List of practice Questions

The maximum number of points into which 4 circles and 4 straight lines intersect, is

Given that $ \alpha_1, \alpha_2, \alpha_3 $ are the roots of $ 3x^3 - x^2 - 10x + 8 = 0 $, then the value of $ \alpha_1^2 + \alpha_2^2 + \alpha_3^2 $ is

The value of the integral \[ \int_0^0 9 [x - 2\lfloor x \rfloor] \, dx \] where $ [.] $ denotes the greatest integer function, is

The number of solutions of the equation \[ 3 \sin^2 x - 7 \sin x + 2 = 0 \] in the interval $ [0, 5\pi] $ is

If \[ a = \gamma \hat{i} + \delta \hat{j} + 2\zeta \hat{k}, \, b = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k}, \, r \times a = b \times a \, r \times b = a \times b \] then a unit vector in the direction of $ r $ is

From the bottom of a pole of height h, the angle of elevation of the top of a tower is $ \alpha $. The pole subtends an angle $ \beta $ at the top of the tower. The height of the tower is

If the numbers $a_1, a_2, \ldots, a_n$ are different from zero and form an arithmetic progression, then $$ \frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3} + \dots + \frac{1}{a_n} $$ is equal to

If $n$ is an integer, compute the value of the fraction \[ \frac{(1 + \theta)^n}{(1 - \theta)^{n-2}} \]

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If $ A $ is a square matrix such that $ A^2 = A $ and $ B = I $, then $ AB + BA + I - (I - A)^2 $ is equal to:

Calculate

\[ \begin{vmatrix} x & y & x + y \\ y & x + y & x \\ x + y & x & y \end{vmatrix} \]

Let $ f(x) = (x^3 + 2)^{30} $. If $ f^{(n)}(x) $ is a polynomial of degree $ 20 $, where $ f^{(n)}(x) $ denotes the $ n + h $-order derivative of $ f(x) $, then the value of $ n $ is:

The set of values of $ x $ satisfying the system of inequalities $ 5x + 2 < 3x + 8 $ and $ \frac{x^2}{x-1} < 4 $ is:

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If $ |z^2 - 1| = |z|^2 + 1 $, then $ z $ lies on a

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If $ f(x) = [x \sin n\pi x] $, then which of the following is incorrect?

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Find the distance from the point A (2, 3, -1) to the given straight line. \[ x = 3t + 5, \quad y = 2t, \quad z = -2t - 25 \]

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The limit \[ \lim_{x \to 0} \frac{1 - \cos 2x}{x \tan 4x} \]

The number of real roots of $$ (6 - x)^4 + (8 - x)^4 = 16 $$

The sum \[ \sum_{k=0}^5 \left( 5C_k \right)^2 \] is equal to

If \[ f(x) = \begin{cases} \frac{1 - \sin x}{(n - 2x)^2} & \text{if} \quad x \neq \frac{\pi}{2} \log (\sin x) \cdot \log \left( 1 + \frac{\pi}{4x + x^2} \right) & \text{if} \quad x = \frac{\pi}{2} \end{cases} \] is continuous at $ x = \frac{\pi}{2} $, then $ k $ is equal to

The length of the axes of the conic \[ 9x^2 + 4y^2 - 6x + 4y + 1 = 0 \] are

A differentiable function $ f(x) $ has a relative minimum at $ x = 0 $, then the function $ y = f(x) + ax + b $ has a relative minimum at $ x = 0 $ for

The solution of the differential equation \[ (x^2 - yx^2) \frac{dy}{dx} + y^2 + xy^2 = 0 \] is

The equation of a straight line passing through the point of intersection of $ x - y + 1 = 0 $ and $ 3x + y - 5 = 0 $ and perpendicular to one of them, is:

The mid-point of the chord $2x + y - 4 = 0$ of the parabola $y^2 = 4x$ is

Gas is being pumped into a spherical balloon. Then, the rate at of $30 \, \text{ft}^3/\text{min}$ which the radius increases when it reaches the value 15 ft, is