MCQ (Single Correct Answer)
For the system of linear equations $$\alpha x+y+z=1,x+\alpha y+z=1,x+y+\alpha z=\beta$$, which one of the following statements is NOT correct?...
If $$A = {1 \over 2}\left[ {\matrix{
1 & {\sqrt 3 } \cr
{ - \sqrt 3 } & 1 \cr
} } \right]$$, then :
Let $$S$$ denote the set of all real values of $$\lambda$$ such that the system of equations
$$\lambda x+y+z=1$$
$$x+\lambda y+z=1$$
$$x+y+\lambda z=1...
For the system of linear equations
$$x+y+z=6$$
$$\alpha x+\beta y+7 z=3$$
$$x+2 y+3 z=14$$
which of the following is NOT true ?...
Let $$A = \left( {\matrix{
1 & 0 & 0 \cr
0 & 4 & { - 1} \cr
0 & {12} & { - 3} \cr
} } \right)$$. Then the sum of the diagonal element...
For $\alpha, \beta \in \mathbb{R}$, suppose the system of linear equations
$$
\begin{aligned}
& x-y+z=5 \\
& 2 x+2 y+\alpha z=8 \\
& 3 x-y+4 z=\beta
\...
If $P$ is a $3 \times 3$ real matrix such that $P^T=a P+(a-1) I$, where $a>1$, then :
Let the system of linear equations
$$x+y+kz=2$$
$$2x+3y-z=1$$
$$3x+4y+2z=k$$
have infinitely many solutions. Then the system
$$(k+1)x+(2k-1)y=7$$
$$(2...
Let $$A=\left(\begin{array}{cc}\mathrm{m} & \mathrm{n} \\ \mathrm{p} & \mathrm{q}\end{array}\right), \mathrm{d}=|\mathrm{A}| \neq 0$$ and $$\mathrm{|A...
The set of all values of $$\mathrm{t\in \mathbb{R}}$$, for which the matrix $$\left[ {\matrix{
{{e^t}} & {{e^{ - t}}(\sin t - 2\cos t)} & {{e^{ - t...
Let $$\alpha$$ and $$\beta$$ be real numbers. Consider a 3 $$\times$$ 3 matrix A such that $$A^2=3A+\alpha I$$. If $$A^4=21A+\beta I$$, then
Consider the following system of equations
$$\alpha x+2y+z=1$$
$$2\alpha x+3y+z=1$$
$$3x+\alpha y+2z=\beta$$
for some $$\alpha,\beta\in \mathbb{R}$$. ...
Let A, B, C be 3 $$\times$$ 3 matrices such that A is symmetric and B and C are skew-symmetric. Consider the statements
(S1) A$$^{13}$$ B$$^{26}$$ $$-...
Let $$A = \left[ {\matrix{
{{1 \over {\sqrt {10} }}} & {{3 \over {\sqrt {10} }}} \cr
{{{ - 3} \over {\sqrt {10} }}} & {{1 \over {\sqrt {10} }}...
Let $$x,y,z > 1$$ and $$A = \left[ {\matrix{
1 & {{{\log }_x}y} & {{{\log }_x}z} \cr
{{{\log }_y}x} & 2 & {{{\log }_y}z} \cr
{{{\log }_z}...
Let S$$_1$$ and S$$_2$$ be respectively the sets of all $$a \in \mathbb{R} - \{ 0\} $$ for which the system of linear equations
$$ax + 2ay - 3az = 1$$...
Let A be a 3 $$\times$$ 3 matrix such that $$\mathrm{|adj(adj(adj~A))|=12^4}$$. Then $$\mathrm{|A^{-1}~adj~A|}$$ is equal to
If the system of equations
$$x+2y+3z=3$$
$$4x+3y-4z=4$$
$$8x+4y-\lambda z=9+\mu$$
has infinitely many solutions, then the ordered pair ($$\lambda,\mu$...
The number of square matrices of order 5 with entries from the set {0, 1}, such that the sum of all the elements in each row is 1 and the sum of all t...
If A and B are two non-zero n $$\times$$ n matrices such that $$\mathrm{A^2+B=A^2B}$$, then
Let $$\alpha$$ be a root of the equation $$(a - c){x^2} + (b - a)x + (c - b) = 0$$ where a, b, c are distinct real numbers such that the matrix $$\lef...
Which of the following matrices can NOT be obtained from the matrix $$\left[\begin{array}{cc}-1 & 2 \\ 1 & -1\end{array}\right]$$ by a single elementa...
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 infinitely many solutions, the...
Let A and B be two $$3 \times 3$$ non-zero real matrices such that AB is a zero matrix. Then
Let $$\mathrm{A}$$ and $$\mathrm{B}$$ be any two $$3 \times 3$$ symmetric and skew symmetric matrices respectively. Then which of the following is NOT...
Let the matrix $$A=\left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{array}\right]$$ and the matrix $$B_{0}=A^{49}+2 A^{98}$$. If $$B_{n...
Let $$A=\left(\begin{array}{rr}4 & -2 \\ \alpha & \beta\end{array}\right)$$.
If $$\mathrm{A}^{2}+\gamma \mathrm{A}+18 \mathrm{I}=\mathrm{O}$$, then $$...
Let $$A=\left(\begin{array}{cc}1 & 2 \\ -2 & -5\end{array}\right)$$. Let $$\alpha, \beta \in \mathbb{R}$$ be such that $$\alpha A^{2}+\beta A=2 I$$. T...
$$
\text { Let } A=\left[\begin{array}{l}
1 \\
1 \\
1
\end{array}\right] \text { and } B=\left[\begin{array}{ccc}
9^{2} & -10^{2} & 11^{2} \\
12^{2} &...
If the system of linear equations.
$$8x + y + 4z = - 2$$
$$x + y + z = 0$$
$$\lambda x - 3y = \mu $$
has infinitely many solutions, then the distance...
