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Solution to the differential equation $\frac{x+\frac{x^3}{3!}+\frac{x^5}{5!}+\dots \dots }{1+\frac{x^2}{2!}+\frac{x^4}{4!}+\dots \dots }=\frac{dx-dy}{dx+dy}$ is

Find the equation to the straight line parallel to $3x-4y+6=0$ and passing through the middle point of the join of points $(2,3)$ and $(4,-1).$

A common tangent to $9x^2-16y^2=144and{x}^2+y^2=9$ is,

Find a and b such that 2a + 4ib and 2i represent the same complex numbers.

Suppose we have four boxes A, B, C and D containing coloured marbles as given below

$\begin{array}{cccc}& & Marblecolour& \\ Box& Red& White& Black\\ A& 1& 6& 3\\ B& 6& 2& 2\\ C& 8& 1& 1\\ D& 0& 6& 4\end{array}$

One of the boxes has been selected at random and a single marble is drawn from it. If the marble is red what is the probability that it was drawn from box A? box B? , box C?

The coefficient of $x^5$ in the expansion of $(1+x{)}^{21}+(1+x{)}^{22}+...+(1+x{)}^{30}$ is

Given z is a complex number satisfying $z^2-z-|z{|}^2+\frac{64}{|z{|}^5}=0$ and Re(z) $\ne \frac{1}{2}$ (where Re(z) denotes real part of z), then |z| is less than

Cards are drawn from a pack of 52 cards one by one. The probability that exactly 10 cards will be drawn before the first ace is:

The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line $4x-5y=20$ to the circle $x^2+y^2=9$ is

The number arrangements of the letters of the word INDEPENDENCE is

A bird is sitting on the top of a vertical pole $20m$ high and its elevation from a point $O$ on the ground is ${45}^{\circ }$. It flies off horizontally straight away from the point $O$. After one second, the elevation of the bird from $O$ is reduced to ${30}^{\circ }$. Then the speed (in $m/s$) of the bird is

'n' whole numbers are randomly chosen and multiplied, then probability that

$$\begin{array}{cc}ColumnI& ColumnII\\ \left(A\right)thelastdigitis1,3,7,or9& \left(P\right)\frac{8^n-4^n}{{10}^n}\\ \left(B\right)thelastdigitis2,4,6,r8& \left(Q\right)\frac{5^n-4^n}{{10}^n}\\ \left(C\right)thelastdigitis5& \left(R\right)\frac{4^n}{{10}^n}\\ \left(D\right)thelastdigitiszero& \left(S\right)\frac{{10}^n-8^n-5^n+4^n}{{10}^n}\end{array}$$

The domain of definition of the function $f\left(x\right)=\sqrt{x-1}+\sqrt{3-x}$ is

An unbiased coin is tossed. If the outcome is head then a pair of unbiased dice is rolled and the sum of the numbers obtained on them is noted. If the toss of the coin results in tail then a card from a well shuffled pack of nine cards numbered $1,2,\cdots 9$ is randomly picked and the number on the card is noted. The probability that the noted number is either $7$ or $8$ is :

We wish to select 6 persons from 8, but if the person A is chosen, then B must be chosen. In how many ways can the selection be made ?

If A = {1, 2, 3, 4, 5, 6, 7, 8, 9} Relation R from A to A by R = {(x, y):y = x + 3}. Find Domain, Co domain and Range of R.

