Maharashtra HSC Mathematics and Statistics (Commerce) Board Paper March 2024 with Solutions

Q.1 (A)(i) Which of the following is not a statement?

Q.1 (A)(ii) If \(x + y + z = 3,\ x + 2y + 3z = 4,\ x + 4y + 9z = 6\) then \((y, z) =\) _____.

Q.1 (A)(iii) If \(y = \log\left(\dfrac{e^x}{x^2}\right)\), then \(\dfrac{dy}{dx} = ?\)

Q.1 (A)(iv) The value of \(\displaystyle\int \frac{dx}{\sqrt{1-x}}\) is _____.

Q.1 (A)(v) \(\displaystyle\int \frac{dx}{(x-8)(x+7)} =\) _____.

Q.1 (A)(vi) The differential equation of \(y = k_1 e^x + k_2 e^{-x}\) is :

Q.1 (B)(i) State whether true or false: \(\displaystyle\int_a^b f(x)\,dx = \int_a^b f(t)\,dt\)

Q.1 (B)(ii) State whether true or false: For \(\displaystyle\int \frac{x-1}{(x+1)^3}\,e^x\,dx = e^x f(x) + c,\ f(x) = (x+1)^2\)

Q.1 (B)(iii) State whether true or false: Order and degree of a differential equation are always positive integers.

Q.1 (C)(i) The slope of tangent at any point \((a, b)\) is called as ______.

Q.1 (C)(ii) If \(f'(x) = \dfrac{1}{x} + x\) and \(f(1) = \dfrac{5}{2}\) then \(f(x) = \log x + \dfrac{x^2}{2} +\) _____.

Q.1 (C)(iii) A solution of differential equation which can be obtained from the general solution by giving particular values to the arbitrary constants is called _____ solution.

Q.2 (A)(i) Examine whether the following statement pattern is tautology, a contradiction or a contingency. \(\sim p \rightarrow (p \rightarrow \sim q)\)

Q.2 (A)(ii) Find \(\dfrac{dy}{dx}\) if, \(x = e^{3t},\ y = e^{(4t+5)}\)

Q.2 (A)(iii) If \(A = \begin{bmatrix} 7 & 3 & 0 \\ 0 & 4 & -2 \end{bmatrix},\ B = \begin{bmatrix} 0 & -2 & 3 \\ 2 & 1 & -4 \end{bmatrix}\) then find \(A^T + 4B^T\)

Q.2 (B)(i) Consider the following statements

(a) If D is dog, then D is very good.
(b) If D is very good, then D is dog.
(c) If D is not very good, then D is not a dog.
(d) If D is not a dog, then D is not very good.

Identify the pairs of statements having the same meaning. Justify.

Q.2 (B)(ii) Determine the minimum value of the function : \(f(x) = 2x^3 - 21x^2 + 36x - 20\).

Q.2 (B)(iii) Find the area of the regions bounded by the line \(y = -2x\), the X-axis and the lines \(x = -1\) and \(x = 2\).

Q.3 (A)(i) Find \(\dfrac{dy}{dx}\) if \(y = x^{e^x}\)

Q.3 (A)(ii) If \(f'(x) = 4x^3 - 3x^2 + 2x + k,\ f(0) = 1\) and \(f(1) = 4\), find \(f(x)\).

Q.3 (A)(iii) Obtain the differential equation whose general solution is \(x^3 + y^3 = 35ax\)

Q.3 (B)(i) Find the inverse of \(\begin{bmatrix} 3 & 1 & 5 \\ 2 & 7 & 8 \\ 1 & 2 & 5 \end{bmatrix}\) by adjoint method.

Q.3 (B)(ii) The consumption expenditure \(E_c\) of a person with income \(x\), is given by \(E_c = 0.0006x^2 + 0.003x\). Find average propensity to consume (APC), marginal propensity to consume (MPC) when his income is ₹ 200. Also find his marginal propensity to save (MPS).

Q.3 (C)(i)

Complete the following activity :

\(\displaystyle\int_0^2 \frac{dx}{4 + x - x^2} = \int_0^2 \frac{dx}{-x^2 + \square + \square}\)

\(\displaystyle= \int_0^2 \frac{dx}{-x^2 + x + \frac{1}{4} - \square + 4}\)

\(\displaystyle= -\int_0^2 \frac{dx}{\left(x - \frac{1}{2}\right)^2 - \left(\square\right)^2}\)

\(\displaystyle= \frac{1}{\sqrt{17}}\log\left(\frac{20 + 4\sqrt{17}}{20 - 4\sqrt{17}}\right)\)

Q.3 (C)(ii)

The rate of growth of population is proportional to the number of inhabitants. If the population doubles in 25 years and the present population is 1,00,000, when will the city have population 4,00,000?

Solution : Let ‘P’ be the population at time ‘t’. Since rate of growth of population is proportional to the no. of inhabitants : \(\dfrac{dP}{dt} \propto P\)

\(\therefore\) Differential equation can be written as \(\dfrac{dP}{dt} = kP\), where \(k\) is constant of proportionality.