Let A be a 2 $$\times$$ 2 matrix with det (A) = $$-$$ 1 and det ((A + I) (Adj (A) + I)) = 4. Then the sum of the diagonal elements of A can be :...
The number of real values of $$\lambda$$, such that the system of linear equations
2x $$-$$ 3y + 5z = 9
x + 3y $$-$$ z = $$-$$18
3x $$-$$ y + ($$\lamb...
The number of $$\theta \in(0,4 \pi)$$ for which the system of linear equations
$$
\begin{aligned}
&3(\sin 3 \theta) x-y+z=2 \\
&3(\cos 2 \theta) x+4 ...
Let $$A = \left[ {\matrix{
1 & { - 2} & \alpha \cr
\alpha & 2 & { - 1} \cr
} } \right]$$ and $$B = \left[ {\matrix{
2 & \alpha \cr ...
Let A and B be two square matrices of order 2. If $$det\,(A) = 2$$, $$det\,(B) = 3$$ and $$\det \left( {(\det \,5(det\,A)B){A^2}} \right) = {2^a}{3^b}...
Let $$A = \left( {\matrix{
2 & { - 1} \cr
0 & 2 \cr
} } \right)$$. If $$B = I - {}^5{C_1}(adj\,A) + {}^5{C_2}{(adj\,A)^2} - \,\,.....\,\, ...
If the system of linear equations
2x + y $$-$$ z = 7
x $$-$$ 3y + 2z = 1
x + 4y + $$\delta$$z = k, where $$\delta$$, k $$\in$$ R has infinitely many s...
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, the matrix A2 + A3 + ...... +...
If the system of linear equations
$$2x + 3y - z = - 2$$
$$x + y + z = 4$$
$$x - y + |\lambda |z = 4\lambda - 4$$
where, $$\lambda$$ $$\in$$ R, has n...
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 _____________.
Let $$f(x) = \left| {\matrix{
a & { - 1} & 0 \cr
{ax} & a & { - 1} \cr
{a{x^2}} & {ax} & a \cr
} } \right|,\,a \in R$$. Then the sum ...
Let A and B be two 3 $$\times$$ 3 matrices such that $$AB = I$$ and $$|A| = {1 \over 8}$$. Then $$|adj\,(B\,adj(2A))|$$ is equal to
Let the system of linear equations $$x + 2y + z = 2$$, $$\alpha x + 3y - z = \alpha $$, $$ - \alpha x + y + 2z = - \alpha $$ be inconsistent. Then $$...
If the system of equations
$$\alpha$$x + y + z = 5, x + 2y + 3z = 4, x + 3y + 5z = $$\beta$$
has infinitely many solutions, then the ordered pair ($$\...
Let A be a 3 $$\times$$ 3 invertible matrix. If |adj (24A)| = |adj (3 adj (2A))|, then |A|2 is equal to :
The ordered pair (a, b), for which the system of linear equations
3x $$-$$ 2y + z = b
5x $$-$$ 8y + 9z = 3
2x + y + az = $$-$$1
has no solution, is :...
The system of equations
$$ - kx + 3y - 14z = 25$$
$$ - 15x + 4y - kz = 3$$
$$ - 4x + y + 3z = 4$$
is consistent for all k in the set...
Let A be a 3 $$\times$$ 3 real matrix such that
$$A\left( {\matrix{
1 \cr
1 \cr
0 \cr
} } \right) = \left( {\matrix{
1 \cr
1 ...
Let $$A = \left[ {\matrix{
0 & { - 2} \cr
2 & 0 \cr
} } \right]$$. If M and N are two matrices given by $$M = \sum\limits_{k = 1}^{10} {{A...
Let the system of linear equations
x + y + $$\alpha$$z = 2
3x + y + z = 4
x + 2z = 1
have a unique solution (x$$^ * $$, y$$^ * $$, z$$^ * $$). If ($$\...
The number of values of $$\alpha$$ for which the system of equations :
x + y + z = $$\alpha$$
$$\alpha$$x + 2$$\alpha$$y + 3z = $$-$$1
x + 3$$\alpha$$...
Let S = {$$\sqrt{n}$$ : 1 $$\le$$ n $$\le$$ 50 and n is odd}.
Let a $$\in$$ S and $$A = \left[ {\matrix{
1 & 0 & a \cr
{ - 1} & 1 & 0 \cr
...
Consider the system of linear equations$$-$$x + y + 2z = 03x $$-$$ ay + 5z = 12x $$-$$ 2y $$-$$ az = 7Let S1 be the set of all a$$\in$$R for which the...
Let $${J_{n,m}} = \int\limits_0^{{1 \over 2}} {{{{x^n}} \over {{x^m} - 1}}dx} $$, $$\forall$$ n > m and n, m $$\in$$ N. Consider a matrix $$A = {[{...
If $$\alpha$$ + $$\beta$$ + $$\gamma$$ = 2$$\pi$$, then the system of equations x + (cos $$\gamma$$)y + (cos $$\beta$$)z = 0(cos $$\gamma$$)x + y + (c...
If the following system of linear equations2x + y + z = 5x $$-$$ y + z = 3x + y + az = bhas no solution, then :
If $${a_r} = \cos {{2r\pi } \over 9} + i\sin {{2r\pi } \over 9}$$, r = 1, 2, 3, ....., i = $$\sqrt { - 1} $$, then the determinant $$\left| {\matrix{
...
Let $$A = \left( {\matrix{
{[x + 1]} & {[x + 2]} & {[x + 3]} \cr
{[x]} & {[x + 3]} & {[x + 3]} \cr
{[x]} & {[x + 2]} ...
Let A(a, 0), B(b, 2b + 1) and C(0, b), b $$\ne$$ 0, |b| $$\ne$$ 1, be points such that the area of triangle ABC is 1 sq. unit, then the sum of all pos...
Let [$$\lambda$$] be the greatest integer less than or equal to $$\lambda$$. The set of all values of $$\lambda$$ for which the system of linear equat...
If the matrix $$A = \left( {\matrix{
0 & 2 \cr
K & { - 1} \cr
} } \right)$$ satisfies $$A({A^3} + 3I) = 2I$$, then the value of K ...
Let $$A = \left( {\matrix{
1 & 0 & 0 \cr
0 & 1 & 1 \cr
1 & 0 & 0 \cr
} } \right)$$. Then A2025 $$-$$ A2020 is...
Let $$\theta \in \left( {0,{\pi \over 2}} \right)$$. If the system of linear equations$$(1 + {\cos ^2}\theta )x + {\sin ^2}\theta y + 4\sin 3\,\thet...
If $$A = \left( {\matrix{
{{1 \over {\sqrt 5 }}} & {{2 \over {\sqrt 5 }}} \cr
{{{ - 2} \over {\sqrt 5 }}} & {{1 \over {\sqrt 5 }}} \c...
Let A and B be two 3 $$\times$$ 3 real matrices such that (A2 $$-$$ B2) is invertible matrix. If A5 = B5 and A3B2 = A2B3, then the value of the determ...
Let $$A = \left[ {\matrix{
1 & 2 \cr
{ - 1} & 4 \cr
} } \right]$$. If A$$-$$1 = $$\alpha$$I + $$\beta$$A, $$\alpha$$, $$\beta$$ $$...
The number of distinct real roots of $$\left| {\matrix{
{\sin x} & {\cos x} & {\cos x} \cr
{\cos x} & {\sin x} & {\cos x} \cr...
If $$P = \left[ {\matrix{
1 & 0 \cr
{{1 \over 2}} & 1 \cr
} } \right]$$, then P50 is :
The values of a and b, for which the system of equations 2x + 3y + 6z = 8x + 2y + az = 53x + 5y + 9z = bhas no solution, are :...
The values of $$\lambda$$ and $$\mu$$ such that the system of equations $$x + y + z = 6$$, $$3x + 5y + 5z = 26$$, $$x + 2y + \lambda z = \mu $$ has no...
Let A = [aij] be a real matrix of order 3 $$\times$$ 3, such that ai1 + ai2 + ai3 = 1, for i = 1, 2, 3. Then, the sum of all the entries of the matrix...
The value of k $$\in$$R, for which the following system of linear equations3x $$-$$ y + 4z = 3,x + 2y $$-$$ 3z = $$-$$26x + 5y + kz = $$-$$3,has infin...
Let $$A = \left[ {\matrix{
2 & 3 \cr
a & 0 \cr
} } \right]$$, a$$\in$$R be written as P + Q where P is a symmetric matrix and Q is...
Let the system of linear equations 4x + $$\lambda$$y + 2z = 02x $$-$$ y + z = 0$$\mu$$x + 2y + 3z = 0, $$\lambda$$, $$\mu$$$$\in$$R.has a non-trivial ...
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 P such that PAP$$-$$1 = B". Th...
The solutions of the equation $$\left| {\matrix{
{1 + {{\sin }^2}x} & {{{\sin }^2}x} & {{{\sin }^2}x} \cr
{{{\cos }^2}x} & {1 + {{...
Let $$A + 2B = \left[ {\matrix{
1 & 2 & 0 \cr
6 & { - 3} & 3 \cr
{ - 5} & 3 & 1 \cr
} } \right]$$ and $$2A - ...
If x, y, z are in arithmetic progression with common difference d, x $$\ne$$ 3d, and the determinant of the matrix $$\left[ {\matrix{
3 & {4\sq...
The system of equations kx + y + z = 1, x + ky + z = k and x + y + zk = k2 has no solution if k is equal to :
If $$A = \left( {\matrix{
0 & {\sin \alpha } \cr
{\sin \alpha } & 0 \cr
} } \right)$$ and $$\det \left( {{A^2} - {1 \over 2}I} \ri...
Let $$A = \left[ {\matrix{
i & { - i} \cr
{ - i} & i \cr
} } \right],i = \sqrt { - 1} $$. Then, the system of linear equations $${...
Consider the following system of equations :x + 2y $$-$$ 3z = a2x + 6y $$-$$ 11z = bx $$-$$ 2y + 7z = c,where a, b and c are real constants. Then the ...
Let A be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of A2 is 1, then the possible number of such matrices...
The value of $$\left| {\matrix{
{(a + 1)(a + 2)} & {a + 2} & 1 \cr
{(a + 2)(a + 3)} & {a + 3} & 1 \cr
{(a + 3)(a + 4)} &a...
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 performing the operation R2 $$ \to $$ 2...
If for the matrix, $$A = \left[ {\matrix{
1 & { - \alpha } \cr
\alpha & \beta \cr
} } \right]$$, $$A{A^T} = {I_2}$$, then the va...
The following system of linear equations2x + 3y + 2z = 93x + 2y + 2z = 9x $$-$$ y + 4z = 8
Let A and B be 3 $$\times$$ 3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then the system of linear equations (A2B2 ...
For the system of linear equations:$$x - 2y = 1,x - y + kz = - 2,ky + 4z = 6,k \in R$$,consider the following statements :(A) The system has unique s...
The system of linear equations
3x - 2y - kz = 10
2x - 4y - 2z = 6
x+2y - z = 5m
is inconsistent if :
Let $$\theta = {\pi \over 5}$$ and $$A = \left[ {\matrix{
{\cos \theta } & {\sin \theta } \cr
{ - \sin \theta } & {\cos \theta } \c...
Let m and M be respectively the minimum and maximum values of
$$\left| {\matrix{
{{{\cos }^2}x} & {1 + {{\sin }^2}x} & {\sin 2x} \cr
{...
