List of top Mathematics Questions asked in PSEB XII

Name of the inequality \( |\vec{a} + \vec{b}| \leq |\vec{a}| + |\vec{b}| \) is:

Find the shortest distance between the following pairs of lines: \( \vec{r} = \hat{i} - 4\hat{j} + 5\hat{k} + \mu(5\hat{i} + 9\hat{j} + \hat{k}) \) and \( \vec{r} = 2\hat{i} + 8\hat{j} - 6\hat{k} + \lambda(3\hat{i} - 2\hat{j} + \hat{k}) \).

Adjacent sides of a parallelogram are given by \( 6\hat{i} - \hat{j} + 5\hat{k} \) and \( \hat{i} + 5\hat{j} - 2\hat{k} \). Find the area of parallelogram.

If \( A = \begin{bmatrix} 3 & 2 \\ 4 & 7 \end{bmatrix} \) and \( f(x) = x^2 - 10x + 13 \), then show that \( f(A) = O \) and using this result find \( A^{-1} \).

Express \( \begin{bmatrix} 5 & 1 \\ 3 & 7 \end{bmatrix} \) as the sum of a symmetric matrix and a skew-symmetric matrix.

Ajay, Sameer and Meenal have Rs. 20/- each and some footballs, basketballs and volleyballs in their shops. In a week, Ajay sold 3 footballs and a volleyball but he bought 2 basketballs for his shop and he has Rs. 35/- now. In same duration, Sameer sold 2 basketballs and 2 volleyballs but he bought a football for his shop and he has Rs. 95/- now. Similarly, Meenal sold 2 footballs and a basketball but she bought 3 volleyballs for her shop and she has Rs. 15/- now. Find the cost of a football, a basketball and a volleyball by the help of matrices.

If \( X = \begin{bmatrix} 3 & 1 \\ 2 & -1 \end{bmatrix} \) and \( 2X - Y = \begin{bmatrix} 5 & 10 \\ 3 & -5 \end{bmatrix} \), then find the matrix \( Y \).

Using determinants, find the value of \( k \) if the area of the triangle formed by the points \( (-3, 6), (-4, 4) \) and \( (k, -2) \) is 12 sq. units.

If \( A = \begin{bmatrix} 2 & 1 & 3 \\ 4 & 1 & 0 \end{bmatrix} \), \( B = \begin{bmatrix} 1 & -1 \\ 0 & 2 \\ 5 & 0 \end{bmatrix} \) then verify that \( (AB)' = B'A' \)

Evaluate \( \int \frac{x^2}{x^4+1} dx \).

Find the height of the right circular cone of maximum volume, which is inscribed in a sphere of radius 12cm.

Find the angle between the lines \( \vec{r} = 3\hat{i} + 8\hat{j} + 3\hat{k} + \mu(\hat{i} + 2\hat{j} - \hat{k}) \) and \( \vec{r} = -3\hat{i} + 9\hat{j} - \hat{k} + \lambda(5\hat{i} + 3\hat{j} + 4\hat{k}) \).

Prove that for any two non-zero vectors \( \vec{a} \) and \( \vec{b} \), \( |\vec{a} + \vec{b}| \leq |\vec{a}| + |\vec{b}| \). Also write the name of this inequality.

Evaluate \( \int \frac{3x+2}{(x^2+1)(x-2)}dx \).

Solve: \( x \log x \frac{dy}{dx} + y = \frac{2}{x} \log x \).

Solve the following linear programming problem graphically: Maximize and minimize \( Z = 4x + 2y - 7 \) subject to the constraints \( x + 3y \le 60, x + y \ge 10, x - y \le 0, x \ge 0, y \ge 0 \).
Correct Answer: Max Value \( Z = 73 \) at \( (15, 15) \); Min Value \( Z = 13 \) at \( (5, 5) \). (Based on standard corner point analysis)

A laboratory blood test is 99% effective in detecting a certain disease when it is in fact present. However, the test also yields a false positive result for 0.5% of the healthy person tested. If 0.1% of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive?

Find the value of \( p \) if the vectors \( p\hat{i} - 8\hat{j} + 5\hat{k} \) and \( 5\hat{i} + 2\hat{j} - 3\hat{k} \) are perpendicular to each other.

Prove that the function, \( f: \mathbb{R} \to \mathbb{R}, f(x) = \frac{5x+3}{4} \) is one-one and onto.

If \( u = x^y, v = y^x \) and quantity \( y \) remains 3 times the quantity \( x \) then find that amongst quantities \( u \) and \( v \), which changes more rapidly with respect to quantity \( x \) when \( x = 1 \). (Take \( \log_e 3 = 1.09 \))

If \( x = \frac{1-t^2}{1+t^2} , y = \frac{2t}{1+t^2} \) then prove that \( \frac{dy}{dx} + \frac{x}{y} = 0 \).
Correct Answer: Proved

Find the points on the curve \( y = 2x^3 - 3x^2 - 12x + 15 \) where rate of change of dependent variable with respect to the independent variable vanishes. What do we call these type of points?

Evaluate \( \int \frac{2x-3}{x^2+1} dx \).