11th Maths Monthly Test- 3 (Matrices and Determinants)

St. Britto Hr. Sec. School - Madurai

11th Maths Monthly Test- 3 (Matrices and Determinants)-Aug 2020

Time : 2.00 Hours

Part - A

Generic 2 - Answer All Question

Without expanding the determinant, prove that \(\begin{vmatrix} s & a^2 & b^2+c^2 \\ s & b^2 &c^2+a^2 \\ s & c^2 & a^2+b^2 \end{vmatrix}=0\)

Evaluate \(\begin{vmatrix} 2014& 2017 &0 \\ 2020 & 2023 &1 \\ 2023 &2026 &0 \end{vmatrix}\)

Without expanding evaluate the determinant \(\left| \begin{matrix} 41 & 1 & 5 \\ 79 & 7 & 9 \\ 29 & 5 & 3 \end{matrix} \right| \)

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

If a, b, c and x are positive real numbers, then show that \(\begin{vmatrix} (a^x+a^{-x})^2 &(a^x-a^{-x})^2 &1 \\ (b^x+b^{-x})^2 & (b^x-b^{-x})^2 & 1 \\ (c^x+c^{-x})^2 & (c^x-c^{-x})^2 & 1 \end{vmatrix}\)is zero.

If (k, 2), (2, 4) and (3, 2) are vertices of the triangle of area 4 square units then determine the value of k.

Prove that \(\begin{bmatrix} sec^2 \theta & tan ^2 \theta & 1 \\ tan^2 \theta & sec^2 \theta & -1 \\ 38 & 36 & 2 \end{bmatrix}=0\)

If A=\(\begin{bmatrix} 1 & a \\ 0 & 1 \end{bmatrix}\), then compute A⁴.

Find the area of the triangle whose vertices are (0, 0), (1, 2) and (4, 3).

Construct an m × n matrix A= [a_ij], where a _ij is given by
\(a_{ij}={|3i-4j|\over 4}with \ m=3,n=4\)

Consider the matrix Aa=\(\begin{bmatrix} cos \alpha & -sin\alpha \\ sin\alpha & cos\alpha \end{bmatrix}\)
Find all possible real values of α satisfying the condition \(A\alpha +A^T_{\alpha}=I\)

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

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

Identify the singular and non-singular matrix
(i) \(\left[ \begin{matrix} 1 & 4 & 9 \\ 4 & 9 & 16 \\ 9 & 16 & 25 \end{matrix} \right] \) (ii) \(\left[ \begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ -2 & -4 & -6 \end{matrix} \right] \)

Part - B

Generic 3 - Answer All Question

Solve for x if [x 2 -1] \(\begin{bmatrix} 1&1 &2 \\ -1 & -4 &1 \\ -1 &-1 &-2 \end{bmatrix}\)\(\begin{bmatrix} x \\ 2 \\ 1 \end{bmatrix}\)=0

Prove that \(\begin{vmatrix} 1& a & a^2-bc \\1 &b &b^2-ca \\ 1 & c & c^2-ab \end{vmatrix}=0.\)

Show that f(x)f(y)=f(x+y) where f(x)=\(\begin{bmatrix} cos \ x & -sin \ x & 0 \\ sin x & cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}\).

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

Show that \(\begin{vmatrix} 0 & c &b \\ c & 0 &a \\ b & a & 0 \end{vmatrix}^2=\begin{vmatrix} b^2+c^2 & ab & ac \\ ab & c^2+a^2 & bc \\ ab & bc & a^2+b^2 \end{vmatrix}\)

Give your own examples of matrices satisfying the following conditions in each case:
A and B such that AB \(\neq\) BA.

Express the following matrices as the sum of a symmetric matrix and a skew-symmetric matrix:
\(\begin{bmatrix} 3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \end{bmatrix}\)

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).

Show that \(\begin{vmatrix} a^2+x^2& ab & ac \\ ab & b^2+x^2 & bc \\ ac &bc &c^2+x^2 \end{vmatrix}\) is divisible by x⁴.