The values of $$\lambda $$ and $$\mu $$ for which the system of linear equations
x + y + z = 2
x + 2y + 3z = 5
x + 3y + $$\lambda $$z = $$\mu $$
has i...
If a + x = b + y = c + z + 1, where a, b, c, x, y, z
are non-zero distinct real numbers, then
$$\left| {\matrix{
x & {a + y} & {x + a} \cr...
If the system of linear equations
x + y + 3z = 0
x + 3y + k2z = 0
3x + y + 3z = 0
has a non-zero solution (x, y, z) for some k $$ \in $$ R,
then x + $...
Let $$\lambda \in $$ R . The system of linear equations
2x1
- 4x2 + $$\lambda $$x3 = 1
x1 - 6x2 + x3 = 2
$$\lambda $$x1 - 10x2 + 4x3 = 3
is inconsist...
Suppose the vectors x1, x2 and x3 are the solutions of the system of linear equations, Ax = b when the vector b on the right side is equal to b1, b2 a...
If the system of equations
x+y+z=2
2x+4y–z=6
3x+2y+$$\lambda $$z=$$\mu $$
has infinitely many solutions, then
If $$A = \left[ {\matrix{
{\cos \theta } & {i\sin \theta } \cr
{i\sin \theta } & {\cos \theta } \cr
} } \right]$$, $$\left( {\thet...
Let A be a 3 $$ \times $$ 3 matrix such that
adj A = $$\left[ {\matrix{
2 & { - 1} & 1 \cr
{ - 1} & 0 & 2 \cr
1 & { -...
If $$\Delta $$ = $$\left| {\matrix{
{x - 2} & {2x - 3} & {3x - 4} \cr
{2x - 3} & {3x - 4} & {4x - 5} \cr
{3x - 5} & {...
Let a, b, c $$ \in $$ R be all non-zero and satisfy
a3 + b3 + c3 = 2. If the matrix
A = $$\left( {\matrix{
a & b & c \cr
b & c &am...
Let A = {X = (x, y, z)T: PX = 0 and
x2 + y2 + z2 = 1} where
$$P = \left[ {\matrix{
1 & 2 & 1 \cr
{ - 2} & 3 & { - 4} \cr
...
Let S be the set of all $$\lambda $$ $$ \in $$ R for which the system
of linear equations
2x – y + 2z = 2
x – 2y +
$$\lambda $$z = –4
x +
$$\lambda $$...
Let A be a 2 $$ \times $$ 2 real matrix with entries from
{0, 1} and |A|
$$ \ne $$ 0. Consider the following two
statements :
(P) If A $$ \ne $$ I2
, ...
The following system of linear equations
7x + 6y – 2z = 0
3x + 4y + 2z = 0
x – 2y – 6z = 0, has
If the matrices A = $$\left[ {\matrix{
1 & 1 & 2 \cr
1 & 3 & 4 \cr
1 & { - 1} & 3 \cr
} } \right]$$,
B = adjA...
If for some $$\alpha $$ and $$\beta $$ in R, the intersection of the
following three places
x + 4y – 2z = 1
x + 7y – 5z = b
x + 5y + az = 5
is a line ...
The system of linear equations
$$\lambda $$x + 2y + 2z = 5
2$$\lambda $$x + 3y + 5z = 8
4x + $$\lambda $$y + 6z = 10 has
If $$A = \left( {\matrix{
2 & 2 \cr
9 & 4 \cr
} } \right)$$ and $$I = \left( {\matrix{
1 & 0 \cr
0 & 1 \cr
} }...
For which of the following ordered pairs ($$\mu $$, $$\delta $$),
the system of linear equations
x + 2y + 3z = 1
3x + 4y + 5z = $$\mu $$
4x + 4y + 4z ...
Let A = [aij] and B = [bij] be two 3 × 3 real matrices such that bij = (3)(i+j-2)aji, where i, j = 1, 2, 3.
If the determinant of B is 81, then the de...
Let $$\alpha $$ be a root of the equation x2 + x + 1 = 0 and the matrix A = $${1 \over {\sqrt 3 }}\left[ {\matrix{
1 & 1 & 1 \cr
1 &am...
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 non-zero distinct; has a non-ze...
A value of $$\theta \in \left( {0,{\pi \over 3}} \right)$$, for which
$$\left| {\matrix{
{1 + {{\cos }^2}\theta } & {{{\sin }^2}\theta } &...
If A is a symmetric matrix and B is a skew-symmetric matrix such that A + B = $$\left[ {\matrix{
2 & 3 \cr
5 & { - 1} \cr
} } \rig...
If $$B = \left[ {\matrix{
5 & {2\alpha } & 1 \cr
0 & 2 & 1 \cr
\alpha & 3 & { - 1} \cr
} } \right]$$ is the ...
Let $$\lambda $$ be a real number for which the system of linear equations x + y + z = 6, 4x + $$\lambda $$y – $$\lambda $$z = $$\lambda $$ – 2,
3x + ...
The sum of the real roots of the equation
$$\left| {\matrix{
x & { - 6} & { - 1} \cr
2 & { - 3x} & {x - 3} \cr
{ - 3} &am...
If $${\Delta _1} = \left| {\matrix{
x & {\sin \theta } & {\cos \theta } \cr
{ - \sin \theta } & { - x} & 1 \cr
{\cos \the...
If the system of linear equations
x + y + z = 5
x + 2y + 2z = 6
x + 3y + $$\lambda $$z = $$\mu $$, ($$\lambda $$, $$\mu $$ $$ \in $$ R), has infinitel...
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 $${x \over y} + {y \over z} +...
The total number of matrices
$$A = \left( {\matrix{
0 & {2y} & 1 \cr
{2x} & y & { - 1} \cr
{2x} & { - y} & 1 \cr...
