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The integrating factor of the differential equation $\cos x\frac{dy}{dx}+\sin x\cdot y=x\cdot \sec x,(\cos x\ne 0)$ is

Answer each of the following questions in one word or one sentence or as per exact requirement of for question:

$Inany\Delta ABC,\text{find the value of}\sum a(sinB-sinC)$

If the parabolas $y^2=4b(x-c)$ and $y^2=8ax$ have a common normal, then which one of the follwing is a valid choice for the ordered triplets $(a,b,c)$?

In how many ways 15 chocolates can be distributed among 3 boys so that each one gets at least one chocolate no two boys gets equal number of chocolates.

If A ={1,2,3,4}, define relations on A which have properties of being
(iii)reflexive,symmetric and transitive.

If the distance between the points $(5,-2)$ and $(1,a)$ is $5$ units, then the value of $a$ can be

The series of natural numbers is divided into groups as follows ; (1), (2,3),(4,5,6), (7,8,9,10) and so on. The sum of the numbers in the Nth group is

In a chess tournment, where the participants were to play one game with another. Two chess players fell ill, having played three games each. If the total number of games played is $84$, the number of participants at the beginning was:

Integrate the function.
$\int (x^2+1)logxdx.$

If A and B are two matrices of order '3' such that $3A+4B{B}^T=I$ and $B^{-1}=A^T$, then identify which of the following statements is/are correct?

Length of intercepts made by circle $x^2+y^2-10x-8y+4=0$ on the $X$ and $Y$ axes respectively are

If $H_1,H_2,\dots \dots ,H_{20}$ be 20 harmonic means between $2$ and $3$, then the value of $\frac{H_1+2}{H_1-2}+\frac{H_{20}+3}{H_{20}-3}$ is

If $\cos \alpha +\cos \beta +\cos \gamma =0=\sin \alpha +\sin \beta +\sin \gamma =0$ then $\cos (2\alpha -\beta -\gamma )+\cos (2\beta -\gamma -\alpha )+\cos (2\gamma -\alpha -\beta )=$

Integrate the following functions.
$\int \frac{1}{(1-tanx)}dx.$

Compute the following:
$\left(iii\right)\left[\begin{array}{ccc}-1& 4& -6\\ 8& 5& 16\\ 2& 8& 5\end{array}\right]+\left[\begin{array}{ccc}12& 7& 6\\ 8& 0& 5\\ 3& 2& 4\end{array}\right]$

The principal amplitude of $(2-i)(1-2i{)}^2$ is in the interval :

Which of the following is always true if $|x-2|$ = $x-2$

Find the coefficient of $x^{8}in(1+x{)}^8(1-x{)}^{-2}$.

Equation of the ellipse with foci ($\pm$4,0) and length of latusrectum$\frac{20}{3}$is

Let $f:[0,1]\to [0,1]$ be a continuous function such that $x^2+\left(f\right(x){)}^2\le 1$ for all $x\in [0,1]$ and $\underset{0}{\overset{1}{\int }}f\left(x\right)dx=\frac{\pi }{4}$, then $\underset{\frac{1}{2}}{\overset{\frac{1}{\sqrt{2}}}{\int }}\frac{f\left(x\right)}{1-x^2}dx$ equals

If ${\tan }^2{45}^0-{\cos }^2{30}^0=x\sin {45}^0\cos {45}^0,thenx=?$

If a matrix has 6 elements, the number of possible orders it can have are

A hyperbola intersects an ellipse $x^2+9y^2=9$ orthogonally. The eccentricity of the hyperbola is reciprocal of that of ellipse. If the axes of the hyperbola are along coordinate axes, then

A series of concentric ellipses E₁, E₂, ................, E_n are drawn such that E_n touches the extremities of the major axis of _En-1 and the foci of E_n coincide with the extremities of minor axis of _En-1. If the eccentricity of the ellipses is independent of n, then the value of the eccentricity is

Evaluate the following :

(i) ${\sum }_{n=1}^{11}(2+3^n)$

(ii) ${\sum }_{k=1}^n(2^k+3^{k-1})$

(iii) ${\sum }_{n=2}^{10}{4}^n$

If 3rd, 4th, 5th and 6th terms in the expansion of $(x+\alpha {)}^n$ be respectively a,b,c and d. prove that $\frac{b^2-ac}{c^2-bd}=\frac{5a}{3c}.$

If the coefficients of 2nd, 3rd and 4th terms in the expansion of $(1+x{)}^n$ are in A.P., then find the value of

Let $\frac{3\pi }{4}<\theta <\pi$ and $\sqrt{2\cot \theta +\frac{1}{{\sin }^2\theta }}=K-\cot \theta ,$ then $K$ equals

Write the coordinates of the centre of the circle passing through (0, 0), (4, 0) and (0, -6).

How many four digit different numbers, greater than 5000 can be formed with the digits 1,2,5,9,0 when repetition of digits is not allowed?

If $y=lo{g}_{sinx}\left(tanx\right)then\frac{dy}{dx}atx=\frac{\pi }{4}$ is

If p,q be two A.M.'s and G be one G.M. between two numbers, then $G^2=$

If $\tan A-\tan B=x$ and $\cot B-\cot A=y$, where $x,y\ne 0$, then $\cot (A-B)$ is

Prove that:

(sin 3x + sin x) sin x + (cos 3x-cos x) cos x = 0

Let $P$ be a variable point on the ellipse $\frac{x^2}{100}+\frac{y^2}{64}=1$ with foci $F_1$ and $F_2$. If $A$ is the area of triangle $P{F}_1{F}_2$, then the maximum possible value of $A$ is

If P(2n-1,n): P(2n+1,n-1)= 22:7 find n.

The domain of the function $f\left(x\right)=\sqrt{(x^2-5x+6)}+\sqrt{(8-x^2+2x)}$ is

The minor of the element $3$ in the determinant $\left|\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right|$ is .......

__

A line with direction ratios (2, 1, 2) intersects the lines $\overline{r}$ =$-j+\lambda (i+j+k)$ and $\overline{r}$ = $-i+\mu (2i+j+k)$ at A and B, then AB =

A person goes to office by car or scooter or bus or train, probability of which are $\frac{1}{7},\frac{3}{7},\frac{2}{7}$ and $\frac{1}{7}$ respectively. Probability that he reaches office late, if he takes car, scooter, bus or train is $\frac{2}{9},\frac{1}{9},\frac{4}{9}$ and $\frac{1}{9}$ respectively. Given that he reached office in time, the probability that he travelled by a car is

The coordinates of a moving point particle in a plane at time $t$ is given by $x=a(t+\sin t),y=a(1-\cos t).$ The magnitude of acceleration of the particle is