots of the equation $$\left| {\matrix{ x & { - 6} & { - 1} \cr 2 & { - 3x} &am...
JEE Main 2019 (Online) 10th April Evening Slot
Let $$\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
If the system of linear equations x + y + z = 5 x + 2y + 2z = 6 x + 3y + $$\lambda $$z = $$\mu $$, ($$\lambda $$, $$\mu ...
JEE Main 2019 (Online) 10th April Morning Slot
If $${\Delta _1} = \left| {\matrix{ x & {\sin \theta } & {\cos \theta } \cr { - \sin \theta } & { - ...
JEE Main 2019 (Online) 10th April Morning Slot
The total number of matrices $$A = \left( {\matrix{ 0 & {2y} & 1 \cr {2x} & y & { - 1} \cr ...
JEE Main 2019 (Online) 9th April Evening Slot
If the system of equations 2x + 3y – z = 0, x + ky – 2z = 0 and 2x – y + z = 0 has a non-trival solution (x, y, z), then...
JEE Main 2019 (Online) 9th April Evening Slot
If $$\left[ {\matrix{ 1 & 1 \cr 0 & 1 \cr } } \right]\left[ {\matrix{ 1 & 2 \cr 0 & ...
JEE Main 2019 (Online) 9th April Morning Slot
Let $$\alpha $$ and $$\beta $$ be the roots of the equation x2 + x + 1 = 0. Then for y $$ \ne $$ 0 in R, $$$\left| {\mat...
JEE Main 2019 (Online) 9th April Morning Slot
Let the number 2,b,c be in an A.P. and A = $$\left[ {\matrix{ 1 & 1 & 1 \cr 2 & b & c \cr 4...
JEE Main 2019 (Online) 8th April Evening Slot
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
Let $$A = \left( {\matrix{ {\cos \alpha } & { - \sin \alpha } \cr {\sin \alpha } & {\cos \alpha } \cr ...
JEE Main 2019 (Online) 8th April Morning Slot
The set of all values of $$\lambda $$ for which the system of linear equations x – 2y – 2z = $$\lambda $$x x + 2y + z = ...
JEE Main 2019 (Online) 12th January Evening Slot
If   A = $$\left[ {\matrix{ 1 & {\sin \theta } & 1 \cr { - \sin \theta } & 1 & {\sin \...
JEE Main 2019 (Online) 12th January Evening Slot
Let P = $$\left[ {\matrix{ 1 & 0 & 0 \cr 3 & 1 & 0 \cr 9 & 3 & 1 \cr } } \right...
JEE Main 2019 (Online) 12th January Morning Slot
An ordered pair ($$\alpha $$, $$\beta $$) for which the system of linear equations (1 + $$\alpha $$) x + $$\beta $$y + ...
JEE Main 2019 (Online) 12th January Morning Slot
Let A and B be two invertible matrices of order 3 $$ \times $$ 3. If det(ABAT) = 8 and det(AB–1) = 8, then det (BA–1 BT...
JEE Main 2019 (Online) 11th January Evening Slot
If  $$\left| {\matrix{ {a - b - c} & {2a} & {2a} \cr {2b} & {b - c - a} & {2b} \cr ...
JEE Main 2019 (Online) 11th January Evening Slot
Let A = $$\left( {\matrix{ 0 & {2q} & r \cr p & q & { - r} \cr p & { - q} & r \cr ...
JEE Main 2019 (Online) 11th January Morning Slot
If the system of linear equations 2x + 2y + 3z = a 3x – y + 5z = b x – 3y + 2z = c where a, b, c are non zero real numbe...
JEE Main 2019 (Online) 11th January Morning Slot
The number of values of $$\theta $$ $$ \in $$ (0, $$\pi $$) for which the system of linear equations x + 3y + 7z = 0 $$-...
JEE Main 2019 (Online) 10th January Evening Slot
Let A = $$\left[ {\matrix{ 2 & b & 1 \cr b & {{b^2} + 1} & b \cr 1 & b & 2 \cr ...
JEE Main 2019 (Online) 10th January Evening Slot
Let  d $$ \in $$ R, and  $$A = \left[ {\matrix{ { - 2} & {4 + d} & {\left( {\sin \theta } \rig...
JEE Main 2019 (Online) 10th January Morning Slot
If the system of equations x + y + z = 5 x + 2y + 3z = 9 x + 3y + az = $$\beta $$ has infinitely many solutions, then $$...
JEE Main 2019 (Online) 10th January Morning Slot
If   $$A = \left[ {\matrix{ {{e^t}} & {{e^{ - t}}\cos t} & {{e^{ - t}}\sin t} \cr {{e^t}}...