The range of the function $f\left(x\right)=|x-1|$ is

The integral of the function $x\sin x$ is

Sum of the common roots of $z^{2006}+z^{100}+1=0$
and $z^3+2z^2+2z+1=0$ is

Without using derivatives find the maximum value of function g(x)=|x+2|-1

Column Matching
$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(I)}& \text{Let volume of a tetrahedron ABCD is}\frac{81}{2}\text{cube unit and}& \text{(P)}& \text{9}\\ & \text{volume of parallelepiped whose three coterminous edges are line}& & \\ & \text{segments joining centroid of any face of the tetrahedron with}& & \\ & \text{centroid of its other three faces is V cubic units, then V is}& & \\ \text{(II)}& \text{If the image of the point (1, 0, 1) in the plane x-y-z=1 is (a,b,c),}& \text{(Q)}& \text{8}\\ & \text{then the value of 3(a-b+c) is}& & \\ \text{(III)}& \text{Locus of all the points which are at a distance of 3 units}& \text{(R)}& \text{3}\\ & \text{from the line}\overrightarrow{r}=\lambda (\hat{i}+\hat{j}+\hat{k})\text{is given by}& & \\ & x^2+y^2+z^2-xy-yz-zx=\frac{k}{2}\text{, then the value of}\frac{k}{9}\text{is}& & \\ \text{(IV)}& \overrightarrow{a},\overrightarrow{b}\text{and}\overrightarrow{c}\text{are such that}|\overrightarrow{a}|=\sqrt{3},|\overrightarrow{b}|=2,|\overrightarrow{c}|=\sqrt{6}.& \text{(S)}& \text{7}\\ & \text{If}\overrightarrow{a}\cdot \overrightarrow{b}<0,\overrightarrow{c}\text{is perpendicular to both}\overrightarrow{a}\&\overrightarrow{b}& & \\ & \text{and}\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\text{form the coterminous edges of a tetrahedron}& & \\ & \text{of unit volume, then the angle between}\overrightarrow{a}\&\overrightarrow{b}\text{is}\frac{p\pi }{q}\text{(where p \& q}& & \\ & \text{are coprime numbers), then the value of}(p+q)\text{is}& & \end{array}$

Which of the following is only "CORRECT" combination?

If the roots of the quadratic equation $a(x-1{)}^2+2b(x-2)+c(x-1)+4=0$ are imaginary, where $a,b,c\in R$ and $b>2$, then

In the first paragraph of the passage, the author's attitude toward the literary critics mentioned can best be described as:

The first order, first degree, linear differential equation among the following is .

The co-ordinates of the extremities of the latus rectum of the parabola $5y^2=4x$ are

A cylindrical tank is filled by pumping water from a cuboidal tank of dimensions 200cm × 150cm × 95 cm. The radius of the cylindrical tank is 60cm and height is 95cm. Find the height(in m) of the water left in the cuboidal tank after the cylindrical tank is completely filled. (Take $\pi$ = 3.14)

Expand ${(x^3-\frac{2}{x^2})}^6$ using binomial expansion.

Evaluate the following limit:
$\underset{x\to 0}{\text{lim}}\frac{(x+1{)}^5-1}{x}$

The value of $\cos 1^{\circ }\cdot \cos 2^{\circ }\cdot \cos 3^{\circ }\cdots \cos {180}^{\circ }$ is

A telephone company in town has 500 subscribers on its list and collects fixed charges of 300 per subscriber per year. The company proposes to increase the annual subscription and it is believed that for every increase of 1, one subscriber will discontinue the service. Find what increase will bring maximum profit.

The values of x satisfying $lo{g}_3(x^2+4x+12)=2are$

Let O be the origin and OX, OY, OZ be three unit vectors in the directions of the sides QR, RP, PQ respectively, of a triangle PQR.
If the triangle PQR varies, then the minimum value of $\cos (P+Q)+\cos (Q+R)+\cos (R+P)$ is

Equation of the circle which touches the lines x = 0, y = 0 and 3x + 4y = 4 is

If $(2^n+1)\theta =\pi ,\text{then}{2}^{n}cos\theta cos2\theta \cdots cos{2}^2\theta \cdots 2^{n-1}\theta =$

If f(x+y)=f(x) + f(y) + 2xy - 6 for all $x,yϵR$ and f'(0) = 2, then y = f(x) will be

The range of the function $f\left(x\right){=}^{7-x}{P}_{x-3}$ is

Let $z$ be an imaginary complex number satisfying $|z-1|=1.$ If $\alpha =2z,$ $\beta =2\alpha$ and $\gamma =2\beta ,$ then the value of $|z{|}^2+|\alpha {|}^2+|\beta {|}^2+|\gamma {|}^2+|z-2{|}^2+|\alpha -4{|}^2+|\beta -8{|}^2+|\gamma -16{|}^2$ is

One of the two events must happen. Given that the chance of one is two -third of the other, find the odds in favour of the other.