List of top Mathematics Questions asked in CBSE CLASS XII

A man goes to buy fruits from the market. The shopkeeper informs him that 4 apples, 3 oranges and 2 bananas cost Rs.60, 2 apples, 4 oranges and 6 bananas cost Rs.90, whereas 6 apples, 2 oranges and 3 bananas cost Rs.70. Using matrix method, find the cost of one fruit of each kind.

Sketch the curve \( \{(x, y) : 100x^2 + 25y^2 = 2500\ \) and find the area of the region enclosed by it, using integration.}

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 differential of \( x^{\cot x} + \frac{2x^2-3}{2x^2-x+2} \) with respect to \( x \).

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

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

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.

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 \( P(A \cup B | C) \)

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

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'.

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 \(\Delta ABC\).

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

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

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 :

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

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.

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

The corner points of the feasible region determined by the system of linear constraints are \( (0, 0), (0, 40), (20, 40), (60, 20) \) and \( (60, 0) \). If the objective function of an LPP is \( Z = 4x + 3y \), then the maximum value is :

The general solution for the differential equation \( \frac{dy}{dx} = e^{3x-y} \) is :

An ant is observed crawling on a sheet of paper along a straight line given by equation \( y = 2x - 4 \). Area of the surface covered by the ant bounded by y-axis, x-axis and \( x = 1 \) is :

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