Let $$\alpha $$ and $$\beta $$ be the roots of the equation
x2 + x + 1 = 0. Then for y $$ \ne $$ 0 in R,
$$$\left| {\matrix{
{y + 1} & \alpha ...
If $$\left[ {\matrix{
1 & 1 \cr
0 & 1 \cr
} } \right]\left[ {\matrix{
1 & 2 \cr
0 & 1 \cr
} } \right]$$$$\left...
Let the number 2,b,c be in an A.P. and
A = $$\left[ {\matrix{
1 & 1 & 1 \cr
2 & b & c \cr
4 & {{b^2}} & {{c^2}} ...
Let $$A = \left( {\matrix{
{\cos \alpha } & { - \sin \alpha } \cr
{\sin \alpha } & {\cos \alpha } \cr
} } \right)$$, ($$\alpha $$ ...
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 = 0
has a non-trivial solutio...
If A = $$\left[ {\matrix{
1 & {\sin \theta } & 1 \cr
{ - \sin \theta } & 1 & {\sin \theta } \cr
{ - 1} & ...
The set of all values of $$\lambda $$ for which the system of linear equations
x – 2y – 2z = $$\lambda $$x
x + 2y + z = $$\lambda $$y
– x – y = $$\lam...
An ordered pair ($$\alpha $$, $$\beta $$) for which the system of linear equations
(1 + $$\alpha $$) x + $$\beta $$y + z = 2
$$\alpha $$x + (1 + $$\...
Let P = $$\left[ {\matrix{
1 & 0 & 0 \cr
3 & 1 & 0 \cr
9 & 3 & 1 \cr
} } \right]$$ and Q = [qij] be two 3 $$ ...
If $$\left| {\matrix{
{a - b - c} & {2a} & {2a} \cr
{2b} & {b - c - a} & {2b} \cr
{2c} & {2c} & {c - a...
Let A and B be two invertible matrices of order 3 $$ \times $$ 3. If det(ABAT) = 8 and det(AB–1) = 8,
then det (BA–1 BT) is equal to :
...
If the system of linear equations
2x + 2y + 3z = a
3x – y + 5z = b
x – 3y + 2z = c
where a, b, c are non zero real numbers, has more one solution, the...
Let A = $$\left( {\matrix{
0 & {2q} & r \cr
p & q & { - r} \cr
p & { - q} & r \cr
} } \right).$$ ...
Let A = $$\left[ {\matrix{
2 & b & 1 \cr
b & {{b^2} + 1} & b \cr
1 & b & 2 \cr
} } \right]$$ where b > 0...
The number of values of $$\theta $$ $$ \in $$ (0, $$\pi $$) for which the system of linear equations
x + 3y + 7z = 0
$$-$$ x + 4y + 7z = 0
(sin3$$\the...
If the system of equations
x + y + z = 5
x + 2y + 3z = 9
x + 3y + az = $$\beta $$
has infinitely many solutions, then $$\beta $$ $$-$$ $$\alpha $$ equ...
Let d $$ \in $$ R, and
$$A = \left[ {\matrix{
{ - 2} & {4 + d} & {\left( {\sin \theta } \right) - 2} \cr
1 & {\le...
If the system of linear equations
x $$-$$ 4y + 7z = g
3y $$-$$ 5z = h
$$-$$2x + 5y $$-$$ 9z = k
is consiste...
If $$A = \left[ {\matrix{
{{e^t}} & {{e^{ - t}}\cos t} & {{e^{ - t}}\sin t} \cr
{{e^t}} & { - {e^{ - t}}\cos t - ...
If $$A = \left[ {\matrix{
{\cos \theta } & { - \sin \theta } \cr
{\sin \theta } & {\cos \theta } \cr
} } \right]$$, then the matri...
The system of linear equations
x + y + z = 2
2x + 3y + 2z = 5
2x + 3y + (a2 – 1) z = a + 1 then
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 solution, is :
Let A = $$\left[ {\matrix{
1 & 0 & 0 \cr
1 & 1 & 0 \cr
1 & 1 & 1 \cr
} } \right]$$ and B = A20. Then the sum ...
If $$\left| {\matrix{
{x - 4} & {2x} & {2x} \cr
{2x} & {x - 4} & {2x} \cr
{2x} & {2x} & {x - 4} \cr
} } \righ...
If the system of linear equations
x + ky + 3z = 0
3x + ky - 2z = 0
2x + 4y - 3z = 0
has a non-zero solution (x, y, z), then $${{xz} \over {{y^2}}}$$ i...
Suppose A is any 3$$ \times $$ 3 nonsingular matrx and ( A $$-$$ 3I) (A $$-$$ 5I) = O where I = I3 and O = O3. If $$\alpha $$A + $$\beta $$A-1 = 4I, t...
If the system of linear equations
x + ay + z = 3
x + 2y + 2z = 6
x + 5y + 3z = b
has no solution, then :
Let $$A$$ be a matrix such that $$A.\left[ {\matrix{
1 & 2 \cr
0 & 3 \cr
} } \right]$$ is a scalar matrix and |3A| = 108.
Then A2...
Let S be the set of all real values of k for which the systemof linear equations
x + y + z = 2
2x + y $$-$$ z = 3
3x + 2y + kz = 4
has a unique soluti...
For two 3 × 3 matrices A and B, let A + B = 2BT and 3A + 2B = I3, where BT is
the transpose of B and I3 is 3 × 3 identity matrix. Then :...
Let A be any 3 $$ \times $$ 3 invertible matrix. Then which one of the following is not always true ?
The number of real values of $$\lambda $$ for which the system of linear equations
2x + 4y $$-$$ $$\lambda $$z = 0
4x + $$\lambda $$y + 2z = 0
$$\lamb...
If
$$S = \left\{ {x \in \left[ {0,2\pi } \right]:\left| {\matrix{
0 & {\cos x} & { - \sin x} \cr
{\sin x} & 0 & {\cos x} \cr...
If $$A = \left[ {\matrix{
2 & { - 3} \cr
{ - 4} & 1 \cr
} } \right]$$,
then adj(3A2 + 12A) is equal to
If S is the set of distinct values of 'b' for which the following system of linear equations
x + y + z = 1
x + ay + z = 1
ax + by + z = 0
has no solu...
Let A be a 3 $$ \times $$ 3 matrix such that A2 $$-$$ 5A + 7I = 0
Statement - I :
A$$-$$1 = $${1 \over 7}$$ (5I $$-$$ A).
Statement - II ...
If A = $$\left[ {\matrix{
{ - 4} & { - 1} \cr
3 & 1 \cr
} } \right]$$,
then the determinant of the matrix (A2016 − 2...
If P = $$\left[ {\matrix{
{{{\sqrt 3 } \over 2}} & {{1 \over 2}} \cr
{ - {1 \over 2}} & {{{\sqrt 3 } \over 2}} \cr
} } \right],A =...
The number of distinct real roots of the equation,
$$\left| {\matrix{
{\cos x} & {\sin x} & {\sin x} \cr
{\sin x} & {\cos x} &am...
If $$A = \left[ {\matrix{
{5a} & { - b} \cr
3 & 2 \cr
} } \right]$$ and $$A$$ adj $$A=A$$ $${A^T},$$ then $$5a+b$$ is equal to :
The system of linear equations
$$\matrix{
{x + \lambda y - z = 0} \cr
{\lambda x - y - z = 0} \cr
{x + y - \lambda z = 0} \cr
} $$
...
The set of all values of $$\lambda $$ for which the system of linear equations:
$$\matrix{
{2{x_1} - 2{x_2} + {x_3} = \lambda {x_1}} \cr
{2{x_...
If $$A = \left[ {\matrix{
1 & 2 & 2 \cr
2 & 1 & { - 2} \cr
a & 2 & b \cr
} } \right]$$ is a matrix satisfying...
If $$\alpha ,\beta \ne 0,$$ and $$f\left( n \right) = {\alpha ^n} + {\beta ^n}$$ and
$$$\left| {\matrix{
3 & {1 + f\left( 1 \right)} & {1...
If $$A$$ is a $$3 \times 3$$ non-singular matrix such that $$AA'=A'A$$ and
$$B = {A^{ - 1}}A',$$ then $$BB'$$ equals:
If $$P = \left[ {\matrix{
1 & \alpha & 3 \cr
1 & 3 & 3 \cr
2 & 4 & 4 \cr
} } \right]$$ is the adjoint of a $...
Let $$A = \left( {\matrix{
1 & 0 & 0 \cr
2 & 1 & 0 \cr
3 & 2 & 1 \cr
} } \right)$$. If $${u_1}$$ and $${u_2}$...
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 determinant of $$\left( {{P^2} + {Q^2}}...
Let $$A$$ and $$B$$ be two symmetric matrices of order $$3$$.
Statement - 1: $$A(BA)$$ and $$(AB)$$$$A$$ are symmetric matrices.
Statement - 2: $$A...
The number of values of $$k$$ for which the linear equations
$$4x + ky + 2z = 0,kx + 4y + z = 0$$ and $$2x+2y+z=0$$ possess a non-zero solution is ...
Let $$A$$ be a $$\,2 \times 2$$ matrix with non-zero entries and let $${A^2} = I,$$
where $$I$$ is $$2 \times 2$$ identity matrix. Define
$$Tr$$$$(A...
Consider the system of linear equations;
$$$\matrix{
{{x_1} + 2{x_2} + {x_3} = 3} \cr
{2{x_1} + 3{x_2} + {x_3} = 3} \cr
{3{x_1} + 5{x_2}...
The number of $$3 \times 3$$ non-singular matrices, with four entries as $$1$$ and all other entries as $$0$$, is
Let $$A$$ be a $$\,2 \times 2$$ matrix
Statement - 1 : $$adj\left( {adj\,A} \right) = A$$
Statement - 2 :$$\left| {adj\,A} \right| = \left| A \right|$...
Let $$a, b, c$$ be such that $$b\left( {a + c} \right) \ne 0$$ if
$$\left| {\matrix{
a & {a + 1} & {a - 1} \cr
{ - b} & {b + 1} &a...
Let $$A$$ be $$a\,2 \times 2$$ matrix with real entries. Let $$I$$ be the $$2 \times 2$$ identity matrix. Denote by tr$$(A)$$, the sum of diagonal ent...
Let $$a, b, c$$ be any real numbers. Suppose that there are real numbers $$x, y, z$$ not all zero such that $$x=cy+bz,$$ $$y=az+cx,$$ and $$z=bx+ay.$$...
Let $$A$$ be a square matrix all of whose entries are integers.
Then which one of the following is true?
If$$D = \left| {\matrix{
1 & 1 & 1 \cr
1 & {1 + x} & 1 \cr
1 & 1 & {1 + y} \cr
} } \right|$$ for $$x \ne 0,y ...
Let $$A = \left| {\matrix{
5 & {5\alpha } & \alpha \cr
0 & \alpha & {5\alpha } \cr
0 & 0 & 5 \cr
} } \right...
If $$A$$ and $$B$$ are square matrices of size $$n\, \times \,n$$ such that
$${A^2} - {B^2} = \left( {A - B} \right)\left( {A + B} \right),$$ then wh...
Let $$A = \left( {\matrix{
1 & 2 \cr
3 & 4 \cr
} } \right)$$ and $$B = \left( {\matrix{
a & 0 \cr
0 & b \cr
} ...
If $${A^2} - A + 1 = 0$$, then the inverse of $$A$$ is
The system of equations
$$\matrix{
{\alpha \,x + y + z = \alpha - 1} \cr
{x + \alpha y + z = \alpha - 1} \cr
{x + y + \alpha \,z = \al...
