List of practice Questions

A motor can pump 7560 kg of water per hour from a well of depth 100 m. If the efficiency of the pump is 70%, then the power of the pump is:

If the potential energy of a particle of mass \(0.1 \, \text{kg}\) moving along the x-axis is \(5x(x - 4)\), then the speed of the particle is maximum at a position of:

The general solution of the differential equation \((1 + \sin^2 x) \, \frac{dy}{dx} + \sin 2x = 0\) is?

The displacement (\(x\)) and time (\(t\)) graph of a particle moving along a straight line is shown in the figure. The average velocity of the particle in the time of 10s is:

Evaluate \[ \int \sin^3 x \cos^2 x \, dx = ? \]

Evaluate \[ \int_0^\pi \left( \sin^3 x \cos^3 x + \sin^4 x \cos^4 x + \sin^3 x \cos^3 x \right) dx = ? \]

Evaluate \[ \int \frac{1}{x^4 + 1} \, dx = ? \]

Evaluate \[ \int (\log 2x)^3 \, dx = ? \]

If \(y = \sin^{-1} \left(\frac{2x}{1 + x^2}\right)\) and \(\left(\frac{d^2 y}{dx^2}\right)_{x=2} = k\), then find \(25k\).

If \(f(x) = x^{\sec^{-1} x}\), then find \(f'(2)\).

If the tangent to the curve \(xy + ax + by = 0\) at \((1,1)\) makes an angle \(\tan^{-1} 2\) with X-axis, then find \(\frac{ab}{a+b}\).

If the displacement \(S\) of a particle travelling along a straight line in \(t\) seconds is given by \[ S = 2t^3 + 2t^2 - 2t - 3, \] then the time taken (in seconds) by the particle to change its direction is?

If the image of the point \(A(1,1,1)\) with respect to the plane \(4x + 2y + 4z + 1 = 0\) is \(B(\alpha, \beta, \gamma)\), then find \(\alpha + \beta + \gamma\).

Evaluate \[ \lim_{x \to \infty} \frac{(3 - x)^{25} (6 + x)^{35}}{(12 + x)^{38} (9 - x)^{22}} = ? \]

If a tangent having slope \(\frac{1}{3}\) to the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a>b)\) is normal to the circle \((x+1)^2 + (y+1)^2 = 1\), then \(a^2\) lies in the interval?

The lines \(L_1: y - x = 0\) and \(L_2: 2x + y = 0\) intersect the line \(L_3: y + 2 = 0\) at points \(P\) and \(Q\) respectively. The bisector of the angle between \(L_1\) and \(L_2\) divides the segment \(PQ\) internally at \(R\). Consider: Statement-I: \(PR : RQ = 2\sqrt{2} : \sqrt{5}\).
Statement-II: In any triangle, bisector of an angle divides that triangle into two similar triangles.
Which statement(s) is/are correct?

If \[ 2x^2 + 3xy - 2y^2 - 5x + 2fy - 3 = 0 \] represents a pair of straight lines, then one of the possible values of \(f\) is?

Two dice are thrown and the sum of the numbers appeared on the dice is noted. If A is the event of getting a prime number as their sum and B is the event of getting a number greater than 8 as their sum, then find \(P(A \cap \overline{B})\).

Let \(X \sim B(n, p)\) with mean \(\mu\) and variance \(\sigma^2\). If \(\mu = 2\sigma^2\) and \(\mu + \sigma^2 = 3\), then find \(P(X \leq 3)\).

If \(A(\cos \alpha, \sin \alpha)\), \(B(\sin \alpha, -\cos \alpha)\), and \(C(1, 2)\) are the vertices of \(\triangle ABC\), then find the locus of its centroid.

Line \(L_1\) passes through the points \(\mathbf{i} + \mathbf{j}\) and \(\mathbf{k} - \mathbf{i}\). Line \(L_2\) passes through the point \(\mathbf{j} + 2\mathbf{k}\) and is parallel to the vector \(\mathbf{i} + \mathbf{j} + \mathbf{k}\). If \(\mathbf{x}i + \mathbf{y}j + \mathbf{z}k\) is the point of intersection of the lines \(L_1\) and \(L_2\), then find \((y - x) =\)

A and B are two independent events of a random experiment and \(P(A)>P(B)\). If the probability that both A and B occur is \(\frac{1}{6}\) and neither of them occurs is \(\frac{1}{3}\), then the probability of the occurrence of B is:

In \(\triangle ABC\), if the line joining the circumcentre and incentre is parallel to \(BC\), then find \( \cos B + \cos C \).

In a triangle \(ABC\), if \(r_1 : r_2 = 3 : 4\) and \(r_1 : r_3 = 2 : 3\), then find the ratio \(a : b : c\).