Class 12 Business Mathematics

      If  A is a non-singular matrix, then \( A^{-1} \) equal to

    A) \( A \cdot (\text{adj } A) \)        B) \( (\text{adj } A) \cdot A \)       C)\( \frac{1}{|A|} (\text{adj } A) \)   D) \( \frac{1}{|A|} (A)^T \)

What is the value of \( \begin{vmatrix} 4 & 3 & 5 \\ 0 & 0 & 7 \\ 8 & 6 & -1 \end{vmatrix} \)?

   A) 0         B) -2      C) 2              D) 4

Which of the following matrix is satisfied Hawkin's-Simon condition?

     A)\( \begin{pmatrix} -0.5 & 0.3 \\ 0.7 & -0.2 \end{pmatrix} \)           B)\( \begin{pmatrix} -0.5 & -0.2 \\ 0.7 & 0.3 \end{pmatrix} \)

    C) \( \begin{pmatrix} 0.3 & 0.7 \\ -0.2 & -0.5 \end{pmatrix} \)             D) \( \begin{pmatrix} 0.3 & -0.5 \\ -0.2 & 0.7 \end{pmatrix} \)

Three numbers are in the ratio of 1:4:12. If one is added to the first number, the resulting number together with other two numbers form a G.P. The three original numbers are:

A) 4, 12, 36                  B) 5, 10, 20 

 C) 6, 12, 24                D) 3, 12, 36

a) Write the associative property of matrix addition. 

b) Define symmetric matrix. 

c) If $A$ is a square matrix then what is the value of \( A \times (\text{adj } A) \)? 

d) Write the condition in Cramer's rule that the system of linear equations cannot be solved. 

e) What is the Leontief's technology matrix if\( X = \text{output vector} \), \( A = \text{input coefficient matrix} and I \) the unit matrix. 

(a) Given the input-output table for the two sector economy

(a) Calculate the input-output coefficient matrix. 

(b) Calculate the Leontief matrix \( (I - A) \), where I being unit matrix of order 2. 

(c) Test Hawkins-Simon condition for the viability of the system.

a) Find two geometric mean between 9 and 243. 

b) A sum doubles itself in 5 years by compound interest. How long does it take to be four times the sum? 

c) A loan of Rs. 2,00,000 is to be paid in 10 yearly instalment. Find the amount of each installment if the rate of interest is 15% p.a.

If A is a square matrix then \( A - A^T \) is

 A) a diagonal matrix      B) Symmetric matrix     C) Skew-symmetric matrix    D) Scalar matrix

Two matrices A and B can be multiplied to get AB if

A) Both matrices are square of different order.

B) Both matrices are rectangular of same order.

C) No. of rows of A = no. of columns of B.

D) No. of columns of A = no. of rows of B.

If the matrix of technical coefficients is \( A = \begin{bmatrix} 0.1 & 0.2 \\ 0.3 & 0.4 \end{bmatrix} \)  then Leontief matrix is

 A) \( \begin{bmatrix} 0.1 & 0.2 \\ 0.3 & 0.4 \end{bmatrix} \)

 B) \( \begin{bmatrix} 0.9 & 0.2 \\ 0.3 & 0.6 \end{bmatrix} \)

 C)\( \begin{bmatrix} 0.9 & -0.2 \\ -0.3 & 0.6 \end{bmatrix} \)

D) \( \begin{bmatrix} 0.1 & -0.8 \\ -0.7 & 0.4 \end{bmatrix} \)

Which one of the following is the sum of the series\( 1 + \frac{1}{2} + \frac{1}{4} + \dots to \infty \)

A)\( \frac{2}{3} \)    B) \( \frac{1}{2} \)     C) 1       D) 2

Which one of the following is the value of 'n' in a binomial distribution for which the mean is 4 and variance is 3.

 A) 4       B) 12      C) 16        D) 20

If \( A = \begin{bmatrix} 1 & 0 & 2 \\ -2 & 1 & 3 \\ 4 & 5 & 6 \end{bmatrix} \)

a) Write the order of \( A^T. \) 

b) What types does matrix form when \( (A + A^T) \)? 

c) What is the determinant of matrix A, if A is singular? 

d) Write the formula for finding \( A^{-1} \) 

 e) Under which condition \( A^{-1} \) does not exist. 

If \( A = \begin{bmatrix} 0 & 3 \\ -3 & 0 \end{bmatrix} \)be a square matrix, which of the following relations is true?

