List of practice Questions

If the length of a rod is measured as 830600 mm, then the number of significant figures in the measurement is

A particle initially at rest is moving along a straight line with an acceleration of 2 ms⁻². At a time of 3 s after the beginning of motion, the direction of acceleration is reversed. The time from the beginning of the motion in which the particle returns to its initial position is

If a body projected with a velocity of 19.6 ms⁻¹ reaches a maximum height of 9.8 m, then the range of the projectile is (Neglect air resistance)

$\int e^x(x^3-2x^2+3x-4)dx =$

$\int(1+\tan^2 x)(1+2x\tan x)dx =$

$\int \frac{x\text{Tan}^{-1}x}{(1+x^2)^2}dx =$

$\int \frac{\log x}{(1+x)^2}dx = $

$\int_0^{\pi/2} \frac{1}{5\cos^2 x + 16\sin^2 x + 8\sin x \cos x} dx =$

$\int_4^{18} \frac{1}{(x+2)\sqrt{x-3}}dx = $

If $[\cdot]$ denotes the greatest integer function, then $\int_1^2 [x^2] dx =$

If the normal drawn at the point P on the curve $y^2 = x^2-x+1$ makes equal intercepts on the coordinate axes, then the equation of the tangent drawn to the curve at P is

If a balloon flying at an altitude of 30 m from an observer at a particular instant is moving horizontally at the rate of 1 m/s away from him, then the rate at which the balloon is moving away directly from the observer at the 40th second is (in m/s)

The approximate value of $\sqrt{6560}$ is

A real valued function
$f:[4, \infty) \to \mathbb{R}$ is defined as $f(x) = (x^2+x+1)^{(x^2-3x-4)}$, then f is

If a normal is drawn at a variable point P(x, y) on the curve $9x^2+16y^2-144=0$, then the maximum distance from the centre of the curve to the normal is

If $\{x\}=x-[x]$ where $[x]$ is the greatest integer $\le x$ and $\lim_{x\to 0^+} \frac{\text{Cos}^{-1}(1-\{x\}^2)\text{Sin}^{-1}(1-\{x\})}{\{x\}-\{x\}^3} = \theta$, then $\tan\theta=$

For $a\neq0$ and $b\neq0$, if the real valued function $f(x) = \frac{\sqrt[4]{625+4x}-5}{\sqrt[4]{625+5bx}-5}$ is continuous at $x=0$, then $f(0) =$

If $y^3=x$ then the value of $\frac{dy}{dx}$ at $x=1$ is

The values of x at which the real valued function $f(x)=7|2x+1|-19|3x-5|$ is not differentiable is

If $y=(1-x^2)\text{Tanh}^{-1}x$ then $\frac{d^2y}{dx^2}=$

If $f(x) = \log_{(x-1)^2}(x^2-3x+2)$, $x \in \mathbb{R}-[1,2]$ and $x\neq0$, then $f'(3)=$

If $\theta$ is the acute angle between the tangents drawn from the point (1,5) to the parabola $y^2 = 9x$ then

Let P be a point on the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ and let the perpendicular drawn through P to the major axis meet its auxiliary circle at Q. If the normals drawn at P and Q to the ellipse and the auxiliary circle respectively meet in R, then the equation of the locus of R is

The midpoint of the chord of the ellipse $x^2+\frac{y^2}{4}=1$ formed on the line $y=x+1$ is

If the tangent drawn at the point $P(3\sqrt{2}, 4)$ on the hyperbola $\frac{x^2}{9}-\frac{y^2}{16}=1$ meets its directrix at $Q(\alpha, \beta)$ in the fourth quadrant then $\beta = $