If $${a_1},{a_2},{a_3},........,{a_n},.....$$ are in G.P., then the determinant
$$$\Delta \left| {\matrix{
{\log {a_n}} & {\log {a_{n + 1}}} &...
If $${a^2} + {b^2} + {c^2} = - 2$$ and
f$$\left( x \right) = \left| {\matrix{
{1 + {a^2}x} & {\left( {1 + {b^2}} \right)x} & {\left( {1 +...
Let $$A = \left( {\matrix{
1 & { - 1} & 1 \cr
2 & 1 & { - 3} \cr
1 & 1 & 1 \cr
} } \right).$$ and $$10$$ $$B...
Let $$A = \left( {\matrix{
0 & 0 & { - 1} \cr
0 & { - 1} & 0 \cr
{ - 1} & 0 & 0 \cr
} } \right)$$. The only c...
If $${a_1},{a_2},{a_3},.........,{a_n},......$$ are in G.P., then the value of the determinant
$$\left| {\matrix{
{\log {a_n}} & {\log {a_{n +...
If $$A = \left[ {\matrix{
a & b \cr
b & a \cr
} } \right]$$ and $${A^2} = \left[ {\matrix{
\alpha & \beta \cr
\beta ...
If $$1,$$ $$\omega ,{\omega ^2}$$ are the cube roots of unity, then
$$\Delta = \left| {\matrix{
1 & {{\omega ^n}} & {{\omega ^{2n}}} \c...
If the system of linear equations
$$x + 2ay + az = 0;$$ $$x + 3by + bz = 0;\,\,x + 4cy + cz = 0;$$
has a non - zero solution, then $$a, b, c$$....
If $$a>0$$ and discriminant of $$\,a{x^2} + 2bx + c$$ is $$-ve$$, then
$$\left| {\matrix{
a & b & {ax + b} \cr
b & c & {bx ...
Numerical
Let A be a $n \times n$ matrix such that $|\mathrm{A}|=2$. If the determinant of the matrix
$\operatorname{Adj}\left(2 \cdot \operatorname{Adj}\left(...
Let A be a symmetric matrix such that $$\mathrm{|A|=2}$$ and $$\left[ {\matrix{
2 & 1 \cr
3 & {{3 \over 2}} \cr
} } \right]A = \left[ {\ma...
Let $$\mathrm{A_1,A_2,A_3}$$ be the three A.P. with the same common difference d and having their first terms as $$\mathrm{A,A+1,A+2}$$, respectively....
Let $$X=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$$ and $$A=\left[\begin{array}{ccc}-1 & 2 & 3 \\ 0 & 1 & 6 \\ 0 & 0 & -1\end{array}\right]$...
The number of matrices of order $$3 \times 3$$, whose entries are either 0 or 1 and the sum of all the entries is a prime number, is __________....
Let p and p + 2 be prime numbers and let
$$
\Delta=\left|\begin{array}{ccc}
\mathrm{p} ! & (\mathrm{p}+1) ! & (\mathrm{p}+2) ! \\
(\mathrm{p}+1) ! & (...
Let $$A=\left[\begin{array}{cc}1 & -1 \\ 2 & \alpha\end{array}\right]$$ and $$B=\left[\begin{array}{cc}\beta & 1 \\ 1 & 0\end{array}\right], \alpha, \...
Consider a matrix $$A=\left[\begin{array}{ccc}\alpha & \beta & \gamma \\ \alpha^{2} & \beta^{2} & \gamma^{2} \\ \beta+\gamma & \gamma+\alpha & \alpha+...
Let $$S$$ be the set containing all $$3 \times 3$$ matrices with entries from $$\{-1,0,1\}$$. The total number of matrices $$A \in S$$ such that the s...
The number of matrices $$A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$$, where $$a, b, c, d \in\{-1,0,1,2,3, \ldots \ldots, 10\}$$, such ...
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 some $$n \in \mathbb{N}, A^{n}=...
Let $$A=\left(\begin{array}{rrr}2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0\end{array}\right)$$ and $$B=A-I$$. If $$\omega=\frac{\sqrt{3} i-1}{2}$$, then ...
Let $$M = \left[ {\matrix{
0 & { - \alpha } \cr
\alpha & 0 \cr
} } \right]$$, where $$\alpha$$ is a non-zero real number an $$N = \sum\li...
If the system of linear equations $$2x - 3y = \gamma + 5$$, $$\alpha x + 5y = \beta + 1$$, where $$\alpha$$, $$\beta$$, $$\gamma$$ $$\in$$ R has inf...
Let $$A = \left( {\matrix{
{1 + i} & 1 \cr
{ - i} & 0 \cr
} } \right)$$ where $$i = \sqrt { - 1} $$. Then, the number of elements in the s...
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 entries of A is a prime number p, ...
The positive value of the determinant of the matrix A, whose
Adj(Adj(A)) = $$\left( {\matrix{
{14} & {28} & { - 14} \cr
{ - 14} & {14} & {28} ...
Let $$X = \left[ {\matrix{
0 & 1 & 0 \cr
0 & 0 & 1 \cr
0 & 0 & 0 \cr
} } \right],\,Y = \alpha I + \beta X + \gamma {X^2}$$ and $$Z = ...
Let $$A = \left( {\matrix{
2 & { - 2} \cr
1 & { - 1} \cr
} } \right)$$ and $$B = \left( {\matrix{
{ - 1} & 2 \cr
{ - 1} & 2 \cr
...
Let A be a 3 $$\times$$ 3 matrix having entries from the set {$$-$$1, 0, 1}. The number of all such matrices A having sum of all the entries equal to ...
