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Evaluate the integrals using substitution.
${\int }_0^{\frac{\pi }{2}}\frac{sinx}{1+co{s}^{2}x}dx.$

If the function $f\left(x\right)=\{\begin{array}{c}\frac{A^2{x}^3+x^2-(A+2)x+A}{x-2},forx\ne 2\\ 2,forx=2\end{array}$ is continuous at x=2, then

Let $f:R\to R$ be defined as $f\left(x\right)=\frac{x^2-x+4}{x^2+x+4}$. Then the range of the function $f\left(x\right)$ is

Let $S$ be the area bounded by $y=e^{|\cos 4x|},x=0,y=0$ and $x=\pi$, Then

The line x+y+2 = 0 is a tangent to a parabola $y^2=4ax$ at point A, it intersects the directrix at B and tangent at vertex at C, then (AC $\times$ BC) =

Common tangents are drawn to parabola $y^2=4x$ and ellipse $3x^2+8y^2=48$ touching the parabola at A and B and the ellipse at C and D. Area of quadrilateral ABCD is ?

If the direction cosines of a line are k, k and k,then

(a) k>0 (b)0<k<1

(c) k=1 (d)$k=\frac{1}{\sqrt{3}}or-\frac{1}{\sqrt{3}}$

If the area bounded by the parabola $y^2$ = 16ax and the line y = 4mx is $\frac{a^2}{12}$ sq.units, then using integration, find the value of m.

If $\int \left(\frac{x-1}{x+1}\right)\frac{dx}{\sqrt{x^3+x^2+x}}=2ta{n}^{-1}\sqrt{f\left(x\right)}+C$, find f(x).

Let `head` means 1 and `tail` means 2 and coefficients of the equation $a{x}^2+bx+c=0$ are chosen by tossing a fair coin. The probability that the roots of the equation are non-real, is equal to

Differentiate $(x^2-5x+8)(x^3+7x+9)$in three ways mentioned below.

(a) By using product rule.

(b) By expanding the product to obtain a single polynomial.

(c) By logarithmic differentiation.

Do they all given the same answer?

Value of sin 18

A five digit number is formed using the digits $0,1,2,3,4$ and $5$ without repetition. The probability that number is divisible by $6$?

How to convert 4800 angstrom to m with steps

The value of $\sum _{k=1}^{13}\frac{1}{\sin (\frac{\pi }{4}+\frac{(k-1)\pi }{6})\sin (\frac{\pi }{4}+\frac{k\pi }{6})}$ is equal to

In a battle 70% of the combatants lost eye, 80% an ear, 75% an arm, 85% a leg, x% lost all of four limbs. The minimum value of x.

Please do explain in detail

A real number α is said to be a root of ax²+bx+c =0

One ticket is selected at random from 100 tickets numbered 00, 01, 02, ...... 98, 99. If X and Y denote the sum and the product of the digits on the tickets, then $P\left(\frac{X=9}{Y=0}\right)$ equals

What do the following figure represent in (i),(ii) and (iii) ?

The point $P(3,6)$ is first reflected on the line $y=x$ and then the image point $Q$ is again reflected on the line $y=-x$ to get the image point $Q^{\prime }.$ Then the circumcentre of the $△PQ{Q}^{\prime }$ is

Find the equation whose roots are $2x_1+3and2x_2+3,if{x}_{1}and{x}_2$ are the roots of $x^2+6x+7=0$.

Find the equations of the straight lines which pass through the origin and trisect the portion of the straight line 2x + 3y = 6 which is intercepted between the axes.

If v is the variance and $\sigma$ is the standard deviation , then

If one root of $a{x}^2+bx+c=0$ is reciprocal of one root of $a_1{x}^2+b_{1}x+c_1=0$, then which of the following condition is correct?

The value of $\underset{n\to \infty }{lim}\prod _{r=2}^n\frac{r^3+1}{r^3-1}$ is

(Here, $\prod$ stands for the product.)

The radius of the base of a cone is increasing at the rate of $3cm/min$ and the altitude is decreasing at the rate of $4cm/min.$ The rate of change of lateral surface area when the radius is $7cm$ and the altitude is $24cm$ is

$f\left(x\right)=\frac{\left[x\right]+1}{\left\{x\right\}+1}forf:[0,\frac{5}{2})\to (\frac{1}{2},3]$, where [.] represents G.I.F. and {.} represents F.P.F., then

The value of c$ϵ$(0,2) in LMVT for f(x) = $x(x-2{)}^2$ in [0,2] is

Evaluate:

${lim}_{n\to \infty }\frac{1.2+2.3+3.4+...+n(n+1)}{n^3}$

Some of zeros of a polynomial with real coefficients are $\omega +2{\omega }^2,1+2\omega ,1+2{\omega }^2,\omega$ and $1-\omega -{\omega }^2,$ where $\omega$ is the cube root of unity. Then the minimum degree of the polynomial is

A point $p$ is selected randomly from the interior of the circle, then the probability that it is closer to the center of the circle rather than its boundary is :