List of practice Questions

An equilateral triangle is inscribed in the parabola \( y^2 = 4ax \), one of whose vertices is at the vertex of the parabola, the length of each side of the triangle is:

If \( f(2) = 4 \) and \( f'(2) = 1 \), then \[ \lim_{x \to 2} \frac{x f(2) - 2f(x)}{x - 2} \text{ is equal to:} \]

What is the least value of \( k \) such that the function \( x^2 + kx + 1 \) is strictly increasing on \( (1, 2) \)?

The maximum value of \( \left| \frac{1}{x} \right| \) is:

If \( u = \tan^{-1} \left( \frac{x^3 + y^2}{x + y} \right) \), then \( \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} \) is:

The value of the integral \( \int_0^{\frac{\pi}{2}} \log (\tan x) \, dx \) is:

The region of the Argand plane defined by \( |z - 1| + |z + 1| \leq 4 \) is:

The value of the sum \( \sum_{n=1}^{13} (i^n + i^{n+1}) \), where \( i = \sqrt{-1} \), equals:

If \( \sin \theta, \cos \theta, \tan \theta \) are in G.P., then \( \cos^2 \theta + \cos \theta + 3 \cos \theta - 1 \) is equal to:

In a triangle ABC, \( 5 \cos C + 6 \cos B = 4 \) and \( 6 \cos A + 4 \cos C = 5 \), then:

In a model, it is shown that an arc of a bridge is semielliptical with major axis horizontal. If the length of the base is 9m and the highest part of the bridge is 3m from horizontal, the best approximation of the height of the arch, 2m from the center of the base is:

The number of real tangents through \( (3, 5) \) that can be drawn to the ellipses \( 3x^2 + 5y^2 = 32 \) and \( 25x^2 + 9y^2 = 450 \) is:

If the normal to the rectangular hyperbola \( xy = c^2 \) at the point \( (ct, c/t) \) meets the curve again at \( (ct', c/t') \), then:

If \( e^x = y + \sqrt{1 + y^2} \), then the value of y is:

Consider an infinite geometric series with the first term and common ratio. If its sum is 4 and the second term is \( \frac{3}{4} \), then:

If \( \alpha \) and \( \beta \) are the roots of the equation \( ax^2 + bx + c = 0 \), then the value of \( \alpha^3 + \beta^3 \) is:

The volume of the tetrahedron with vertices \( P(1, 2, 0), Q(2, 1, -3), R(1, 0, 1) \), and \( S(3, -2, 3) \) is:

If \( \mathbf{a} = i + 2j + 3k \), \( \mathbf{b} = i + 2j + k \), and \( \mathbf{c} = 3i + j \), then \( \mathbf{a} + \mathbf{b} \) is at right angle to \( \mathbf{c} \), then \( a + b \) and \( t \) are equal to:

An equation of the plane passing through the line of intersection of the planes \( x + y + z = 6 \) and \( 2x + 3y + 4z = 5 \), and passing through \( (1, 1, 1) \) is:

The length of the shortest distance between the lines \( \mathbf{r} = 3i + 5j + 7k + \lambda(2i - 2j + 3k) \) and \( \mathbf{r} = -i - j + k + \mu(7i - 6j + k) \) is:

Activation of benzene ring in aniline can be decreased by treating with:

The value of \( x \), for which the matrix \( A \) is singular, is: \[ A = \begin{pmatrix} 2 & x & -1 & 2 \\ 1 & x & 2x^2 \\ 1 & \frac{1}{x} & 2 \end{pmatrix} \]

The values of \( \alpha \) for which the system of equation \( x + y + z = 1 \), \( x + 2y + 4z = \alpha \), \( x + 4y + 10z = \alpha^2 \) is consistent are given by:

The non-integer roots of \( x^4 - 3x^3 - 2x^2 + 3x + 1 = 0 \) are:

The O–H stretching vibration of alcohols absorbs in the region 3700–3500 cm\(^{-1}\). The O–H stretching of carboxylic acids absorbs in the region: