List of practice Questions

A fruit box contains 6 apples and 4 oranges. A person picks out a fruit three times with replacement. Find:
(i) The probability distribution of the number of oranges he draws.
(ii) The expectation of the number of oranges.

Sketch a graph of $ y = x^2 $. Using integration, find the area of the region bounded by $ y = 9 $, $ x = 0 $, and $ y = x^2 $.

The probability that a student buys a colouring book is 0.7, and a box of colours is 0.2. The probability that she buys a colouring book, given that she buys a box of colours, is 0.3. Find:
(i) The probability that she buys both the colouring book and the box of colours.
(ii) The probability that she buys a box of colours given she buys the colouring book.

Find the particular solution of the differential equation: \[ x \sin^2 \left( \frac{y}{x} \right) \, dx + x \, dy = 0 \quad \text{given that} \quad y = \frac{\pi}{4}, \text{ when } x = 1. \]

Find: \[ \int \frac{2x}{(x^2 + 3)(x^2 - 5)} \, dx \]

If $\vec{a} + \vec{b} + \vec{c} = \vec{0}$ such that $|\vec{a}| = 3$, $|\vec{b}| = 5$, $|\vec{c}| = 7$, then find the angle between $\vec{a}$ and $\vec{b}$.

In the Linear Programming Problem (LPP), find the point/points giving the maximum value for $ Z = 5x + 10y$ subject to the constraints:
\[x + 2y \leq 120 \\ x + y \geq 60 \\ x - 2y \geq 0 \\ x \geq 0, y \geq 0\]

Evaluate: \[ \int_1^5 \left( |x-2| + |x-4| \right) \, dx \]

A man needs to hang two lanterns on a straight wire whose end points have coordinates A (4, 1, -2) and B (6, 2, -3). Find the coordinates of the points where he hangs the lanterns such that these points trisect the wire AB.

Find the value of $ a $ for which } $ f(x) = \sqrt{3} \sin x - \cos x - 2ax + 6 $ $\text{ is decreasing in } \mathbb{R}$.

Let $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ be three vectors such that $\mathbf{a} \times \mathbf{b} = \mathbf{a} \times \mathbf{c} \text{ and } \mathbf{a} \times \mathbf{b} \neq 0 \text{ Show that } \mathbf{b} = \mathbf{c}$.

Find a vector of magnitude 5 which is perpendicular to both the vectors $ 3\hat{i} - 2\hat{j} + \hat{k} \text{ and } 4\hat{i} + 3\hat{j} - 2\hat{k} $.

Differentiate $\frac{\sin x}{\sqrt{\cos x}}$ $\text{ with respect to } x$.

Surface area of a balloon (spherical), when air is blown into it, increases at a rate of 5 mm²/s. When the radius of the balloon is 8 mm, find the rate at which the volume of the balloon is increasing.

Assertion (A): Let $ A = \{ x \in \mathbb{R} : -1 \leq x \leq 1 \} $. If $ f : A \to A $ be defined as $ f(x) = x^2 $, then $ f $ is not an onto function.
Reason (R): If $ y = -1 \in A $, then $ x = \pm \sqrt{-1} \notin A $.

Find the domain of the function $ f(x) = \cos^{-1}(x^2 - 4) $.

If $y = 5 \cos x - 3 \sin x, \text{ prove that } \frac{d^2y}{dx^2} + y = 0$.

Assertion (A): The shaded portion of the graph represents the feasible region for the given Linear Programming Problem (LPP).
Reason (R): The region representing $ Z = 50x + 70y $ such that $ Z < 380 $ does not have any point common with the feasible region.

The respective values of $ |\vec{a}| $ and} $ |\vec{b}| $, if given \[ (\vec{a} - \vec{b}) \cdot (\vec{a} + \vec{b}) = 512 \quad \text{and} \quad |\vec{a}| = 3 |\vec{b}|, \] are:

The sum of the order and degree of the differential equation \[ \left( 1 + \left( \frac{dy}{dx} \right)^2 \right) \frac{d^2y}{dx^2} = \left( \frac{dy}{dx} \right)^3 \] is:

Let $ \vec{a} $ be a position vector whose tip is the point (2, -3). If $ \overrightarrow{AB} = \vec{a} $, where coordinates of A are (–4, 5), then the coordinates of B are:

For a Linear Programming Problem (LPP), the given objective function $ Z = 3x + 2y $ is subject to constraints: \[ x + 2y \leq 10, \] \[ 3x + y \leq 15, \] \[ x, y \geq 0. \]

The correct feasible region is:

The integrating factor of the differential equation \[ e^{-\frac{2x}{\sqrt{x}}} \, \frac{dy}{dx} = 1 \] is:

The area of the region enclosed between the curve $ y = |x| $, x-axis, $ x = -2 $} and $ x = 2 $ is:

The integral \[ \int_0^{\frac{\pi}{2}} \cos x \cdot e^{\sin x} \, dx \] is equal to: