Null Matrix

Q. If $A=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$, then which of the following is/are equal to $A^{2020}$

1. $I$
2. $O$
3. $A$
4. $A^2$

Q.

What is a null matrix give an example?

Q. For the matrix $A=\left[\begin{array}{ccc}1& 1& 1\\ 1& 2& -3\\ 2& -1& 3\end{array}\right]$ show that $A^3-6A^2+5A+11I=0.$ Hence find $A^{-1}.$

Q. What is a matrix whose all entries are zero called ?

1. Vertical Matrix
2. Column Matrix
3. Null Matrix
4. Square Matrix

Q.

If $1³+2³+.......+9³=2025$, then $(0.11)³+(0.22)³+......+(0.99)³$will be-

1. $0.2695$

2. $2.695$

3. $3.695$

4. $0.3695$

Q.

Are all zero matrices equal?

Q.

Find $P^{-1}$, if it exists, given $P=\left[\begin{array}{cc}10& -2\\ -5& 1\end{array}\right]$

Q.

$1^{13}+2^{13}+3^{13}+.....+{60}^{13}$ is divisible by:

1. $61$

2. $63$

3. $65$

4. $59$

Q. If matrix $A=\left[\begin{array}{cc}6& 11\\ 2& 4\end{array}\right],$ then the value of determinant of $A^{2015}-6A^{2014}$ is

1. $2^{2016}$
2. $(-11)\times 2^{2015}$
3. $-2^{2015}\times 7$
4. $(-9)\times 2^{2014}$

Q. If $A=\left[\begin{array}{ccc}2& 2& 2\\ 2& 2& 2\\ 2& 2& 2\end{array}\right]$, then $A^3-35A=$

1. $A$
2. $2A$
3. $3A$
4. $4A$

Q.

Which of the following statements is correct about a null matrix?

1. It has more than one entries as zero

2. It has only one non-zero entry

3. It has all the entries as zero

4. It is represented as $O_{mxn}$ where mxn is the order of the matrix

Q. Suppose g is a differentiable function such that $g\left(x\right)+g^{\prime }(x{)}^{\prime }1$ for all $x$ and $g\left(0\right)=0$, then the largest possible value of $g\left(1\right)$ is

1. $e^{-1}$
2. $e^{-2}$
3. $1-e^{-2}$
4. $1-e^{-1}$

Q. The value of the sum $\sum _{k=1}^{\infty }\frac{6^k}{\left(3^k{2}^k\right)\left(3^{k+1}{2}^{k+1}\right)}$ is

1. 1
2. 3
3. 2
4. 0

Q. Let $A$ be a non singular square matrix of order $3\times 3$. Then $\left|\begin{array}{c}adjA\end{array}\right|$ is equal to

1. $3\left|\begin{array}{c}A\end{array}\right|$
2. $\left|\begin{array}{c}A\end{array}\right|$
3. ${\left|\begin{array}{c}A\end{array}\right|}^2$
4. ${\left|\begin{array}{c}A\end{array}\right|}^3$

Q. If $\left[\begin{array}{cc}2a+b& a-2b\\ 5c-d& 4c+3d\end{array}\right]=\left[\begin{array}{cc}4& -3\\ 11& 24\end{array}\right]$, then $b$ is

1. $1$
2. $2$
3. $-2$
4. $-3$

Q. Using Binomial theorem, evaluate $(101{)}^4$.

Q.

Which of the following is a null matrix?

Q. If $A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ a& b& -1\end{array}\right]$, then $A^2$=

1. Unit matrix
2. Null matrix
3. A
4. -A

Q. This section contains 1 Matrix Match type question, which has 2 Columns (Column I and Column II). Column I has four entries (A), (B), (C) and (D), Column II has four entries (P), (Q), (R) and (S). Match the entries in Column I with the entries in Column II. Each entry in Column I may match with one or more entries in Column II.

इस खण्ड में 1 मैट्रिक्स मिलान प्रकार का प्रश्न है, जिसमें 2 कॉलम (कॉलम I तथा कॉलम II) हैं। कॉलम I में चार प्रविष्टियाँ (A), (B), (C) तथा (D) हैं, कॉलम II में चार प्रविष्टियाँ (P), (Q), (R) तथा (S) हैं। कॉलम I में दी गयी प्रविष्टियों का मिलान कॉलम II में दी गयी प्रविष्टियों के साथ कीजिए। कॉलम I में दी गयी प्रत्येक प्रविष्टि का मिलान कॉलम II में दी गयी एक या अधिक प्रविष्टियों के साथ हो सकता है।

Match the Columns and choose the appropriate answer.

