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A picture frame P of weight W is hung by two strings as shown in the figure. The total upward force on the strings is

If $a_1,a_2,a_3,\dots \dots a_r$ are in GP, then prove that the determinant $\left|\begin{array}{ccc}{a}_{r+1}& a_{r+5}& a_{r+9}\\ a_{r+7}& a_{r+11}& a_{r+}\\ a_{r+11}& a_{r+17}& a_{r+21}\end{array}\right|$ is independent of r.

If,
for, −1 < x
<1, prove that

Length of latus rectum of the parabola whose focus is at $(2,3)$ and directrix is the line $x–4y+3=0$ is

Which of the following function is a Periodic function -

For non-zero real parameters $a,b$ and $x\in R,$ if the range of $f\left(x\right)=2|x-a|+b$ and $g\left(x\right)=3|x-b|+a$ is same, then point $(a,b)$ lies on

If (a,b) is positive integral solution of the equation $7x^2-2xy+3y^2=27$ , then max. b + min. a =

The value of the following integral is

${\int }_0^{1}xlnxdx$

If $f\left(x\right)$ is invertible and twice differentiable function satisfying $f^{\prime }\left(x\right)=\stackrel{f\left(x\right)}{\underset{0}{\int }}{f}^{-1}\left(t\right)dt,\forall x\in R$ and $f^{\prime }\left(0\right)=1$, then $f^{\prime }\left(1\right)$ can be

A circle of radius ‘5’ touches the coordinate axes in the first quadrant. If the circle makes one complete roll on x-axis along the positive direction, then its equation in new position is

The equation of common tangent to the hyperbola $9x^2-16y^2=144$ and circle $x^2+y^2=9$ is

There are two perpendicular straight lines touching the parabola $y^2=4a(x+a)$ and $y^2=4b(x+b)$, then the point of intersection of these two lines lie on the line given by

If angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ${30}^{\circ }$ and their magnitudes are respectively $\sqrt{3}$ and $4$ units, then the value of $\overrightarrow{a}\cdot \overrightarrow{b}$ is

The number of integral solutions of $x+y+z=0$ with $x\ge -5,y\ge -5,z\ge -5$ is

The principal solution(s) for $\tan x=-1$ is/are

Let f,
g and h be functions from R to R. Show
that

For which of the following values of $x$, $5$th term will be the numerically greatest term in the expansion of ${(1+\frac{x}{3})}^{10}$.

Express the following complex numbers in the form $r(cos\theta +isin\theta )$.

$\left(i\right)1+itan\theta \left(ii\right)tan\alpha -i\left(iii\right)1-sin\alpha +icos\alpha \left(iv\right)\frac{1-i}{cos\frac{\pi }{3}+isin\frac{\pi }{3}}$

The area bounded by the curves $y=|x|-1$ and $y=-|x|+1$ is

How many number of four digits can be formed with the ditis 1,2,3,4,5 if the digits can be repeated in the same number?

Let M be 2 $\times$ 2 symmetric matrix with integer entries. Then M is invertible if

If $\frac{x^2}{36}-\frac{y^2}{k^2}=1$ is a hyperbola then which of the following statement can be true?

If
verify
that A³ − 6A² + 9A −
4I = O and hence find A⁻¹

The value of the integral $\int \frac{dx}{\sin (x-a)\sin (x-b)}$ is:

The last two digits of the number $(23{)}^{14}$ are