What is the value of \( \begin{vmatrix} 1 & -1 & -1 \\ -1 & 0 & 1 \\ 0 & 1 & 0 \end{vmatrix} \)? 

a) 0      b) 1      c) -1       d) 3

Which of the following entries will make the matrix \( \begin{pmatrix} ... & 0.1 \\ -0.2 & 0.3 \end{pmatrix} \)satisfy Hawkin's-Simon conditions?

 a) -0.2     b) 0       c) 0.5      d) 0.01

What is the geometric mean between 1 and 64?
a) 5   b) 7    c) 8      d) 9

\( 3x + 3y = 6 \) and \( x + y = 2 \)are two simultaneous equations.

 i. Write the above system of equations in the matrix form AX = B where A, X and B have their usual meanings. 

 ii. Find the determinant of A. 

iii. Does the inverse of A exist? 

iv. Does the system have unique solution? 

If \( A = \begin{pmatrix} 1 & 2 & -3 \\ 2 & -1 & 4 \\ 3 & 1 & 3 \end{pmatrix} \), find \( A^{-1} \)and verify that \( A \cdot A^{-1} = I_3 \) 

It is given that A, B, C are three matrices with order \( 2 \times 2 \), \( 2 \times 3 \) and \( 3 \times 3 \) respectively. In multiplication of two matrices, which one of the following exist?

A) AC     B) CA       C) AB      D) BA

The equations x + 2y + 5 = 0 and x - 3y - 5 = 0 have

A) Unique solution     B) Only two solutions    C) No solution     D) Infinitely many solutions

Let \( \text{AX} = \text{B} \) be the corresponding matrix equation of the system of linear equations, then the system of equations is consistent and has a unique solution if

A) |A| = 0     B) 2|A| = 0

C) 3|A| = 0     D) |A| \neq 0

Three geometric means between 2 and 32 are

A) 2, 4, 8    B) 4, 8, 16    C) 4, 6, 9    D) 8, 12, 18

Consider the following system of linear equations \( a_1x + b_1y = c_1 \) and\( a_2x + b_2y = c_2 \).

a) Express the system of linear equations in the matrix form AX = B. 

b) Write the determinant of matrix A. 

c) When does the system have a unique solution? 

 d) Write the matrix of cofactors. 

e) Write \( A^{-1} \)in terms of \( |A| \) and adjoint (A). 

The technology matrix of an economic system of two industries is

(a) Find Leontief's technology matrix. 

(b) Test whether the given system is viable as per Hawkins Simon conditions. 

(c) Find the total outputs of each industry if the final demand for two industries are 4 and 32 units respectively. 

Find the absolute maxima and absolute minima of the function\( f(x) = x^3 - 3x^2 + 5 \) on \( [-2, 2] \) Also find the point of inflection. 

For a distribution, the distance of the median from first quartile is equal to the distance of the third quartile from the median. The coefficient of skewness is

A) -0.5    B) 0       C) 0.75    D) 1

X  and Y are two dependent events. Which one of the following is \( P(X \cap Y) \)?

 A) \( P(X) \cdot P(Y) \)         B) \( P(X) \cdot P(Y/X) \)

 C) \( P(X) \cdot P(X/X) \)    D) \( P(X/Y) \cdot P(Y/X) \)

a) Write the average revenue (AR) in terms of total revenue (R) and output (Q). 

b) Write the formula of elasticity of supply \( \varepsilon_s \) 

c) Write the condition of maximization of a function. 

d) If C be the total cost and R be the total revenue, write the break-even condition in terms of cost and revenue. 

e) What is the condition for finding stationary point for the function g(x)? 

Calculate the Karl Pearson's coefficient of correlation between the height (X) and weight (Y) for the following data.

Also interpret the result.

In a cap factory, machine A, B and C manufacture respectively 30%, 35% and 45% out of their output 2%, 3% and 4% defectives respectively are found. A cap is drawn at random to be defective, what is the probability that it is manufactured by the machine A?

a) Using the line of regression, estimate the weight of a baby at the age of 4 months and age of the baby when weight is 6 kg, from the following table. 

b) A fair coin is tossed 4 times. What is the probability of getting at least 2 heads

The Bowley's coefficient of skewness is based on

A) median, standard deviation and mean

B) lower quartile, upper quartile and median

C) quartiles and standard deviation

D) percentile and standard deviation

a) Write multiplicative law of probability for independent events A and B. 

b) Write conditional probability of event P given that event Q has already occurred. 

c) In a factory producing radios, there are three machines producing 500, 1000, 1500 radios per hour respectively. These machines produce 3%, 5%, 8% defective radios. One radio selected random from a hour production of the three machines and found to be defective. What is the probability that the radio is produced from the third machine?

