List of practice Questions

Let \( f: \mathbb{R} \to \mathbb{R}, g: \mathbb{R} \to \mathbb{R} \) be two functions such that \[ f(x) = 2x - 3, \quad g(x) = x^3 + 5. \] \text{The function} \( (f \circ g)^{-1}(x) \) is equal to:

The inverse of the statement \[ (p \land \neg q) \to r \] \text{is:}

A wire 34 cm long is to be bent in the form of a quadrilateral of which each angle is 90°. What is the maximum area which can be enclosed inside the quadrilateral?

The equation of the chord of the hyperbola \( 25x^2 - 16y^2 = 400 \) that is bisected at point \( (5, 3) \) is:

Value of \[ \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{1}{1 + \sqrt{\cot x}} dx \]

The domain of the function \[ f(x) = \sqrt{\frac{1}{|x-2| - (x-2)}} \] \text{is:}

Six identical coins are arranged in a row. The number of ways in which the number of tails is equal to the number of heads is:

If \( \sin y = x \sin(a + y) \), then \( \frac{dy}{dx} \) is equal to:

Evaluate \[ \int \left( 27e^{9x} + e^{12x} \right)^{1/3} dx \]

The area under the curve \( y = | \cos x - \sin x | \), where \( 0 \leq x \leq \frac{\pi}{2} \), and the x-axis is:

The conic represented by \[ x = 2(\cos t + \sin t), \quad y = 5(\cos t - \sin t) \] \text{is}

At how many points between the interval \( (-\infty, \infty) \) is the function \( f(x) = \sin x \) is not differentiable?

If vector equation of the line \( \frac{x-2}{2} = \frac{2y-5}{-3} = z+1 \), is \[ \mathbf{r} = \left( 2\hat{i} + \frac{5}{2} \hat{j} - k \right) + \lambda \left( 2\hat{i} - \frac{3}{2} \hat{j} + p \hat{k} \right) \] then \( p \) is equal to:

Evaluate \( \lim_{x \to 2} \frac{\sqrt{x+7} - 3}{\sqrt{x-3} - 2} \):

The radius of a right circular cylinder increases at the rate of 0.1 cm/min, and the height decreases at the rate of 0.2 cm/min. The rate of change of the volume of the cylinder, in cm\(^3\)/min, when the radius is 2 cm and the height is 3 cm is:

If \( a \), \( b \), and \( c \) are in A.P., then the value of: \[ |x + 1| + |x + 2| + |x + 3| + |x + b| + |x + c| \]

The solution of \( \sin x = \frac{-\sqrt{3}}{2} \) is:

The equation \( y^2 + 3 = 2(2x + y) \) represents a parabola with the vertex at:

If \( \omega = \frac{-1 + \sqrt{3}i}{2} \), then \( (3 + \omega + 3\omega^2) \) is:

If the lines \( 3x - 4y + 4 = 0 \) and \( 6x - 8y - 7 = 0 \) are tangents to a circle, then radius of the circle is:

The position vector of A and B are: \[ 2\hat{i} + 2\hat{j} + \hat{k} \quad \text{and} \quad 2\hat{i} + 4\hat{j} + 4\hat{k} \] The length of the internal bisector of \( \triangle AOB \) is:

If \[ A = \begin{bmatrix} 0 & c & -b \\ -c & 0 & a \\ b & -a & 0 \end{bmatrix}, \quad B = \begin{bmatrix} a^2 & ab & ac \\ ab & b^2 & bc \\ ac & bc & c^2 \end{bmatrix}, \] then \( AB \) is equal to:

A circle has radius 3 and its center lies on the line \( y = x - 1 \). The equation of the circle, if it passes through (7, 3), is:

A coordination complex compound of cobalt has the molecular formula containing five ammonia molecules, one nitro group and two chlorine atoms for one cobalt atom. One mole of this compound produces three mole ions in an aqueous solution; on reacting with excess of \( \text{AgNO}_3 \), \( \text{AgCl} \) is precipitated. The ionic formula for this complex would be:

An endothermic reaction with high activation energy for the forward reaction is given by the diagram: