11th Maths Monthly Test -1 (Matrices and Determinants)

St. Britto Hr. Sec. School - Madurai

11th Maths Monthly Test -1 (Matrices and Determinants)-Aug 2020

Time : 2.30 Hours

Part - A

Multiple Choice Question - Answer All Question

The matrix A satisfying the equation \(\begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix}\)A=\(\begin{bmatrix} 1 & 1 \\ 0 & -1 \end{bmatrix}\)is

\(\begin{bmatrix} 1 & 4 \\ -1 & 0 \end{bmatrix}\)
\(\begin{bmatrix} 1 & -4 \\ 1 & 0 \end{bmatrix}\)
\(\begin{bmatrix} 1 & 4 \\ 0 & -1 \end{bmatrix}\)
\(\begin{bmatrix} 1 & -4 \\ 1 & 1 \end{bmatrix}\)

Choose the correct statement

Matrix addition is not associative
Matrix addition is not commutative
Matrix multiplication is associative
Matrix multiplication is commutative

A root of the equation \(\begin{vmatrix} 3-x&-6 &3 \\ -6 & 3-x & 3 \\ 3 &3 &-6-x \end{vmatrix}=0 \ is\)

6
3
0
-6

If A=\(\begin{bmatrix} 1 & -1 \\ 2 &-1 \end{bmatrix}\),B=\(\begin{bmatrix} a & 1 \\ b &-1 \end{bmatrix}\) and (A+B)²=A²+B², then the values of a and b are

a = 4, b =1
a =1, b = 4
a = 0, b = 4
a = 2, b = 4

If A +I =\(\begin{bmatrix} 3& -2 \\ 4 & 1 \end{bmatrix},\) then (A+ I )(A-I ) is equal to

\(\begin{bmatrix} -5& -4 \\ 8 & -9 \end{bmatrix}\)
\(\begin{bmatrix} -5& 4 \\ -8 & 9 \end{bmatrix}\)
\(\begin{bmatrix} 5& 4 \\ 8 & 9 \end{bmatrix}\)
\(\begin{bmatrix} -5& -4 \\ -8 & -9 \end{bmatrix}\)

Assertion (A): The inverse of \(\\ \\ \\ \begin{bmatrix} 2 & -1 \\ -4 & 2 \end{bmatrix}\) does not exist.
Reason (R): The matrix is non-singular

Both A and R are true and R is the correct explanation of A
Both A and R are true and R is not a correct explantion of A
A is true but R is false
A is false but R is true

Which one of the following is not true about the matrix \(\begin{bmatrix} 1 &0 &0 \\ 0 & 0 &0 \\ 0 & 0 & 5 \end{bmatrix}?\)

a scalar matrix
a diagonal matrix
an upper triangular matrix
a lower triangular matrix

If A is a matrix 3 x 3, then \({ { (A }^{ 2 }) }^{ -1 }\)=

\(\frac { 1 }{ { A }^{ 2 } } \)
A^-2
\({ { (A }^{ -1 }) }^{ 2 }\)
I

Find the odd one out of the following:

\(\begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix}\)
\(\begin{bmatrix} 0 & \frac { -7 }{ 2 } \\ \frac { 7 }{ 2 } & 0 \end{bmatrix}\)
\(\begin{bmatrix} 0 & 3.2 \\ -3.2 & 0 \end{bmatrix}\)
\(\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\)

On using elementary row operation R₁⟶R₁-3R₂ in the following matrix equation \(\begin{pmatrix} 4 & 2 \\ 3 & 3 \end{pmatrix}=\begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}\begin{pmatrix} 2 & 0 \\ 1 & 1 \end{pmatrix}\)

\(\begin{pmatrix} -5 & -7 \\ 3 & 3 \end{pmatrix}=\begin{pmatrix} 1 & -7 \\ 0 & 3 \end{pmatrix}\begin{pmatrix} 2 & 0 \\ 1 & 1 \end{pmatrix}\)
\(\begin{pmatrix} -5 & -7 \\ 3 & 3 \end{pmatrix}=\begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}\begin{pmatrix} -1 & -3 \\ 1 & 1 \end{pmatrix}\)
\(\begin{pmatrix} -5 & -7 \\ 3 & 3 \end{pmatrix}=\begin{pmatrix} 1 & 2 \\ 1 & -7 \end{pmatrix}\begin{pmatrix} 2 & 0 \\ 1 & 1 \end{pmatrix}\)
\(\begin{pmatrix} 4 & 2 \\ -5 & -7 \end{pmatrix}=\begin{pmatrix} 1 & 2 \\ -3 & -3 \end{pmatrix}\begin{pmatrix} 2 & 0 \\ 1 & 1 \end{pmatrix}\)

