List of practice Questions

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 \( 2x^2 \frac{dy}{dx} = y^2 + 2xy \).

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

Assertion (A) : One of the particular solutions of the differential equation \(\frac{dy}{dx} = e^{x+y}\) can be \(e^x + e^{-y} = -2\).
Reason (R) : \(e^x + e^{-y} = C\) is the general solution of the differential equation \(\frac{dy}{dx} = e^{x+y}\).

Assertion (A) : The vectors \(\vec{a}\) and \((-2\vec{a})\), where \(\vec{a} \neq \vec{0}\) are collinear vectors.
Reason (R) : \(\vec{a} \cdot (-2\vec{a}) = 0\).

If \(x = t + \frac{1}{t}\) and \(y = t - \frac{1}{t}\), find \(\frac{dy}{dx}\) at \(t = 2\).

Find the sub-interval(s) of \((0, \frac{\pi}{2})\) in which \(f(x) = \tan x - 4x\) is increasing.

Find the value of \(\sin [\cot^{-1} \sqrt{2} (\cos (\tan^{-1} 1))]\).

If for two unit vectors \(\vec{a}\) and \(\vec{b}\), \(|\vec{a} + 2\vec{b}| = |2\vec{a} - \vec{b}|\), then find the angle between \(\vec{a}\) and \(\vec{b}\).

For any two vectors \( \vec{a} \) and \( \vec{b} \), which of the following statements is always true ?

If \( (\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b}) = 198 \) and \( |\vec{a}| = 10|\vec{b}| \), then :

If \(l_1, m_1, n_1\) and \(l_2, m_2, n_2\) are direction cosines of lines \(L_1\) and \(L_2\) respectively and \(\theta\) is the acute angle between them, then :

Direction ratios of lines \(l_1\) and \(l_2\) are \(\) and \(\) respectively. The values of \(p\) and \(q\) for which \(l_1\) and \(l_2\) are parallel are respectively :

The rate of change of volume of a sphere with respect to its diameter, when its radius is 5 cm, is :

\(\int \frac{dx}{2^x + 2^{-x}}\) is equal to :