Class 12 Business Mathematics Question Paper 2080 (2023)

Consider \(A = \begin{pmatrix} 1 & 2 \\ 0 & 5 \end{pmatrix}\) and \(B = (2 \quad 5)\). What is the order of \(BA\)?
A) \(1 \times 2\)    B) \(2 \times 2\)    C) \(2 \times 1\)    D) \(1 \times 1\)

While we solve \(2x + 5y - 12 = 0\) and \(5x - 2y - 1 = 0\) using Cramer's rule, which is the solution of \(x\)?
A) \(\dfrac{\begin{vmatrix} 12 & 5 \\ 1 & -2 \end{vmatrix}}{\begin{vmatrix} 2 & 5 \\ 5 & -2 \end{vmatrix}}\)    B) \(\dfrac{\begin{vmatrix} 5 & 12 \\ -2 & 1 \end{vmatrix}}{\begin{vmatrix} 2 & 5 \\ 5 & -2 \end{vmatrix}}\)    C) \(\dfrac{\begin{vmatrix} 12 & 2 \\ 1 & 5 \end{vmatrix}}{\begin{vmatrix} 2 & 5 \\ 5 & -2 \end{vmatrix}}\)    D) \(\dfrac{\begin{vmatrix} 2 & 5 \\ 12 & 5 \\ 1 & -2 \end{vmatrix}}{\begin{vmatrix} 2 & 5 \\ 5 & -2 \end{vmatrix}}\)

Consider \(A = \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix}\), \(B = \begin{bmatrix} -1 & 1 \\ -1 & 1 \end{bmatrix}\) and \(C = \begin{bmatrix} -1 & 1 \\ 1 & -1 \end{bmatrix}\), which one of the following is null matrix?
A) \(A + B\)    B) \(B + C\)    C) \(A + C\)    D) \((A + B) + C\)

In the figure, \(f\) is continunes on \([a, b]\), \(f\) has minimum value at a certain point. Which is this point?
A) A    B) B
C) C    D) A and B

What is the integrating factor of differential equation \(\dfrac{dy}{dx} + \dfrac{y}{x} = x^3\)?
A) \(\dfrac{1}{x}\)    B) \(x\)    C) \(\log x\)    D) \(-x\)

Let \(a\) and \(b\) are two positive numbers. What is their geometric mean (GM)?
A) \(\dfrac{a + b}{2}\)    B) \(\pm\sqrt{ab}\)    C) \(\sqrt{ab}\)    D) \(-\sqrt{ab}\)

What is the compound interest of Rs. 1000 for 2 years at 10% compounded yearly?
A) Rs. 200    B) Rs. 210    C) Rs. 1200    D) Rs. 1210

There are 8 varieties of monkeys in a certain zoo. There are 2 monkeys in the first variety and each variety has 3 times more monkeys than its preceeding variety. Which one is the expression for finding the number of monkeys in 8th variety?
A) \(2 + (8-1) \times 3\)    B) \(2 \times (3)^{8-1}\)    C) \(4[4 + (8-1) \times 3]\)    D) \(\dfrac{2(3^8 - 1)}{2}\)

Which one of the following point belongs to region of \(x - 2y \leq 2\)?
A) \((5, 0)\)    B) \((0, -5)\)    C) \((0, 0)\)    D) \((3, 0)\)

If mean \((\bar{x}) = 14\), mode \((m_o) = 12\) and standard deviation \((\sigma) = 4\), what is the Karl Pearson's coefficient of correclation?
A) \(-\dfrac{1}{3}\)    B) \(-\dfrac{1}{2}\)    C) \(\dfrac{1}{3}\)    D) \(\dfrac{1}{2}\)

Let A and B be any two events of a sample space S. A and B are dependent, which one of the following is equal to \(P(A \cap B)\)?
A) \(P(A) \times P(B)\)      B) \(P(A) \times P(B/A)\)
C) \(P(A/B) \times P(B/A)\)   D) \(P(A) \times P(A/B)\)

Consider the matrix \(A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}\)

a) Write the minor of \(a_{12}\).                      1
b) Write the co-factor of \(a_{12}\).                     1
c) Write the determinant \(|A|\) in terms of minors and cofactors of \(R_1\).   1
d) What is the condition of singularity of matrix A?           1
e) What is the order of the matrix \(A^T\)?                1

The cost of 3kg of tomato and 2kg potato is Rs. 260. The cost of 4kg tomato and 3kg potato is Rs. 360. Find the rate of each kind of item using inverse matrix method. [5]

a) A differential equation \(\dfrac{dy}{dx} + py = Q\) is given.
i) What is the order of differential equation?             1
ii) What does \(Q\) represent?                     1
b) Let \(R = p \cdot q\) be a revenue function for price \(p\) and quantity \(q\).
i) What is marginal revenue?                    1
ii) What does \(\left(\dfrac{-p}{q} \cdot \dfrac{dq}{dp}\right)\) represent?                 1
iii) What is the impact on marginal revenue when \(\lvert E_d \rvert = 1\) as we know marginal revenue \(= p\left(1 - \dfrac{1}{\lvert E_d \rvert}\right)\)?                1

a) Show that maximum value is less than the minimum value for \(f(x) = x + \dfrac{36}{x}\) \((x \neq 0,\ x \in R)\). [5]

Find the area of the region in the first quadrant that is bounded above \(y = \sqrt{x}\) by the x-axis and the line \(y = x - 2\). [5]

Using simplex method, maximize \(Z(x, y) = 7x + 5y\) subject to the constraints \(4x + 3y \leq 12\), \(x + 2y \leq 6\), \(x,\ y \geq 0\). [5]

Compute the regression equation \(y\) on \(x\) for the following. [5]

Assume that half of the population in a village is vagetarian. The sample of 8 individuals to see whether they are vegetarians, how many investigators would you expect to report that 75% or less than 75% of the sample were vagetarians? [5]

a) Examine the increasing or decreasing of any quadratic function at \(x = 0\), \(x = 1\) and \(x = -3\).                                  3
b) The cost function is \(C = 4q - q^2 + 2q^3\). Show that the minimum average cost is equal to marginal cost when average cost is minimum. Is such situation always exist? Give reason.                                               5

a) A machine, the life of which is estimated to be 10 years, costs Rs. 30,000. Calculate the scrap value at the end of its life, the rate of compound depreciation is charged at 10% per year.                                      4
b) How much should be paid now to secure an annuity of Rs. 2600 to begin at once and to continue for 20 years at 5% p.a.?                                                                4

a) Calculate the mean, standard deviation, median and coefficient of skewness for the following data.                                        4