A block of mass '$m$' (as shown in figure) moving with kinetic energy $E$ compresses a spring through a distance 25 cm when, its speed is halved. The value of spring constant of used spring will be $nE$ N$^{-1}$ for $n =$ \_\_\_\_. [Figure]
If the minimum value of $f(x) = \frac{5x^2}{2} + \frac{\alpha}{x^5}$, $x > 0$, is 14, then the value of $\alpha$ is equal to (1) 32 (2) 64 (3) 128 (4) 256
For $t \in (0, 2\pi)$, if $ABC$ is an equilateral triangle with vertices $A(\sin t, -\cos t)$, $B(\cos t, \sin t)$ and $C(a, b)$ such that its orthocentre lies on a circle with centre $\left(1, \frac{1}{3}\right)$, then $a^2 - b^2$ is equal to (1) $\frac{8}{3}$ (2) $8$ (3) $\frac{77}{9}$ (4) $\frac{80}{9}$
Let $C$ be the centre of the circle $x^2 + y^2 - x + 2y = \frac{11}{4}$ and $P$ be a point on the circle. A line passes through the point $C$, makes an angle of $\frac{\pi}{4}$ with the line $CP$ and intersects the circle at the points $Q$ and $R$. Then the area of the triangle $PQR$ (in unit$^2$) is (1) 2 (2) $2\sqrt{2}$ (3) $8\sin\frac{\pi}{8}$ (4) $8\cos\frac{\pi}{8}$
If the tangents drawn at the points $P$ and $Q$ on the parabola $y^2 = 2x - 3$ intersect at the point $R(0, 1)$, then the orthocentre of the triangle $PQR$ is (1) $(0, 1)$ (2) $(2, -1)$ (3) $(6, 3)$ (4) $(2, 1)$
Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation $\cos^{-1}x - 2\sin^{-1}x = \cos^{-1}(2x)$ is equal to (1) 0 (2) 1 (3) $\frac{1}{2}$ (4) $-\frac{1}{2}$