Part - B

Generic 2 - Answer All Question

Prove that \(\begin{vmatrix} 1&1 &1 \\ x & y& z\\ x^2& y^2 &z^2 \end{vmatrix}\)=(x - y) (y - z) (z - x).
Evaluate :\(\begin{vmatrix} cos \theta & sin \theta \\ -sin \theta & cos \theta \end{vmatrix}\)

Using properties of determinant, show that\(\triangle =\left| \begin{matrix} { cosec }^{ 2 }\theta & -{ cot }^{ 2 }\theta & 1 \\ { cot }^{ 2 }\theta & -cose{ c }^{ 2 }\theta & -1 \\ 42 & 40 & 2 \end{matrix} \right| =0\)

Find the area of the triangle whose vertices are (- 2, - 3), (3, 2), and (- 1, - 8)

Solve the following problems by using Factor Theorem :
Show that \(\begin{vmatrix} b+c & a-c & a-b \\ b-c & c+a & b-a \\ c-b & c-a & a+b \end{vmatrix}=8abc\)

If A is a skew-symmetric matrix of odd order n, then |A| =0.

Evaluate :\(\begin{vmatrix} 2 & 4 \\ -1 & 2 \end{vmatrix}\)

Prove that \(\begin{vmatrix} a^2 & bc & ac+c^2 \\ a^2+ab & b^2 & ac \\ ab & b^2+bc & c^2 \end{vmatrix}=4a^2b^2c^2\)

If A=\(\begin{bmatrix} 4 & 2 \\ -1 & x \end{bmatrix}\) and such that (A- 2I)(A-3I)=0, find the value of x.

If the area of the triangle with vertices (- 3, 0), (3, 0) and (0, k) is 9 square units, find the values of k.

Part - C

Generic 3 - Answer All Question

Prove that \(LHS=\left| \begin{matrix} -{ a }^{ 2 } & ab & ac \\ ab & -{ b }^{ 2 } & bc \\ ac & bc & -{ c }^{ 2 } \end{matrix} \right| ={ 4a }^{ 2 }{ b }^{ 2 }{ c }^{ 2 }\)

Identify the singular and non-singular matrices:\(\begin{bmatrix} 0&a-b &k \\ b-a & 0 &5 \\ -k & -5 & 0 \end{bmatrix}\)

If A =\(\begin{bmatrix} 4 & 6 & 2 \\ 0 & 1 & 5 \\ 0 & 3 & 2 \end{bmatrix}\) and B= \(\begin{bmatrix} 0 & 1 & -1 \\ 3 & -1 & 4 \\ -1 & 2 & 1 \end{bmatrix}\)
verify(A+B)^T=A^T+B^T

If A is a 3 × 4 matrix and B is a matrix such that both A^TB and BA^Tare defined, what is the order of the matrix B?

Identify the singular and non-singular matrices:\(\begin{bmatrix} 1&2 &3 \\ 4 & 5 &6 \\ 7 & 8 & 9 \end{bmatrix}\)
If a, b, c are all positive, and are p^th, q^th and r^th terms of a G.P., show that \(\begin{vmatrix} log a & p & 1 \\ log b & q & 1 \\ log c & r & 1 \end{vmatrix}=0.\)

Using factor method show that \(\left| \begin{matrix} 1 & a & { a }^{ 2 } \\ 1 & b & { b }^{ 2 } \\ 1 & c & { c }^{ 2 } \end{matrix} \right| \) = (a - b) (b - c) (c - a).

If A = \(\begin{bmatrix} {1\over 2}&\alpha \\ 0 & {1\over2} \end{bmatrix}\) ,prove that \(\sum^n_{k=1}det(A^k)={1\over3}(1-{1\over 4^n}).\)

If AB=A and BA=B, then show that A²=A and B²=B.

If A =\(\begin{bmatrix} 4 & 6 & 2 \\ 0 & 1 & 5 \\ 0 & 3 & 2 \end{bmatrix}\) and B= \(\begin{bmatrix} 0 & 1 & -1 \\ 3 & -1 & 4 \\ -1 & 2 & 1 \end{bmatrix}\)
verify(A-B)^T=A^T-B^T

Part - D

Generic 5 - Answer All Question

If A=\(\left| \begin{matrix} 0 & 1 \\ -1 & 0 \end{matrix} \right| \) find x and y such that (xI +yA)² =A

Without expanding, evaluate the following determinants:
\(\begin{vmatrix} x+y& y+z &z+x \\ z & x & y \\ 1 & 1 &1 \end{vmatrix}\)

Prove that \(\begin{vmatrix} 1 &x^2 &x^3 \\ 1 & y^2 &y^3 \\1 &z^2 &z^3 \end{vmatrix}\) =(x-y)(y-z)(z-x)(xy+yz+zx).
Factorise \(\left| \begin{matrix} a & b & c \\ { a }^{ 2 } & { b }^{ 2 } & { c }^{ 2 } \\ bc & ca & ab \end{matrix} \right| \) .

If \(A=\left[ \begin{matrix} 1 & -1 \\ 2 & -1 \end{matrix} \right] \) and \(B=\left[ \begin{matrix} x & 1 \\ y & -1 \end{matrix} \right] \) and (A + B)²=A² + B², find X and.

Express the matrix A=\(\begin{bmatrix} 1 & 3 & 5 \\ -6 & 8 & 3 \\ -4 & 6 & 5 \end{bmatrix}\)as the sum of a symmetric and a skew-symmetric matrices.

Solve : \(X+2Y=\left[ \begin{matrix} 4 & 6 \\ -8 & 10 \end{matrix} \right] \) ;\(X-Y=\left[ \begin{matrix} 4 & 6 \\ -8 & 10 \end{matrix} \right] \)

If \(A=\left[ \begin{matrix} 3 & -2 \\ 4 & -2 \end{matrix} \right] \),find k so that A² = kA - 2I.

Without expanding, evaluate the following determinants:
\(\begin{vmatrix} 2& 3 &4 \\ 5 & 6 & 8 \\ 6x & 9x &12x \end{vmatrix}\)

Prove that \(\left| \begin{matrix} { a }^{ 2 }+\lambda & ab & ac \\ ab & { b }^{ 2 }+\lambda & bc \\ ac & bc & { c }^{ 2 }+\lambda \end{matrix} \right| ={ \lambda }^{ 2 }\left( { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }+\lambda \right) \)

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11th Maths Monthly Test -1 (Matrices and Determinants)

St. Britto Hr. Sec. School - Madurai

11th Maths Monthly Test -1 (Matrices and Determinants)-Aug 2020

Part - A

Multiple Choice Question - Answer All Question

The matrix A satisfying the equation \(\begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix}\)A=\(\begin{bmatrix} 1 & 1 \\ 0 & -1 \end{bmatrix}\)is

Choose the correct statement

A root of the equation \(\begin{vmatrix} 3-x&-6 &3 \\ -6 & 3-x & 3 \\ 3 &3 &-6-x \end{vmatrix}=0 \ is\)

If A=\(\begin{bmatrix} 1 & -1 \\ 2 &-1 \end{bmatrix}\),B=\(\begin{bmatrix} a & 1 \\ b &-1 \end{bmatrix}\) and (A+B)2=A2+B2, then the values of a and b are

If A +I =\(\begin{bmatrix} 3& -2 \\ 4 & 1 \end{bmatrix},\) then (A+ I )(A-I ) is equal to

Assertion (A): The inverse of \(\\ \\ \\ \begin{bmatrix} 2 & -1 \\ -4 & 2 \end{bmatrix}\) does not exist. Reason (R): The matrix is non-singular

Which one of the following is not true about the matrix \(\begin{bmatrix} 1 &0 &0 \\ 0 & 0 &0 \\ 0 & 0 & 5 \end{bmatrix}?\)

If A is a matrix 3 x 3, then \({ { (A }^{ 2 }) }^{ -1 }\)=

Find the odd one out of the following:

On using elementary row operation R1⟶R1-3R2 in the following matrix equation \(\begin{pmatrix} 4 & 2 \\ 3 & 3 \end{pmatrix}=\begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}\begin{pmatrix} 2 & 0 \\ 1 & 1 \end{pmatrix}\)

Part - B

Generic 2 - Answer All Question

Prove that \(\begin{vmatrix} 1&1 &1 \\ x & y& z\\ x^2& y^2 &z^2 \end{vmatrix}\)=(x - y) (y - z) (z - x).

