M.C.Q (1 Marks)

MCQ 1511 Mark

If $\text{A}=\begin{bmatrix}a^2 &\text{amp; ab}&\text{amp; ac} \\\text{ab}&\text{amp; }\text{b}^2&\text{amp;}\text{ bc}\\\text{ac}&\text{amp;}\text{bc}&\text{amp;}\text{c}^2 \end{bmatrix}$and $\text{a}^2+\text{b}^2+\text{c}^3=1$ then $\text{A}^2=$

A
$2\text{A}$
✓
$\text{A}$
C
$3\text{A}$
D
$\frac{1}{2}\text{A}$

Answer

Correct option: B.

$\text{A}$

$\text{A}^2=\begin{bmatrix}\text{a}^2 &\text{amp; ab}&\text{amp; ac} \\\text{ab}&\text{amp; }\text{b}^2&\text{amp;}\text{ bc}\\\text{ac}&\text{amp;}\text{bc}&\text{amp;}\text{c}^2 \end{bmatrix}\begin{bmatrix}\text{a}^2 &\text{amp; ab}&\text{amp; ac} \\\text{ab}&\text{amp; }\text{b}^2&\text{amp;}\text{ bc}\\\text{ac}&\text{amp;}\text{bc}&\text{amp;}\text{c}^2 \end{bmatrix}=\text{A}$

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MCQ 1521 Mark

The order the matrix is $\begin{bmatrix}2&\text{amp; }3&\text{amp; }4\\9&\text{amp; }8&\text{amp; }7\end{bmatrix}$ is :

A
$4 \times 3$
B
$3 \times 2$
✓
$2 \times 3$
D
$3 \times 1$

Answer

Correct option: C.

$2 \times 3$

If $A$ is a matrix with mm rows and $n$ columns.
Then the order of a matrix is nothing but a size of a matrix, which is given by $m \times n.$
Since, in the given matrix, there are $2$ rows and $3$ columns.
So, order of given matrix will be $2 \times 3$.

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MCQ 1531 Mark

If $\begin{bmatrix}\text{r}+4&\text{amp; 6}\\3&\text{amp; 3}\end{bmatrix}=\begin{bmatrix}{5}&\text{amp;}\text{ r}+5\\\text{r+2}&\text{amp; 4}\end{bmatrix}$ then $\text{r}=$

✓
1
B
2
C
3
D
-1

Answer

Correct option: A.

We know that two matrices are equal iff their corresponding elements are equal.

Thus comparing corresponding elements we get, for the first entry of.

the given matrices r + 4 = 5 and r is satisfying other equations which are involving r ⇒ r = 1

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MCQ 1541 Mark

If a matrix is of order $2 \times 3,$ then the number of elements in the matrix is:

A
$5$
✓
$6$
C
$2$
D
$3$

Answer

Correct option: B.

$6$

Given a matrix $2\times3$
$\Rightarrow \begin{bmatrix} { \text{a} }_{11} &\text{amp; } {\text{a} }_{12} &\text{amp; } { \text{a} }_{13} \\ { \text{a} }_{21} &\text{amp; } {\text{a} }_{22} &\text{amp; } {\text{a} }_{23} \end{bmatrix}$
Clearly there are $6 $ elements.

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MCQ 1551 Mark

If $A$ is a matrix of order $m\times n$ and $B$ is a matrix such that $AB^T$ and $B^TA$ are both defined, then the order of matrix $B$ is:

✓
$m\times n$
B
$n\times n$
C
$n\times m$
D
$m\times n$

Answer

Correct option: A.

$m\times n$

$A$ is $m \times n$ matrix and $AB^T$ is defined then
number of columns in $A =$ number of rows in$ B^T =n$
$B^TA$ is also defined then number of columns in $B^T =$ number of rows in $A = m$
Order of $B$ is $m\times n$

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MCQ 1561 Mark

Choose the correct answer from the given four options.If $\begin{bmatrix}2\text{x}+\text{y}&4\text{x}\\5\text{x}-7&4\text{x}\end{bmatrix}=\begin{bmatrix}7&7\text{y}-13\\\text{y}&\text{x}+6\end{bmatrix},$ then the value of x + y is:

A
x = 3, y = 1
✓
x = 2, y = 3
C
x = 2, y = 4
D
x = 3, y = 3

Answer

Correct option: B.

x = 2, y = 3

We have, $\begin{bmatrix}2\text{x}+\text{y}&4\text{x}\\5\text{x}-7&4\text{x}\end{bmatrix}=\begin{bmatrix}7&7\text{y}-13\\\text{y}&\text{x}+6\end{bmatrix}$
⇒ 4x = x + 6 ⇒ x = 2

and 4x = 7y - 13

⇒ 8 = 7y - 13

⇒ y = 3

$\therefore$ x + y = 2 + 3 = 5

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MCQ 1571 Mark

If $\text{A}=\begin{bmatrix}1&2&\text{x}\\0&1&0\\0&0&1\end{bmatrix}$ and $\text{B}=\begin{bmatrix}1&-2&\text{y}\\0&1&0\\0&0&1\end{bmatrix}$ and $AB = I_3,$ then $x + y$ equals :

✓
$0$
B
$-1$
C
$2$
D
None of these.

