List of top Mathematics Questions asked in CBSE CLASS XII

(iii) (b) If it is given that the area of the window is \(100\ \text{m}^2\), then find the expression for its perimeter in terms of x.

(iii) (a) Find the dimensions of the rectangle that will allow the maximum light through the window.

(ii) From (i), write an expression for the area of the window as a function of x only.

(iii) (b) If a bird flies across the top of the swimming pool with its path represented by \(\frac{x+4}{2}=\frac{y+3}{8}=\frac{z-5}{-18}\), then find the shortest distance between its path and the path of student A.

Let \( \vec{\alpha}, \vec{\beta}, \vec{\gamma} \) be three non-zero vectors, such that \( \vec{\alpha} \times \vec{\beta} = \vec{\gamma} \) and \( \vec{\beta} \times \vec{\gamma} = \vec{\alpha} \). Show that \( \vec{\alpha}, \vec{\beta}, \vec{\gamma} \) are mutually perpendicular vectors such that \( |\vec{\beta}| = 1 \) and \( |\vec{\alpha}| = |\vec{\gamma}| \).

Find projection of vector \( \vec{\alpha} \) on \( \vec{\beta} \). Give reasons to support your answer.

(ii) Write the vector equation of the path of student B swimming parallel to the path of student A, if at a certain instant he passes through the point with coordinates \((2, 4, -1)\).

(i) If the path of student A is represented by line \(\frac{x+5}{1}=\frac{y+3}{4}=\frac{z-6}{-3p}\), find the value of p.

(iii) (a) Find the foot of the perpendicular drawn from \((2, 4, -1)\) to the path represented by student A.

If points \( A(3, -1, 2) \), \( B(6, -3, 6) \), \( C(4, -3, 1) \) and \( D(1, -1, -3) \) are vertices of a quadrilateral ABCD, write the position vectors of A, B, C and D and show that ABCD represents a parallelogram. Also, find the area of the parallelogram ABCD.

Evaluate :

\[ \int_{\pi/6}^{\pi/3} \frac{\sin x + \cos x}{\sqrt{\sin 2x}}\, dx \]

Express the matrix \(A = \begin{bmatrix} 2 & 4 \\ 5 & 6 \end{bmatrix}\) as the sum of a symmetric matrix and a skew symmetric matrix.

Two lines are represented by equations \(5x - 7y - 2 = 0\) and \(7x - 5y - 3 = 0\) in the xy-plane. Using matrix method, find their point of intersection.

Solve the following Linear Programming Problem graphically :

Minimize \(Z = 4x + y\)

subject to

\(x + y \geq 3\)

\(x + 2y \geq 4\)

\(x, y \geq 0\).

Using integration, find the area bounded by the curve \( 9x^2 + 16y^2 = 144 \) and the lines \( x = 2 \), \( x = -2 \).

Using integration, find the area bounded by the curve \( y = x^2 \) and the line \( y = 2 \).

Solve the differential equation \( x^2\,dy + y(x + y)\,dx = 0 \), given that \( y = 1 \) when \( x = 1 \).

If \(P(A) = \frac{3}{8}\), \(P(B) = \frac{1}{2}\) and \(P(A \cap B) = \frac{1}{4}\), find \(P(\overline{A} / \overline{B})\).

If R is a relation on set of natural numbers N such that

\( R = \{(x, y) : x, y \in N,\ x + 2y = 25\} \),

then find the Domain and Range of the relation R. Also, determine if it is an equivalence relation.

Differentiate \(y = e^{\cos^{-1}\sqrt{1 - x^2}}\) with respect to \(x\) and hence show that \(\frac{dy}{dx} = \frac{y}{\sqrt{1 - x^2}}\).

Find the values of \(a\) and \(b\), if \(y = \frac{ax - b}{(x - 1)(x - 4)}\) has a critical point \((3, 1)\).

Find the equations of the lines AC and BD, if coordinates of points A, B, C and D respectively, are \((3, -4, 2)\), \((-1, -2, 0)\), \((4, 1, -1)\) and \((4, 0, 5)\). Show that the lines AC and BD are perpendicular to each other.

Find the equation of a line passing through the point \((1, -1, 2)\) and parallel to x-axis. Also, find the shortest distance between this line and the line \(\vec{r} = 2\hat{i} + \hat{j} - \hat{k} + \mu(3\hat{i} - 5\hat{j} + 2\hat{k})\).

Determine whether the function \(f : \mathbb{R} \to \mathbb{R}\) defined by \(f(x) = 2x^3 - 7\), \(\forall\, x \in \mathbb{R}\) (Real numbers) is bijective or not.

A box contains cards numbered 1 to 22. Two cards are drawn simultaneously without replacement and their numbers are noted. What is the probability that both numbers are even ?

Determine \(a\) and \(b\) if \(f(x)\) is a continuous function, where

\[ f(x) = \begin{cases} 2, & x \le 3 \\ ax + b, & 3 < x < 5 \\ 5, & x \ge 5 \end{cases} \]