In a college, it has been found that 40% of the students withdraw without completing the mathematics course if there were 10 students have registered for the course of this semester.
a) What is the probability that none will withdraw? 
b) What is the probability that at least one will withdraw?

c) What is the probability at most one will withdraw?

The following table shows the relation between the price (x) and demand (y) of certain items.

a) Calculate the correlation coefficient by Karl Pearson's method. 

 b) The mean and variance of a binomial distribution are 7.5 and 1.875 respectively. Find P(2) and P(0).

If 'P' be the present value '\( P_T \)' the present value of future return, 'T' the time of cash flow, 'i' the discount rate per annum then which of the following is applied to find the present value of future return.

 a) \( P = \frac{P_T}{(1+i)^T} \)

b) \( P = \frac{P_T}{(1-i)^T} \)

c) \( P_T = \frac{P}{(1+i)^T} \)

 d) \( P_T = \frac{P}{(1-i)^T} \)

In a frequency distribution the difference between mean \( (\bar{X} ) \) and mode \( ( M_o ) \)is 1.4 and coefficient of skewness \( S_K(P) \) is 0.625. What is the value of standard deviation?

a) 1.24     b) 2.24     c) 1.34     d) 2.34

 If P(A) and P(B) are probability of events A and B respectively. Which of the following condition is true?

a) \( P(A/B) = \frac{P(A \cap B)}{P(A)} \)

b) \( P(A/B) = \frac{P(A \cap B)}{P(B)} \)

 c) \( P(A/B) = \frac{P(A)}{P(A \cap B)} \)

d) \( P(A/B) = \frac{P(B)}{P(A \cap B)} \)

The height (ft) and weight (Kgs) of five students of a class are given below:

Find the correlation coefficient between the height and weight. Also interpret the result.

A binomial distribution has 5 independent trials. If the probabilities of 1 and 3 success are \( \frac{1}{4} \) and \( \frac{144}{625} \) respectively, find p and q. Also find the probability of 2 success.

a) Regression equations in two variables X and Y are 3X + 2Y - 26 = 0 and 6X + Y - 31 = 0.

 i. Find the mean values of X and Y. 

ii. Find two regression coefficients. 

 b) In a season, 70% visitors went to travel Janakpur and 30% visitors to Lumbini. It is estimated that 35% Janakpur visitors and 65% Lumbini visitors are Nepali. A visitor is selected at random and found to be Nepali.

i. What is the probability that the visitor is Janakpur visitor? 

 ii. What is the probability that the visitor is Lumbini visitor? 

Two regression coefficient are 0.80 and 1.20, what would be the value of the coefficient of correlation ?

 A) \( \sqrt{0.096} \)   B) \( \sqrt{0.96} \)   C) \( \sqrt{9.60} \)    D) \( \sqrt{96} \)

If A and B are two dependent events, then

 A) \( P(B/A) = P(A).P(B) \)    B) \( P(B/A) = \frac{P(B)}{P(A)} \)     C) \( P(B/A) = \frac{P(A \cap B)}{P(A)} \)     D) \( P(B/A) = \frac{P(A \cap B)}{P(B)} \)

The equation of two regression lines are 3x + 4y = 65, 3x + y = 32.

a) Find the mean values of x and y. 

b) Obtain the regression coefficients. 

c) Determine the correlation coefficient between x and y. 

The probability of hitting a target is found to be 0.25. If ten hits are made, find the probability that 

a) None will hit the target.

b) Exactly two will hit the target.

The following table shows the relation between price (X) and demand (Y) of certain items.

a) Calculate the correlation coefficient by using Karl Pearson's method and interpret the result. 
b) Find the equation of regression line *Y on X*. 
c) The mean and variance of binomial distribution are 7.5 and 1.875 respectively. Find the number of trials (n). 

If $f(x) = \( x^2 - 4x \)  the function \( f(x) \) is increasing at

   A) \( x = 0 \)    B) x = 1     C) x = 2     D) x = 4

Which one of the following is the degree of differential equation $\frac{d^4y}{dx^4} = \sqrt{\frac{d^3y}{dx^3}}$?

   A) 1         B) 2        C) 3              D) 4

a) If the revenue function is \( R = Q - 3Q^2 \) and the cost function is \( C = Q^2 - 2Q \), find the value of the maximum profit. 

b) It is given that the demand function \( P = 3Q + 6 \). Find the elasticity of demand at Q = 3. 

a) It is given that \( R(x) = 2x^2 + 6x + 5 \); where R(x) is the revenue function. Find the marginal revenue at x = 10. 

b) Find the area under the curve \( y^2 = 4x \) between x = 2 to \( x = 5 \)using fundamental theorem of calculus. 

a) Give an example of first order linear differential equation with constant coefficient and constant term. Solve it.

b) A function \( f(x) = \frac{1}{x} \)defined in (-8, 0). f'(x) is decreasing in the interval (-8, 0). Justify it. 

c) Derivative and anti derivative are inverse of each other. Justify with example.