JEE Main 2019 (Online) 9th January Evening Slot
If the system of linear equations x $$-$$ 4y + 7z = g        3y $$-$$ 5z = h $$-$$2x...
JEE Main 2019 (Online) 9th January Evening Slot
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
If $$A = \left[ {\matrix{ {\cos \theta } & { - \sin \theta } \cr {\sin \theta } & {\cos \theta } \cr ...
JEE Main 2019 (Online) 9th January Morning Slot
Let A = $$\left[ {\matrix{ 1 & 0 & 0 \cr 1 & 1 & 0 \cr 1 & 1 & 1 \cr } } \right...
JEE Main 2018 (Online) 16th April Morning Slot
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
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
Suppose A is any 3$$ \times $$ 3 nonsingular matrx and ( A $$-$$ 3I) (A $$-$$ 5I) = O where I = I3 and O = O3. If $$\alp...
JEE Main 2018 (Online) 15th April Evening Slot
Let S be the set of all real values of k for which the systemof linear equations x + y + z = 2 2x + y $$-$$ z = 3 3x + 2...
JEE Main 2018 (Online) 15th April Morning Slot
Let $$A$$ be a matrix such that $$A.\left[ {\matrix{ 1 & 2 \cr 0 & 3 \cr } } \right]$$ is a scalar m...
JEE Main 2018 (Online) 15th April Morning Slot
If the system of linear equations x + ky + 3z = 0 3x + ky - 2z = 0 2x + 4y - 3z = 0 has a non-zero solution (x, y, z), t...
JEE Main 2018 (Offline)
If $$\left| {\matrix{ {x - 4} & {2x} & {2x} \cr {2x} & {x - 4} & {2x} \cr {2x} & {2x} &...
JEE Main 2018 (Offline)
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
If $$S = \left\{ {x \in \left[ {0,2\pi } \right]:\left| {\matrix{ 0 & {\cos x} & { - \sin x} \cr {\sin ...
JEE Main 2017 (Online) 8th April Morning Slot
The number of real values of $$\lambda $$ for which the system of linear equations 2x + 4y $$-$$ $$\lambda $$z = 0 4x + ...
JEE Main 2017 (Online) 8th April Morning Slot
Let A be any 3 $$ \times $$ 3 invertible matrix. Then which one of the following is not always true ?
JEE Main 2017 (Online) 8th April Morning Slot
If S is the set of distinct values of 'b' for which the following system of linear equations x + y + z = 1 x + ay + z =...
JEE Main 2017 (Offline)
If $$A = \left[ {\matrix{ 2 & { - 3} \cr { - 4} & 1 \cr } } \right]$$, then adj(3A2 + 12A) is equal ...
JEE Main 2017 (Offline)
If    A = $$\left[ {\matrix{ { - 4} & { - 1} \cr 3 & 1 \cr } } \right]$$, then the determ...
JEE Main 2016 (Online) 10th April Morning Slot
Let A be a 3 $$ \times $$ 3 matrix such that A2 $$-$$ 5A + 7I = 0 Statement - I :   A$$-$$1 = $${1 \over 7}$$...
JEE Main 2016 (Online) 10th April Morning Slot
The number of distinct real roots of the equation, $$\left| {\matrix{ {\cos x} & {\sin x} & {\sin x} \cr ...
JEE Main 2016 (Online) 9th April Morning Slot
If P = $$\left[ {\matrix{ {{{\sqrt 3 } \over 2}} & {{1 \over 2}} \cr { - {1 \over 2}} & {{{\sqrt 3 } \ov...
JEE Main 2016 (Online) 9th April Morning Slot
The system of linear equations $$\matrix{ {x + \lambda y - z = 0} \cr {\lambda x - y - z = 0} \cr {x + y -...
JEE Main 2016 (Offline)
The system of linear equations $$\matrix{ {x + \lambda y - z = 0} \cr {\lambda x - y - z = 0} \cr {x + y -...
JEE Main 2016 (Offline)
If $$A = \left[ {\matrix{ {5a} & { - b} \cr 3 & 2 \cr } } \right]$$ and $$A$$ adj $$A=A$$ $${A^T},$...
JEE Main 2016 (Offline)
If $$A = \left[ {\matrix{ 1 & 2 & 2 \cr 2 & 1 & { - 2} \cr a & 2 & b \cr } } \r...
JEE Main 2015 (Offline)
The set of all values of $$\lambda $$ for which the system of linear equations: $$$\matrix{ {2{x_1} - 2{x_2} + {x_3} ...
JEE Main 2015 (Offline)
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)
If $$\alpha ,\beta \ne 0,$$ and $$f\left( n \right) = {\alpha ^n} + {\beta ^n}$$ and $$$\left| {\matrix{ 3 & {1...
