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

Q.1 (A)(i) Which of the following sentences is a statement in logic:

Q.1 (A)(ii) If \(y = 2x^2 + \log 2 + 5\) then \(\dfrac{dy}{dx} =\) ____.

Q.1 (A)(iii) If \(x = 2at^2\), \(y = 4at\) then \(\dfrac{dy}{dx} =\) ____.

Q.1 (A)(iv) The equation of tangent to the curve \(y = x^2 + 4x + 1\) at \(P(-1, -2)\) is ____.

Q.1 (A)(v) \(\displaystyle\int_{-2}^{3} \frac{dx}{x + 5} =\) ____.

Q.1 (A)(vi) The order and degree of the differential equation \(\dfrac{d^2x}{dt^2} + \left(\dfrac{dx}{dt}\right)^2 + 8 = 0\) are ____.

Q.1 (B)(i) State whether true or false: Every identity matrix is a scalar matrix.

Q.1 (B)(ii) State whether true or false: The rate of change of demand \((x)\) of a commodity w.r.t. its price \((y)\) is \(\dfrac{dy}{dx}\).

Q.1 (B)(iii) State whether true or false: The integrating factor of \(\dfrac{dy}{dx} - y = x\) is \(e^x\).

Q.1 (C)(i) If \(y = x \cdot \log x\) then \(\dfrac{d^2y}{dx^2} =\) ____.

Q.1 (C)(ii) If the marginal revenue \(R_m = 40\) and elasticity of demand \(\eta\) is 5, then the average revenue \(R_A\) will be ____.

Q.1 (C)(iii) Area of the region bounded by \(y = x^4\), \(x = 1\), \(x = 5\) and the X-axis will be ____.

Q.2 (A)(i) Write converse, inverse and contrapositive of the following statement :

If the train reaches on time then I can catch the connecting flight.

Q.2 (A)(ii) If \(A = \begin{bmatrix} 1 & 2 \\ -1 & -2 \end{bmatrix}\), \(B = \begin{bmatrix} 2 & a \\ -1 & b \end{bmatrix}\) and if \((A + B)^2 = A^2 + B^2\), find value of \(a\) and \(b\).

Q.2 (A)(iii) Find \(\dfrac{dy}{dx}\), if \(y = (x)^x + (a)^x\).

Q.2 (B)(i) Evaluate : \(\displaystyle\int \frac{x}{4x^4 - 20x^2 - 3}\,dx\)

Q.2 (B)(ii) Evaluate : \(\displaystyle\int_1^3 \log x\,dx\)

Q.2 (B)(iii) In a certain culture of bacteria, their rate of increase is proportional to the number present. If it is found that the number doubles in 4 hours, find the number of times the bacteria are increased in 12 hours.

Q.3 (A)(i) A metal wire of 36 cm length is bent to form a rectangle. Find its dimensions when its area is maximum.

Q.3 (A)(ii) Evaluate : \(\displaystyle\int \frac{2x + 1}{x(x - 1)(x - 4)}\,dx\)

Q.3 (A)(iii) Find the area of region bounded by \(y^2 = 25x\) and the line \(x = 4\).

Q.3 (B)(i) Using the truth table, verify : \(\sim(p \rightarrow \sim q) \equiv p \wedge \sim(\sim q) \equiv p \wedge q\)

Q.3 (B)(ii) Solve the following equations by method of inversion: \(x + y + z = 1\), \(x - y + z = 2\), \(x + y - z = 3\)

Q.3 (C)(i) The cost \(C\) for producing \(x\) articles is given as \(C = x^3 - 16x^2 + 47x\). For what values of \(x\) the average cost is decreasing?

Solution : Given \(C = x^3 - 16x^2 + 47x\)

Average cost \(C_A = \dfrac{C}{x}\)

\(\therefore C_A = \square\)

Differentiating w.r.t. \(x\), we get

\(\dfrac{d}{dx}(C_A) = \square\)

We know that \(C_A\) is decreasing if \(\dfrac{d}{dx}(C_A)\ \square\ 0\)

\(\therefore 2x - 16 \lt 0\)

\(\therefore 2x \lt 16\)

\(\therefore x \lt \square\)

\(\therefore\) Average cost is decreasing for \(x \in (0, 8)\)

Q.3 (C)(ii) Solve the differential equation : \(y - x\dfrac{dy}{dx} = 0\)

Solution : Given equation is \(y - x\dfrac{dy}{dx} = 0\)

Separating the variables we get

\(\dfrac{dx}{\square} = \dfrac{dy}{\square}\)

Integrating we get,

\(\displaystyle\int \frac{dx}{\square} = \int \frac{dy}{\square} + c\)

\(\therefore \log x = \square + c\)

\(\therefore \log x - \log y = \log c_1\), where \(c = \log c_1\)

\(\therefore \log\left(\dfrac{x}{y}\right) = \log c_1\)

\(\therefore \dfrac{x}{\square} = c_1\)

Hence the required solution is \(x = c_1y\).

Q.4 (A)(i) The date on which the period of the bill expires is called ____.

Q.4 (A)(ii) A person insured a property of ₹ 4,00,000. The rate of premium is ₹ 35 per thousand p.a. The amount of annual premium is ____.

Q.4 (A)(iii) Paasche's Price Index Number is given by ____.

