Inequality

Q.

Using first principle, find the derivative of $tan\sqrt{x}.$

Q.

The value of i + $i^2$ + $i^3$ + $i^4$ is __

Q.

Write the following sets in the set-builder form:

(i) {3, 6, 9, 12} (ii) {2, 4, 8, 16, 32}

(iii) {5, 25, 125, 625} (iv) {2, 4, 6, ........}

(v) {1, 4, 9, ........, 100}

Q. If a, b, c are real numbers such that a + 2b + c = 4 then max. value of ab + bc + ca is

1. 2
2. 4
3. 6
4. 8

Q.

If $\cos \theta =\frac{3}{5}$ and $\cos \varphi =\frac{4}{5}$, where $\theta$ and $\varphi$ are positive acute angles, then $\cos \frac{(\theta -\varphi )}{2}=?$

1. $\frac{7}{\sqrt{2}}$

2. $\frac{7}{5\sqrt{2}}$

3. $\frac{7}{\sqrt{5}}$

4. $\frac{7}{2\sqrt{5}}$

Q.

Prove that sin $({40}^{\circ }+\theta )$ cos$({10}^{\circ }+\theta )$ - cos $({40}^{\circ }+\theta )$ sin $({10}^{\circ }+\theta )$ = $\frac{1}{2}$

Q. Suppose $x$ and $y$ are natural numbers, then the number of ordered pairs $(x,y)$ which satisfy $x+y\le 5$ is

1. $25$
2. $16$
3. $9$
4. $10$

Q.

Find the minimum value of 10 $co{s}^{2}x$ - 6 sinxcos x + 2 $si{n}^{2}x$

__

Q.

If $f$and $g$ are differentiable functions in $\left(0,1\right)$ satisfying $f\left(0\right)=2=g\left(1\right),g\left(0\right)=0$and $f\left(1\right)=6,$ then for some $c\in \left[0,1\right]$

1. $2f\left(c\right)=g\left(c\right)$

2. $2f\left(c\right)=3g\left(c\right)$

3. $f\left(c\right)=g\left(c\right)$

4. $f\left(c\right)=2g\left(c\right)$

Q.

If (a-b) sin ($\theta +\varphi$) = (a+b) sin ($\theta +\varphi$) and a tan$\left(\frac{\theta }{2}\right)$ - btan$\left(\frac{\varphi }{2}\right)$ = C, the the value of sin$\varphi$ is equal

1. 

2. 

3. 

4. 

Q.

Observe the following statements

$\left(i\right)In\Delta ABC,bco{s}^2\frac{C}{2}+cco{s}^2\frac{B}{2}=s$

$\left(ii\right)In\Delta ABC,cot\frac{A}{2}=\frac{b+c}{a}\Rightarrow B={90}^0$

Which of the following is correct?

1. Both I and II are true

2. I is true, II is false

3. I is false, II is true

4. Both I and II are false

Q.

Prove the following:

sin (n+1)x sin (n+2)x + cos (n+1)x cos (n+2)x = cos x

Q. If $\sqrt{\frac{1+i}{1-i}}=(a+ib)$ then show that $(a^2+b^2)=1$.

Q.

In the following, state whether A = B or not :

(i) A = {a, b, c, d} B = {d, c, b, a}

(ii) A = {4, 8, 12, 16} B = {8, 4, 16, 18}

(iii) A = {2, 4, 6, 8, 10} B = {x : x is a positive even integer and $x\le 10$

(iv) A = {x : x is a multiple of 10} B = {10, 15, 20, 25, 30, ...........}

Q.

If a, b, c, d are in G.P., prove that $(a^n+b^n),(b^n+c^n),(c^n+d^n)$ are in G.P.

Q.

Negation of $2+3=5$ and $8<10$ is

1. $2+3\ne 5$ and $8<10$

2. $2+3=5$ and $8\nless 10$

3. $2+3\ne 5$ or $8\nless 10$

4. None of these

Q. The integral value(s) of $x$ which satisfies the inequality $4x+5\le 2x+17$ is/are

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

Q.

The possible values of $x$ are given by $x+8\le 6$. What is the greatest possible value of $7x$? Explain your reasoning.

Q. In $8$ people, there are more girls than boys. How many girl could be there?

1. $5$
2. $6$
3. $7$
4. $8$

Q.

Prove that $0!=1.$

Q. $li{m}_{x\to 0}\left(\frac{3x+|x|}{7x-5|x|}\right)$=___

1. $\frac{1}{6}$
2. 2
3. does not exist
4. $\frac{3}{7}$

Q.

$co{s}^2{48}^0-si{n}^2{12}^0$ =

1. 

2. 

3. 

4. 

Q.

If z = x + iy, $z^{\frac{1}{3}}$ = a - ib and $\frac{x}{a}$ - $\frac{y}{b}$ = $\lambda$$(a^2-b^2)$, then $\lambda$ is equal to

1. 3

2. 4

3. 1

4. 2

Q.

If x, y, z are in A.P., then $3^{-x}$, $3^{-y}$, $3^{-z}$are in :

1. A.P.

2. G.P.

3. H.P.

4. None

Q. 21.Ravi obtained 70 and 75 marks in first two unit test. Find the minimum marks heshould get in the third test to have an average of at least 60 marks.

Q. 3(-2) < 5(2-3)

Q. If $y=3^{x-1}+3^{-x-1}$ (x real), then the least value of y is

1. 2
2. 6
3. $\frac{2}{3}$
4. None of these

Q.

Consider the set of bivariate data shown in the table.

$x$	$0$	$1$	$2$	$3$	$4$	$5$	$6$
$y$	$5$	$12.5$	$20$	$23.2$	$23.4$	$24.1$	$24.8$

Compute ${\left(\sum _{}y\right)}^2$

Q.

2 cos A = x + $\frac{1}{x}$ , 2cosB = y + $\frac{1}{y}$ . Find 2 cos(A-B)

1. 

2. 

3. 

4. 

Q.

Write down the truth set of each of the following open sentences:

(i) $p\left(x\right):x+5<9,x\in N.$

(ii) $p\left(x\right):x+3<3,x\in N.$

(iii) $p\left(x\right):x+5>7,x\in R.$

(iv) $P\left(x\right):2x^2+5x-3=0,x\in 1$