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The ratio of coefficient of $x^{15}$ to the term independent of $x$ in the expansion of ${(x^2+\frac{1}{2x})}^{15}$ is

Find the domain and range of the function $f\left(x\right)=\frac{|x-4|}{x-4}$.

Let Z be a complex Numbers satisfying the equation $z^2$ - (11 + i) z+k+3i = 0 where k $ϵ$ R and i = $\sqrt{(-i)}$, suppose equation has one real and one non-real root. Find the non-real root.

If $\cos y\cos (\frac{\pi }{2}-x)-\cos (\frac{\pi }{2}-y)\cos x+\sin y\cos (\frac{\pi }{2}-x)+\cos x\sin (\frac{\pi }{2}-y)=0$, then which of the following is correct

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) $f\left(x\right)=\sqrt{\frac{x-2}{3-x}}$

The solution set of the equation $ta{n}^{-1}x-co{t}^{-1}x=co{s}^{-1}(2-x)$ is

Let $a,b$ be the roots of the equation $x^2-x+\sin 2\theta \cos \theta =x\sin \theta (2-\sin \theta )$ where $\frac{\pi }{4}<\theta <\frac{\pi }{2}$ and a>b. Find the value of $a+\frac{b}{a+\frac{b}{a+\frac{b}{a+...}}}$

IF cot X =(1-n)cot(x-y), then prove that

tan y= n tan x/ (sec 2x - n tan 2x)

A point moves such that the sum of square of it's distances from the planes $x-z=0,x-2y+z=0$ and $x+y+z=0$ is $36.$ Then the locus of the point is

Let $z=x+iy$ be a complex number such that $|z|=1$, where $i=\sqrt{-1}$. Match List - I with List - II.



$\begin{array}{cccc}\text{List-I}& & \text{List - II}& \\ \text{(I)}& \text{Re}\left(\frac{iz}{1+z^2}\right)\text{is equal to}& \text{(P)}& 0\\ \text{(II)}& \text{Im}\left(\frac{iz}{1+z^2}\right)\text{can be equal to}& \text{(Q)}& 1\\ \text{(III)}& \text{Number of integers NOT in the}& \text{(R)}& \frac{1}{2}\\ & \text{range of}\text{Im}\left(\frac{iz}{1+z^2}\right)\text{is equal to}& & \\ \text{(IV)}& \frac{1}{2\pi }\arg \left(\frac{iz}{1+z^2}\right)\text{is equal to}& \text{(S)}& -\frac{1}{2}\\ & (\text{where}-\pi <\arg (z)\le \pi )& & \\ & & \text{(T)}& -\frac{1}{4}\\ & & \text{(U)}& \frac{1}{4}\end{array}$



Which of the following is only CORRECT combination?

If the parabola $y^2=4ax$ passes through (-3,2), then length of its latus rectum is

${lim}_{x\to 1}(\frac{2}{1-x^2}+\frac{1}{x-1})=$___

Question 10

Students of a class voted for their favourite colour and a pie chart was prepared based on the data collected.

Which of the following is a reasonable conclusion for the given data?

(a) $\frac{1}{20}$th student voted for blue colour

(b) Green is the least popular colour

(c) The number of students who voted for red colour is two times the number of students who voted for yellow colour

(d) Number of students liking together yellow and green colour is approximately the same as those for red colour.

Evaluate $\underset{0}{\overset{1}{\int }}x{e}^{x}dx$

Mukul has earned an average of $4,000$ dollars for the first eleven months of the year. If he justifies his staying in the US based on his ability to earn at least $5000$ dollars per month for the entire year, then how much should he earn (in dollars) in the last month to achieve his required average for the whole year?

Let R be a relation in N defined by $(x,y)ϵR\iff x+2y=8$. Express R and $R^{-1}$ as sets of ordered pairs.

Let $OPQR$ be a square and $M$ and $N$ be the midpoints of the sides $PQ$ and $QR$ respectively. The ratio of the area of square to the triangle $OMN$ is

Which among the following shapes is NOT a polygon?

For each of the following compound statements first identify the connecting words and then break it into component statements.

(i) All rational numbers are real and all real numbers are not complex.

(ii) Square of an integer is positive or negative.

(iii) The sand heats up quickly in the Sun and does not cool down fast at night.

(iv) x = 2 and x = 3 are the roots of the equation 3x² – x – 10 = 0.

$Ifacurvesatisfiesydx–xdy+3x^2$ $y^2{e}^{x^3}$ dx = 0 and y(0) = 1 then y(1)=

If $I=\underset{-20\pi }{\overset{20\pi }{\int }}\left|\sin x\right|\left[\sin x\right]dx$, where $[\cdot ]$ denotes the greatest integer function, then the value of $I$ is

The feasible region of a LPP is shown in the figure. If $Z=3x-y,$ then the maximum value of $Z$ occurs at

The value of the integral ${\int }_a^{a+\frac{\pi }{2}}(|sinx|+|cosx|)dxis$

A function f(x) satisfies the relation ${\int }_2^{x}f\left(t\right)=\frac{x^2}{2}+{\int }_x^2{t}^{2}f\left(t\right)dt$
Then answer the following
The value of ${\int }_{-2}^{2}f\left(x\right)$ dx is

The value of $\underset{x\to 1}{lim}{(2-x)}^{\tan \left(\frac{\pi x}{2}\right)}$ is

Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that :

(i) $A\times C\subset B\times D$

(ii) $A\times (B\cap C)=(A\times B)\cap (A\times C)$

$ta{n}^{-1}\left[\frac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}+\sqrt{1-x^2}}\right]=$

Write the first five terms of the following sequence and obtain the corresponding series:

Given f(x) = $a{x}^2+bx+c$

If f(m) = a - b + c

f(n) = 4a + 2b + c

f(p) = a + b + c

Then, m + n + p = __

The interval(s) of $x$ which satisfies the inequality $\frac{1}{x+2}<\frac{1}{3x+7}$ is/are

Prove the following by using the principle of mathematical induction for all n ∈ N:

and f '(1) = 15, then the value of n is

If m is the A.M. of two distinct real numbers I and n(l, n > 1) and $G_1,G_2$ and $G_3$ are three geometric means between l and n, then $G_1^4+2G_2^4+G_3^4$ equals:

Using elementary transformations, find the inverse of the followng matrix.

$\left[\begin{array}{cc}2& 1\\ 7& 4\end{array}\right]$