JEE Main 2014 (Offline)
If $$P = \left[ {\matrix{ 1 & \alpha & 3 \cr 1 & 3 & 3 \cr 2 & 4 & 4 \cr } } \...
JEE Main 2013 (Offline)
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
Let $$A = \left( {\matrix{ 1 & 0 & 0 \cr 2 & 1 & 0 \cr 3 & 2 & 1 \cr } } \right...
AIEEE 2012
The number of values of $$k$$ for which the linear equations $$4x + ky + 2z = 0,kx + 4y + z = 0$$ and $$2x+2y+z=0$$ pos...
AIEEE 2011
Let $$A$$ and $$B$$ be two symmetric matrices of order $$3$$. Statement - 1: $$A(BA)$$ and $$(AB)$$$$A$$ are symmetric...
AIEEE 2011
The number of $$3 \times 3$$ non-singular matrices, with four entries as $$1$$ and all other entries as $$0$$, is
AIEEE 2010
Consider the system of linear equations; $$$\matrix{ {{x_1} + 2{x_2} + {x_3} = 3} \cr {2{x_1} + 3{x_2} + {x_3} ...
AIEEE 2010
Let $$A$$ be a $$\,2 \times 2$$ matrix with non-zero entries and let $${A^2} = I,$$ where $$I$$ is $$2 \times 2$$ ident...
AIEEE 2010
Let $$a, b, c$$ be such that $$b\left( {a + c} \right) \ne 0$$ if $$\left| {\matrix{ a & {a + 1} & {a - 1} \...
AIEEE 2009
Let $$A$$ be a $$\,2 \times 2$$ matrix Statement - 1 : $$adj\left( {adj\,A} \right) = A$$ Statement - 2 :$$\left| {adj\,...
AIEEE 2009
Let $$A$$ be a square matrix all of whose entries are integers. Then which one of the following is true?
AIEEE 2008
Let $$a, b, c$$ be any real numbers. Suppose that there are real numbers $$x, y, z$$ not all zero such that $$x=cy+bz,$$...
AIEEE 2008
Let $$A$$ be $$a\,2 \times 2$$ matrix with real entries. Let $$I$$ be the $$2 \times 2$$ identity matrix. Denote by tr$$...
AIEEE 2008
Let $$A = \left| {\matrix{ 5 & {5\alpha } & \alpha \cr 0 & \alpha & {5\alpha } \cr 0 &amp...
AIEEE 2007
If$$D = \left| {\matrix{ 1 & 1 & 1 \cr 1 & {1 + x} & 1 \cr 1 & 1 & {1 + y} \cr ...
AIEEE 2007
Let $$A = \left( {\matrix{ 1 & 2 \cr 3 & 4 \cr } } \right)$$ and $$B = \left( {\matrix{ a & 0...
AIEEE 2006
If $$A$$ and $$B$$ are square matrices of size $$n\, \times \,n$$ such that $${A^2} - {B^2} = \left( {A - B} \right)\le...
AIEEE 2006
If $${a^2} + {b^2} + {c^2} = - 2$$ and f$$\left( x \right) = \left| {\matrix{ {1 + {a^2}x} & {\left( {1 + {b^2}...
AIEEE 2005
If $${a_1},{a_2},{a_3},........,{a_n},.....$$ are in G.P., then the determinant $$$\Delta \left| {\matrix{ {\log {a_...
AIEEE 2005
The system of equations $$\matrix{ {\alpha \,x + y + z = \alpha - 1} \cr {x + \alpha y + z = \alpha - 1} \cr...
AIEEE 2005
If $${A^2} - A + 1 = 0$$, then the inverse of $$A$$ is
AIEEE 2005
If $${a_1},{a_2},{a_3},.........,{a_n},......$$ are in G.P., then the value of the determinant $$\left| {\matrix{ {\...
AIEEE 2004
Let $$A = \left( {\matrix{ 0 & 0 & { - 1} \cr 0 & { - 1} & 0 \cr { - 1} & 0 & 0 \c...
AIEEE 2004
Let $$A = \left( {\matrix{ 1 & { - 1} & 1 \cr 2 & 1 & { - 3} \cr 1 & 1 & 1 \cr ...
AIEEE 2004
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
If $$1,$$ $$\omega ,{\omega ^2}$$ are the cube roots of unity, then $$\Delta = \left| {\matrix{ 1 & {{\omega ^...
AIEEE 2003
If $$A = \left[ {\matrix{ a & b \cr b & a \cr } } \right]$$ and $${A^2} = \left[ {\matrix{ \alpha...
AIEEE 2003
If $$a>0$$ and discriminant of $$\,a{x^2} + 2bx + c$$ is $$-ve$$, then $$\left| {\matrix{ a & b & {ax + b}...
AIEEE 2002

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