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The ratio in which the line $3x+4y+2=0$ divides the distance between the lines $3x+4y=0$ and $3x+4y-5=0$ is

Number of binary operation on the set { a, b } is

The coefficient of $x^8{y}^{10}$ in the expansion of $(x+y{)}^{18}$ is

The value of $\underset{-\pi /2}{\overset{\pi /2}{\int }}\frac{dx}{\left[x\right]+\left[\sin x\right]+4},$ where $\left[t\right]$ denotes the greatest integer less than or equal to $t$, is :

If a triangle has its orthocentre at $(1,1)$ and circumcentre at $(\frac{3}{2},\frac{3}{4}),$ then the coordinates of the centroid of the triangle are

If cos 2x = cos 60^o cos 30^o + sin 60^o sin 30^o then x is equal to

The unit vector in ZOX plane and making angle ${45}^{\circ }$ and ${60}^{\circ }$ respectively with $\overrightarrow{a}=2\hat{i}+2\hat{j}-\hat{k}$ and $\overrightarrow{b}=0\hat{i}+\hat{j}-\hat{k}$

If $\frac{si{n}^{4}A}{a}+\frac{co{s}^{4}A}{b}=\frac{1}{a+b},$ then $\frac{si{n}^{8}A}{a^3}+\frac{co{s}^{8}A}{b^3}=$

General solution of $(x+y{)}^2\frac{dy}{dx}=a^2,a\ne 0$ is ($c$ is arbitrary constant)

Directions:- Study the following information carefully to answer the question given below.

Six persons - A, B, C, D, E and F - are sitting in two cars namely X and Y but not necessarily in the same order. Out of the six persons, two are driving the cars. There are three persons in each car and one person must be on the front seat beside the person who is driving the car. In car X, D is neither driving nor sitting in the back seat. F is sitting on the back seat in Car Y. C is in the driver's seat but not in the Car X. A is neither driving nor travelling in the Car Y. E is not in the driver's seat in any car.

Which of the following groups of three persons are travelling in the Car X?

If $z+\frac{1}{z}=2\cos \theta$, where $z$ is complex number and $i=\sqrt{-1}$, then $\frac{z^n-1}{z^n+1}$ is

For all complex numbers $z_1,z_2$ satisfying $|z_1|=20$ and $|z_2-5-12i|=4$, the minimum value of $|z_1-z_2|$ is

The adjoint of the matrix $\left[\begin{array}{ccc}1& 2& 3\\ 1& 1& 1\\ 2& 3& 4\end{array}\right]$ is

The centre of the conic section $2x^2+4y^2+2xy+4x-6y+17=0$ is

The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are Rs 80, Rs 60 and Rs 40 each subject respectively. Find the total amount of the bookshop will receive from selling the books using matrix algebra.

Integrate the function.
$\int x(logx{)}^{2}dx$

Determine order and degree (when defined) of differential equations.
${\left(\frac{ds}{dt}\right)}^4+3s\frac{d^{2}s}{d{t}^2}=0.$

Evaluate the definite integrals.
${\int }_1^2\frac{5x^2}{x^2+4x+3}dx$

The rate of change of the area of a circle with respect to its radius r at r = 6 cm is

(a) $10\pi$

(b) $12\pi$

c) $8\pi$

d) $11\pi$

Differentiate given problems w.r.t.x.

$\frac{co{s}^{-}1\frac{x}{2}}{\sqrt{2x+7}}-2<x<2$.

Evaluate the definite integrals.
${\int }_0^1(x{e}^x+sin\frac{\pi .x}{4})dx$

Prove the following:
$\frac{\text{sin x - sin y}}{\text{cos x + cos y}}=\text{tan}\left(\frac{x-y}{2}\right)$

If $\overrightarrow{p}=\frac{\overrightarrow{b}\times \overrightarrow{c}}{\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]},\overrightarrow{q}=\frac{\overrightarrow{c}\times \overrightarrow{a}}{\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]}$ and $\overrightarrow{r}=\frac{\overrightarrow{a}\times \overrightarrow{b}}{\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]}$, where $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ are three non-coplanar vectors, then the value of the expression $(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})\cdot (\overrightarrow{p}+\overrightarrow{q}+\overrightarrow{r})$ is

Represent to solution set of each of the following in equations graphically in two dimensional plane :

$x+2y-y\le 0$

The equation of the image of circle $x^2+y^2-2x-4y=0$ in the line $3x-4y+30=0$ is

Suppose the parabola $(y–k{)}^2=4(x–h)$, with vertex $A$, passes through $O=(0,0)$ and $L=(0,2)$. Let $D$ be an end point of the latus rectum. Let the $y$-axis intersect the axis of the parabola at $P$. Then $\angle PDA$ is equal to

Solve the following quadratics $5x^2-6x+2=0$

If A + B + C =$\pi$. Prove that

cos 2A + cos 2B + cos 2C = -1 -4 cos A cos B cos C.

A card is drawn at random from a pack of 52 cards: Find the probability that the card drawn is :

(i) a black king

(ii) either a black card or a king

(iii) black and a king

(iv) a jack, queen or a king

(v) neither a heart nor a king

(vI) spade or an ace

(vii) neither an ace nor a king

(viii) a diamond card

(ix) not a diamond card

(x) a black card

(xi) not an ace

(xii) not a black card.

Three dice are thrown at the same time. Find the probability of getting three two's, if it is known that the sum of the numbers on the dice was six.

The point of intersection of normals to the parabola $y^2=4x$ at the points whose ordinates are $4$ and $6$ is

Find the area of the region bounded by the curves $y=x^2+2,y=x,x=0$ and x = 3.

Verify that tan30°=tan60°-tan30°/1+tan60°×tan30°

If $f:[-5,5]\to R$ is a differentiable function and if $f\left(x\right)$ does not vanish anywhere, then prove that $\int (-5)\ne \int \left(5\right)$.

Choose the correct answer
The value of $\hat{i}.(\hat{j}\times \hat{k})+\hat{j}.(\hat{k}\times \hat{i})+\hat{k}(\hat{i}\times \hat{j})$ is
a) Zero
b) - 1
c) 1
d) 3

Differentiate the following functions with respect to x

$2\sqrt{cot\left(x^2\right)}$

If $x\left[\begin{array}{c}2\\ 3\end{array}\right]+y\left[\begin{array}{c}-1\\ 1\end{array}\right]=\left[\begin{array}{c}10\\ 5\end{array}\right]$, then find the values of x and y.

Sum of the series $\sum _{r=0}^n(-1{)}^r{}^n{C}_r\cdot [i^{5r}+i^{6r}+i^{7r}+i^{8r}],$ is