List of top Applied Mathematics Questions

Write the system of linear equations in \( x \) and \( y \) formed from the given situation.
Amrita buys a car for which she makes a down payment of ₹ 2,50,000 and the balance is to be paid in 2 years by monthly installments of ₹ 25,448 each. If the financer charges interest at the rate of 20\% p.a., find the actual price of the car.
[Given \( \left(\frac{61}{60}\right)^{-24} = 0.67253 \)]
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 \( k \).
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.
What will be the computer system's book value at the end of the first year and a half?
Give a linear equation that describes the computer system's book value at the end of \( t \)th year, where \( 0 \leq t \leq 3 \).
What is the rate of depreciation?
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 \( \sqrt{5} = 2.24 \), \( t_{19}(0.05) = 1.729 \)]
Given that the scores of a set of candidates on an IQ test are normally distributed, if the IQ test has a mean of 100 and a standard deviation of 10, determine the probability that a candidate who takes the test will score between 90 and 110.
[Given \( P(Z<1) = 0.8413 \) and \( P(Z<-1) = 0.1587 \)]
Find the differential equation of all circles in the first quadrant which touch both the coordinate axes.
Evaluate \( \int_{0}^{1} \frac{e^{-x}}{1 + e^x} \, dx \).
Evaluate \( \int_{0}^{2} x^2 \, dx \) and hence show the region on the graph whose area it represents.
A container has 50 litres of juice in it. 5 litres of juice is taken out and is replaced by 5 litres of water. This process is repeated 4 more times. Determine the quantity of juice in the container after final replacement.
[Use \( (0.9)^5 = 0.59049 \)]

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.

Assume an investment’s starting value is ₹ 20,000 and it grows to ₹ 50,000 in 3 years. Calculate CAGR (Compounded Annual Growth Rate).
[Use: \( (2.5)^{1/3} = 1.355 \)]

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: \[ 2x_1 + 3x_2 = 5 \] \[ 11x_1 - 5x_2 = 6 \]