The functions f(x) is increasing on a domain D if

A) f'(x) < 0 for all x in D 

 B) f'(x) = 0 for all x in D

C) f'(x) > 0 for all x in D

D) f'(x) doesn't exist for all x in D

Which one of the following differential equation has the solution $xy = c$?

A) ydy + xdx = 0     B) xdy + ydx = 0

 C) xdy - ydx = 0      D) ydy - xdx = 0

a) Write the relation between marginal revenue and total revenue. 

b) Define stationary point for a function. 

c) What is monotonic function? Write it. 

d) For a function f(x), f''(x) > 0 at a point and f'(x) = 0, identify the value of f(x) at that point. 

 e) State fundamental theorem of calculus. 

a) Find the area bounded by the curve \( y^2 = 4x \), the x-axis and the line x = 4. 

 b) The demand and supply function in a competitive market are given by the equation\( Q_d = 240 - 3p \) and \( Q_s = 5P - 150 \) respectively. If the rate of change of the price adjustment proportional to the excess of the demand is given by $\frac{dp}{dt} = \( 0.05(Q_d - Q_s) \). Solve the obtained differential equation given that the initial price \( (P_0) = \text{Rs. } 50 \). 

a) In perfect competition, demand and supply function of commodity are given by \( P_d = 40 - x^2 \) and \( P_s = 3x^2 + 8x + 8 \). Discuss the nature of consumer surplus and producer surplus at the market equilibrium price. 

b) Write a homogeneous differential equation of first order and first degree. Also solve it. 

c) Price of elasticity is related to demand and total revenue. Justify it.

What is the integrating factor of the differential equation \( \frac{dy}{dx} - \frac{y}{x} = e^x \)?

a)\( -\frac{1}{x} \)

b) x

c) log x

d) \( \frac{1}{x} \)

a) Write average cost (AC) in terms of total cost (C) and output (Q) 

 b) What does \( \int_{a}^{b} f(x) dx \) represent geometrically? 

 c) Write the degree of differential equation\( \frac{dy}{dx} + 2y = 5 \) 

 d) If P is demand function and Q is output, Write the corresponding revenue function. 

e) Write the formula of elasticity of demand (\( E_d \)). 

The birth rate population in a certain country is given by $\frac{dp}{dt} = 0.15(6000 - P)$ where $t$ is the time in years. ($P = 0$ when $t = 0$). a. Solve the above differential equation.

b. Calculate the population after 10 years. 

a) A rectangular garden requires 144m of fencing wire for its complete fencing. What will be its shape if its area is maximize? Justify its reason.

b) Find the area between curves \( y = x^3 \) and x-axis, when x = -2 and y = 8.

The function\( f(x) = x^2 - 2x \) is decreasing at

A) x < 1      B) x > 1       C) x > 2       D) x > 3

The area bounded by the curve \( y = 4x^2 \), x-axis and two ordinates x = 0 and x = a is

A) \( \frac{a^3}{3} \)      B) \( \frac{2a^3}{3} \)  C) \( a^3 \)     D) \( \frac{4a^3}{3} \)

a) Solve the differential equation \( \frac{dy}{dx} + y = 2 \). 

b) The demand and supply function of a product are given by \( Q_d = 32 - 0.5P \) and \( Q_s = -8 + 0.3P \) and the rate of adjustment of price when the market is out of equilibrium is\( \frac{dP}{dt} = 0.25(Q_d - Q_s). \) Determine and obtain differential equation to get a function for P, in terms of 't'. Also find the value of P when t = 5, by taking integration constant 15. 

In a simplex method a constraint \( x + 2y \le 6 \) is written as \( x + 2y + v = 6 \). What is the 'v' known as?

  A) Surplus variable        B) Slack variable       C) Pivot element     D) Negative number

Using Simplex method, maximize W = 5x + 2y subject to the constraints,

\( 3x + 5y \le 15 \), \( 5x + 2y \le 10 \), \( x \ge 0, y \ge 0 \). 

In simplex method,\( x + 2y \le 4 \) is written as x + 2y + k = 4. What is k known as?

A) Basic variable      B) Slack variable      C) Surplus variable    D) Pivot element

Solve the following linear programming problem by Simplex method to maximize P = 50x + 60y subject to \( 3x + 4y \le 36, \) \(  9x + 4y \le 60 \), \( x, y \ge 0 \). 