Give your own examples of matrices satisfying the following conditions in each case
A and B such that AB=0 and BA\(\neq\)0.

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 (3A)^T= 3A^T

Prove that \(\left| \begin{matrix} a-b-c & 2a & 2a \\ 2b & b-c-a & 2b \\ 2c & 2c & c-a-b \end{matrix} \right| =\left( a+b+c \right) ^{ 3 }\)

If A^T=\(\begin{bmatrix} 4 & 5 \\ -1 & 0 \\ 2 & 3 \end{bmatrix}\) and B= \(\begin{bmatrix} 2 & -1&1 \\7 & 5&-2 \end{bmatrix}\),verify (A-B)^T=A^T-B^T

If A^T=\(\begin{bmatrix} 4 & 5 \\ -1 & 0 \\ 2 & 3 \end{bmatrix}\) and B= \(\begin{bmatrix} 2 & -1&1 \\7 & 5&-2 \end{bmatrix}\),verify (B^T)^T=B

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11th Maths Monthly Test- 3 (Matrices and Determinants)

St. Britto Hr. Sec. School - Madurai

11th Maths Monthly Test- 3 (Matrices and Determinants)-Aug 2020

Part - A

Generic 2 - Answer All Question

Without expanding the determinant, prove that \(\begin{vmatrix} s & a^2 & b^2+c^2 \\ s & b^2 &c^2+a^2 \\ s & c^2 & a^2+b^2 \end{vmatrix}=0\)

Evaluate \(\begin{vmatrix} 2014& 2017 &0 \\ 2020 & 2023 &1 \\ 2023 &2026 &0 \end{vmatrix}\)

Without expanding evaluate the determinant \(\left| \begin{matrix} 41 & 1 & 5 \\ 79 & 7 & 9 \\ 29 & 5 & 3 \end{matrix} \right| \)

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

If a, b, c and x are positive real numbers, then show that \(\begin{vmatrix} (a^x+a^{-x})^2 &(a^x-a^{-x})^2 &1 \\ (b^x+b^{-x})^2 & (b^x-b^{-x})^2 & 1 \\ (c^x+c^{-x})^2 & (c^x-c^{-x})^2 & 1 \end{vmatrix}\)is zero.

If (k, 2), (2, 4) and (3, 2) are vertices of the triangle of area 4 square units then determine the value of k.

Prove that \(\begin{bmatrix} sec^2 \theta & tan ^2 \theta & 1 \\ tan^2 \theta & sec^2 \theta & -1 \\ 38 & 36 & 2 \end{bmatrix}=0\)

If A=\(\begin{bmatrix} 1 & a \\ 0 & 1 \end{bmatrix}\), then compute A4.

Find the area of the triangle whose vertices are (0, 0), (1, 2) and (4, 3).

Construct an m × n matrix A= [aij], where a ij is given by \(a_{ij}={|3i-4j|\over 4}with \ m=3,n=4\)

Consider the matrix Aa=\(\begin{bmatrix} cos \alpha & -sin\alpha \\ sin\alpha & cos\alpha \end{bmatrix}\) Find all possible real values of α satisfying the condition \(A\alpha +A^T_{\alpha}=I\)

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

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

Identify the singular and non-singular matrix (i) \(\left[ \begin{matrix} 1 & 4 & 9 \\ 4 & 9 & 16 \\ 9 & 16 & 25 \end{matrix} \right] \) (ii) \(\left[ \begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ -2 & -4 & -6 \end{matrix} \right] \)

Part - B

Generic 3 - Answer All Question

Solve for x if [x 2 -1] \(\begin{bmatrix} 1&1 &2 \\ -1 & -4 &1 \\ -1 &-1 &-2 \end{bmatrix}\)\(\begin{bmatrix} x \\ 2 \\ 1 \end{bmatrix}\)=0

Prove that \(\begin{vmatrix} 1& a & a^2-bc \\1 &b &b^2-ca \\ 1 & c & c^2-ab \end{vmatrix}=0.\)

Show that f(x)f(y)=f(x+y) where f(x)=\(\begin{bmatrix} cos \ x & -sin \ x & 0 \\ sin x & cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}\).