\(\therefore \dfrac{dP}{P} = k\cdot dt\). On integrating we get \(\square = kt + c\) .....(i)

(i) When \(t = 0,\ P = 1{,}00{,}000\), \(\therefore\) from (i) \(\log 1{,}00{,}000 = k(0) + c\), \(\therefore c = \square\), \(\therefore \log\left(\dfrac{P}{1{,}00{,}000}\right) = kt\) .....(ii)

(ii) When \(t = 25,\ P = 2{,}00{,}000\) as population doubles in 25 years, \(\therefore\) from (ii) \(\log 2 = 25k\), \(\therefore k = \square\), \(\therefore \log\left(\dfrac{P}{1{,}00{,}000}\right) = \left(\dfrac{1}{25}\log 2\right)\cdot t\)

(iii) \(\therefore\) when \(P = 4{,}00{,}000\): \(\log\left(\dfrac{4{,}00{,}000}{1{,}00{,}000}\right) = \left(\dfrac{1}{25}\log 2\right)\cdot t\), \(\therefore \log 4 = \left(\dfrac{1}{25}\log 2\right)\cdot t\), \(\therefore t = \square\) years

Q.4 (A)(i) The difference between face value and present worth is called _____.

Q.4 (A)(ii) In an ordinary annuity, payments or receipts occur at _____.

Q.4 (A)(iii) \(b_{xy}\) and \(b_{yx}\) are _____.

Q.4 (A)(iv) Dorbish-Bowley's Price Index Number is given by _____.

Q.4 (A)(v) Objective function of L.P.P. is _____.

Q.4 (A)(vi) To use the Hungarian method, a profit maximization assignment problem requires _____.

Q.4 (B)(i) State whether true or false: Broker is an agent who gives a guarantee to seller that the buyer will pay the selling price of goods.

Q.4 (B)(ii) State whether true or false: \(\sum \dfrac{p_0q_0}{p_1q_1}\times 100\) is the Value Index Number by simple aggregate method.

Q.4 (B)(iii) State whether true or false: The optimum value of the objective function of L.P.P. occurs at the center of the feasible region.

Q.4 (C)(i) The banker's discount is always _____ than the true discount.

Q.4 (C)(ii) The cost of living index number using Weighted Relative Method is given by _____.

Q.4 (C)(iii) The time interval between starting the first job and completing the last job including the idle time (if any) in a particular order by the given set of machines is called _____.

Q.5 (A)(i) Deepak's salary was increased from ₹ 4,000 to ₹ 5,000. The sales being the same, due to reduction in the rate of commission from 3% to 2%, his income remains unchanged. Find his sales.

Q.5 (A)(ii) For a bivariate data, the regression co-efficient of Y on X is 0.4 and the regression co-efficient of X on Y is 0.9. Find the value of variance of Y if variance of X is 9.

Q.5 (A)(iii) The following table shows the index of industrial production for the period from 1976 to 1985, using the year 1976 as the base year. Obtain the trend values for the following data using 4 yearly centered moving averages :

Q.5 (B)(i) If for the following data, Walsh's Price Index Number is 150, find ‘x’ :

Q.5 (B)(ii) A toy manufacturing company produces five types of toys. Each toy has to go through three machines A, B and C in the order ABC. The time required in hours for each process is given in the following table :

Find the total elapsed time and also find idle time for machine B.

Q.5 (B)(iii) A random variable X has the following probability distribution :

Find (a) \(k\) (b) \(P(X \lt 3)\) (c) \(P(X \gt 6)\)

Q.6 (A)(i) The building is insured for 75% of its value. The annual premium at 0.70 percent amounts to ₹ 2,625. If the building is damaged to the extent of 60% due to fire, how much can be claimed under the policy?

Q.6 (A)(ii) Three new machines \(M_1, M_2, M_3\) are to be installed in a machine shop. There are four vacant places A, B, C, D. Due to limited space, machine \(M_2\) can not be placed at B. The cost matrix (in hundred ₹) is as follows :

Determine the optimum assignment schedule and find the minimum cost.

Q.6 (A)(iii) The eggs are drawn successively with replacement from a lot containing 10% defective eggs. Find the probability that there is at least one defective egg in the lot of 10 eggs.

Q.6 (B)(i) Following table shows the all India infant mortality rates (per ’000) for years 1980 to 2010 :

Fit the trend line to the above data by the method of least squares.

Q.6 (B)(ii) Minimize : \(z = 6x + 2y\)

Subject to : \(x + 2y \ge 3,\ x + 4y \ge 4,\ 3x + y \ge 3,\ x \ge 0,\ y \ge 0\)

Q.6 (C)(i)

For a bivariate data \(\bar{x} = 10,\ \bar{y} = 12,\ V(X) = 9,\ \sigma_y = 4\) and \(r = 0.6\). Estimate \(y\) when \(x = 5\)

Solution : Line of regression of Y on X is \(Y - \bar{y} = \square\,(X - \bar{x})\)

\(\therefore Y - 12 = r\cdot\dfrac{\sigma_y}{\sigma_x}(X - 10)\)

\(\therefore Y - 12 = 0.6 \times \dfrac{4}{\square}(X - 10)\)

\(\therefore\) When \(x = 5\): \(Y - 12 = \square\,(5 - 10)\)

\(\therefore Y - 12 = -4\)

\(\therefore Y = \square\)

Q.6 (C)(ii)

If \(X \sim P(m)\) with \(P(X = 1) = P(X = 2)\) then find the mean and \(P(X = 2)\). Given \(e^{-2} = 0.1353\)

Solution : Since \(P(X = 1) = P(X = 2)\)

\(\therefore \dfrac{e^{\square}m^1}{1!} = \dfrac{e^{-m}m^2}{\square}\)

\(\therefore m = \square\)

\(\therefore P(X = 2) = \dfrac{e^{-2}\cdot 2^2}{2!} = \square\)