List of practice Questions

Find the foot of the perpendicular from the point \( (0, 2, 3) \) on the line \( \frac{x+3}{5} = \frac{y-1}{2} = \frac{z+4}{3} \) and hence find the length of the perpendicular.

Find the value of \( p \) if the shortest distance between the lines \( \vec{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}) \) and \( \vec{r} = (p\hat{i} - \hat{j} - \hat{k}) + \mu(2\hat{i} + \hat{j} + 2\hat{k}) \) is \( \frac{3}{\sqrt{2}} \) units.

Find : \( \int \frac{x - \sin x}{1 - \cos x} \, dx \)

Evaluate : \( \int_0^2 \frac{1}{\sqrt{x^2 + 2x + 3}} \, dx \)

Solve the differential equation \( (x + 2y^3) \frac{dy{dx} = y \).}

Solve the following Linear Programming Problem graphically :
Maximize \( Z = \frac{2x}{5} + \frac{3y}{10} \)
subject to constraints
\( 2x + y \leq 1000 \)
\( x + y \leq 800 \)
\( x, y \geq 0 \).

Let three toys A, B and C be placed in the same straight line. If the position vectors of A, B and C are \( 55\hat{i} - 2\hat{j} \), \( 5\hat{i} + 8\hat{j} \) and \( a\hat{i} - 52\hat{j} \) respectively, find the value of 'a'.

If \( \vec{a}, \vec{b} \) and \( \vec{c} \) are unit vectors, then prove that \( |\vec{a} - \vec{b}|^2 + |\vec{b} - \vec{c}|^2 + |\vec{c} - \vec{a}|^2 \leq 9 \).

A die is rolled. Consider events : \( A = \{1, 2, 5\}, B = \{3, 5\}, C = \{2, 3, 4, 5\} \) and hence find : (i) \( P(A|C) \) and \( P(C|A) \) (ii) \( P(A \cap B | C) \) and \( P(A \cup B | C) \)

A box contains 6 cards numbered 1 to 6. A student is asked to pick up two cards, one by one after replacement and note down the numbers on the cards. Let A be the event of getting sum of the numbers on two cards as 10, and B, the event of a number other than 4 on the first card selected.
Find P(A and B) and find whether the events A and B are independent events or not.

Differentiate \(x^x\) with respect to \(x \log x\).

If \(y = P \cos ux + Q \sin ux\), show that \(\frac{d^2y}{dx^2} + u^2y = 0\).

Determine the values of \(x\) for which \(f(x) = \frac{x - 3{x + 1}, x \neq -1\) is an increasing function.}

Three honey bees were found flying along the vectors \(\vec{a} = 2\hat{i} - 3\hat{j} + \hat{k}\), \(\vec{b} = 4\hat{j} - 2\hat{k}\) and \(\vec{c} = 3\hat{i} + 2\hat{k}\) respectively.
Find the value of \(\lambda\) such that the path for \(\vec{a} + \lambda\vec{b}\) is perpendicular to \(\vec{c}\).

If \(A, B\) and \(C\) be three non-collinear points such that \(\vec{AB} = \hat{i} + 2\hat{j} - \hat{k}\) and \(\vec{AC} = 2\hat{i} - 3\hat{j}\), then find the area of \(\Delta ABC\).

Find the angle between the following pair of lines :
\(\frac{x - 2}{3} = \frac{y + 5}{2} = \frac{1 - z}{-6}\) and \(\frac{x - 7}{1} = \frac{y}{2} = \frac{6 - z}{-2}\).

A spherical balloon loses its volume due to escape of air from it in such a way that decrease of volume at any instant is proportional to its surface area. Show that the radius is decreasing at a constant rate.

If position vector \( \vec{p} \) of a point \( (24, n) \) is such that \( |\vec{p}| = 25 \), then the value of \( n \) is :

If vectors \( \vec{a} = 3\hat{i} + 2\hat{j} + \lambda\hat{k} \) and \( \vec{b} = 2\hat{i} - 4\hat{j} + 5\hat{k} \), represent the two strips of the Red Cross sign placed outside a doctor's clinic, then the value of \( \lambda \) is :

If \( 3P(A) = P(B) = \frac{3{5} \) and \( P(A|B) = \frac{2}{3} \), then \( P(A \cup B) \) is :}

Assertion (A) : A relation R on the set {1, 2, 3\ defined as R = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3)} is an equivalence relation.
Reason (R) : A relation that is reflexive, symmetric and transitive is an equivalence relation.}

Assertion (A) : Consider a Linear Programming Problem with minimise Z = x + 2y subject to constraints 2x + y \( \geq \) 3, x + 2y \( \geq \) 6, x, y \( \geq \) 0 which gives minimum Z at infinitely many points. The corner points of feasible region are (0, 3) and (6, 0).
Reason (R) : If two corner points produce the same minimum value of the objective function, then every point on the line segment joining the points will give the same minimum value.

Evaluate \(\sin \left[\tan^{-1} \tan\left(\frac{3\pi}{4}\right)\right]\).

\( \int \frac{1}{\sqrt{1 + \cos 2x}} \, dx \) is equal to :

The value of \( \int_{-5}^{-1} \frac{1}{x} \, dx \) is equal to :