Inequality

Q. Given $n^4<{10}^n$ for a fixed positive integer $n\ge 2$, then $(n+1{)}^4<{10}^{n+1}$

1. True
2. False

Q. The function $f:\text{N}\to \text{R}$ is defined by $f\left(\text{n}\right)=2^{\text{n}}$. Find the range of the function.

1. The set of all even positive integers
2. $\text{N}$
3. $\text{R}$
4. A subset of the set of all even positive integers

Q.

Evaluate $\sin {25}^0\cos {65}^0+\cos {25}^0\sin {65}^0$

Q. Find the missing frequencies in the following frequency distribution whose mean is $50$ and total frequency is $150$.

$x$	$10$	$30$	$50$	$70$	$90$	$Total$
$f$	$17$	$f_1$	$32$	$f_2$	$19$	$120$

Q. Question 7
$If\Delta ABC\sim \Delta DEF$,
AB = 4 cm, DE = 6 cm, EF = 9 cm and FD = 12 cm, then find the perimeter of $\Delta ABC$.

Q. The sum of the cubes of the legs of a right angled triangle is less than the cube of teh hypotenuse.

1. True
2. False

Q. A.M. of the square root of products taken two together of n +ive quantities is less than A.M. of the n quantities itself.

1. True
2. False

Q. $(a_1^2+a_2^2+\dots \dots a_n^2)(b_1^2+b_2^2+\dots \dots +b_n^2)\ge (a_1{b}_1+a_2{b}_2+\dots \dots +a_n{b}_n{)}^2$

1. True
2. False

Q. Solve the following pairs of equations graphically.
3x + y = 12
x - 3y = -6
Determine the co-ordinate of the vertices of the triangle so formed by the Y- axis line and X - axis , Also find its area.

Q. What is the least value of x possible if -1 < x ≤ 1 and x is a real number?

1. (a) -1
2. (b) 1
3. (c) -0.99
4. (d) None of these

Q. If x, y, z are three + ive real numbers, then minimum value of $\frac{y+z}{x}+\frac{z+x}{y}+\frac{x+y}{z}$ is

1. 1
2. 6
3. 3
4. 2

Q. The daily sale of kerosene (in litres) in a ration shop for six days is as follows : $75$, $120$, $12$, $50$, $70.5$ and $140.5$ The average daily sale is

1. $150$
2. $10$
3. $142$
4. $78$

Q. If a > b > 1, then $a^n-b^n>n(a-b)$ for every + ive integer $n\ge 2$.

1. True
2. False

Q. Find the value of '$x$'
$\frac{x+1}{x-1}-\frac{x-1}{x+1}=\frac{5}{6}$ where $x\ne 1,-1$

Q. If $x=\frac{\sqrt{b^2+ab}+\sqrt{b^2-ab}}{\sqrt{b^2+ab}-\sqrt{b^2-ab}}$ then find the value of ${ax}^2-2bx$

Q.

The largest interval for which $x^{22}-x^{19}+x^{14}-x^5+1>0$ is

1. $(-\infty ,\infty )$

2. $(-\infty ,1)$

3. $(1,\infty )$

4. $(0,1)$

Q. If $abc$ are in H.P. then prove that $\frac{1}{b-a}+\frac{1}{b-c}=\frac{1}{a}+\frac{1}{c}$

Q. If a, b, c are all +ive real numbers which one of the following holds good?

1. (b + c) (c + a) (a + b) > 8abc
   or (1 - a) (1 - b) (1 - c) > 8abc if a + b + c = 1
2. (a + b + c) (bc + ca + ab) > 9abc
   or $(a+b+c)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})>9$
3. $(\frac{a}{e}+\frac{b}{f}+\frac{c}{g})(\frac{e}{a}+\frac{f}{b}+\frac{g}{c})>g$
4. If $x_i>0,(i=1,2,\dots \dots n)$, then
   $(x_1+x_2+\dots \dots +x_n)(\frac{1}{x_1)}+\frac{1}{x_2}+\dots \dots +\frac{1}{x_n})\ge n^2$
5. If $a_i^s$ are all +ive real numbers, then
   $(1+a_1+a_1^2)(1+a_2+a_2^2)\dots \dots (1+a_n+a_n^2)\ge 3^n{a}_1{a}_2\dots a_n$

Q. If the arithmetic average of date given below be $165$ rupees, find the missing area

Monthly Wages in $\left(Rs\right)$	$100$	$150$	$200$	__	$300$	$500$
Number of Labourers	$30$	$20$	$15$	$10$	$4$	$1$

Q. $\underset{\alpha \to \frac{\pi }{4}}{lim}\frac{\sin \alpha -\cos \alpha }{\alpha -\frac{\pi }{4}}=$

1. $\sqrt{2}$
2. $\frac{1}{\sqrt{2}}$
3. $1$
4. None of these

Q.

In a right-angled triangle, If $cotx$ = 4, what is the value of $tanx$?

1. 0.5

2. 2

3. 0.25

4. 1