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Let $n\ge 2$ be a natural number and $0<\theta <\frac{\pi }{2}$.

Then $\int \frac{{({\sin }^n\theta -\sin \theta )}^{\frac{1}{n}}\cos \theta }{{\sin }^{n+1}\theta }d\theta$ is equal to :

(where $C$ is constant of integration)

$\int \frac{x-sinx}{1+cosx}dx=xtan\left(\frac{x}{2}\right)+plog\left|sec\left(\frac{x}{2}\right)\right|+c\Rightarrow p=$

If $I=\underset{0}{\overset{x}{\int }}\left[\cos t\right]dt$, where $x\in \left[\right(4n+1)\frac{\pi }{2},(4n+3\left)\frac{\pi }{2}\right],n\in N$ and $[\cdot ]$ represents greatest integer function, then the value of $I$ is

If $f\left(x\right)=(x+1)(x^4+x^3+2)(x^3+4x^2+2x+3),$ then ${\frac{dy}{dx}|}_{x=-1}$ is

The product of slope of tangents from point $(0,1)$ to the circle $x^2+y^2-2x+4y=0$ is

The point of intersection of the tangents of the circle $x^2+y^2=10$, drawn at end points of the chord x + y = 2 is

If $\left[t\right]$ denotes the greatest integer $\le t,$ then the value of $\underset{0}{\overset{\pi /8}{\int }}[\sec x+2[\tan x+3[\cos x+4[\sin x\left]\right]]]dx$ is

The locus of a point which divides the join of A(-1, 1) and a variable point P on the circle $x^2+y^2=4$ in the ratio 3:2, is

Find the coordinates of the point where the line through (3, -4, -5) and (2, -3, 1) crosses the planes 2x + y + z = 7

The equation of the parabola whose focus is $(4,-3)$ and vertex is $(4,-1)$

The area of the smaller region lying above the x-axis and included between the circle x² + y² = 2x and the parabola y² = x.

A steel beam of breadth 120 mm and height 750 mm is loaded as shown in the figure. Assume $E_{steel}=200Gpa$.

The beam is subjected to a maximum bending moment of

If one of the diameters of the curve $x^2+y^2-4x-6y+9=0$ is a chord of a circle with centre $(1,1)$, the radius of this circle is

If $P\left(A\right)=\frac{6}{11},P\left(B\right)=\frac{5}{11}$ and $P(A\cup B)=\frac{7}{11}$, then the value of $P\left(A/B\right)$ is:

f(x) = x is called

If an integer p is chosen at random in the interval $0\le p\le 5$, the probability that the roots of the equation $x^2+px+\frac{p}{4}+\frac{1}{2}=0$ are real is:

Suppose $f$ and $g$ are differentiable functions on $(0,\infty )$ such that $f\text{'}\left(x\right)=-\frac{g\left(x\right)}{x}$ and $g\text{'}\left(x\right)=-\frac{f\left(x\right)}{x}$, for all $x>0$. Further, $f\left(1\right)=3$ and $g\left(1\right)=-1$.



$g\left(\frac{1}{10}\right)$ is equal to

(a,b) is the mid point of the chord $\overline{AB}$ of the circle $x^2+y^2=r^2$. The tangent at A,B meet a C. then area of $\Delta ABC$

Find the equation of straight line that passes through the point (1, 2) and perpendicular to the line 4x + 5y + 3 = 0.

The
number of all possible matrices of order 3 ×
3 with each entry 0 or 1 is:

(A) 27

(B) 18

(C) 81

(D) 512

If ${\log }_{2}x+{\log }_{2}y\ge 6$, then the least value of $xy$ is

Find the equation for the ellipse that satisfies the given conditions: Vertices (0, ±13), foci (0, ±5)

If $f:R\to R$ and $g:R\to R$ are given by f(x) = |x| and g(x) = [x], then $g\left(f\right(x\left)\right)\le f\left(g\right(x)$ is true for -

${lim}_{x\to 0}\frac{a^x-a^{-x}}{x}$

If $\alpha$ and $\beta$ $(\alpha >\beta )$ are the roots of the equation $x^2-\sqrt{2}x+\sqrt{3-2\sqrt{2}}=0,$ then the value of $({\cos }^{-1}\alpha +{\tan }^{-1}\alpha +{\tan }^{-1}\beta )$ is equal to

If $3+\frac{1}{4}(3+P)+\frac{1}{4^2}(3+2P)+\frac{1}{4^3}(3+3P)+\dots =8,$ then the value of $P$ is

What is the probability of $\text{not}$ getting purple in the spinner?