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$\int \frac{1}{x^2-4x+8}dx=$

The sum of the series $1+2\times 3+3\times 5+4\times 7+\dots \text{upto}{11}^{\text{th}}\text{term}$ is :

If $z=\frac{(1+i)(1+2i)(1+3i)\dots \dots \dots (1+ni)}{(1-i)(2-i)(3-i)\dots \dots \dots (n-i)}$, where $i=\sqrt{-1},n\in N$, then principal argument of $z$ can be -

The solution of primitive equation $-y\frac{dy}{dx}=x\sqrt{1-y^2}$ is $y=y\left(x\right)$, where $y\left(x\right)$ is non-constant. If $y\left(0\right)=1$, then which of the following is/are correct

Let P be an interior point of a triangle ABC. Let Q and R be the reflections of P in AB and AC, respectively. If Q, A, R are collinear then $\angle$A equals.

The points of extremum of the function $F\left(x\right)={\int }_1^x{e}^{-t^2/2}(1-t^2)$ dt are

The three points with position vectors $\overline{a}+\overline{b},\overline{a}-\overline{b}$ and $\overline{a}+\lambda \overline{b}$ are collinear for

The domain of definition of $f\left(x\right)=\sqrt{4x-x^2}$ is

Let $a,b,c$ be three complex numbers whose modulus is $1$. If $a+b\cos \alpha +c\sin \alpha =0$, where $\alpha \in (0,\frac{\pi }{2})$, then

If the axes are rotated through an angle of ${90}^{\circ }$ in any direction then the transformed equation of $x^2=4ay$ can be

$\text{If}z=cos\frac{\pi }{4}+isin\frac{\pi }{6},\text{then}$

define differentiation and integration

The value of ${\cos }^2{10}^{\circ }-\cos {10}^{\circ }\cos {50}^{\circ }+{\cos }^2{50}^{\circ }$ is equal to

If$a,\frac{1}{b},c$ and $\frac{1}{p},q,\frac{1}{r}$form two arithmetic progression of the same common difference, then a, q, c are in A.P. if

The expression for $n^{th}$ term of an AP, with first term $a$ and common difference $d$, is

Two systems of rectangular axis have the same origin. If a plane cuts them at distances a,b,c and a',b;c', respectively from the origin, then prove that $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=\frac{1}{a^{\prime 2}}+\frac{1}{b^{\prime 2}}+\frac{1}{c^{\prime 2}}$.

Find the second order derivative of the given functions.

$x^{3}logx$

Prove that $\left|\begin{array}{ccc}{a}^2& bc& ac+c^2\\ a^2+ab& b^2& ac\\ ab& b^2+bc& c^2\end{array}\right|=4a^2{b}^2{c}^2$

For given binary operation $\ast$ defined below, determine whether $\ast$ is binary, commutative or associative.
(iii)On Q, define $a\ast b=\frac{ab}{2}$

Evaluate the integrals using substitution.
${\int }_0^1{\sin }^{-1}\left(\frac{2x}{1+x^2}\right)dx$

Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector $3\hat{i}+2\hat{j}-2\hat{k}$.

Number of $4$ digit numbers using digits $0,1,2,3,4,5$ which are divisble by $11$, when each digit is used at most once is

What is the meaning of fair trial and how does inclusion of fair trial influence the result?

The common solution set of $3x-7<5+x$ and $11-5x\le 1$ is

Let f(x) =

$|x|=\{\begin{array}{cc}\frac{3\times (x-1)}{x^2-3x+2}& \text{text{for } ~~~ xneq 1,2}\\ -3& \text{text{for } ~~~ x=1}\\ 4& \text{text{for} ~~~ x=2}\end{array}$. Then f(x) is continuous

A chord is drawn on the parabola $y^2=8x$ through the vertex of the parabola and a point P. If P(x',y') satisfies the condition $y^{\prime 2}=8x^{\prime }$,then what is the midpoint of the chord.

Evaluate $\frac{d}{dx}\left(sin\right(e^{x-2}-1\left)\right)$

Let $A\subset Z$ and a function $f:A\to B$ be defined as $f\left(x\right)=\sqrt{\frac{|x|-1}{|x|+1}}-\sqrt{\frac{2+|x|}{2-|x|}}.$ Then

A line has intercepts $a$ and $b$ on the coordinate axes. When the axes are rotated through an angle $\alpha$ in anticlockwise direction, keeping the origin fixed, the line makes equal intercepts on the coordinate axes. Then the value of $\cot \alpha$ is

cot^{​​​​-1} (x) + tan^{​​​​-1} (3) = π/2 , then x = ?

If ${}^{2n+1}{P}_{n-1}:{}^{2n-1}{P}_n=3:5$, then the value of $n$ is :

If the sum of the coefficients in the expansion of ${(1-3x+10x^2)}^n$ is $a$ and if the sum of the coefficients in the expansion of ${(1+x^2)}^n$ is $b$, then

If $f:[0,2]\to A,f\left(x\right)=\left[x^2\right]-[x{]}^2$ is a real valued function, then minimum elements required in set $A$ is
(where [.] denotes greatest integer function)

Equation of the ellipse with vertices (-4,3) (8,3) and e=$\frac{5}{6}$ is

Two spheres P and Q, of same colour having radii 8 cm and 2 cm are maintained at temperatures ${127}^0$ C and ${527}^0$ C respectively. The ratio of energy radiated by P and Q is

The following figure is a graph of

The equation of the circle passing through (1, 1) and the points of intersection of $x^2+y^2+13x-3y=0$ and $2x^2+2y^2+4x-7y-25=0$ is

If the circle $x^2+y^2=a^2$ intersects the hyperbola $xy=c^2$ at four points
$P(x_1,y_1),Q(x_2,y_2),R(x_3,y_3),andS(x_4,y_4)$, then

a>0, ${\int }_{-\pi }^{\pi }\frac{si{n}^{2}x}{1+a^x}dx=$