Answer

Correct option: A.

$0$

Given: $AB = I_3$
$\Rightarrow\begin{bmatrix}1&2&\text{x}\\0&1&0\\0&0&1\end{bmatrix}\begin{bmatrix}1&-2&\text{y}\\0&1&0\\0&0&1\end{bmatrix}=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$
$\Rightarrow\begin{bmatrix}1&0&\text{y+x}\\0&1&0\\0&0&1\end{bmatrix}=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$
The corresponding elements of two equal matrices are equal.
$\therefore\ \text{y}+\text{x}=0$

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MCQ 1581 Mark

Choose the correct answer from the given four options.If $\text{A}=\frac{1}{\pi}\begin{bmatrix}\sin^{-1}(\text{x}\pi)&\tan^{-1}\Big(\frac{\text{x}}{\pi}\Big)\\\sin^{-1}\Big(\frac{\text{x}}{\pi}\Big)&\cot^{-1}(\pi\text{x})\end{bmatrix}$ and $\text{B}=\frac{1}{\pi}\begin{bmatrix}-\cos^{-1}(\text{x}\pi)&\tan^{-1}\Big(\frac{\text{x}}{\pi}\Big)\\\sin^{-1}\Big(\frac{\text{x}}{\pi}\Big)&\tan^{-1}(\pi\text{x})\end{bmatrix}$ then A - B is:

A
$\text{I}$
B
$0$
C
$2\text{I}$
✓
$\frac{1}{2}\text{I}$

Answer

Correct option: D.

$\frac{1}{2}\text{I}$

We have, $\text{B}=\begin{bmatrix}-\frac{1}{\pi}\cos^{-1}\text{x}\pi&\frac{1}{\pi}\tan^{-1}\frac{\text{x}}{\pi}\\\frac{1}{\pi}\sin^{-1}\frac{\text{x}}{\pi}&-\frac{1}{\pi}\tan^{-1}\pi\text{x}\end{bmatrix}$
and $\text{A}=\begin{bmatrix}\frac{1}{\pi}\sin^{-1}\text{x}\pi&\frac{1}{\pi}\tan^{-1}\frac{\text{x}}{\pi}\\\frac{1}{\pi}\sin^{-1}\frac{\text{x}}{\pi}&\frac{1}{\pi}\cot^{-1}\pi\text{x}\end{bmatrix}$

$\therefore\ \text{A}-\text{B}=\begin{bmatrix}\frac{1}{\pi}(\sin^{-1}\text{x}\pi+\cos^{-1}\text{x}\pi)&0\\0&\frac{1}{\pi}\big(\cot^{-1}\text{x}\pi+\tan^{-1}\pi\text{x}\big)\end{bmatrix}$

$=\begin{bmatrix}\frac{1}{\pi}\Big(\frac{\pi}{2}\Big)&0\\0&\frac{1}{\pi}\Big(\frac{\pi}{2}\Big)\end{bmatrix}$

$=\frac{1}{2}\begin{bmatrix}1&0\\0&1\end{bmatrix}$

$=\frac{1}{2}\text{I}$

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MCQ 1591 Mark

The total number of matrices formed with the help of 6 different numbers are:

A
6
B
6!
C
2(6!)
✓
4(6!)

Answer

Correct option: D.

4(6!)

No.of numbers in Matrix is 6

The possible orientations of Matrix is.

1 × 6, 2 × 3, 3 × 2, 6 × 1

The numbers in each orientation can be arranged in 6! ways.

$\implies$The total possibilities are 4(6!).

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MCQ 1601 Mark

If order of a matrix is 3 × 3, then it is a?

✓
square matrix
B
rectangular matrix
C
unit matrix
D
None of these

Answer

Correct option: A.

square matrix

Since, order of given matrix is 3 × 3.

$\therefore$ No of rows = No. of columns

So, given matrix is a square matrix.

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MCQ 1611 Mark

A matrix having mm rows and nn columns with m = n is said to be a?

A
rectangular matrix
✓
square matrix
C
identity matrix
D
scalar matrix

Answer

Correct option: B.

square matrix

A matrix having mm rows and nn columns with m = n, means number of rows are equal to number of columns.
$\therefore$ given matrix is square matrix.

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MATHS M.C.Q (1 Marks)