Definition of a Determinant

Q.

What is the formula of $x^2+y^2$?

Q.

What is The Difference Between Determinant And Matrix?

Q. A square matrix P satisfies $P^2=I-P,$ where I is an identity matrix of order as order of P. if $P^n=5I-8P,thenn=$

1. 6
2. 7
3. 4
4. 5

Q.

If $\tan \left(\alpha \right)=\frac{m}{m+1},\tan \left(\beta \right)=\frac{1}{2m+1}$ then $\left(\alpha +\beta \right)$ is equal to

1. $\frac{\pi }{3}$

2. $\frac{\pi }{4}$

3. $0$

4. $\frac{\pi }{2}$

Q.

If the minimum and the maximum values of the function $f:\left[\frac{\mathrm{\pi }}{4},\frac{\mathrm{\pi }}{2}\right]\to R$ defined by $f\left(\theta \right)=\left|\begin{array}{ccc}-{\sin }^2\left(\theta \right)& -1-{\sin }^2\left(\theta \right)& 1\\ -{\cos }^2\left(\theta \right)& -1-{\cos }^2\left(\theta \right)& 1\\ 12& 10& -2\end{array}\right|$ are $m$ and $M$ respectively, then the ordered pair $\left(m,M\right)$ is equal to

1. $\left(0,4\right)$

2. $\left(-4,0\right)$

3. $\left(-4,4\right)$

4. $\left(0,2\sqrt{2}\right)$

Q. Which of the following is correct? A. Determinant is a square matrix. B. Determinant is a number associated to a matrix. C. Determinant is a number associated to a square matrix. D. None of these

Q.

If $n\left(A\right)$ denotes the number of elements in set $A$ and if $n\left(A\right)=4$, $n\left(B\right)=5$ and $n(A\cap B)=3$, then $n\left[(A\times B)\cap (B\times A)\right]=$

1. $8$

2. $9$

3. $10$

4. $11$

Q.

If $y=e^{(1+\log x)}$, then the value of $\frac{dy}{dx}$ is equals to ?

1. $e$

2. $1$

3. $0$

4. $\log x{e}^{\log x}$

Q.

If $AandB$ are square matrices of order $3$ such that $\left|A\right|=–1,\left|B\right|=3,$ then $\left|3AB\right|$ is equal to

1. $–9$

2. $–81$

3. $–27$

4. $81$

Q.

The determinant of a matrix is always positive.

1. True

2. False

Q.

If $\sin (\alpha -\beta )=\frac{1}{2}$and $\cos (\alpha +\beta )=\frac{1}{2}$, where $\alpha$ and $\beta$ are positive acute angles, then

1. $\alpha =45°,\beta =15°$

2. $\alpha =15°,\beta =45°$

3. $\alpha =60°,\beta =15°$

4. None of these

Q.

If A is a matrix then IAI represents the modulus of the matrix

1. True

2. False

Q.

How many numbers between $1$ and $100$ (inclusive) are divisible by $3$ or $4$?

Q. The number of three digit numbers of the form xyz such that x < y and $z\le y$ is
If zero is included it will be at ${}^9{C}_2$ no. s
If zero is excluded \( \left\{$$\begin{array}{ccc}x,y,zalldiff.& \Rightarrow & {}^9{C}_3\times 2!\\ x=z<y& \Rightarrow & {}^9{C}_2\\ x<y=z& \Rightarrow & {}^9{C}_{2}N{o}^{\prime }s\end{array}$$\right\)
Total number of ways = 276
Alternative
y can be from 2 to 9 so total number of ways =${\sum }_{r=2}^9(r^2-1)=276$

1. 276
2. 285
3. 240
4. 244

Q.

Prove: $\frac{(cotA+\tan B)}{(cotB+\tan A)}=cotA\tan B$

Q.

$limi{t}_{x\to \frac{\mathrm{\pi }}{2}}\frac{\left(1-\tan \left({\displaystyle \frac{x}{2}}\right)\right)\left(1-\sin \left(x\right)\right)}{\left(1+\tan \left(\frac{x}{2}\right)\right){\left(\mathrm{\pi }-2\mathrm{x}\right)}^3}=$

1. $\frac{1}{8}$

2. $0$

3. $\frac{1}{32}$

4. $\infty$

Q.

