List of practice Questions

If \( \frac{dy}{dx} - y \log_e 2 = 2^{\sin x} (\cos x - 1) \log_e 2 \), then \( y \) is:

The solution of the differential equation \( (x + 1)\frac{dy}{dx} - y = e^{3x}(x + 1)^2 \) is:

If the area bounded by the curves \( y = ax^2 \) and \( x = ay^2 \) (where \( a>0 \)) is 3 sq. units, then the value of \( a \) is:

The area of the region bounded by the curves \( x = y^2 - 2 \) and \( x = y \) is:

The area enclosed by the curves \( y = x^3 \) and \( y = \sqrt{x} \) is:

If \( a, c, b \) are in GP, then the area of the triangle formed by the lines \( ax + by + c = 0 \) with the coordinate axes is equal to:

The line \(y = mx\) bisects the area enclosed by lines \(x = 0\), \(y = 0\), and \(x = \frac{3}{2}\) and the curve \(y = 1 + 4x - x^2\). Then, the value of \(m\) is:

The value of \( \int_0^{\frac{\pi}{2}} \frac{\sin\left( \frac{\pi}{4} + x \right) + \sin\left( \frac{3\pi}{4} + x \right)}{\cos x + \sin x} \, dx \) is:

Evaluate the integral: \[ \int \frac{x^2 (x \sec^2 x + \tan x)}{(x \tan x + 1)^2} dx \]

Evaluate the integral: \[ \int_{5}^{9} \frac{\log 3x^2}{\log 3x^2 + \log (588 - 84x + 3x^2)} dx \]

The value of definite integral \( \int_0^{\pi/2} \log(\tan x) dx \) is:

The value of \( \int_0^\infty \frac{dx}{(x^2 + a^2)(x^2 + b^2)} \) is:

The value of \( \int e^{\tan \theta} (\sec \theta - \sin \theta) \, d\theta \) is:

Evaluate the integral: \[ \int \sqrt{x + \sqrt{x^2 + 2}} \, dx. \]

Evaluate the integral: \[ \int \frac{x^3 - 1}{x^3 + x} dx \]

The population \( p(t) \) at time \( t \) of a certain mouse species satisfies the differential equation:
\[ \frac{d p(t)}{dt} = 0.5p(t) - 450. \] If \( p(0) = 850 \), then the time at which the population becomes zero is:

The point of inflexion for the curve \(y = (x - a)^n\), where \(n\) is odd integer and \(n \ge 3\), is:

The altitude of a cone is 20 cm and its semi-vertical angle is \(30^\circ\). If the semi-vertical angle is increasing at the rate of \(2^\circ\) per second, then the radius of the base is increasing at the rate of:

If the angle made by the tangent at the point \((x_0, y_0)\) on the curve \(x = 12(t + \sin t \cos t)\), \(y = 12(1 + \sin t)^2\), with \(0<t<\frac{\pi}{2}\), with the positive x-axis is \(\frac{\pi}{3}\), then \(y_0\) is equal to:

The maximum volume (in cu. units) of the cylinder which can be inscribed in a sphere of radius 12 units is:

Consider the function \( f(x) = \frac{|x - 1|}{x^2} \). Then \( f(x) \) is:

The maximum area of a rectangle inscribed in a circle of diameter \( R \) is:

At \( x = \frac{\pi^2}{4} \), \( \frac{d}{dx} \left( \tan^{-1}(\cos\sqrt{x}) + \sec^{-1}(e^x) \right) = \)

If \( y = \tan^{-1}\left( \frac{\sqrt{x} - x}{1 + x^{3/2}} \right) \), then \( y'(1) \) is equal to:

If \( x\sqrt{1 + y} + y\sqrt{1 + x} = 0 \), then find \( \frac{dy}{dx} \).