List of practice Questions

If points (2, 3), (0, 4) and (p, 2) are collinear, then the value of p is :

Differential of \(e^{e^x}\) with respect to x is :

The surface area of a sphere when its volume changes at the same rate as its radius is :

If \(f(x) = \begin{cases \frac{\sin x}{x} + \cos x, & x \neq 0 \\ k, & x = 0 \end{cases}\) is continuous at \(x = 0\), then the value of \(k\) is :}

A survey was conducted to find out the success rate of students who qualified the entrance examination by dropping a year after class XII. \includegraphics[width=0.5\linewidth]{Q38.png} As per the data collected, 40% students appearing in the examination were dropouts and the remaining students were regular students of class XII.
Of the dropouts, 5% qualify the examination while 10% of the regular students qualify the examination.
Based on the above information, answer the following questions. (i). Find the probability that a student selected at random is a regular student.

A student is selected at random from a group of dropout students. What is the probability that the student will not qualify the examination ?

A student selected at random qualified the examination. Find the probability that student is not a dropout.

A student selected at random did not qualify the examination. Find the probability that the student was a regular student.

The following graph represents :

Show that $f : \mathbb{R} \to \mathbb{R}$ defined as $f(x) = \frac{x}{\sqrt{1+x^2}}$ is one-one but not onto.}

Solve the following Linear Programming Problem graphically :
Maximise $Z = 600x + 400y$
subject to the constraints
$x + 2y \leq 12$
$4x + 5y \geq 20$
$2x + y \leq 12$
$x, y \geq 0$

On the inauguration day of a new showroom, a lucky draw was organized and some vouchers of Rs 1,000 and Rs 500 were given to the lucky draw winners.

A total of 60 vouchers were given on the day. The number of Rs 1,000 vouchers added to 3 times the number of Rs 500 vouchers, gives 100. Express the given information as a system of linear equations in two variables. Hence, find the number of vouchers of each type by matrix method.

Given that $P = \begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix}$, $Q = \begin{bmatrix} 5 & 2 \\ 7 & 4 \end{bmatrix}$ and $R = \begin{bmatrix} 2 & 5 \\ 3 & 8 \end{bmatrix}$, find a matrix $S$ such that $PQ - RS$ is a null matrix.}

Represent the equations of lines $l_1$ and $l_2$ in vector form and check whether they are intersecting or not.
$l_1 : \frac{x+3}{-3} = \frac{y-1}{1} = \frac{z-5}{5}$
$l_2 : \frac{x+1}{-1} = \frac{2-y}{-2} = \frac{z-5}{5}$}

Opposite sides of a square are along the lines :
$\vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k})$
$\vec{r} = 3\hat{i} + 3\hat{j} - 5\hat{k} + \mu(2\hat{i} + 3\hat{j} + 6\hat{k})$
Find the area of the square if direction ratios of other pair of opposite sides of the square are given by $\langle -3, 6, p \rangle$. Also, find the value of $p$.}

Find the general solution of the differential equation \( 2x^2 \frac{dy}{dx} = y^2 + 2xy \).

Find a particular solution of the differential equation \( (x + 1) \frac{dy{dx} = 2 e^{-y} - 1 \), given that \( y = 0 \) when \( x = 0 \).}

If \( (\sin x)^y = y^{\cos x \), then find \( \frac{dy}{dx} \).}

If the lines \(\frac{x - 3}{1} = \frac{1 - y}{1} = \frac{z + 2}{p}\) and \(\frac{2 - x}{3} = \frac{y + 1}{5} = \frac{z + 56}{2p}\) are perpendicular to each other, then find the value(s) of \(p\).

Find the vector equation of a line passing through the origin and perpendicular to both the lines \(\vec{r} = 2\hat{i} - \hat{j} + 2\hat{k} + \lambda(3\hat{i} + 4\hat{j} + 2\hat{k})\) and \(\vec{r} = \mu(\hat{i} - \hat{j} + \hat{k})\).

Find : \( \int \frac{dx}{x^{1/2} + x^{1/3}} \)

Find : \( \int \tan^{-1} \left( \sqrt{\frac{1-x}{1+x}} \right) \, dx \)

Evaluate : \( \int_{0}^{\pi} \frac{\sin^{2026} x}{\sin^{2026} x + \cos^{2026} x} \, dx \)

Find : \( \int \frac{\cos x}{(2 + \sin x)(4 + \sin x) \, dx \)}

Find : \( \int \frac{x + 3}{x^2 + 4x + 5} \, dx \)