कॉलमों का मिलान करें और सही उत्तर चुनें।

Match the value of integration from Column-I to Column-II

कॉलम-I के समाकलनों के मान का कॉलम-II के साथ मिलान कीजिए।

	Column-I कॉलम-I		Column –II कॉलम-II
(A)	$\int e^x\cdot (\sin x+\cos x)dx$	(P)	$e^x\cos x+c$
(B)	$\int e^x(\cos x-\sin x)dx$	(Q)	$-e^x\sin x+c$
(C)	$\int e^x(-\sin x-\cos x)dx$	(R)	$-e^x\cos x+c$
(D)	$\int e^x(\sin x-\cos x)dx$	(S)	$e^x\cdot \sin x+c$

1. A-RS, B-PQ, C-PQR, D-RS
2. A-RS, B-P, C-PQ, D-RS
3. A-PS, B-PQ, C-Q, D-RS
4. A-S, B-P, C-Q, D-R

Q. If $A=\left[\begin{array}{cc}2& 2\\ 9& 4\end{array}\right]$ and $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, then $10A^{-1}$ is equal to:

1. $6I-A$
2. $A-6I$
3. $4I-A$
4. $A-4I$

Q. If $5A=\left[\begin{array}{cc}3& -4\\ 4& x\end{array}\right]$ and $A{A}^T=A^{T}A=I$ then $x=?$

1. $2$
2. $-2$
3. $3$
4. $-3$

Q. If $A=\left[\begin{array}{cc}2& -1\\ -1& 2\end{array}\right]$ and $I$ is the unit matrix of order $2,$ then $A^2$ is equal to

1. $4A-3I$
2. $3A-4I$
3. $A-I$
4. $A+I$

Q. If $A=\left[\begin{array}{cc}3& -2\\ 4& -2\end{array}\right]$ and $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, find $k$ so that $A^2=kA-2I$

Q. If $A=\left[\begin{array}{cc}\alpha & 0\\ 0& 1\end{array}\right]\text{and}B=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, then the value of $\alpha$ for which $A^2=B$ is:

1. $1$
2. $2$
3. $0$
4. No real values

Q. If $A=\left[\begin{array}{ccc}1& -3& -4\\ -1& 3& 4\\ 1& -3& -4\end{array}\right]$, then $A^2=$

1. $-A$
2. Null matrix
3. $2A$
4. $A$

Q. If $A=\left[\begin{array}{ccc}-1& -3& -3\\ 3& 1& -3\\ 3& -3& 1\end{array}\right]$ then adj (A) is

1. $=4\left[\begin{array}{ccc}-2& 3& 3\\ -3& 2& -3\\ -3& 3& 2\end{array}\right]$
2. $=4\left[\begin{array}{ccc}-2& 3& 3\\ 3& 2& -3\\ -3& -3& 2\end{array}\right]$
3. $=4\left[\begin{array}{ccc}-2& -3& 3\\ -3& 2& -3\\ -3& -3& 2\end{array}\right]$
4. $=4\left[\begin{array}{ccc}-2& 3& 3\\ -3& 2& -3\\ -3& -3& 2\end{array}\right]$

Q. Find the adjoint of the matrix $A=\left[\begin{array}{ccc}1& 2& 3\\ 1& 3& 5\\ 1& 5& 12\end{array}\right]$.
If $\text{Adjoint A:}\left[\begin{array}{ccc}a& -9& 1\\ b& 9& -2\\ 2& c& 1\end{array}\right]$, find the value of $abc$.

1. 213
2. 231
3. 321
4. 312

Q. Using properties of determinants prove that :
$\left|\begin{array}{ccc}1& 1& 1\\ a& b& c\\ bc& ca& ab\end{array}\right|=(a-b)(b-c)(c-a)$.

Q. If $A=\left[\begin{array}{ccc}-1& 0& 2\\ 3& 1& 4\end{array}\right],B=\left[\begin{array}{ccc}0& -2& 5\\ 1& -3& 1\end{array}\right]$ and $C=\left[\begin{array}{ccc}1& -5& -2\\ 6& 0& -4\end{array}\right],$ then $2A-3B+4C$ is

1. $\left[\begin{array}{ccc}2& -14& -19\\ 11& 17& -11\end{array}\right]$
2. $\left[\begin{array}{ccc}2& 11& -19\\ -14& 11& -11\end{array}\right]$
3. $\left[\begin{array}{ccc}2& -14& -19\\ 27& 11& -11\end{array}\right]$
4. $\left[\begin{array}{ccc}-14& -14& -2\\ 17& 11& -11\end{array}\right]$

Q. What is $\left[\begin{array}{ccc}x& y& z\end{array}\right]\left[\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right]$ equal to?

1. $\left[\begin{array}{ccc}ax& hy& gz\\ hx& by& fz\\ gx& fy& cz\end{array}\right]$
2. $\left[\begin{array}{ccc}ax+hy+gz& hx+by+fz& gx+fy+cz\end{array}\right]$
3. $\left[\begin{array}{ccc}a& h& g\\ hx& by& fz\\ g& f& c\end{array}\right]$
4. $\left[\begin{array}{ccc}ax+hy+gz& h+b+f& g+f+c\end{array}\right]$