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If a and d are two complex numbers, then the sum

to (n+1) terms of the following series

a$C_0$ - (a + d)$C_1$ + (a + 2d)$C_2$ - ........... is

If the line $\frac{x-2}{3}=\frac{y+1}{2}=\frac{z-1}{-1}$ intersects the plane $2x+3y-z+13=0$ at a point $P$ and the plane $3x+y+4z=16$ at a point $Q,$ then $PQ$ is equal to :

If $\int \frac{dx}{x^3(1+x^6{)}^{2/3}}=xf\left(x\right)(1+x^6{)}^{1/3}+C$ where $C$ is a constant of integration, then the function $f\left(x\right)$ is equal to :

The cartesian product $A\times A$ has 9 elements among which are found (-1, 0) and (0, 1). Find the set A and the remaining elements of $A\times A$.

If ${}^n{C}_4={}^n{C}_5$, find ${}^n{C}_3$

The value of $cos\frac{2\pi }{7}+cos\frac{4\pi }{7}+cos\frac{6\pi }{7}$ is

If the probability that a student is not a swimmer is $\frac{1}{5}$, then the probability that out of 5 students one is swimmer is

Given two sets $X$ and $Y$ such that $n\left(X\right)=20,n\left(Y\right)=25\text{and}n\left(XUY\right)=40,$ then $n(X-Y)=$

Find the value of $tan{1}^{\circ }\times tan{2}^{\circ }\times tan{88}^{\circ }timestan{89}^{\circ }$

Find the domain of the real function, $f\left(x\right)=\frac{1}{\sqrt{x+|x|}}$

$\int \left[f\right(x\left)g^{\prime \prime }\right(x)-f"(x\left)g\right(x\left)\right]dx$ is equal to

The pole of $3x+4y-45=0$ with respect to the circle $x^2+y^2-6x-8y+5=0$ is

Let A, B and C be the sets such that $A\cup B=A\cup C$ and $A\cap B=A\cap C.$ Show that $B=C.$

Normality of 1% (w/w) $H_{2}S{O}_4$ solution is nearly

The function

$f\left(x\right)=\{\begin{array}{cc}|x-3|,& x\ge 1\\ \frac{x^2}{4}-\frac{3x}{2}+\frac{13}{4},& x<1\end{array}$,

is

If the system of linear equations

$x+ay+z=3$

$x+2y+2z=6$

$x+5y+3z=b$

has no solution, then:

Suppose there are 10 students in your class. You want to select three out of them. How many samples are possible?

A series in G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying odd places, then the common ratio will be equal to

Let $f\left(x\right)=x^2+6x+c,c\in R.$ If $f\left(f\right(x\left)\right)=0$ has exactly three distinct real roots, then the value of $c$ can be

Evaluate the following limit:
$\underset{x\to 0}{\text{lim}}\frac{\text{sin ax + bx}}{\text{ax+sin bx}},a,b,a+b\ne 0$

The value of ${\int }_0^{\pi }\frac{1}{5+4\cos x}\mathrm{dx}$ is

A man of height 2m walks directly away from a lamp of height 5m, on a level road at 3 m/s. The rate at which the length of his shadow is increasing is

The remainder when $2^{2003}$ is divided by $17$ is

Angle between the tangents drawn from the origin to the parabola $y^2=4a(x-a)$ is

Express the complex numbers in the form of a + ib:

$\left(5i\right)(-\frac{3}{5}i)$

The equation of the locus of a point whose distance from (a, 0) is equal to its distance from y-axis, is

A variable circle passes through the fixed point A(p, q) and touches the x-axis. The locus of the other end of the diameter through A is

If $z$ is a complex number of unit modulus and argument $\theta$, then arg$\left(\frac{1+z}{1+\overline{z}}\right)$ is equal to

If the ratio of area of triangle inscribed in the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ to that of triangle formed by the corresponding points on the auxiliary circle is $\frac{1}{2}$, then the eccentricity of the ellipse is

If $\alpha ,\beta ,\gamma$ be the roots of the equation $(x-a)(x-b)(x-c)=d,d\ne 0$, then the roots of the equation $(x-\alpha )(x-\beta )(x-\gamma )+d=0$ are

If $S_n=\frac{1}{1\times 3}+\frac{1}{3\times 5}+\frac{1}{5\times 7}+\cdots n\text{terms}$, then $S_{\infty }$ is

$\overrightarrow{V}=2\hat{i}+\hat{j}-\hat{k}$ and $\overrightarrow{W}=\hat{i}+3\hat{k}$. If $\overrightarrow{U}$ is a unit vector, then the maximum value of the scalar triple product $\left[\overrightarrow{U}\overrightarrow{V}\overrightarrow{W}\right]$ is

If complex numbers $(-3+iy{x}^2)$ and $(x^2+y+4i)$ are conjugates of each other, where $x,y\in R,$ then $(x,y)$ can be

The value of $(1+i{)}^{2002}$ is

A survey shows that in a city, $45$% citizens like tea, whereas $65$% citizens like coffee. If $x$% like both tea and coffee, then

$9^n$ + 7 is divisible by ________.

If $\sqrt{2}$ and $3i$ are two roots of a biquadratic equation with rational coefficients, then its equation is, (where $i^2=-1$)

If tan x =n tany, $n\in R^{+}$, then maximum value of $se{c}^2(x-y)=$___

If ${\log }_{0.04}(x-1)\ge {\log }_{0.2}(x-1),$ then

If ${\log }_{0.5}\left(\frac{3-x}{x+2}\right)<0$, then $x$ can be

If $lo{g}_{12}27=a,thenlo{g}_{6}16=$

If m is a natural such that $m\le 5,$ then the probability that the quadratic equation $x^2+mx+\frac{1}{2}+\frac{m}{2}=0$ has real roots is

Let $A$ and $B$ be two invertible matrices of order $3\times 3$. If $\text{det}\left(AB{A}^T\right)=8$ and $\text{det}\left(A{B}^{-1}\right)=8$, then $\text{det}\left(B{A}^{-1}{B}^T\right)$ is equal to :