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Differentiate the following functions with respect to x :

$a_0{x}^n+a_1{x}^{n-1}+a_2{x}^{n-2}+....+a_{n-1}x+a_n.$

If for all real values of $x,\frac{4x^2+1}{64x^2-96xsin\alpha +5}<\frac{1}{32}$, then $\alpha$ lies in the interval.

The distance between the plane $\overrightarrow{r}.(\hat{i}+5\hat{j}+\hat{k})=5$
and the line $\overrightarrow{r}=2\hat{i}-2\hat{j}+3\hat{k}+\lambda (\hat{i}-\hat{j}+4\hat{k})$ is

Let two matrices $A=\left[\begin{array}{ccc}1& 2\omega & {\omega }^2\\ 3& -1& 5\\ 2& i& \omega \end{array}\right]$ and $B=\left[\begin{array}{cc}2& 3\\ 1& i\\ 2& \omega \end{array}\right],$ where $i^2=-1$ and $\omega$ is a cube root of unity. Then which of the following statement(s) is(are) correct ?

If $\cos \left(2{\sin }^{-1}x\right)=\frac{1}{9}$, then $x=$

Sum upto $n$ terms of the series $y_n=1+(1+x)+(1+x+x^2)+(1+x+x^2+x^3)+\cdots +(1+x+x^2+\cdots +x^n)$ is true for $n\in N,$ then $y_n$ is

Let f (x) be a function such that f (x) = x - [x], where [x] is the greatest integer less than or equal to x. Then the number of solutions of the equation $f\left(x\right)+f\left(\frac{1}{x}\right)=1$ is (are)

Evaluate $\int \frac{\cos x}{\cos (x-a)}dx$

(where $C$ is constant of integration)

The value of $\int \left(\cos x\tan x\right)dx$ is

(where $C$ is constant of integration)

The domain of the function $f\left(x\right)=\sqrt{\frac{|1-x|(|x|-1)}{(4-|x|)|x-2|}}$ is

Determine the domain and range of the relation R defined by

(i) $R=\left\{\right(x,x+5):xϵ(0,1,2,3,4,5\left)\right\}$

(ii) $R=\left\{\right(x,x^3):x$ is a prime number less than 10}.

An alternating current is given by $I=i_{1}cos\omega t+i_{2}sin\omega t$. The rms current is given by

Find out the wrong number in the series given below :

$3,15,34,63,99,143$

If $f\left(x\right)$ is polynomial of degree $3$ with leading coefficient $2$ and $f\left(1\right)=4,f\left(2\right)=7,f\left(3\right)=12$ and $f\left(5\right)=11\alpha -1,$ then $\alpha =$

Find the values of x for
whichis
an increasing function.

Let f and g be real functions, defined by $f\left(x\right)=\sqrt{x+2}$ and $g\left(x\right)=\sqrt{4-x^2}$. Find

(i) $(f+g)\left(x\right)$

(ii) $(f-g)\left(x\right)$

(iii) $\left(fg\right)\left(x\right)$

(iv) $\left(ff\right)\left(x\right)$

(v) $\left(gg\right)\left(x\right)$

(vi) $\left(\frac{f}{g}\right)\left(x\right)$

For a given set $X=\{2,4,6\},$ tap the bubbles having a subset of $X.$

If the tangent to $y^2=4ax$ at the point $(a{t}^2,2at)$ where $|t|>1$ is a normal to $x^2-y^2=a^2$ at the point $(a\sec \theta ,a\tan \theta ),$ then

If the angle of elevation of a cloud from a point $P$ which is $25\text{m}$ above a lake be ${30}^{\circ }$ and the angle of depression of reflection of the cloud in the lake from $P$ be ${60}^{\circ }$, then the height of the cloud (in meters) from the surface of the lake is :

If the asymptotes of the hyperbola $(x+y+1{)}^2-(x-y-3{)}^2=5$ cuts each other at A and the coordinate axes at B and C, then radius of the circle passing through the points A, B, C is

Find the angle between X-axis and the line joining the points (3, -1) and (4, -2).

The conditional $(p\wedge q)\Rightarrow p$ is ___.

Question 110

The following data represents the approximate percentage of water in varioous oceans. Prepare a pie chart of the given data.

$\begin{array}{cc}\text{Ocean}& \text{Percentage of water}\\ \text{Pacific}& 40\%\\ \text{Atlantic}& 30\%\\ \text{Indian}& 20\%\\ \text{Others}& 10\%\end{array}$

In a class, for a competition, atleast 3 students are needed. Which of the following graph shows the inequality for this situation?

If $x\sin \left(\frac{y}{x}\right)dy=[y\sin \left(\frac{y}{x}\right)-x]dx,x>0\text{and}y\left(1\right)=\frac{\pi }{2}$ then the value of $\cos \left(\frac{y}{x}\right)$ is

If $z=x+iy$ and $Re\left(z^2\right)=0$, then the locus of $z$ can be

For certain values of $a,m$ and $b,$ the function

$f\left(x\right)=\{\begin{array}{cc}3,& x=0\\ -x^2+3x+a,& 0<x<1\\ mx+b,& 1\le x\le 2\end{array}$ satisfies the hypothesis of the mean value theorem for the interval $[0,2].$ Then the value of $a+b+m$ is

The set of values of $x$ which satisfy the inequality ${\cos }^{-1}\left(\frac{4xf\left(x\right)}{\pi }\right)\le \frac{\pi }{3};$ where $f\left(x\right)=({\sin }^{-1}x{)}^2+\pi {\cos }^{-1}x-({\cos }^{-1}x{)}^2$ is

The number of ways in which the letters of the word TRIANGLE can be arranged such that two vowels do not occur together is

Let $f:R\to R$ be a function such that $f\left(2\right)=4$ and $f^{\prime }\left(2\right)=1.$ Then, the value of $\underset{x\to 2}{lim}\frac{x^{2}f\left(2\right)-4f\left(x\right)}{x-2}$ is equal to

In a GP, first term is a and the common ratio is r. If A and H are the arithmetic means and harmonic means respectively for the first n terms of GP. Them A * H is equal to

Number of $2$-digit numbers (having different digits), which are divisible by $5$ is