List of top Mathematics Questions asked in CBSE CLASS XII

A bacteria sample of a certain number of bacteria is observed to grow exponentially in a given amount of time. Using the exponential growth model, the rate of growth of this sample of bacteria is calculated. The differential equation representing the growth is:

\[ \frac{dP}{dt} = kP, \] where \( P \) is the bacterial population.

Based on this, answer the following:

Self-study helps students to build confidence in learning. It boosts the self-esteem of the learners. Recent surveys suggested that close to 50% learners were self-taught using internet resources and upskilled themselves. A student may spend 1 hour to 6 hours in a day upskilling self. The probability distribution of the number of hours spent by a student is given below:

\[ P(X = x) = \begin{cases} kx^2 & {for } x = 1, 2, 3, \\ 2kx & {for } x = 4, 5, 6, \\ 0 & {otherwise}. \end{cases} \]

Based on the above information, answer the following:

A scholarship is a sum of money provided to a student to help him or her pay for education. Some students are granted scholarships based on their academic achievements, while others are rewarded based on their financial needs.

Every year a school offers scholarships to girl children and meritorious achievers based on certain criteria. In the session 2022–23, the school offered monthly scholarships of ₹3,000 each to some girl students and ₹4,000 each to meritorious achievers in academics as well as sports.

In all, 50 students were given the scholarships, and the monthly expenditure incurred by the school on scholarships was ₹1,80,000.

Based on the above information, answer the following questions:

Using integration, find the area of the region enclosed between the curve \( y = \sqrt{4 - x^2} \), and the lines \( x = -1 \), \( x = 1 \), and the \( x \)-axis.

The perimeter of a rectangular metallic sheet is 300 cm. It is rolled along one of its sides to form a cylinder. Find the dimensions of the rectangular sheet so that the volume of the cylinder so formed is maximum.

It is given that the function \( f(x) = x^4 - 62x^2 + ax + 9 \) attains a local maximum value at \( x = 1 \). Find the value of \( a \), and hence obtain all other points where the given function \( f(x) \) attains local maximum or local minimum values.

Let \( A = \mathbb{R} - \{3\} \) and \( B = \mathbb{R} - \{a\} \). Find the value of \( a \) such that the function \( f: A \to B \) defined by \( f(x) = \frac{x-2}{x-3} \) is onto. Also, check whether the given function is one-one or not.

Two vertices of the parallelogram ABCD are given as \( A(-1, 2, 1) \) and \( B(1, -2, 5) \). If the equation of the line passing through \( C \) and \( D \) is: \[ \frac{x - 4}{1} = \frac{y + 7}{-2} = \frac{z - 8}{2}, \] then find the distance between sides \( AB \) and \( CD \). Hence, find the area of parallelogram ABCD.

Find the equation of the line passing through the point of intersection of the lines \[ \frac{x - 1}{1} = \frac{y - 1}{2} = \frac{z - 2}{3} \] and \[ \frac{x - 1}{0} = \frac{y}{-3} = \frac{z - 7}{2}, \] and perpendicular to these given lines.

Evaluate: \[ \int_{0}^{\pi/4} \frac{1}{\sin x + \cos x} \, dx. \]

Evaluate: \[ \int \frac{2 + \sin 2x}{1 + \cos 2x} e^x \, dx. \]

Find: \[ I = \int \frac{\sqrt{x}}{(x+1)(x-1)} \, dx \]

Find the absolute maximum and absolute minimum values of the function \( f(x) = x^2 + \frac{2}{x} \) on the interval \([1, 2]\).

Find the intervals in which the function \( f(x) = \frac{\log x}{x} \) is strictly increasing or strictly decreasing.

Find the value of \( a \) and \( b \) so that the function \( f(x) \), defined as: \[ f(x) = \begin{cases} \frac{x - 2}{|x - 2|} + a, & x < 2, \\ a + b, & x = 2, \\ \frac{x - 2}{|x - 2|} + b, & x > 2, \end{cases} \] is a continuous function.

If \( x \cos(p + y) + \cos p \sin(p + y) = 0 \), prove that \( \cos p \frac{dy}{dx} = -\cos^2(p + y) \), where \( p \) is a constant.

The chances of \( P \), \( Q \), and \( R \) getting selected as CEO of a company are in the ratio \( 4 : 1 : 2 \), respectively. The probabilities for the company to increase its profits from the previous year under the new CEO \( P \), \( Q \), or \( R \) are \( 0.3 \), \( 0.8 \), and \( 0.5 \), respectively.
If the company increased the profits from the previous year, find the probability that it is due to the appointment of \( R \) as CEO.

Solve the following linear programming problem graphically:
Maximize \( z = x + y \), subject to constraints:
\[ 2x + 5y \leq 100, \quad 8x + 5y \leq 200, \quad x \geq 0, \quad y \geq 0. \]

Find the position vector of point \( C \) which divides the line segment joining points \( A \) and \( B \) having position vectors \( \vec{a} = \hat{i} + 2\hat{j} - \hat{k} \) and \( \vec{b} = -\hat{i} + \hat{j} + \hat{k} \), respectively, in the ratio 4:1 externally. Further, find \( |\vec{AB}| : |\vec{BC}| \).

Given \( \frac{d}{dx}F(x) = \frac{1}{\sqrt{2x - x^2}} \) and \( F(1) = 0 \), find \( F(x) \).

Check the differentiability of \( f(x) = \begin{cases} x^2 + 1, & 0 \leq x < 1 \\ 3 - x, & 1 \leq x \leq 2 \end{cases} \) at \( x = 1 \).