There are m simultaneous linear equations and n unknown variables (n>m) then which of the following is the numbers of basic solution.

a) m + n      b) n - m     c) m - n     d) n

Using simplex method, maximize \( (z) = 6x - 9y \), subject to the constraints \( x + y \le 20 \), \( 2x - 3y \le 6 \), \( x, y \ge 0 \). 

The inequality in a simplex method is \( x + 4y \leq 5 \). It is written as x + 4y + m = 5, the variable m is called.

 A) Surplus      B) Slack     C) Basic      D) Pivot

Using Simplex method, solve the following linear programming problem:

Minimize \( P = 3x + 4y \) subject to \( 2x + y \leq 4 \), \( x + 2y \leq 4, x, y \geq 0 \)

Which one of the following represents net present value?

 A) Total future returns of the investment     

 B) Total investments

 C) Difference of total present value of future returns and the total investment

  D) Total present value of future returns

The initial cost of a bicycle is Rs. 50,000 and the rate of compound depreciation is 10% p.a. diminishing balance, then scrap value at the end of third year is

  A) Rs. 36,200    B) Rs. 36,360     C) Rs. 36,400         D) Rs. 36,450

If a computer costing Rs. 1,00,000 will depreciate to scrap value of Rs. 81,000 in 2 years. The rate of depreciation is

A) 5%       B) 10%       C) 19%      D) 23.45%

Which one of the following is the amount of annuity of Rs. 1,400 for a year at 8% p.a.?

A) Rs. 1400     B) Rs. 1512     C) Rs. 1521    D) Rs. 2511

a) Prove that. 

a)  Machinery costing Rs 4,50,000 has an estimated life of 15 years. If its cost is reduced by Rs. 3,00,000 in 15 years, find the rate of depreciation. 
b) Manoj borrows Rs. 4368 which he promises to pay in 6 annual instalments, each instalment being treble of the preceding one. Find amount of the First instalment. 
c) The compound interest on a certain sum at a certain rate is Rs. 204 in 2 years and the simple interest at the same rate is Rs. 300 in 3 years. Find the sum and rate of interest

On calculation of Elasticity of demand if \( E_d = -2.25 \) then a) Demand is unit elastic b) Demand is elastic c) Demand is inelastic d) Demand is viable

The original cost of machine is Rs. 50000. It depreciates at the rate of 20% per annum then what is the value of machine after 2 years.

 a) Rs. 10000      b) Rs. 20000      c) Rs. 25000     d) Rs. 32000

The demand function is \( P_d = 196 - x^2 \) and the supply function is \( P_s = x^2 + 4x + 126 \). Find the consumer and producer surplus at market equilibrium.

a) The ages in years of three students are in G.P such that the sum and the product of their ages are 21 and 64 respectively. Find their ages in years.

 b) A donor decides to grant scholarship to the student of medicine Rs. 50000 yearly for 5 years. What sum of money should be deposited in the bank with the compound rate of interest 8% per year so that the sum is sufficient to grant the scholarship at the end of each year.

A project would normally be accepted if its net present value (NPV) is...

A) Positive real number   B) Negative real number    C) Zero   D) Always one

The recent price of a car is Rs. 10,00,000. If its price reduces by 10% yearly, after how many years its price will be Rs. 7,29,000 ?
A) 1 year     B) 2 year     C) 3 years     D) 4 years

The demand equation for a certain commodity is \( P = \frac{Q^2}{3} - 10Q + 75 \). Find the value of Q and the corresponding value of P that maximizes the total revenue.

The population growth rate of a country is 2.5% per year.
a) Model the situation using differential equation. 
b) What will be the population after 1 year? 
c) If the current population of the country is 20 million, what will be the population after 20 years.

a) Evaluate : \( \int_{0}^{1} (x^2 + 2x + 1) dx \)

b) The supply function for a good is given by \( \text{P} = 60 + 0.75\text{Q} \)

i) Calculate the elasticity of supply at \( \text{P} = \text{Rs. } 100 \)

ii) Interpret the result. 

iii) Calculate the percentage change in quantity supplied at \( \text{P} = \text{Rs. } 100 \) if the price of good increased by \( 15\% \). 

a) Three numbers are in the ratio 1:2:3. If 2, 4 and 11 are added to them respectively, the resulting numbers are in G.P. Find the original three numbers. 

b) Find the compound interest on Rs. 10,000 for 3 years, the rate for the first year being \( 5\% \) p.a., for the second year \( 6\% \) p.a., and for the third year is $8\%$ per annum. 

c) Find the present value of an annuity due of Rs. 5,000 per year for 10 years at \( 9\% \) p.a.