Evaluate :\(\begin{vmatrix} cos \theta & sin \theta \\ -sin \theta & cos \theta \end{vmatrix}\)

Using properties of determinant, show that\(\triangle =\left| \begin{matrix} { cosec }^{ 2 }\theta & -{ cot }^{ 2 }\theta & 1 \\ { cot }^{ 2 }\theta & -cose{ c }^{ 2 }\theta & -1 \\ 42 & 40 & 2 \end{matrix} \right| =0\)

Find the area of the triangle whose vertices are (- 2, - 3), (3, 2), and (- 1, - 8)

Solve the following problems by using Factor Theorem : Show that \(\begin{vmatrix} b+c & a-c & a-b \\ b-c & c+a & b-a \\ c-b & c-a & a+b \end{vmatrix}=8abc\)

If A is a skew-symmetric matrix of odd order n, then |A| =0.

Evaluate :\(\begin{vmatrix} 2 & 4 \\ -1 & 2 \end{vmatrix}\)

Prove that \(\begin{vmatrix} a^2 & bc & ac+c^2 \\ a^2+ab & b^2 & ac \\ ab & b^2+bc & c^2 \end{vmatrix}=4a^2b^2c^2\)

If A=\(\begin{bmatrix} 4 & 2 \\ -1 & x \end{bmatrix}\) and such that (A- 2I)(A-3I)=0, find the value of x.

If the area of the triangle with vertices (- 3, 0), (3, 0) and (0, k) is 9 square units, find the values of k.

Part - C

Generic 3 - Answer All Question

Prove that \(LHS=\left| \begin{matrix} -{ a }^{ 2 } & ab & ac \\ ab & -{ b }^{ 2 } & bc \\ ac & bc & -{ c }^{ 2 } \end{matrix} \right| ={ 4a }^{ 2 }{ b }^{ 2 }{ c }^{ 2 }\)

Identify the singular and non-singular matrices:\(\begin{bmatrix} 0&a-b &k \\ b-a & 0 &5 \\ -k & -5 & 0 \end{bmatrix}\)

If A =\(\begin{bmatrix} 4 & 6 & 2 \\ 0 & 1 & 5 \\ 0 & 3 & 2 \end{bmatrix}\) and B= \(\begin{bmatrix} 0 & 1 & -1 \\ 3 & -1 & 4 \\ -1 & 2 & 1 \end{bmatrix}\) verify(A+B)T=AT+BT

If A is a 3 × 4 matrix and B is a matrix such that both ATB and BAT are defined, what is the order of the matrix B?

Identify the singular and non-singular matrices:\(\begin{bmatrix} 1&2 &3 \\ 4 & 5 &6 \\ 7 & 8 & 9 \end{bmatrix}\)

If a, b, c are all positive, and are pth, qth and rth terms of a G.P., show that \(\begin{vmatrix} log a & p & 1 \\ log b & q & 1 \\ log c & r & 1 \end{vmatrix}=0.\)

Using factor method show that \(\left| \begin{matrix} 1 & a & { a }^{ 2 } \\ 1 & b & { b }^{ 2 } \\ 1 & c & { c }^{ 2 } \end{matrix} \right| \) = (a - b) (b - c) (c - a).

If A = \(\begin{bmatrix} {1\over 2}&\alpha \\ 0 & {1\over2} \end{bmatrix}\) ,prove that \(\sum^n_{k=1}det(A^k)={1\over3}(1-{1\over 4^n}).\)

If AB=A and BA=B, then show that A2=A and B2=B.

If A =\(\begin{bmatrix} 4 & 6 & 2 \\ 0 & 1 & 5 \\ 0 & 3 & 2 \end{bmatrix}\) and B= \(\begin{bmatrix} 0 & 1 & -1 \\ 3 & -1 & 4 \\ -1 & 2 & 1 \end{bmatrix}\) verify(A-B)T=AT-BT

Part - D

Generic 5 - Answer All Question

If A=\(\left| \begin{matrix} 0 & 1 \\ -1 & 0 \end{matrix} \right| \) find x and y such that (xI +yA)2 =A

Without expanding, evaluate the following determinants: \(\begin{vmatrix} x+y& y+z &z+x \\ z & x & y \\ 1 & 1 &1 \end{vmatrix}\)

Prove that \(\begin{vmatrix} 1 &x^2 &x^3 \\ 1 & y^2 &y^3 \\1 &z^2 &z^3 \end{vmatrix}\) =(x-y)(y-z)(z-x)(xy+yz+zx).

Factorise \(\left| \begin{matrix} a & b & c \\ { a }^{ 2 } & { b }^{ 2 } & { c }^{ 2 } \\ bc & ca & ab \end{matrix} \right| \) .