Let $$S = \left\{ {\left( {\matrix{
{ - 1} & a \cr
0 & b \cr
} } \right);a,b \in \{ 1,2,3,....100\} } \right\}$$ and let $${T_n} = \{ A \i...
The number of elements in the set $$\left\{ {A = \left( {\matrix{
a & b \cr
0 & d \cr
} } \right):a,b,d \in \{ - 1,0,1\} \,and\,{...
If the system of linear equations2x + y $$-$$ z = 3x $$-$$ y $$-$$ z = $$\alpha$$3x + 3y + $$\beta$$z = 3has infinitely many solution, then $$\alpha$$...
Let A be a 3 $$\times$$ 3 real matrix. If det(2Adj(2 Adj(Adj(2A)))) = 241, then the value of det(A2) equal __________.
If $$A = \left[ {\matrix{
1 & 1 & 1 \cr
0 & 1 & 1 \cr
0 & 0 & 1 \cr
} } \right]$$ and M = A + A2 + A3 + ........
For real numbers $$\alpha$$ and $$\beta$$, consider the following system of linear equations :x + y $$-$$ z = 2, x + 2y + $$\alpha$$z = 1, 2x $$-$$ y ...
Let $$f(x) = \left| {\matrix{
{{{\sin }^2}x} & { - 2 + {{\cos }^2}x} & {\cos 2x} \cr
{2 + {{\sin }^2}x} & {{{\cos }^2}x} & {\c...
Let $$M = \left\{ {A = \left( {\matrix{
a & b \cr
c & d \cr
} } \right):a,b,c,d \in \{ \pm 3, \pm 2, \pm 1,0\} } \right\}$$. Defi...
Let $$A = \left[ {\matrix{
0 & 1 & 0 \cr
1 & 0 & 0 \cr
0 & 0 & 1 \cr
} } \right]$$. Then the number of 3 $$\t...
Let $$A = \{ {a_{ij}}\} $$ be a 3 $$\times$$ 3 matrix, where $${a_{ij}} = \left\{ {\matrix{
{{{( - 1)}^{j - i}}} & {if} & {i < j,} \cr ...
Let $$A = \left( {\matrix{
1 & { - 1} & 0 \cr
0 & 1 & { - 1} \cr
0 & 0 & 1 \cr
} } \right)$$ and B = 7A20 $$-...
Let a, b, c, d in arithmetic progression with common difference $$\lambda$$. If $$\left| {\matrix{
{x + a - c} & {x + b} & {x + a} \cr
...
Let I be an identity matrix of order 2 $$\times$$ 2 and P = $$\left[ {\matrix{
2 & { - 1} \cr
5 & { - 3} \cr
} } \right]$$. Then t...
Let $$A = \left[ {\matrix{
a & b \cr
c & d \cr
} } \right]$$ and $$B = \left[ {\matrix{
\alpha \cr
\beta \cr
} } \ri...
If 1, log10(4x $$-$$ 2) and log10$$\left( {{4^x} + {{18} \over 5}} \right)$$ are in arithmetic progression for a real number x, then the value of the ...
If $$A = \left[ {\matrix{
2 & 3 \cr
0 & { - 1} \cr
} } \right]$$, then the value of det(A4) + det(A10 $$-$$ (Adj(2A))10) is equal ...
Let $$A = \left[ {\matrix{
{{a_1}} \cr
{{a_2}} \cr
} } \right]$$ and $$B = \left[ {\matrix{
{{b_1}} \cr
{{b_2}} \cr
} } \right...
Let $$P = \left[ {\matrix{
{ - 30} & {20} & {56} \cr
{90} & {140} & {112} \cr
{120} & {60} & {14} \cr
} } \ri...
The total number of 3 $$\times$$ 3 matrices A having entries from the set {0, 1, 2, 3} such that the sum of all the diagonal entries of AAT is 9, is e...
If the matrix $$A = \left[ {\matrix{
1 & 0 & 0 \cr
0 & 2 & 0 \cr
3 & 0 & { - 1} \cr
} } \right]$$ satisfies t...
If $$A = \left[ {\matrix{
0 & { - \tan \left( {{\theta \over 2}} \right)} \cr
{\tan \left( {{\theta \over 2}} \right)} & 0 \cr
}...
Let $$A = \left[ {\matrix{
x & y & z \cr
y & z & x \cr
z & x & y \cr
} } \right]$$, where x, y and z are real...
If the system of equationskx + y + 2z = 13x $$-$$ y $$-$$ 2z = 2$$-$$2x $$-$$2y $$-$$4z = 3has infinitely many solutions, then k is equal to _________...
Let P = $$\left[ {\matrix{
3 & { - 1} & { - 2} \cr
2 & 0 & \alpha \cr
3 & { - 5} & 0 \cr
} } \right]$$, wher...
Let M be any 3 $$ \times $$ 3 matrix with entries from the set {0, 1, 2}. The maximum number of such matrices, for which the sum of diagonal elements ...
The sum of distinct values of $$\lambda $$ for which the
system of equations$$\left( {\lambda - 1} \right)x + \left( {3\lambda + 1} \right)y + 2\lam...
If the system of equations
x - 2y + 3z = 9
2x + y + z = b
x - 7y + az = 24, has infinitely many solutions, then a - b is equal to............
Let S be the set of all integer solutions, (x, y, z),
of the system of equations
x – 2y + 5z = 0
–2x + 4y + z = 0
–7x + 14y + 9z = 0
such that 15 $$ \...
Let A = $$\left[ {\matrix{
x & 1 \cr
1 & 0 \cr
} } \right]$$, x $$ \in $$ R and A4 = [aij].
If
a11 = 109, then a22 is equal to ___...
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 AAT is 3, is
If the system of linear equations,
x + y + z = 6
x + 2y + 3z = 10
3x + 2y + $$\lambda $$z = $$\mu $$
has more than two solutions, then $$\mu $$ - $$\l...