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The value of
$(\frac{30}{0})(\frac{30}{10})-(\frac{30}{1})(\frac{30}{11})+(\frac{30}{2})(\frac{30}{12})...+(\frac{30}{20})(\frac{30}{30})$ is; where $(\frac{n}{r}){=}^n{C}_r$

One side of an equilateral triangle is $24$ cm. The midpoints of its sides are joined to form another triangle whose midpoints are in turn joined to form still another triangle this process continues indefinitely. The sum of the perimeters of all the triangles is

The equation of straight lines which are both tangent and normal to the curve $27x^2=4y^3$ are

Let function $f\left(x\right)=\{\begin{array}{c}\cos x+2;x\le 0\\ A\sin x+B;x>0\end{array}$

is continuous everywhere but not differentiable $(A,B\in R)$, then

Axis of a parabola is $y=x$ and vertex and focus are at a distance $\sqrt{2}$ and $2\sqrt{2}$ respectively from the origin. Then, equation of the parabola is

The sum of the roots of equation $z^6+64=0,$ whose real part is positive is

The harmonic conjugate of the point $R(2,4)$ with respect to the points $P(2,2)$ and $Q(2,5)$ is

A real number $a$ is chosen randomly and uniformly from the interval $[-20,18].$ The probability that the roots of the polynomial $x^4+2a{x}^3+(2a-2)x^2+(-4a+3)x-2$ are all real, is given by

The entire graphs of the equation $y=x^2+kx-x+9$ is strictly above the x-axis if and only if

The image of the lines 2x-y =1 in the line x+y=0 is what ?

The area (in sq. units) of the smaller of the two circles that touch the parabola, $y^2=4x$ at the point $(1,2)$ and the $x$-axis is :

If ${\left({\sin }^{-1}x\right)}^2-{\left({\cos }^{-1}x\right)}^2=a;0<x<1,a\ne 0$, then the value of $2x^2-1$ is:

Prove that sin10+sin 30+sin 50+sin 70 =root3÷16

If $\text{A}=\{x:x\text{is a prime number and}x<13\}$, then find the number of proper subsets of set $\text{A}$.​

The values of $m$ for which $y=\frac{m{x}^2+3x-4}{-4x^2+3x+m}$ has range $R$ is

If ${\tan }^{-1}\left(\frac{x-1}{x+1}\right)+{\tan }^{-1}\left(\frac{2x-1}{2x+1}\right)={\tan }^{-1}\frac{23}{36};x>1,$ then the value of $x$ can be

The value of ${\int }_0^a\frac{1}{\sqrt{\mathrm{ax}-x^2}}\mathrm{dx}$ is

The number of quadratic equation(s), with real roots which remain unchanged even after squaring its roots, is

$1^2+3^2+5^{2}5+...+(2n-1{)}^2=\frac{1}{3}n(4n^2-1)$

Express the complex number $\frac{1}{(1+i)}$ in a + ib form where a and b are real. Find the value of $a^2+b^2$.

After striking the floor ball rebounds ${\frac{4}{5}}^{\text{th}}$ of its height from which it has fallen. If it is released from a height of $120$m, then the total distance travelled by the ball (in $\text{m}$) before it comes to rest is

The value(s) of 'p' for which the parabola represented by quadratic function $y=(p-2)x^2+8x+(p+4)$ will remain below X-axis for all real values of x is .

If the function $f\left(x\right)=\lambda |\sin x|+{\lambda }^2|\cos x|+g\left(\lambda \right),\lambda \in R,$ where $g$ is a function of $\lambda ,$ is periodic with fundamental period $\frac{\pi }{2},$ then

The number of real numbers $\lambda$ for which the equality
$\frac{\sin \lambda \alpha }{\sin \alpha }-\frac{\cos \lambda \alpha }{\cos \alpha }=\lambda -1,$
holds for all real $\alpha$ which are not integral multiples of $\frac{\pi }{2}$ is

Let $\alpha$ and $\beta$ be the roots of the equation $x^2+x+1=0$. The equation whose roots are ${\alpha }^{19},{\beta }^7$ is

The equation of the plane containing the line $\frac{x-x_1}{l}=\frac{y-y_1}{m}=\frac{z-z_1}{n}$ is $a(x-x_1)+b(y-y_1)+c(z-z_1)=0,$ then which of the following is true?

If $20$ men are to be seated on a long table having $10$ chairs on either side. $3$ particular men want to sit on one particular side and $5$ particular men want to sit on other side, then number of possible arrangement is

The number of triangles $\Delta ABC,$ such that $b<c\sin B$ is

The percentage of doctors that fall into the 35 to 40 years age group (both inclusive) is at least

If $2\sec \theta \text{cosec}\theta -\cot \theta =3$, then the value of $\tan \theta$ is/are

The range of $\alpha$ for which the points $(\alpha ,\alpha +2)$ and $(\frac{3\alpha }{2},{\alpha }^2)$ lie on opposite sides of the line $2x+3y-6=0$ is

The locus of the mid-point of the line segment joining the focus ot a moving point on the parabola $y^2=4ax$ is another parabola with directrix

If $\int \frac{8x^{43}+13.x^{38}}{(x^{13}+x^5+1{)}^4}dx=\frac{1}{3}.\frac{x^a}{(x^b+x^c+1{)}^3}+C$ where $a,b,cϵN,(a>b>c$ and where C is a constant of integration), then

Number of inflection points for the curve $y=x+2x^4$ is

Two cards are drawn at random and without replacement from a pack of 52 playing cards. Find the probability that both the cards are black.