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

Show that \(\begin{vmatrix} 0 & c &b \\ c & 0 &a \\ b & a & 0 \end{vmatrix}^2=\begin{vmatrix} b^2+c^2 & ab & ac \\ ab & c^2+a^2 & bc \\ ab & bc & a^2+b^2 \end{vmatrix}\)

Give your own examples of matrices satisfying the following conditions in each case: A and B such that AB \(\neq\) BA.

Express the following matrices as the sum of a symmetric matrix and a skew-symmetric matrix: \(\begin{bmatrix} 3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \end{bmatrix}\)

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).

Show that \(\begin{vmatrix} a^2+x^2& ab & ac \\ ab & b^2+x^2 & bc \\ ac &bc &c^2+x^2 \end{vmatrix}\) is divisible by x4.

Give your own examples of matrices satisfying the following conditions in each case A and B such that AB=0 and BA\(\neq\)0.

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 (3A)T= 3AT

Prove that \(\left| \begin{matrix} a-b-c & 2a & 2a \\ 2b & b-c-a & 2b \\ 2c & 2c & c-a-b \end{matrix} \right| =\left( a+b+c \right) ^{ 3 }\)

If AT=\(\begin{bmatrix} 4 & 5 \\ -1 & 0 \\ 2 & 3 \end{bmatrix}\) and B= \(\begin{bmatrix} 2 & -1&1 \\7 & 5&-2 \end{bmatrix}\),verify (A-B)T=AT-BT

If AT=\(\begin{bmatrix} 4 & 5 \\ -1 & 0 \\ 2 & 3 \end{bmatrix}\) and B= \(\begin{bmatrix} 2 & -1&1 \\7 & 5&-2 \end{bmatrix}\),verify (BT)T=B

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If A=\(\begin{bmatrix} 1 & a \\ 0 & 1 \end{bmatrix}\), then compute A⁴.

Construct an m × n matrix A= [a_ij], where a _ij is given by
\(a_{ij}={|3i-4j|\over 4}with \ m=3,n=4\)

Consider the matrix Aa=\(\begin{bmatrix} cos \alpha & -sin\alpha \\ sin\alpha & cos\alpha \end{bmatrix}\)
Find all possible real values of α satisfying the condition \(A\alpha +A^T_{\alpha}=I\)

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

Identify the singular and non-singular matrix
(i) \(\left[ \begin{matrix} 1 & 4 & 9 \\ 4 & 9 & 16 \\ 9 & 16 & 25 \end{matrix} \right] \) (ii) \(\left[ \begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ -2 & -4 & -6 \end{matrix} \right] \)

Give your own examples of matrices satisfying the following conditions in each case:
A and B such that AB \(\neq\) BA.

Express the following matrices as the sum of a symmetric matrix and a skew-symmetric matrix:
\(\begin{bmatrix} 3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \end{bmatrix}\)

Show that \(\begin{vmatrix} a^2+x^2& ab & ac \\ ab & b^2+x^2 & bc \\ ac &bc &c^2+x^2 \end{vmatrix}\) is divisible by x⁴.

Give your own examples of matrices satisfying the following conditions in each case
A and B such that AB=0 and BA\(\neq\)0.

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 (3A)^T= 3A^T

If A^T=\(\begin{bmatrix} 4 & 5 \\ -1 & 0 \\ 2 & 3 \end{bmatrix}\) and B= \(\begin{bmatrix} 2 & -1&1 \\7 & 5&-2 \end{bmatrix}\),verify (A-B)^T=A^T-B^T

If A^T=\(\begin{bmatrix} 4 & 5 \\ -1 & 0 \\ 2 & 3 \end{bmatrix}\) and B= \(\begin{bmatrix} 2 & -1&1 \\7 & 5&-2 \end{bmatrix}\),verify (B^T)^T=B