The value of ${\int }_0^{\sqrt{2}}\left[x^2\right]·dx$, where $\left[x\right]$ is the greatest integer function, is

1. $2-\sqrt{2}$

2. $2+\sqrt{2}$

3. $\sqrt{2}-1$

4. $\sqrt{2}-2$

Q.

How many two-digit numbers are divisible by $4$ ?

Q.

The function $f\left(x\right)=\left(x^2-1\right)\left|x^2-3x+2\right|+\cos \left(\left|x\right|\right)$ is not differentiable at

1. $-1$

2. $0$

3. $1$

4. $2$

Q.

If $\sin \left(\theta \right)+\cos \left(\theta \right)=m$ and $\sec \left(\theta \right)+\mathrm{cosec}\left(\theta \right)=n$, then $n\left(m+1\right)\left(m-1\right)=$

1. $m$

2. $n$

3. $2m$

4. $2n$

Q.

If $\alpha$ and $\beta$ are the roots of the equation $x^2-ax+b=0$ and$A_n={\alpha }^n+{\beta }^n$. Then $A_{n+1}-a{A}_n+b{A}_{n-1}$is equal to

1. $-a$

2. $b$

3. $0$

4. $a-b$

Q. Let $f\left(x\right)=\sqrt{x^2-3x+2}$ and $g\left(x\right)=\sqrt{x}$ be two given functions . If $S$ be the domain of $f\circ g$ and $T$ be the domain of $g\circ f$, then

1. $S=T$
2. $S\cap T=\phi$
3. $S\cap T$ is a singleton
4. $S\cap T$ is an interval

Q.

If a = b + c, then is it true that |a| = |b|+|c|? Justify your answer.

Q. If the four points with position vectors $-2\hat{i}+\hat{j}+\hat{k},\hat{i}+\hat{j}+\hat{k},\hat{j}-\hat{k}$ and $\lambda \hat{j}+\hat{k}$ are coplanar, then $\lambda =$

1. $1$
2. $2$
3. $0$
4. $-1$

Q.

For complex numbers $z_1$, $z_2$ if $\left|z_1\right|=12$ and $\left|z_2-3-4i\right|=5$ then minimum value of $\left|z_1-z_2\right|$ is

1. $0$

2. $2$

3. $7$

4. $17$

Q. The number of distinct real values of $\lambda$ for which vectors $-{\lambda }^2\hat{i}+\hat{j}+\hat{k},\hat{i}-{\lambda }^2\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}-{\lambda }^2\hat{k}$ are coplanar is

Q.

Can We Add Two Determinants

Q. If $A$ is a square matrix of order 4, then $\text{adj(adj}A)=$

1. $|A{|}^{3}A$
2. $|A{|}^{2}A$
3. $|A{|}^2{A}^2$
4. $|A{|}^3$

Q. If ${}^{\prime }{\alpha }^{\prime }$ is a root of $x^4=1$ with negative principal argument, then the principal argument of $\Delta \left(\alpha \right)$ where
$\Delta \left(\alpha \right)=\left|\begin{array}{ccc}1& 1& 1\\ {\alpha }^n& {\alpha }^{n+1}& {\alpha }^{n+3}\\ \frac{1}{{\alpha }^{n+1}}& \frac{1}{{\alpha }^n}& 0\end{array}\right|$ is

1. $-\frac{3\pi }{4}$
   
2. $\frac{5\pi }{14}$
   
3. $-\frac{\pi }{4}$
4. $\frac{\pi }{4}$
   

Q. Let f(x) =max {2 sin x, 1-cos x), $xϵ(0,\pi )$, Then set of points of non-differentiability is:

1. $\left\{\frac{\pi }{2}\right\}$
2. $\{\pi -co{s}^{-1}\frac{3}{5}\}$
   
3. $\left\{co{s}^{-1}\frac{3}{5}\right\}$
4. $\varphi$