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If $\int \frac{\sqrt{x^2+1}}{1-x^2}dx=\frac{1}{\sqrt{2}}f\left(x\right)-g\left(x\right)+C$, where $C$ is a constant of integration, then

If the distance of two lines passing through origin from the point $(x_1,y_1)$ is 'd',then the equation of lines is

The roots of the equation $x^4-8x^2-9$ = 0 are

If a is any vector then $(a\times \hat{i}{)}^2+(a\times \hat{j}{)}^2+(a\times \hat{k}{)}^2$ =

If A = {x : x is a multiple of 3} and , B = {x :x is a multiple of 5}, then A - B is

Constant term in the expansion of ${(x-\frac{1}{x})}^{10}$ is

Find the direction cosines of the line segment joining the two points (1, - 2, 3). and ( 2, 4, 5)

Three faces of a fair die are yellow, two faces are red and one face is blue. The die is tossed three times. The probability that the colours, yellow, red and blue, appear in the first, second and the third tosses respectively, is

In $\Delta ABC,if\angle C={90}^{\circ },\angle A={30}^{\circ },c=20$, then the values of a and b are

$\text{Find the point to which the origin should be dhifted after a translation of axes so that the following equations will have no first degree terms:}?\left(i\right)y^2+x^2-4x-8y+3=0\left(ii\right)x^2+y^2-5x+2y-5=0\left(iii\right)x^2-12x+4=0$

The value of ${lim}_{x\to 0}\frac{(1-cosx)+2sinx-si{n}^{3}x-x^2+3x^4}{ta{n}^{3}x-6si{n}^{2}x+x-5x^3}$ is

The equation of the plane, which bisects the line joining the points $(1,2,3)$ and $(3,4,5)$ at right angles is,

The $6^{\text{th}}$ term in the expansion of ${(2x^2-\frac{1}{3x^2})}^{10}$ is

Find the equations of the diagonals of the square formed by the lines x = 0, y = 0, x = 1 and y = 1.

A chord $ax+y=1$ subtends ${90}^{\circ }$ at the centre of the circle $x^2+y^2=\frac{3}{2}.$ Then the value of $a$ is

The dual of statement $p\vee (q\wedge r)\equiv (p\vee q)\wedge (p\vee r)$ is

If $PQ$ is a normal chord of the parabola $y^2=4ax$ at $P(a{t}^2,2at).$ Then the axis of the parabola divides $\overline{PQ}$ in the ratio

If $z_1,z_2,z_3$ be three non-zero complex number, such that $z_2\ne z_1,a=|z_1|,b=|z_2|andc=|z_3|$ suppose that $\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|=0$, then arg $\left(\frac{z_3}{z_2}\right)$ is equal to

Find the coefficient of $x^r$ in the expansion of $(1-x{)}^{-n}$

Equation of the line passing through the point $(1,2)$ and perpendicular to the line $y=3x-1$ is

TanAtan(A+60)+tanAtan(A-60)+tan(A+60)tan(A-60)=3

A constant force of 5 N is acting on a body. If x(t) is the dispacement at time t, v(t) velocity at time t, m mass of the body, which of the following differential equations represent the situation?

A, B and C invested Rs.3000, Rs.4000 and Rs.5000 in a business and after the registration charges of Rs.300 they were allowed to do their business. If the total profit earned by them is Rs.900, find the share that A would get in the profit?

The coefficient of $x^{53}$ in the expansion of $\sum _{m=0}^{100}{}^{100}{C}_m(x-3{)}^{100-m}{2}^m$ is:

The equation of normal to the hyperbola $3x^2-y^2=1$ having slope $\frac{1}{3}$ is

The co-ordinates of the foot of the perpendicular drawn from the point A(1, 0, 3)to the join of the points B(4, 7, 1) and C(3, 5, 3) are :

If ${\sin }^{2}x-2\sin x-1=0$ has exactly four different solutions in the interval $[0,n\pi ]$ , then possible value(s) of $n$ is/are :

Study the following table carefully and answer the questions given below: No. of marks obtained by five students in five subjects in an Examination
$$\begin{array}{ccccc}& \text{P}& \text{Q}& \text{R}& \text{S}\\ \text{Saturday}& 45& 40& 25& 20\\ \text{Sunday}& 40& 55& 48& 25\\ \text{Monday}& 25& 20& 15& 41\\ \text{Tuesday}& \text{35}& \text{45}& \text{42}& \text{39}\end{array}$$
Q. Items sold by S on Sunday is what percent of item sold by Q on Tuesday?

If $\frac{x}{c}+\frac{y}{d}=1$ be any line passing through point of intersection of lines $\frac{x}{a}+\frac{y}{b}=1\text{and}\frac{x}{b}+\frac{y}{a}=1$ then

Prove that $\frac{\text{sin}3x-\text{sin}x}{\text{cos}2x}=2\text{sin}x$

The points (1, 7), (4, 2), (-1, -1), (-4, 4) are the vertices of a _____

Let $g\left(x\right)=\underset{0}{\overset{x}{\int }}f\left(t\right)dt$, where $f$ is such that $\frac{1}{2}\le f\left(x\right)\le 1$ for $t\in [0,1]$ and $0\le f\left(t\right)\le \frac{1}{2}$ for $t\in [1,2]$ Then, $g\left(2\right)$ satisfies the inequality

If the equations $x^2+y^2+2\lambda x+4=0$ and $x^2+y^2–4\lambda y+8$ = 0 represent real circles then the value of $\lambda$ can be

Differentiate between sub-divided and multiple bar diagram.

Let [∙] represents the greatest integer function and f(x)=[tan² x], then

The point of intersection of the tangents at the point P on the ellipse$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and its corresponding point Q on the auxiliary circle, lies on the line

The equation of the common tangent to $x^2+y^2=2a^2$ and $y^2=8ax$ is

Find the orthocecentre of the triangle with the vertices (-5,-7),(13,2),(-5,6)

If ax+by+c=0 and lx+my+n=0 are asymptotes of a hyperbola, then