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Prove

sinA.sin(60-A).sin(60+A)=sin3A/4

Point of inflection of the function $f\left(x\right)=x^3$ is

Select the option to which the underlined part of the sentence is related.
Carol reminded her brother of all the times when he needed help and she was available forhim.

A rod of length $l$ moves such that its ends $A$ and $B$ always lie on the lines $3x-y+5=0$ and $y+5=0$ respectively. The locus of the point $P,$ which divides $AB$ internally in the ratio $2:1,$ is $l^2=\frac{1}{k}(ax-by-5{)}^2+9(y+5{)}^2.$ Then

The number of distinct solutions of ${\int }_0^x(4t^2+1)e^{2t^2}dt=5x$ for $x\ge 0$ is

The incentre of the triangle formed by the vertices $A(-36,7),B(20,7)$ and $C(0,-8)$ is

In a $\Delta ABC$, if $\left|\begin{array}{ccc}1& a& b\\ 1& c& a\\ 1& b& c\end{array}\right|=0$, then $si{n}^{2}A+si{n}^{2}B+si{n}^{2}C=$

Complete set of values of 'a' such that $\frac{x^2-x}{1-ax}$ attains all real values is :

If x is real and satisfies x + 2 > $\sqrt{x+4}$, then

If $z+\frac{1}{z}=\sqrt{3}$ then ${\sum }_{r=1}^5{(z^r+\frac{1}{z^r})}^2$ =

If the radical axis of the circles $x^2+y^2+2gx+2fy+c=0$ and $2x^2+2y^2+3x+8y+2c=0$ touches the circle $x^2+y^2+2x+2y+1=0$, then

If $x^2+ax+b=0$ has two roots $\alpha$ and ${\alpha }^2$, where $b\ge 4$, then the maximum value of $a$ is

If P (n) is the statement "2" $\le$ "3n", and If P (r) is true, prove that P (r + 1) is true.

The length of the line segments joining focus to the point of intersection of angular bisector of co-ordinate axes (in the first quadrant) and the parabola $y^2=lx$ is

How many elements will be there in the universal relation defined on the set A = {1,4, 9}

If $\beta$ is one of the angles between the normals to the ellipse, $x^2+3y^2=9$ at the points $(3\cos \theta ,\sqrt{3}\sin \theta )$ and $(-3\sin \theta ,\sqrt{3}\cos \theta )$; $\theta \in (0,\frac{\pi }{2});$ then $\frac{2\cot \beta }{\sin 2\theta }$ is equal to

Is it possible to raise a negative number to a fractional power .

If it is possible ,how can I find it because I am not able to do it on the calculator of my tab

General solution of a differential equation is $y=k{e}^x$. Find the particular solution if the curve passes through (0,1)

If the length of latus rectum of a hyperbola $\frac{x^2}{k}-\frac{y^2}{25}=-1$ is $\frac{22}{5}$ units, then its $e$ (eccentricity) is

In $△ABC,\angle B=90°$ and perpendicular from $B$ on $AC$ intersects it at $D.$ If $AC=4BD,$ then the smallest angle of $△ABC$ is

The angle at which the curve $y=x^2$ and the curve $x=\frac{5}{3}\cos t,y=\frac{5}{4}\sin t$ intersect is :

The value of $x,$ satisfying the inequality $x^{{\log }_{2}x}>2$ lies in

f(x) = $x^2$+2x+5. Find the average rate of change of f(x) in the interval [0,2]

Integrate the following functions.
$\int \frac{(1+logx{)}^2}{x}dx.$

A window is in the form of a rectangle surmounted by a semi-circle opening. The perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.

Let m be a given fixed positive integer.

Let $R=\left\{\right(a,b):a,bϵZ\}\text{and}(a-b)\text{is divisible by m}$

Show that R is an equivalence relation on Z.

Find the degree measure of the angle subtended at the centre of a circle of radius 100 cm by an arc of length 22 cm $(\text{use}\pi =\frac{22}{7})$.

Find the general solution of sin3x + cos2x =o

r × $\frac{{}^n{C}_r}{{}^n{C}_{r-1}}$ When n = 1000 and r = 500

If A(z₁), B(z₂) and C(z₃) are three points in the argand plane where

| z₁ + z₂ | = | |z₁| - |z₂| | and | (1 - i)z₁ + iz₃ | = | z₁ | + | z₃ - z₁ | ,

where i = √( -1) then

(a) A, B and C lie on a fixed circle with centre (z₂ + z₃)/2

(b) A, B and C are collinear points

(c) ABC form an equilateral triangle

(d) ABC form an obtuse angle triangle

The ratio of the radius of gyration of a circular disc and a circular ring of the same radii, about the tangential axis perpendicular to the plane.

If $2cos\theta +sin\theta =1,\theta \ne \frac{\pi }{2}$, then the value of $7cos\theta +6sin\theta$ is equal to

The latus rectum of the parabola whose focal chord is PSQ such that SP = 3 and SQ = 2 is

1. 2

A box contains 10 good articles and 6 with defects. One item is draqn at random. The Probability that it is either good of has a defect is

The direction cosines of the line drawn from P(-5, 3, 1) to Q(1,5,-2) is

The value of $\int \frac{dx}{3-2x-x^2}$ will be

The graph of the function $y=f\left(x\right)$ is symmertical about the line $x=2$ , then

Write the value of ${lim}_{x\to \frac{\pi }{2}}\frac{2x-\pi }{cosx}.$

Find the equation of the circle, whose centre is (2, - 3) and radius is 5.