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A function $y=f\left(x\right)$ satisfies the equation $f(x+y)=f\left(x\right)\cdot f\left(y\right)$ where $x,y\in R$. It is known that $f\left(1\right)=25$. If $S=f\left(2\right)+f\left(1\right)+f\left(0\right)+f(-1)+...\infty$, then the value of $\left[\right(f\left(1\right)-1)S{]}^{1/2}$ is

If $co{s}^{-1}\left(\frac{x^2-y^2}{x^2+y^2}\right)=loga$ then $\frac{dy}{dx}$ is equal to

Find the equation of normal to the parabola

$y^2=16x$ at point (4,8)

If a > 0, b > 0, c > 0 are positive read numbers in AP. If $a{x}^2+bx+c=0$ has real roots then

Find the equations of the straight lines which pass through (4, 3) and are respectively parallel and perpendicular to the x-axis.

If $\int \frac{\sin x}{\sin (x-\alpha )}dx=px-q\log |\sin (x-\alpha )|+C$, Then the value of $pq$ is

Solve the given inequality graphically in two-dimensional plane: x + y < 5

The variance of first $20$ natural numbers is

Let $f_p\left(\alpha \right)=e^{\frac{i\alpha }{p^2}}.e^{\frac{2i\alpha }{p^2}}.e^{\frac{3i\alpha }{p^2}}.e^{\frac{4i\alpha }{p^2}}\dots e^{\frac{i\alpha }{p}}$, (where $i=\sqrt{-1}$ and $p\in N$) then ${lim}_{n\to \infty }{f}_n\left(\pi \right)$ is

If ${\log }_{2}x+{\log }_{2}y\ge 6,$ then the least value of $x+y$ is

If $1,\omega ,{\omega }^2,\dots ,{\omega }^{n-1}$ are the $n^{\text{th}}$ roots of unity and $z_1$ and $z_2$ are any two complex numbers, then $\sum _{k=0}^{n-1}|z_1+{\omega }^k{z}_2{|}^2$ is equal to

If $\frac{\tan 2^{\circ }\cdot \tan {2017}^{\circ }\cdot \tan {2019}^{\circ }}{\tan {2019}^{\circ }-\tan {2017}^{\circ }-\tan 2^{\circ }}=a$ such that $a=\tan x,$ where $x$ is an acute angle in degree, then the value of $x$ is

For the equation a $x^2$ + bx + c = 0 having 2 real roots, one root lies between real values $x_1$ & $x_2$ if

There are $(n+1)$ white and $(n+1)$ black balls, each set numbered from $1$ to $n+1$. The number of ways in which the balls can be arranged in a row so that the adjacent balls are of different colours is

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

$f\left(x\right)=\{\begin{array}{cc}\frac{|x-4|}{2(x-4)},& ifx\ne 4\\ 0,& ifx=4\end{array}$

Is the function f(x) continuous at x=0?

If the two curves $y=a^x$ and $y=b^x$ intersect at an angle of $\alpha .$ Then $\tan \alpha =$

Equation of circle passing through the origin and making intercepts of length $10$ units and $12$ units on $x$ axis and $y$ axis respectively, can be

(a) cos x

(b) sin 2x

(c) tan x

(d) cos 3x

Find the value of x which satisfies the following equation.
$\frac{1}{10!}+\frac{1}{11!}=\frac{x}{12!}$

If $a_1,a_2;g_1,g_2$ and $h_1,h_2$ are two arithmetic, geometric and harmonic means respectively between two quantities $a$ and $b$, then $ab$ is equal to

If ${\cos }^{-1}\frac{3}{5}-{\sin }^{-1}\frac{4}{5}={\cos }^{-1}x,$ then $x=$

If vector $\overrightarrow{x}$ satisfying $\overrightarrow{x}\times \overrightarrow{a}+(\overrightarrow{x}\cdot \overrightarrow{b})\overrightarrow{c}=\overrightarrow{d}$ is given by $\overrightarrow{x}=\lambda \overrightarrow{a}+\overrightarrow{a}\times \frac{\overrightarrow{a}\times (\overrightarrow{d}\times \overrightarrow{c})}{(\overrightarrow{a}\cdot \overrightarrow{c})|\overrightarrow{a}{|}^2},$ then the value of $\lambda =$

The solution set of $x^2-16\le 0$ and $x^2-9\ge 0$ is

Let $\omega$ be a complex number such that $2\omega +1=z$ where $z=\sqrt{-3}$. If $\left|\begin{array}{ccc}1& 1& 1\\ 1& -{\omega }^2-1& {\omega }^2\\ 1& {\omega }^2& {\omega }^7\end{array}\right|=3k$, then $k$ is equal to:

For $z=x+iy$, where $x,y\in R$ and $i=\sqrt{-1}$, what is the polar form of representation?

If a,b,c are non-coplanar unit vectors such that $a\times (b\times c)=\frac{b+c}{\sqrt{2}}$, then the angle between a and b is