Matrices and Determinants Questions

MathematicsMatrices and Determinants115 Questions Found

MathematicsAdjoint and Inverse of a MatrixJEE Advanced 2006Moderate

Comprehension Passage

Let

A = 123012001

, and

U_{1}, U_{2}

and

U_{3}

are columns of a

3 \times 3

matrix

U

. If column matrices

U_{1}, U_{2}

and

U_{3}

satisfying

A U_{1} = 100, A U_{2} = 230, A U_{3} = 231

evaluate as directed in the following questions.

Question 1:

The value $∣ U ∣$ is

(A)

(B)

-3

(C)

3/2

(D)

Answer:A

Question 2:

The sum of the elements of the matrix $U^{- 1}$ is

(A)

-1

(B)

(C)

(D)

Answer:B

Question 3:

The value of $[320] U 320$ is

(A)

(B)

5/2

(C)

(D)

3/2

Answer:A

MathematicsSolution of System of Linear Equations (Matrix Method and Cramer's Rule)JEE Advanced 2009Moderate

View in:English Hindi

Comprehension Passage

Let

A

be the set of all

3 \times 3

symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0.

Question 1:

The number of matrices in $A$ is

(A)

(B)

(C)

(D)

Answer:A

Question 2:

The number of matrices $A$ in $A$ for which the system of linear equations $A x y z = 100$ has a unique solution, is

(A)

less than 4

(B)

at least 4 but less than 7

(C)

at least 7 but less than 10

(D)

at least 10

Answer:B

Question 3:

The number of matrices $A$ in $A$ for which the system of linear equations $A x y z = 100$ is inconsistent, is

(A)

(B)

more than 2

(C)

(D)

Answer:B

MathematicsTypes of MatricesJEE Advanced 2010Moderate

View in:English Hindi

Comprehension Passage

Let

p

be an odd prime number and

T_{p}

be the following set of

2 \times 2

matrices :

T_{p} = {A = [a c b a] : a, b, c \in {0, 1, 2, \dots, p - 1}}

Question 1:

The number of $A$ in $T_{p}$ such that $A$ is either symmetric or skew-symmetric or both, and $det (A)$ divisible by $p$ is

(A)

(p - 1)^{2}

(B)

2 (p - 1)

(C)

(p - 1)^{2} + 1

(D)

2 p - 1

Answer:D

Question 2:

The number of $A$ in $T_{p}$ such that the trace of $A$ is not divisible by $p$ but $det (A)$ is divisible by $p$ is

(A)

(p - 1) (p^{2} - p + 1)

(B)

p^{3} - (p - 1)^{2}

(C)

(p - 1)^{2}

(D)

(p - 1) (p - 2)

Answer:C

Question 3:

The number of $A$ in $T_{p}$ such that $det (A)$ is not divisible by $p$ is

(A)

2 p^{2}

(B)

p^{3} - 5 p

(C)

p^{3} - 3 p

(D)

p^{3} - p^{2}

Answer:D

MathematicsSolution of System of Linear Equations (Matrix Method and Cramer's Rule)JEE Advanced 2011Moderate

View in:English Hindi

Comprehension Passage

Let

a, b

and

c

be three real numbers satisfying

[a b c] 187923777 = [000] \dots (E)

Question 1:

If the point $P (a, b, c)$ , with reference to (E), lies on the plane $2 x + y + z = 1$ , then the value of $7 a + b + c$ is

(A)

(B)

(C)

(D)

Answer:D

Question 2:

Let $ω$ be a solution of $x^{3} - 1 = 0$ with $Im (ω) > 0$ , if $a = 2$ with $b$ and $c$ satisfying (E), then the value of $\frac{3}{ω ^{a}} + \frac{1}{ω ^{b}} + \frac{3}{ω ^{c}}$ is equal to

(A)

-2

(B)

(C)

(D)

-3

Answer:A

Question 3:

Let $b = 6$ , with $a$ and $c$ satisfying (E). If $α$ and $β$ are the roots of the quadratic equation $a x^{2} + b x + c = 0$ , then $\sum_{n = 0}^{\infty} (\frac{1}{α} + \frac{1}{β})^{n}$ is

(A)

(B)

(C)

6/7

(D)

\infty

Answer:B

MathematicsSolution of System of Linear Equations (Matrix Method and Cramer's Rule)JEE Advanced 2007Moderate

View in:English Hindi

Consider the following linear equations $a x + b y + cz = 0; b x + cy + a z = 0; c x + a y + b z = 0$ Match the conditions/expressions in Column I with statements in Column II and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS.

List-I

(P)

a + b + c \neq = 0

and

a^{2} + b^{2} + c^{2} = ab + b c + c a

(Q)

a + b + c = 0

and

a^{2} + b^{2} + c^{2} \neq = ab + b c + c a

(R)

a + b + c \neq = 0

and

a^{2} + b^{2} + c^{2} \neq = ab + b c + c a

(S)

a + b + c = 0

and

a^{2} + b^{2} + c^{2} = ab + b c + c a

List-II

(1)

the equations represent planes meeting only at a single point

(2)

the equations represent the line

x = y = z

(3)

the equations represent identical planes.

(4)

the equations represent the whole of the three dimensional space.

Answer:P → 3, Q → 2, R → 1, S → 4

MathematicsAlgebraic Operations on MatricesJEE Advanced 2008Moderate

View in:English Hindi

Match the Statements/Expressions in Column I with the Statements / Expressions in Column II and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS.

List-I

(P)

The minimum value of

\frac{x ^{2} + 2 x + 4}{x + 2}

(Q)

Let A and B be

3 \times 3

matrices of real numbers, where A is symmetric, B is skew-symmetric, and

(A + B) (A - B) = (A - B) (A + B)

. If

(A B)^{t} = (- 1)^{k} A B

, where

(A B)^{t}

is the transpose of the matrix AB, then the possible values of

k

are

(R)

Let

a = lo g_{3} lo g_{3} 2

. An integer

k

satisfying

1 < 2^{- k + 3^{- a}} < 2

, must be less than

(S)

sin θ = cos ϕ

, then the possible values of

\frac{1}{π} (θ \pm ϕ - \frac{π}{2})

are

List-II

(1)

(2)

(3)

(4)

Answer:P → 3, Q → 4, R → 4, P → 3

MathematicsProperties of DeterminantsJEE Advanced 1984Moderate

View in:English Hindi

If $α$ be a repeated root of a quadratic equation $f (x) = 0$ and $A (x), B (x)$ and $C (x)$ be polynomials of degree 3, 4 and 5 respectively, then show that $A (x) A (α) A^{'} (α) B (x) B (α) B^{'} (α) C (x) C (α) C^{'} (α)$ is divisible by $f (x)$ , where prime denotes the derivatives.

MathematicsProperties of DeterminantsJEE Advanced 2001Moderate

View in:English Hindi

Let $a, b, c$ be real numbers with $a^{2} + b^{2} + c^{2} = 1$ . Show that the equation $a x - b y - c b x + a y c x + a b x + a y - a x + b y - c cy + b c x + a cy + b - a x - b y + c = 0$ represents a straight line.

MathematicsSolution of System of Linear Equations (Matrix Method and Cramer's Rule)JEE Advanced 1979Moderate

View in:English Hindi

For what value of $k$ do the following system of equations possess a non trivial (i.e., not all zero) solution over the set of rationals $Q$ ? $x + k y + 3 z = 0, 3 x + k y - 2 z = 0, 2 x + 3 y - 4 z = 0$ . For that value of $k$ , find all the solutions for the system.

Answer:

k = 33/2

; solutions are

x = b, y = - 2 b /15, z = 2 b /5

for

b \in Q

MathematicsSolution of System of Linear Equations (Matrix Method and Cramer's Rule)JEE Advanced 1980Easy

View in:English Hindi

Find the solution set of the system $x + 2 y + z = 1; 2 x - 3 y - w = 2; x \geq 0; y \geq 0; z \geq 0; w \geq 0$ .

Answer:

x = 1, y = 0, z = 0, w = 0

...12

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value ∣U∣ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of the elements of the matrix U−1 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of [3​2​0​]U​320​​ is

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of matrices in A is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of matrices A in A for which the system of linear equations A​xyz​​=​100​​ has a unique solution, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of matrices A in A for which the system of linear equations A​xyz​​=​100​​ is inconsistent, is

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of A in Tp​ such that A is either symmetric or skew-symmetric or both, and det(A) divisible by p is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of A in Tp​ such that the trace of A is not divisible by p but det(A) is divisible by p is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of A in Tp​ such that det(A) is not divisible by p is

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the point P(a,b,c), with reference to (E), lies on the plane 2x+y+z=1, then the value of 7a+b+c is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let ω be a solution of x3−1=0 with Im(ω)>0, if a=2 with b and c satisfying (E), then the value of ωa3​+ωb1​+ωc3​ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let b=6, with a and c satisfying (E). If α and β are the roots of the quadratic equation ax2+bx+c=0, then ∑n=0∞​(α1​+β1​)n is

List-I

List-II

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b,c be real numbers with a2+b2+c2=1. Show that the equation ​ax−by−cbx+aycx+a​bx+ay−ax+by−ccy+b​cx+acy+b−ax−by+c​​=0 represents a straight line.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find the solution set of the system x+2y+z=1;2x−3y−w=2;x≥0;y≥0;z≥0;w≥0.

