Method of Intervals

Q. If A is a square matrix such that A² = A, then write the value of 7A − (I + A)³, where I is the identity matrix.

Q. If A is a square matrix such that A² = I, then (A − I)³ + (A + I)³ − 7A is equal to

(a) A
(b) I − A
(c) I + A
(d) 3A

Q. All $x$ satisfying the inequality $({\cot }^{-1}x{)}^2-7({\cot }^{-1}x)+10>0$, lie in the interval :

1. $(\cot 5,\cot 4)$
2. $(\cot 2,\infty )$
3. $(-\infty ,\cot 5)\cup (\cot 4,\cot 2)$
4. $(-\infty ,\cot 5)\cup (\cot 2,\infty )$

Q. If $f\left(x\right)$ is defined on $(0,1)$, then the domain of $g\left(x\right)=f\left(e^x\right)+f\left({\log }_e|x|\right)$ is

1. $(-1,e)$
2. $(1,e)$
3. $(-e,-1)$
4. $(-e,1)$

Q. If $A=\left[\begin{array}{ccc}2& 0& 1\\ 2& 1& 3\\ 1& -1& 0\end{array}\right]$, find A² − 5A + 4I and hence find a matrix X such that A² − 5A + 4I + X = 0.

Q. If $S$ is the set of all real values of $x$ such that $\frac{2x-1}{2x^3+3x^2+x}$ is positive, then $S$ contains

1. $(-\infty ,-\frac{3}{2})$
2. $(-\frac{1}{2},0)$
3. $(\frac{1}{2},3)$
4. $(-\frac{1}{2},\frac{1}{2})$

Q. number of positive integers n for which n^2+ 96 is a perfect square

Q.

For all complex numbers $z_1,z_2$ satisfying $\left|z_1\right|=12$ and $|z_2-3-4i|=5$, the minimum value of $\mathbf{|}{\mathit{z}}_{\mathbf{1}}\mathbf{-}{\mathit{z}}_{\mathbf{2}}\mathbf{|}\mathbf{}$is

1. $4$

2. $3$

3. $1$

4. $2$

Q. The solution set of $\frac{3x^2-7x+8}{x^2+1}\le 2$ is

1. $[1,4]$
2. $[1,2]$
3. $[2,6]$
   
4. $[1,6]$

Q. Let $A=\left[a_{ij}\right]$ be a square matrix of order $3$ such that $a_{ij}=2^{j-i}$, for all $i,j=1,2,3$. Then, the matrix $A^2+A^3+\cdots +A^{10}$ is equal to:

Q. Let $\omega =\frac{\sqrt{3}+i}{2}$ and $P=\{{\omega }^n:n=1,2,3,\cdots \}.$ Further $H_1=\{z\in C:Rez>\frac{1}{2}\}$ and $H_2=\{z\in C:Rez<-\frac{1}{2}\},$ where $C$ is the set of all complex numbers. If $z_1\in P\cap H_1,$ $z_2\in P\cap H_2$ and $O$ represents the origin, then $\angle z_{1}O{z}_2=$

1. $\frac{\pi }{2}$
2. $\frac{\pi }{6}$
3. $\frac{2\pi }{3}$
4. $\frac{5\pi }{6}$

Q. If matrix $A=\left[\begin{array}{cc}2& -2\\ -2& 2\end{array}\right]$ and A² = pA, then write the value of p.

Q. Solution set of $x(2^x-1)(3^x-9)(x-3)<0$ is

1. $(-\infty ,0)\cup (2,3)$
2. $(2,3)$
3. $(-\infty ,2)\cup (3,\infty )$
4. $(-\infty ,0]\cup [3,\infty )$

Q. Let $x$ and $y$ be real numbers satisfying the inequality $5x^2+y^2-4xy+24\le 10x-1.$ Find the value of $x^2+y^2-29.$
(correct answer + 2, wrong answer 0)

Q. Solution set of $(x+1)(x-1{)}^2(x-2)\ge 0$ is

1. $(-\infty ,-1]\cup [2,\infty )$
2. $(-1,2)$
3. $[-1,2]$
4. $(-\infty ,-1]\cup \left\{1\right\}\cup [2,\infty )$

Q. If $A_k=\left[\begin{array}{cc}k& k-1\\ k-1& k\end{array}\right],$ then $|A_1|+|A_2|+\cdots +|A_{2021}|$ is equal to

1. $0$
2. $2020$
3. $(2021{)}^2$
4. $(2020{)}^3$

Q. The domain of the function $f\left(x\right)=\frac{1}{\sqrt{\left(\right[x{]}^2-7\left[x\right]+10)}}$ is $(-\infty ,a)\cup [b,\infty ),$ then $a+b$ is
($\left[x\right]$ denotes the greatest integer less than or equal to $x$)

Q. The integral part of ${(\sqrt{2}+1)}^6$ is

1. 197
2. 196
3. 175
4. 176

Q. The solution set of $x\ge \frac{1}{x}$ is

1. $[-1,0]$
2. $[-1,\infty )$
3. $[-1,0)\cup [1,\infty )$
4. $(-\infty ,-1)$

Q. Solution set of $\frac{x^2-4x+3}{x^2-8x+15}\le 0$ is

1. $[1,5]$
2. $[1,5]-\left\{3\right\}$
3. $[1,5)-\left\{3\right\}$
4. $(1,5)$

Q. The solution set of $x^3-10x^2+21x>0$ is

1. $(0,3)$
2. $(0,\infty )$
3. $(3,7)$
4. $(0,3)\cup (7,\infty )$

Q.

Which of the following is logically equivalent to $~\left(~p\Rightarrow q\right)$?

1. $p\wedge q$

2. $p\wedge ~q$

3. $~p\wedge q$

4. $~p\wedge ~q$

Q.

By using properties of determinants, show that:

Q. The seolution set of $x^2-5x+6\ge 2$ is

1. $[2,3]$
2. $(2,3)$
3. $[1,4]$
4. $(-\infty ,1]\cup [4,\infty )$

Q. The solution set of $3^{x^2-4}\ge 243$ is

1. $(-2,2)$
2. $[-2,2]$
3. $(-\infty ,-2]\cup [2,\infty )$
4. $(-\infty ,-3]\cup [3,\infty )$

Q. All the values of $x$ for which $x^2-5x+6$ is non-negative are

1. $(2,3)$
2. $[2,3]$
3. $(-\infty ,2)\cup (3,\infty )$
4. $(-\infty ,2]\cup [3,\infty )$

Q.

The function$f\left(x\right)={\left[x\right(x-2\left)\right]}^2$is increasing in the set

1. $(-\infty ,0)\cup (2,\infty )$

2. $(-\infty ,1)$

3. $(0,1)\cup (2,\infty )$

4. $(1,2)$

5. $(0,2)$

Q. The solution set of $x^2+1<10$ is

1. $[-3,3]$
2. $(-3,3)$
3. $R$
4. $\varphi \text{(empty set)}$

Q. If x=3 - square root of 5 then the value of square root of x/2 +whole square root of 3x-2 will be equal to

Q. Let $\alpha =\underset{x\in R}{max}\{8^{2\sin 3x}\cdot 4^{4\cos 3x}\}$ and $\beta =\underset{x\in R}{min}\{8^{2\sin 3x}\cdot 4^{4\cos 3x}\}.$
If $8x^2+bx+c=0$ is a quadratic equation whose roots are ${\alpha }^{1/5}$ and ${\beta }^{1/5}$, then the value of $c-b$ is equal to

1. $43$
2. $42$
3. $50$
4. $47$