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$\text{if}si{n}^{-1}x+si{n}^{-1}y=\frac{2\pi }{3}\text{and}co{s}^{-1}x-co{s}^{-1}y=\frac{\pi }{6},\text{then}x=$

If A is 3×4matrix and B is a matrix such that A^TB and B×A^T are both defined then order of B is

Here A^T is transpose of A

The image of the line $20x=15(y-3)=12z$ in the plane $ax+by+cz=d$ is $\frac{100x}{3}+21=25y-54=20z-21$. The distance of the plane from the origin is $\frac{3\sqrt{2}}{20}$. If $a,b,c\&d$ are integer then the value of $a+b+c+d$ is

If the maximum value of $(x+y{)}^2\text{is}\lambda$ and $P(x,y)$ satisfies $x^2+y^2=1$, then the number of tangents that can drawn from $(\lambda ,0)$ to the hyperbola $(x-2{)}^2-y^2=1$ is

$If(1+x{)}^n=C_0+C_{1}x+C_2{x}^2+.......C_{n4}{x}^n....(i)$
then sum of series $C_0+C_k+C_{2k}+$....can be obtained by putting all roots of equation $x^k-1=0$ in (i) & then adding vertically:
for example: sum of $C_0+C_2+C_4.$....can be obtained by putting all roots of equation $x^2=1$
i.e.$x=\pm 1$ in (i)
At $x=1C_0+C_1+C_2..........C_n=2^n$
$x=-1C_0-C_1+C_2-C_3...........=0$
Adding we get $C_0+C_2+C_4.....=2^{n-1}$

Now answer the folloiwng

Sum of values of x which should be substituted in (i) to get the sum of $C_0+C_4+C_8+C_{12}....................$ is

Find the equation of the hyperbola satisfying the give conditions: Vertices (0, ±5), foci (0, ±8)

Solution set of $(x+2)(x-7)<0$ is

If the circle $x^2+y^2-6x-10y+k=0$ does not touch or intersect the coordinate axes, and the point $(1,4)$ is inside the circle, then the range of $k$ is

For all real permissible values of $m,$ if the straight lines $y=mx\pm \sqrt{9m^2-4}$ are tangents to a hyperbola, then equation of the hyperbola is

If the difference between the number of subsets of two finite sets $A$ and $B$ is $120,$ then $n(A\times B)$ is

If $A,B$ and $C$ are three subsets of a non-empty set $X$, then $(A^{\prime }\cap B^{\prime }\cap C)\cup (B\cap C)\cup (A\cap C)$ is equal to

Let $A$ and $B$ be two sets such that $A\cup B=A.$ Then $A\cap B$ is

The function $f:R\to R$ satisfies
$f\left(x^2\right)f^{\prime \prime }\left(x\right)=f^{\prime }\left(x\right)f^{\prime }\left(x^2\right)\forall x\in R,$ given that $f\left(1\right)=1$ and $f^{\prime \prime \prime }\left(1\right)=8$, then

The equation of circle whose two diameters are $2x-3y=5,$ $3x-2y=10$ and having perimeter $6\pi \text{cm},$ is

Let $f\left(x\right)$ be a twice differentiable function and $f^{\prime \prime }\left(0\right)=5$, then $\underset{x\to 0}{lim}\frac{3f\left(x\right)-4f\left(3x\right)+f\left(9x\right)}{x^2}$ is equal to

The value of the integral $\int \frac{3x+5}{x^3-x^2-x+1}dx$ is

(where $C$ is integration constant)

Out of the 240 students in a school, 20 are in class VII. Central angle of class VII (depicted in a pie chart) is .

If α and β are the roots of the equation $x^2+6x+\lambda =0$ and $3\alpha +2\beta =-20$, then λ =

Let m and n be 2 digit numbers. The number of pairs (m,n) such that n can be subtracted from m without borrowing is

The line x + y = 10 divides line segment AB in the ratio a : 1. Find the value of a.

If $\frac{1}{(x-1)(x+2)(2x+3)}$ can be expressed as $\frac{A}{x-1}+\frac{B}{x+2}+\frac{C}{2x+3}$ then what will be the respective values of A, B and C?

The length of tangent to the curve $x=a(\cos t+\log \left(\tan \frac{t}{2}\right)),y=a\sin t,$ at any point is :

The co-ordinates of the point which divides the join of the points (2, –1, 3) and (4, 3, 1) in the ratio 3 : 4 internally are given by

Find the area of the triangle formed by the points A(5,2) B (4,7) and C (7,-4).

If the image of the point P(1, -2, 3) in the plane 2x+3y-4z+22=0 measured parallel to the line $\frac{x}{1}=\frac{y}{4}=\frac{z}{5}$ is Q, then PQ is equal to

Evaluate $\int \frac{x+3}{x^2+5x+4}dx$

(where $C$ is constant of integration)

If $\left|\begin{array}{ccc}1+si{n}^2\theta & co{s}^2\theta & 4sin4\theta \\ si{n}^2\theta & 1+co{s}^2\theta & 4sin4\theta \\ si{n}^2\theta & co{s}^2\theta & 1+4sin4\theta \end{array}\right|=0$ such that $0\le \theta \le \frac{\pi }{2}$ then $\theta$ is

Let $\overrightarrow{a}=\frac{1}{7}(2\hat{i}+3\hat{j}+6\hat{k}),\overrightarrow{b}=\frac{1}{7}(6\hat{i}+2\hat{j}-3\hat{k}),\overrightarrow{c}=c_1\hat{i}+c_2\hat{j}+c-3\hat{k}$ and matrix $A=\left[\begin{array}{ccc}\frac{2}{7}& \frac{3}{7}& \frac{6}{7}\\ \frac{6}{7}& \frac{2}{7}& -\frac{3}{7}\\ c_1& c_2& c_3\end{array}\right]$
If $A{A}^T=I,then\overrightarrow{c}$ is equal to