List of top Applied Mathematics Questions asked in CBSE Class Twelve Board Exam

Determine the values of \( x \) and \( y \).
Find the inverse of matrix \( A \).
Write the system of linear equations, obtained in (i) above, in matrix form \( AX = B \).
Write the system of linear equations in \( x \) and \( y \) formed from the given situation.

On her birthday, Prema decides to donate some money to children of an orphanage home.

Birthday

If there are 8 children less, everyone gets ₹ 10 more. However, if there are 16 children more, everyone gets ₹ 10 less. Let the number of children in the orphanage home be \( x \) and the amount to be donated to each child be \( y \).

Based on the above information, answer the following questions:

A river near a small town floods and overflows twice in every 10 years on an average. Assuming that the Poisson distribution is appropriate, what is the mean expectation? Also, calculate the probability of 3 or less overflows and floods in a 10-year interval.
[Given \( e^{-2} = 0.13534 \)]
Determine the probability that the student studied for at most 2 hours.
Determine the probability that the student studied for at least 2 hours.
Find the value of \( k \).

Let \( X \) denote the number of hours a Class 12 student studies during a randomly selected school day. The probability that \( X \) can take the values \( x_i \), for an unknown constant \( k \):

\[ P(X = x_i) = \begin{cases} 0.1, & {if } x_i = 0, \\ kx_i, & {if } x_i = 1 { or } 2, \\ k(5 - x_i), & {if } x_i = 3 { or } 4. \end{cases} \]
Find the intervals in which the following function \( f(x) \) is strictly increasing or strictly decreasing: \[ f(x) = 20 - 9x + 6x^2 - x^3. \]
Find all the points of local maxima and local minima of the function: \[ f(x) = -\frac{3}{4}x^4 - 8x^3 - \frac{45}{2}x^2 + 105. \]
A cistern has three pipes A, B, and C. Pipes A and B are inlet pipes, whereas C is an outlet pipe. Pipes A and B can fill the cistern separately in 3 hours and 4 hours respectively, while pipe C can empty the completely filled cistern in 1 hour. If the pipes A, B, and C are opened in order at 5, 6, and 7 a.m. respectively, at what time will the cistern be empty?
Quarterly.
Semi-annually.

Find the effective rate which is equivalent to a normal rate of 10% p.a. compounded:

Given:

\[ (1.05)^2 = 1.1025, \quad (1.025)^4 = 1.1038. \]

A recent accounting graduate opened a new business and installed a computer system that costs ₹ 45,200. The computer system will be depreciated linearly over 3 years and will have a scrap value of ₹ 0.

The mean weekly sales of a 4-wheeler was 50 units per agency in 20 agencies. After an advertising campaign, the mean weekly sales increased to 55 units per agency with a standard deviation of 10 units. Test whether the advertising campaign was successful.
[Given \( \sqrt{5} = 2.24 \), \( t_{19}(0.05) = 1.729 \)]
Evaluate \( \int_{0}^{2} x^2 \, dx \) and hence show the region on the graph whose area it represents.
Find the differential equation of all circles in the first quadrant which touch both the coordinate axes.
Evaluate \( \int_{0}^{1} \frac{e^{-x}}{1 + e^x} \, dx \).

A man bought an item for ₹ 12,000. At the end of the year, he decided to sell it for ₹ 15,000. If the inflation rate was 6%, find the nominal and real rate of return.

At 6 % p.a., compounded quarterly, find the present value of a perpetuity of ₹ 600 payable at the end of each quarter.

Find the solution to the following linear programming problem (if it exists) graphically: 

Maximize \( Z = x + y \) 
Subject to the constraints \[ x - y \leq -1, \quad -x + y \leq 0, \quad x, y \geq 0. \]

Using Cramer's rule, solve the following system of equations: \[ 2x_1 + 3x_2 = 5 \] \[ 11x_1 - 5x_2 = 6 \]