List of top Applied Mathematics Questions asked in CBSE CLASS XII

Fit a straight-line trend by the method of least squares for the following data:

\[ \begin{array}{lcccccc} \text{Year:} & 2011 & 2012 & 2013 & 2014 & 2015 & 2016 \\ \text{Production (in tons):} & 210 & 225 & 275 & 220 & 240 & 235 \end{array} \]

Compute the seasonal indices by 4-year moving averages from the given data of production of paper (in thousand tons):

\[ \begin{array}{lcccccccccc} \text{Year:} & 2001 & 2002 & 2003 & 2004 & 2005 & 2006 & 2007 & 2008 & 2009 & 2010 \\ \text{Index:} & 2450 & 1470 & 2150 & 1800 & 1210 & 1950 & 2300 & 2500 & 2480 & 2680 \\ \end{array} \]

If the supply function is $p = 4 - 5x + x^2$, then find the producer’s surplus when price is 18.

Using properties of determinants, prove that
\[ \Delta = \begin{vmatrix} (b + c)^2 & a^2 & a^2 \\ b^2 & (c + a)^2 & b^2 \\ c^2 & c^2 & (a + b)^2 \end{vmatrix} = 2abc(a + b + c)^3 \]

If \( A = \begin{bmatrix} 2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2 \end{bmatrix} \), find \( A^{-1} \) and hence solve the system of equations:

\( 2x - 3y + 5z = 11 \)
\( 3x + 2y - 4z = -5 \)
\( x + y - 2z = -3 \)

Calculate EMI under Flat-Rate System for a loan of ₹ 5,00,000 with 7.5% annual interest rate for 5 years.

A machinist is making engine parts with axle diameter of 0.7 cm. A random sample of 10 parts shows mean diameter 0.742 cm with a standard deviation of 0.04 cm. On the basis of this sample, find if you would say that the work is inferior.
(Given $t_9(0.05) = 2.262$)

Solve the following linear programming problem graphically:
Maximize $z = 50x + 30y$
Subject to:
$2x + y \leq 18$
$3x + 2y \leq 34$
$x \geq 0$, $y \geq 0$

There are two types of fertilizers $F_1$ and $F_2$. $F_1$ consists of 10% nitrogen and 6% phosphoric acid. $F_2$ consists of 5% nitrogen and 10% phosphoric acid.
After testing the soil conditions, a farmer finds that he needs at least 14 kg of nitrogen and 14 kg of phosphoric acid for his crop.
If $F_1$ costs ₹ 6 per kg and $F_2$ costs ₹ 5 per kg, how much of each type of fertilizer should be used so that the cost is minimum.
Formulate a linear programming problem.

The mortality rate for a certain disease is 0.007. Using Poisson distribution, calculate the probability for 2 deaths in a group of 400 people. [Use $e^{-2.8} = 0.0608$]

If the mean and variance of a binomial distribution are $\dfrac{4}{3}$ and $\dfrac{8}{9}$ respectively, then find $P(X = 1)$.

Two numbers are selected at random (without replacement) from the first six positive integers. Let $X$ denote the smaller of the two numbers obtained. Calculate the mathematical expectation of $X$.

Find the units digit in $7^{295}$.

Given that the mean of a normal variate $X$ is 9 and standard deviation is 3, then find:
(i) the z-score of the data point 15
(ii) the data point if its z-score is 4

If a fair coin is tossed 6 times, find the probability of getting at least 4 heads.

Given \( A = \begin{bmatrix} 2 & 0 & 1 \\ 3 & 4 & 5 \\ 0 & 2 & 3 \end{bmatrix} \) and \( B = \begin{bmatrix} 1 & 1 & -5 \\ -5 & 1 & -5 \\ 1 & -2 & 4 \end{bmatrix} \), find \( BA \).

A runs $\dfrac{3}{2}$ times as fast as B. If A gives B a start of 40 m, how far must the winning post be from the starting point, so that A and B reach at the same time?

A boat goes 3.5 km upstream and then returns. Total time taken is 1 hour and 12 minutes. If the speed of the current is 1 km/h, then find the speed of the boat in still water.

Pipe A can fill a tank in 1 hour and Pipe B can fill it in $1\frac{1}{2}$ hours. If both the pipes are opened in the empty tank, how much time will they take to fill the tank?

The cost of Type I sugar is ₹ 25 per kg and Type II sugar is ₹ 35 per kg. If both Type I sugar and Type II sugar are mixed in the ratio 3:2, find the price per kg of the mixture.

Assertion (A): The differential equation representing the family of curves $y = mx$, $m$ being an arbitrary constant, is \[ x \dfrac{dy}{dx} - y = 0 \]
Reason (R): For a family of curves, the differential equation is obtained by differentiating the equation of family with respect to $x$ and then eliminating the arbitrary constant, if any.

Assertion (A): The area of the region bounded by the line $y - 1 = x$, the $x$-axis and the ordinates $x = -1$ and $x = 1$ is 2 square units.
Reason (R): The area of the region bounded by the curve $y = f(x)$, the $x$-axis and the ordinates $x = a$ and $x = b$ is given by \[ \int_{a}^{b} f(x)\, dx. \]