Properties of Inequalities

Q.

If $a\in R$ and the equation $-3{(x-[x\left]\right)}^2+2(x-[x\left]\right)+a^2=0$ (where, $\left[x\right]$denotes the greatest integer $\le x]$ has no integral solution, then all possible values lie in the interval )

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

2. $(1,2)$

3. $(-2,-1)$

4. $(-\infty ,-2)\cup (2,\infty )$

Q. Each of the following defines a relation on N:

(i) x > y, x, y $\in$ N
(ii) x + y = 10, x, y $\in$ N
(iii) xy is square of an integer, x, y $\in$ N
(iv) x + 4y = 10, x, y $\in$ N

Determine which of the above relations are reflexive, symmetric and transitive. [NCERT EXEMPLAR]

Q.

What is the domain of definition of the function $f\left(x\right)=\sqrt{{\log }_{10}\frac{5x-x^2}{4}}$

1. $\left[1,4\right]$

2. $\left[1,0\right]$

3. $\left[0,5\right]$

4. $\left[5,0\right]$

Q. If $A=\left[\begin{array}{cc}3& 1\\ -1& 2\end{array}\right]$, show that A² − 5A + 7I = O use this to find A⁴.

Q.

If $p\left(x\right)=x+3$, then $p\left(x\right)+p(-x)$ is equal to

1. $3$

2. $2x$

3. $0$

4. $6$

Q. Find the rate of change of the area of a circle with respect to its radius $r$ when
(a) $r=3\text{cm}$ (b) $r=4\text{cm}$

Q.

The only statement among the following which is a tautology, is

1. $B\to \left[A\wedge \left(A\to B\right)\right]$

2. $A\wedge \left(A\vee B\right)$

3. $A\vee \left(A\wedge B\right)$

4. $\left[A\wedge \left(A\to B\right)\to B\right]$

Q.

The radius of the circle is $\left|\frac{\left(z-i\right)}{\left(z+i\right)}\right|=5$ is given by

1. $\frac{13}{12}$

2. $\frac{5}{12}$

3. $5$

4. $625$

Q. The solution set of the inequation $\frac{x^2+6x-7}{|x+4|}<0$ is

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

Q.

|x^2 -1| +x+1 =0

Q.

Find the domain of each of the following real valued functions of real variable:
(i) $f\left(x\right)=\sqrt{x-2}$
(ii) $f\left(x\right)=\frac{1}{\sqrt{x^2-1}}$
(iii) $f\left(x\right)=\sqrt{9-x^2}$
(iv) $f\left(x\right)=\sqrt{\frac{x-2}{3-x}}$

Q. Let $f$, be a continuous function in $[0,1]$, then $\underset{n\to \infty }{lim}\sum _{j=0}^n\frac{1}{n}f\left(\frac{j}{n}\right)$ is

1. $\underset{1/2}{\overset{1}{\int }}f\left(x\right)dx$
2. $\underset{0}{\overset{1}{\int }}f\left(x\right)dx$
3. $\frac{1}{2}\underset{0}{\overset{1/2}{\int }}f\left(x\right)dx$
4. $\underset{0}{\overset{1/2}{\int }}f\left(x\right)dx$

Q. Find the rate of change of the area of a circle per second with respect to its radius $r$ when $r=5\text{cm}.$

Q. The solution set of $x^2-16\le 0$ and $x^2-9\ge 0$ is

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

Q. The sum of four integers in AP is 24 and their product is 945. Find the numbers.

Q. If $A=\{x:x^2-4\le 0,x\in Z\}$ and $B=\{y:y^2-9\ge 0,y\in Z\}$, then $n(A\cap B)$ is equal to

1. $1$
2. $4$
3. $0$
4. $10$

Q.

The maximum value of $x{e}^{-x}$ is

1. $e$

2. $\frac{1}{e}$

3. $-e$

4. $-\frac{1}{e}$

Q. The greatest integral value of $x$ which can satisfy the inequality $\frac{x^2-6x+3}{x^2-4x+3}<0$ is

Q.

If -3x+17<-13, then

1. $xϵ(10,\infty )$

2. $xϵ[10,\infty )$

3. $xϵ(-\infty ,10]$

4. $xϵ[-10,10)$

Q. If $\left|\begin{array}{cc}2x& x+3\\ 2\left(x+1\right)& x+1\end{array}\right|=\left|\begin{array}{cc}1& 5\\ 3& 3\end{array}\right|$, then write the value of x.

Q. If $I_1=\underset{0}{\overset{1}{\int }}{(1-x^{50})}^{100}dx,I_2=\underset{0}{\overset{1}{\int }}{(1-x^{50})}^{101}dx$, then

1. $51I_1=50I_2$
2. $101I_1=100I_2$
3. $5051I_1=5050I_2$
4. $5050I_1=5051I_2$

Q. Let $S_1,S_2,S_3,\dots ,S_n$ be squares such that for each $n\ge 1$, the length of a side of $S_n$ equals the length of a diagonal of $S_{n+1}$. If the length of a side of $S_1$ is $10$ cm, then for which of the following values of $n$ is the area of $S_n$ less than $1$ sq. cm?

1. $7$
2. $9$
3. $8$
4. $10$

Q. For $k\ne 0$, if $x>y$, then the inequality which is not always correct is

1. $x+k>y+k$
2. $x-k>y-k$
3. $kx>ky$
4. $\frac{x}{k^2}>\frac{y}{k^2}$

Q.

$\frac{5x}{2}+\frac{3x}{4}\ge \frac{39}{4}$

Q.

If either a = 0 or b = 0, then a.b = 0. But the converse need not to be true. Justify your answer with an example.

Q.

If $f\left(x\right)=k{x}^3-9x^2+9x+3$ is monotonically increasing in each interval, then

1. $k<3$

2. $k\le 3$

3. $k>3$

4. None of these

Q. If $\frac{4}{x}\ge 7$, then the range of $x$ is

1. $(-\infty ,0)$
2. $(-\infty ,\frac{7}{4}]$
3. $(0,\frac{4}{7}]$
4. $(-\infty ,0)\cup [\frac{4}{7},\infty )$

Q.

The domain of $f\left(x\right)=\frac{1}{\sqrt{2x-1}}-\sqrt{1-x^2}$ is

1. $]0.5,1]$

2. $[-1,\infty [$

3. $[1,\infty [$

4. None of these

Q.

Solve the following linear inequations in R.

Solve : 4x-2<8, when

(i) $xϵR$ (ii) $xϵZ$ (iii) $xϵN$

Q.

Find the number of non-zero integral solutions of the equation.