Q.4 (A)(iv) If jobs I, II, III have processing times as 8, 6, 5 on machine M₁ and 8, 3, 4 on machine M₂ in the order M₁-M₂. Then the optimal sequence is ____.

Q.4 (A)(v) If \(E(X) = 4\) and \(X\) follows Poisson's distribution then \(V(X) =\) ____.

Q.4 (A)(vi) Three coins are tossed simultaneously. \(X\) is the number of heads. Then the expected value of \(X\) is ____.

Q.4 (B)(i) State whether true or false: In the regression of \(Y\) on \(X\), \(X\) is the independent variable and \(Y\) is the dependent variable.

Q.4 (B)(ii) State whether true or false: The region represented by the inequalities \(x \le 0,\ y \le 0\) lies in the first quadrant.

Q.4 (B)(iii) State whether true or false: In an assignment problem, if the number of columns are greater than number of rows, then a dummy column is added.

Q.4 (C)(i) If an agent charges 12% commission on the sales of ₹ 48,000 then his total commission is ₹ ____.

Q.4 (C)(ii) The optimal value of the objective function is attained at the ____ points of feasible region.

Q.4 (C)(iii) Given p.d.f. of a continuous random variable \(X\) is, \(f(x) = \dfrac{x}{8}\), for \(0 \lt x \lt 4\); \(= 0\), otherwise. Then \(P(1 \lt x \lt 2) =\) ____.

Q.5 (A)(i) Find the rate of interest compounded annually if an immediate annuity of ₹ 20,000 per year amounts to ₹ 41,000 in 2 years.

Q.5 (A)(ii) Find the Value Index Number using Simple Aggregate Method for the following data :

Q.5 (A)(iii) Five jobs must pass through a lathe and a surface grinder, in that order. The processing times in hours are shown below. Determine the optimal sequence of the jobs. Also find the total elapsed time :

Q.5 (B)(i) A bill was drawn on 14^th April for ₹ 7,000 and was discounted on 6^th July at 5% p.a. The banker paid ₹ 6,930 for the bills. Find the period of the bill.

Q.5 (B)(ii) The following table gives the production of steel (in millions of tons) for years 1976 to 1986 :

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

Q.5 (B)(iii) Solve the following L.P.P. by graphical method :

Maximize : \(z = 7x + 11y\)

Subject to : \(3x + 4y \le 24\), \(5x + 3y \le 30\), \(x \ge 0\), \(y \ge 0\)

Q.6 (A)(i) For a bivariate data, \(\bar{x} = 53\), \(\bar{y} = 28\), \(b_{yx} = -1.2\), \(b_{xy} = -0.3\). Find : (a) Correlation coefficient between \(x\) and \(y\). (b) Estimate \(y\) for \(x = 50\).

Q.6 (A)(ii) Given that \(\Sigma p_0q_0 = 220\), \(\Sigma p_0q_1 = 380\), \(\Sigma p_1q_1 = 350\) and Marshall-Edgeworth's Price Index Number is 150, find Laspeyre's Price Index Number.

Q.6 (A)(iii) The following data gives the production of bleaching powder (in '000 tons) for the years 1962 to 1972 :

Obtain the trend values for the above data using 5 yearly moving averages.

Q.6 (B)(i) Four new machines M₁, M₂, M₃ and M₄ are to be installed in a machine shop. There are five vacant places A, B, C, D and E available. Because of limited space, machine M₂ cannot be placed at C and M₃ cannot be placed at A. The cost matrix is given below :

Find the optimal assignment schedule.

Q.6 (B)(ii) There are 10% defective items in a large bulk of items. What is the probability that a sample of 4 items will include not more than one defective item?

Q.6 (C)(i) The equations of the two regression lines are \(3x + 2y - 26 = 0\) and \(6x + y - 31 = 0\). Obtain the correlation coefficient between \(x\) and \(y\).

Solution :

To find correlation coefficient, we have to find the regression coefficients \(b_{yx}\) and \(b_{xy}\).

Let \(3x + 2y = 26\) be equation of the line of regression of \(y\) on \(x\).

This gives \(y = \square\,x + 13\)

\(\therefore b_{yx} = -\dfrac{3}{2}\)

Now consider \(6x + y = 31\) as equation of the line of regression of \(x\) on \(y\).

This can be written as \(x = \square\,y + \dfrac{31}{6}\)

\(\therefore b_{xy} = -\dfrac{1}{6}\)

Now \(r^2 = \square = 0.25\)

As both \(b_{yx}\) and \(b_{xy}\) are negative,

\(\therefore r = \square\)

Q.6 (C)(ii) The probability distribution of \(X\) is as follows :

Find : (a) \(k\) (b) \(P(X \lt 2)\) (c) \(P(1 \le X \lt 4)\) (d) \(F(2)\)

Solution : The table gives a probability distribution.

\(\therefore \Sigma p_i = 1\)

\(\therefore 0.1 + k + 2k + 2k + k = 1\)

(a) \(k = \square\)

(b) \(P(X \lt 2) = P(X = 0) + P(X = 1) = \square\)

(c) \(P(1 \le X \lt 4) = P(1) + P(2) + P(3) = \square\)

(d) \(F(2) = P(X \le 2) = P(0) + P(1) + P(2) = \square\)