List of practice Questions

A car moving with uniform acceleration covers the distance of 200 m in first 2 seconds and the distance of 220 m in next 4 seconds. The velocity of the car after 7 seconds is
A flywheel is rotating at a rate of 150 rev/minute. If it slows at constant retardation of \( \pi \) rads\(^{-2} \), then the time required for the wheel to come to rest is
If the graph of the anti derivative \( g(x) \) of \( f(x) = \log(\log x) + (\log x)^{-2} \) passes through \( (e, 2023 - e) \) and the term independent of \( x \) in \( g(x) \) is \( k \), then the sum of all the digits of \( k \) is
\( \int 3^{-\log_3 x^2} dx = \)
\( \int_0^{1/2} \frac{x \sin^{-1} x}{\sqrt{1 - x^2}} dx = \)
Given that \( \frac{d}{dx} \left[ \int_0^{\phi(x)} f(t) dt \right] = \phi'(x) f(\phi(x)) \). If \( \int_0^{x^3} f(t) dt = x^2 \sin 2\pi x \), then the value of \( f(8) \) is
\( \lim_{n \to \infty} \frac{1}{n} \left( \frac{1}{e^{1/n}} + \frac{1}{e^{2/n}} + \frac{1}{e^{3/n}} + \dots + \frac{1}{e^{n/n}} \right) = \)
If \( \int_0^{2024\pi} \frac{2023^{\sin^2 x}}{2023^{\sin^2 x} + 2023^{\cos^2 x}} dx = k \), then \( \left( \frac{2k}{\pi} + 1 \right) = \)
The area bounded by the curves \( y - 1 = \cos x \), \( y = \sin x \) and the X-axis between \( x = 0 \) and \( x = \pi \) is
Let \( f(x) = \max\{\cos x, \sin x, 0\} \). If the number of points at which \( f(x) \) is not differentiable in \( (0, 2024\pi) \) is \( 1012k \), then \( k = \)
If \( f(x) = px^3 + qx^2 + rx + t \) attains local minimum and local maximum values at \( x = -2 \) and \( x = 2 \) respectively and \( p \) is a root of \( 9x^2 - 1 = 0 \), then \( p + q + r = \)
If the tangent drawn at \( A(2, 1) \) to the curve \( x = 1 + \frac{1}{y^2} \) meets the curve again at \( B \), then
The points on the curve \( y^2 = x + \sin x \) at which the normal is parallel to the Y-axis lie on
Given that the solid obtained by rotating a rectangle about one of its sides is a cylinder. If the perimeter of a rectangle is 48 cm and the volume of the cylinder formed by rotating it is maximum, then the dimensions of that rectangle is
\( \int \frac{x^2 - 1}{x^3 \sqrt{2x^4 - 2x^2 + 1}} dx = \)
If \( \theta \) is the angle between the asymptotes of the hyperbola \( \frac{x^2}{9} - \frac{(y - 2)^2}{16} = 1 \) and \( \cos \theta = \frac{a^2}{b^2} \), then \( a^2 = \)
Let \( P(h, k) \) be the point of contact of the tangent to the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) which is parallel to the line \( \sqrt{3}x - y + 1 = 0 \). If \( P \) lies in the fourth quadrant then \( 3h^2 - 2k = \)
Let \( f(x) = |x - 3| + |x + 5| \) and \( A = \left\{ a \in \mathbb{R} / \lim_{x \to a} \frac{f(x) - f(a)}{x - a} \text{ exists} \right\} \). Then the number of real numbers which are in \( [-8, 3] \cup [5, \infty) \) but not in \( A \) is
\( \lim_{x \to 0} \left( \frac{(1 + y)^{1/x} - 1}{y} \right) = \)
If \( y = x \log \left( \frac{1}{ax} \right) \), then \( x(1 + x) \frac{d^2 y}{dx^2} + x \frac{dy}{dx} - y = \)
If the circles \( x^2 + y^2 - 2x + 4y + c = 0 \) and \( x^2 + y^2 + 2x - 4y + c = 0 \) have four common tangents, then
The locus of the poles of the tangents to the circle \( x^2 + y^2 - 2x + 2y - 2 = 0 \) with respect to the circle \( x^2 + y^2 = 4 \) is
Let the circle \( S \) be concentric with the circle \( x^2 + y^2 - 2x + ky + 4 = 0 \). If one of the diameters of \( S \) lies along the line \( 3x - 2y + 4 = 0 \) and the length of the diameter is 6, then the radius of the circle \( S \) is
If the length of the chord \( 2x + 3y + k = 0 \) of the circle \( x^2 + y^2 - 6x - 8y + 9 = 0 \) is \( 2\sqrt{5} \), then one of the values of \( k \) is
If \( Q \) is the inverse point of the point \( P(2, 3) \) with respect to the circle \( x^2 + y^2 - 2x - 2y + 1 = 0 \), then the circle with \( PQ \) as diameter is