If \(A=\left[ \begin{matrix} 1 & -1 \\ 2 & -1 \end{matrix} \right] \) and \(B=\left[ \begin{matrix} x & 1 \\ y & -1 \end{matrix} \right] \) and (A + B)2=A2 + B2, find X and.

Express the matrix A=\(\begin{bmatrix} 1 & 3 & 5 \\ -6 & 8 & 3 \\ -4 & 6 & 5 \end{bmatrix}\)as the sum of a symmetric and a skew-symmetric matrices.

Solve : \(X+2Y=\left[ \begin{matrix} 4 & 6 \\ -8 & 10 \end{matrix} \right] \) ;\(X-Y=\left[ \begin{matrix} 4 & 6 \\ -8 & 10 \end{matrix} \right] \)

If \(A=\left[ \begin{matrix} 3 & -2 \\ 4 & -2 \end{matrix} \right] \),find k so that A2 = kA - 2I.

Without expanding, evaluate the following determinants: \(\begin{vmatrix} 2& 3 &4 \\ 5 & 6 & 8 \\ 6x & 9x &12x \end{vmatrix}\)

Prove that \(\left| \begin{matrix} { a }^{ 2 }+\lambda & ab & ac \\ ab & { b }^{ 2 }+\lambda & bc \\ ac & bc & { c }^{ 2 }+\lambda \end{matrix} \right| ={ \lambda }^{ 2 }\left( { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }+\lambda \right) \)

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If A=\(\begin{bmatrix} 1 & -1 \\ 2 &-1 \end{bmatrix}\),B=\(\begin{bmatrix} a & 1 \\ b &-1 \end{bmatrix}\) and (A+B)²=A²+B², then the values of a and b are

Assertion (A): The inverse of \(\\ \\ \\ \begin{bmatrix} 2 & -1 \\ -4 & 2 \end{bmatrix}\) does not exist.
Reason (R): The matrix is non-singular

On using elementary row operation R₁⟶R₁-3R₂ in the following matrix equation \(\begin{pmatrix} 4 & 2 \\ 3 & 3 \end{pmatrix}=\begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}\begin{pmatrix} 2 & 0 \\ 1 & 1 \end{pmatrix}\)

Solve the following problems by using Factor Theorem :
Show that \(\begin{vmatrix} b+c & a-c & a-b \\ b-c & c+a & b-a \\ c-b & c-a & a+b \end{vmatrix}=8abc\)

If A =\(\begin{bmatrix} 4 & 6 & 2 \\ 0 & 1 & 5 \\ 0 & 3 & 2 \end{bmatrix}\) and B= \(\begin{bmatrix} 0 & 1 & -1 \\ 3 & -1 & 4 \\ -1 & 2 & 1 \end{bmatrix}\)
verify(A+B)^T=A^T+B^T

If A is a 3 × 4 matrix and B is a matrix such that both A^TB and BA^Tare defined, what is the order of the matrix B?

If a, b, c are all positive, and are p^th, q^th and r^th terms of a G.P., show that \(\begin{vmatrix} log a & p & 1 \\ log b & q & 1 \\ log c & r & 1 \end{vmatrix}=0.\)

If AB=A and BA=B, then show that A²=A and B²=B.

If A =\(\begin{bmatrix} 4 & 6 & 2 \\ 0 & 1 & 5 \\ 0 & 3 & 2 \end{bmatrix}\) and B= \(\begin{bmatrix} 0 & 1 & -1 \\ 3 & -1 & 4 \\ -1 & 2 & 1 \end{bmatrix}\)
verify(A-B)^T=A^T-B^T

If A=\(\left| \begin{matrix} 0 & 1 \\ -1 & 0 \end{matrix} \right| \) find x and y such that (xI +yA)² =A

Without expanding, evaluate the following determinants:
\(\begin{vmatrix} x+y& y+z &z+x \\ z & x & y \\ 1 & 1 &1 \end{vmatrix}\)

If \(A=\left[ \begin{matrix} 1 & -1 \\ 2 & -1 \end{matrix} \right] \) and \(B=\left[ \begin{matrix} x & 1 \\ y & -1 \end{matrix} \right] \) and (A + B)²=A² + B², find X and.

If \(A=\left[ \begin{matrix} 3 & -2 \\ 4 & -2 \end{matrix} \right] \),find k so that A² = kA - 2I.

Without expanding, evaluate the following determinants:
\(\begin{vmatrix} 2& 3 &4 \\ 5 & 6 & 8 \\ 6x & 9x &12x \end{vmatrix}\)