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value ∣U∣ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of the elements of the matrix U−1 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of [3​2​0​]U​320​​ is

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of matrices in A is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of matrices A in A for which the system of linear equations A​xyz​​=​100​​ has a unique solution, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of matrices A in A for which the system of linear equations A​xyz​​=​100​​ is inconsistent, is

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of A in Tp​ such that A is either symmetric or skew-symmetric or both, and det(A) divisible by p is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of A in Tp​ such that the trace of A is not divisible by p but det(A) is divisible by p is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of A in Tp​ such that det(A) is not divisible by p is

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the point P(a,b,c), with reference to (E), lies on the plane 2x+y+z=1, then the value of 7a+b+c is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let ω be a solution of x3−1=0 with Im(ω)>0, if a=2 with b and c satisfying (E), then the value of ωa3​+ωb1​+ωc3​ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let b=6, with a and c satisfying (E). If α and β are the roots of the quadratic equation ax2+bx+c=0, then ∑n=0∞​(α1​+β1​)n is

List-I

List-II

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b,c be real numbers with a2+b2+c2=1. Show that the equation ​ax−by−cbx+aycx+a​bx+ay−ax+by−ccy+b​cx+acy+b−ax−by+c​​=0 represents a straight line.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find the solution set of the system x+2y+z=1;2x−3y−w=2;x≥0;y≥0;z≥0;w≥0.

The value $∣ U ∣$ is

The sum of the elements of the matrix $U^{- 1}$ is

The value of $[320] U 320$ is

The number of matrices in $A$ is

The number of matrices $A$ in $A$ for which the system of linear equations $A x y z = 100$ has a unique solution, is

The number of matrices $A$ in $A$ for which the system of linear equations $A x y z = 100$ is inconsistent, is

The number of $A$ in $T_{p}$ such that $A$ is either symmetric or skew-symmetric or both, and $det (A)$ divisible by $p$ is

The number of $A$ in $T_{p}$ such that the trace of $A$ is not divisible by $p$ but $det (A)$ is divisible by $p$ is

The number of $A$ in $T_{p}$ such that $det (A)$ is not divisible by $p$ is

If the point $P (a, b, c)$ , with reference to (E), lies on the plane $2 x + y + z = 1$ , then the value of $7 a + b + c$ is

Let $ω$ be a solution of $x^{3} - 1 = 0$ with $Im (ω) > 0$ , if $a = 2$ with $b$ and $c$ satisfying (E), then the value of $\frac{3}{ω ^{a}} + \frac{1}{ω ^{b}} + \frac{3}{ω ^{c}}$ is equal to

Let $b = 6$ , with $a$ and $c$ satisfying (E). If $α$ and $β$ are the roots of the quadratic equation $a x^{2} + b x + c = 0$ , then $\sum_{n = 0}^{\infty} (\frac{1}{α} + \frac{1}{β})^{n}$ is

Let $a, b, c$ be real numbers with $a^{2} + b^{2} + c^{2} = 1$ . Show that the equation $a x - b y - c b x + a y c x + a b x + a y - a x + b y - c cy + b c x + a cy + b - a x - b y + c = 0$ represents a straight line.

Find the solution set of the system $x + 2 y + z = 1; 2 x - 3 y - w = 2; x \geq 0; y \geq 0; z \geq 0; w \geq 0$ .

The value $∣ U ∣$ is

The sum of the elements of the matrix $U^{- 1}$ is

The value of $[320] U 320$ is

The number of matrices in $A$ is

The number of matrices $A$ in $A$ for which the system of linear equations $A x y z = 100$ has a unique solution, is

The number of matrices $A$ in $A$ for which the system of linear equations $A x y z = 100$ is inconsistent, is

The number of $A$ in $T_{p}$ such that $A$ is either symmetric or skew-symmetric or both, and $det (A)$ divisible by $p$ is

The number of $A$ in $T_{p}$ such that the trace of $A$ is not divisible by $p$ but $det (A)$ is divisible by $p$ is

The number of $A$ in $T_{p}$ such that $det (A)$ is not divisible by $p$ is

If the point $P (a, b, c)$ , with reference to (E), lies on the plane $2 x + y + z = 1$ , then the value of $7 a + b + c$ is

Let $ω$ be a solution of $x^{3} - 1 = 0$ with $Im (ω) > 0$ , if $a = 2$ with $b$ and $c$ satisfying (E), then the value of $\frac{3}{ω ^{a}} + \frac{1}{ω ^{b}} + \frac{3}{ω ^{c}}$ is equal to

Let $b = 6$ , with $a$ and $c$ satisfying (E). If $α$ and $β$ are the roots of the quadratic equation $a x^{2} + b x + c = 0$ , then $\sum_{n = 0}^{\infty} (\frac{1}{α} + \frac{1}{β})^{n}$ is

Let $a, b, c$ be real numbers with $a^{2} + b^{2} + c^{2} = 1$ . Show that the equation $a x - b y - c b x + a y c x + a b x + a y - a x + b y - c cy + b c x + a cy + b - a x - b y + c = 0$ represents a straight line.

Find the solution set of the system $x + 2 y + z = 1; 2 x - 3 y - w = 2; x \geq 0; y \geq 0; z \geq 0; w \geq 0$ .