List of practice Questions

A hydraulic lift has two pistons: 3 cm and 5 cm. A 1875 kg mass is lifted. Find force on small piston. \( g = 10\ \text{m/s}^2 \)
In SI units, \( \text{kg m}^2 \text{s}^{-2} \) is equivalent to which of the following?
Two masses \( M_1 \) and \( M_2 \) are arranged as shown. If \( a \) is the acceleration of the system, and \( M_1 \) is doubled, \( M_2 \) halved, then new acceleration is:
Two masses M1 and M2 are arranged as shown
A ball of mass 300 g is dropped from a height of 10 m and penetrates 1.5 m into sand. Find the average resistance by the sand. Take \( g = 10\ \text{m/s}^2 \).
Evaluate: \[ \int_{0}^{\pi/4} e^{\tan^2 \theta} \sin^2 \theta \tan \theta \, d\theta = \]
Evaluate: \[ \int_{\pi/4}^{5\pi/4} \left(\cos t |\sin t| + \sin t |\cos t|\right) dt = \]
If \( f(x) = \max(\sin x, \cos x) \) and \( g(x) = \min(\sin x, \cos x) \), then \[ \int_0^{\pi/2} f(x) dx + \int_0^{\pi/2} g(x) dx = \]
If \( 2x - y + c \log(|x - 2y - 4|) = k \) is the general solution of \[ \frac{dy}{dx} = \frac{2x - 4y - 5}{x - 2y + 2} \] then \( c = \):
If the straight line \( x\cos\alpha + y\sin\alpha = p \) touches the curve \( \left( \frac{x}{a} \right)^n + \left( \frac{y}{b} \right)^n = 2 \) at the point (a, b), and \( \frac{1}{a^2} + \frac{1}{b^2} = \frac{k}{p^2} \), then \( k = \):
A closed cylinder of given volume will have least surface area when the ratio of its height and base radius is:
Two particles P and Q located at the points \( P(t, t^3 - 16t - 3) \), \( Q(t+1, t^3 - 6t - 6) \) are moving in a plane. The minimum distance between the points in their motion is:
If \( f(x) = \int_{x^2}^{\cos^2 x} (2x \tan^2 x - 2x - 6 \tan x) dx \), and \( f(0) = \pi \), then \( f(x) = \):
If $[\cdot]$ denotes greatest integer function, then $\lim_{x \to 3} \dfrac{[-x]}{x} =$
If $l, m$ $(l<m)$ are roots of $ax^2 + bx + c = 0$, then $\lim_{x \to a} \left| \dfrac{ax^2 + bx + c}{ax^2 + bx + c} \right|$ =
Calculate $\dfrac{d}{dx} \left(\lim_{y \to 2} \dfrac{1}{y-2} \left(\dfrac{1}{x} - \dfrac{1}{x+y-2}\right)\right)$
The number of those tangents to the curve $y^2 - 2x^3 - 4y + 8 = 0$ which pass through the point $(1, 2)$ is
The focal distances of the point $\left(\dfrac{4}{\sqrt{5}}, \dfrac{3}{\sqrt{5}}\right)$ on the ellipse $\dfrac{x^2}{4} + \dfrac{y^2}{9} = 1$ are
If the normal to the rectangular hyperbola $x^2 - y^2 = 1$ at the point $P(\tfrac{\pi}{4})$ meets the curve again at $Q(\theta)$, then $\sec^2\theta + \tan\theta =$
If the vertices and foci of a hyperbola are respectively $(\pm3,0)$ and $(\pm4,0)$ then the parametric equations of that hyperbola are
If $x$-coordinate of a point $P$ on the line joining the points $Q(2,2,1)$ and $R(5,2,-2)$ is $4$, then the $y$-coordinate of $P$ is
If $(2, 3, c)$ are the direction ratios of a ray passing through the point $C(5, q, 1)$ and also the midpoint of the line segment joining the points $A(p, -4, 2)$ and $B(3, 2, -4)$, then $c(p + 7q) = $
If the equation of the plane which is at a distance of $\dfrac{1}{\sqrt{3}}$ units from the origin and perpendicular to a line whose directional ratios are $(1, 2, 2)$ is $x + py + qz + r = 0$, then $\sqrt{p^2 + q^2 + r^2} =$
The area of the circle passing through the points $(5, \pm2)$, $(1,2)$ is
A circle has its centre in the first quadrant and passes through $(2, 3)$. If this circle makes intercepts of length 3 and 4 respectively on $x = 2$ and $y = 3$, its equation is
A manager decides to distribute Rs. 20000 between two employees X and Y. He knows X deserves more than Y, but does not know how much more. So, he decides to arbitrarily break Rs. 20000 into two parts and gives X the bigger part. Then, the chance that X gets twice as much as Y or more is