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Let $f\left(x\right)=$$\int \frac{\sqrt{x}}{(1+x{)}^2}dx(x\ge 0)$. Then $f\left(3\right)-f\left(1\right)$ is equal to

If $\sin {21}^{\circ }=\frac{x}{y}$, then $\sec {21}^{\circ }-\sin {69}^{\circ }$ is equal to

Prove the following by using the principle of mathematical induction for all n ∈ N: x²ⁿ – y²ⁿ is divisible by x + y.

The value of the expression $\prod _{r=3}^{r=20}\frac{r-2}{r+1}$ is

If a, b, c are in A.P., then show that:
(i) $a^2(b+c),b^2(c+a),c^2(a+b)$ are also in A.P.
(ii) $b+c-a,c+a-b,a+b-c$ are in A.P.
(iii) $bc-a^2,ca-b^2,ab-c^2$ are in A.P.

The acute angle between the pair of straight lines passing through $(-6,-8)$ and also through the points which divide the line $2x+y+10=0$ enclosed between coordinate axes in the ratio $1:2:2$ in the direction from the point of intersection with the $x-$axis to the point of intersection with $y-$axis is

Two functions $f:R\to R,g:R\to R$ are defined as follows $f\left(x\right)=\{\begin{array}{cc}0,& x\in Q\\ 1,& x\notin Q\end{array}$, $g\left(x\right)=\{\begin{array}{cc}-1& x\in Q\\ 0,& x\notin Q\end{array}$, then $g\left(f\right(e\left)\right)+f\left(g\right(\pi \left)\right)=$

Let $D_1=\left|\begin{array}{ccc}x& a& b\\ -1& 0& x\\ x& 2& 1\end{array}\right|$ and $D_2=\left|\begin{array}{ccc}c{x}^2& 2a& -b\\ x& 2& 1\\ -1& 0& x\end{array}\right|.$ If all the roots of $(x^2-4x-7)(x^2-2x-3)=0$ satisfies the equation $D_1+D_2=0,$ then the value of $a+4b+c$ is

If the vertex of a parabola be at origin and directrix be x+5 = 0 , then its latus rectum is

The number of surjective functions from {2,4,6,8,.....2n} to {1,2} is

If $i{z}^3+z^2-z+i=0,$ then show that $\left|z\right|=1.$

Or

Find the real values of x and y, if $\frac{x-1}{3+i}+\frac{y-1}{3-i}=i.$

If x = 2 cos t – cot 2t, y = 2 sin t – sin 2t, then $\frac{d^{2}y}{d{x}^2}$ at $t=\frac{\pi }{2}$

The equation of chord of ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ whose sum and difference of eccentric angles are $\frac{\pi }{3}$ and $\frac{2\pi }{3}$ respectively is

The differential equation of the family of curves, $x^2=4b(y+b),b\in R$, is:

Evaluate $\underset{x\to 0}{lim}\left(\frac{3^{2x}-1}{2^{3x}-1}\right)$

1+sin2x/1-sin2x=tan^{​​​​2}(45°+x)

It is known that $10\%$ of certain articles manufactured are defective. The probability that in a random sample of $12$ such articles, $9$ are defective is:

The approximate value of $(1.0002{)}^{3000}$ is

The length of projection of the line segment joining the points, (1, 0, -1) and (-1, 2, 2) on the plane x + 3y - 5z = 6 is equal to

The equation of a circle whose radius is $7$ units and $x-$coordinate of the centre is $-2$ and also touches the $x-$axis, is

If f(x, y) = 0 be the solution of differential equation (2y cosec 2x + ln cot y)dx + (ln tan x - 2x cosec 2y)dy = 0 such that $f(\frac{\pi }{4},\frac{\pi }{2})=0$

If ${\int }_0^{\infty }\frac{(\frac{5\pi }{4},\frac{2017\pi }{4})cosx}{x}dx=\frac{\pi }{2},then{\int }_0^{\infty }\frac{1}{x}(1-si{n}^{2}x{)}^{\frac{3}{2}}dx$ is equal to

If $A=\left[\begin{array}{cc}\cos \theta & i\sin \theta \\ i\sin \theta & \cos \theta \end{array}\right],(\theta =\frac{\pi }{24})$ and $A^5=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, Where $i=\sqrt{-1}$, then which one of the following is not true?

Let $a=min\{x^2+2x+3,x\in R\},b=\underset{\theta \to 0}{lim}\frac{1-\cos \theta }{{\theta }^2}$, and $\sum _{r=0}^n{a}^r\cdot b^{n-r}=f\left(n\right),$ then which among the following is/are correct

The lines $x=ay–1=z–2$ and $x=3y–2=bz–2,$ $(ab\ne 0)$ are coplanar, if

The value of $\int \frac{3\sin x+2\cos x}{3\cos x+2\sin x}dx$ is

(where $C$ is integration constant)

If maximum and minimum values of $D=\left|\begin{array}{ccc}1& -\cos \theta & -1\\ \cos \theta & 1& -\cos \theta \\ 1& \cos \theta & 1\end{array}\right|$ are $p$ and $q$ respectively, then the value of $2p+3q$ is

There are twelve seats in a row and six boys and six girls occupy the seats at random. Find the probability that the boys and girls sit alternatively.

The solution of the differential equation $\frac{dy}{dx}+\frac{1}{x}\tan y=\frac{\tan y\sin y}{x^2}$ is