List of practice Questions

In the given sentence, find out which part has an error. The letter of that part will be your answer. If there is no error, mark (d) as your answer. \[ \text{She is a brilliant teacher (a)/ but of her three children (b)/ neither has any merit. (c)/ No error. (d)} \]

If \( f(x) = x^3 + bx^2 + cx + d \) and \( 0<b^2<c \), then in \( (-\infty, \infty) \),

The solution of the differential equation \[ \left(1 + x \sqrt{x^2 + y^2}\right) dx + \left( \sqrt{x^2 + y^2} - 1 \right) y dy = 0 \]

The integrating factor of \[ \frac{dy}{dx} - y = x^4 - 3x \] \text{is:}

It is given that the events \( A \) and \( B \) are such that \[ P(A) = \frac{1}{4}, \quad P(A|B) = \frac{1}{2}, \quad P(B|A) = \frac{2}{3}. \] \text{Then \( P(B) \) is:}

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: