# Mathematics and Statistics Set 1 2016-2017 HSC Commerce 12th Board Exam Question Paper Solution

Mathematics and Statistics [Set 1]
Date: July 2017

(1) All questions are compulsory.
(2) Figure to the right tndtcate full marks.
(3) Answer to every question must be .written on a new page.
(4) Graph paper is necessaryJor L.P.P.
(5) Log table will be provided on demand.
(6) Answer to Section-I and Section-II should be written In two separate answer books.
(7) Question from Section-I attempted in the answer book of Section-II and vice versa wil not be assessed/not given any credit.
vice versa will not be assessed/not given any credit.

SECTION - I
[12]1 | Attempt any Six
[2]1.A
[2]1.A.i

Write the negation of the following statements :
(a) Radha likes tea or coffee.
(b) ∃x cc R such that x + 3 ≥ 10.

Concept: Introduction of Matrices
Chapter: [0.02] Matrices
[2]1.A.ii

If A = [(1,2), (1,3)], find A2 - 3A

Concept: Introduction of Matrices
Chapter: [0.02] Matrices
[2]1.A.iii

Discuss the continuity of the function f at x = 0

If f(x) = (2^(3x) - 1)/tanx, for x ≠ 0

= 1,   for x = 0

Concept: Continuous Function of Point
Chapter: [0.03] Continuity
[2]1.A.iv

Evaluate : int ("x" + 1)/("x"("x" + log"x")dx

Concept: Indefinite Integration - Integral of Standard Functions
Chapter: [0.06] Indefinite Integration
[2]1.A.v

Solve the equations x + y = 4 and 2x - y = 5 using the method of reduction.

Concept: Introduction of Matrices
Chapter: [0.02] Matrices
[2]1.A.vi

If the function f is continuous at = 2, then find f(2) where f(x) = (x^5 - 32)/(x - 2), for ≠ 2.

Concept: Continuous Function of Point
Chapter: [0.03] Continuity
[2]1.A.vii

Find the elasticity of demand, if the marginal revenue is 50 and price is Rs 75.

Concept: Application of Derivatives to Economics
Chapter: [0.013999999999999999] Applications of Derivatives
[2]1.A.viii

Differentiate
log (1 + x2) w.r.t. tan-1 (x)

Concept: Logarithmic Differentiation
Chapter: [0.04] Differentiation
[14]2 | Attempt any Two
[6]2.A | Attempt any two
[3]2.A.i

If   p : It is raining
q : It is humid

Write the following statements in symbolic form:

(a) It is raining or humid.
(b) If it is raining then it is humid.
(c) It is raining but not humid.

Concept: Mathematical Logic - Statement Patterns and Logical Equivalence
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[3]2.A.ii

Using truth table, examine whether the following statement pattern is tautology, contradiction or contingency: p ∨ [∼(p ∧ q)]

Concept: Mathematical Logic - Statement Patterns and Logical Equivalence
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[3]2.A.iii

Evaluate ; int x^2/(x^6 - 4x^3 + 13) dx

Concept: Indefinite Integration - Integration by Parts
Chapter: [0.06] Indefinite Integration
[8]2.B | Attempt any two
[4]2.B.i

If the function f is continuous at x = 0

Where f(x) = 2sqrt(x^3 + 1) + a,  for x < 0,
= x^3 + a + b,  for x > 0
and f (1) = 2, then find a and b.

Concept: Continuous Function of Point
Chapter: [0.03] Continuity
[4]2.B.ii

For manufacturing x units, labour cost is 150 - 54x, processing cost is x2 and revenue R = 10800x - 4x. Find the value of x for which Total cost is decreasing.

Concept: Increasing and Decreasing Functions
Chapter: [0.013999999999999999] Applications of Derivatives [0.05] Applications of Derivative

For manufacturing x units, labour cost is 150 – 54x and processing cost is x2. Price of each unit is p = 10800 – 4x2. Find the values of x for which Revenue is increasing.

Concept: Increasing and Decreasing Functions
Chapter: [0.013999999999999999] Applications of Derivatives [0.05] Applications of Derivative
[4]2.B.iii

The total cost C for producing x units is Rs (x2 + 60x + 50) and the price is Rs (180 - x) per unit. For how many units the profit is maximum.

Concept: Continuous Function of Point
Chapter: [0.03] Continuity
[14]3 | Attempt any Two
[6]3.A
[3]3.A.i

If y = sin -1 ((8x)/(1 + 16x^2)), find (dy)/(dx)

Concept: Derivatives of Functions in Parametric Forms
Chapter: [0.04] Differentiation
[3]3.A.ii

If y = 5+ xx, Find (dy)/(dx).

Concept: Applications of Definite Integrals
Chapter: [0.07] Definite Integrals
[3]3.A.iii

int (2x + 1)/((x + 1) ( x - 2)) dx

Concept: Indefinite Integration - Integration by Parts
Chapter: [0.06] Indefinite Integration
[8]3.B | Attempt any Two
[4]3.B.i

Find the inverse of matrix A by using adjoint method; where A = [(1, 0, 1), (0, 2, 3), (1, 2, 1)]

Concept: Inverse of Matrix
Chapter: [0.012] Matrices [0.02] Matrices
[4]3.B.ii

Find the area of the region bounded by the curve (parabola) y2 = 4x and the line

Concept: Indefinite Integration - Integral of Standard Functions
Chapter: [0.06] Indefinite Integration
[4]3.B.iii

Evaluate int_0^1 (x(sin^-1 x)^2)/sqrt(1 - x^2) dx

Concept: Applications of Definite Integrals
Chapter: [0.07] Definite Integrals
SECTION - II
[12]4 | Attempt any six
[2]4.A
[2]4.A.i

In a class 60% students arc boys and 40% are girls. By admitting 16 boys and 8 girls, the ratio of boys to girls becomes 8 : 5. Find the original number of boys and girls in the class.

Concept: Mathematical Logic - Statements - Introduction in Logic
Chapter: [0.01] Mathematical Logic
[2]4.A.ii

An agent charges 12% commission on the sales. How much does he earn if the total sales amounts to ~ 36,000? How muuch does the seller get from this sale?

Concept: Commission, Brokerage and Discount
Chapter: [0.09] Commission, Brokerage and Discount
[2]4.A.iii

Find the Age-Specific Death Rate (SDR) for the following data :

 Age groups(in years) Population Number of deaths 0 - 10 11,000 220 10 - 20 12,000 240 20 - 60 9,000 180 60 and above 2,000 90
Concept: Commission, Brokerage and Discount
Chapter: [0.09] Commission, Brokerage and Discount
[2]4.A.iv

If the Crude Death Rate (CDR) for the following data is 13.4 per thousand, find x:

 Age groups(in years) Population Number of deaths 0 - 20 40,000 350 20 - 65 65,000 650 65 and above 15,000 x
Concept: Commission, Brokerage and Discount
Chapter: [0.09] Commission, Brokerage and Discount
[2]4.A.v

If a fair die is rolled four times,. what is the probability that two of the rolls will show '1'.

Concept: Probability Distribution of a Discrete Random Variable
Chapter: [0.14] Random Variable and Probability Distribution
[2]4.A.vi

Solve the inequation -8 ≤ 3x - 5 < 7 and represent it on the real number line.

Concept: Probability Distribution - Distribution Function of a Continuous Random Variable
Chapter: [0.14] Random Variable and Probability Distribution
[2]4.A.vii

Sketch the graph of | y | ≤ 3 in a coordinate plane.

Concept: Concept of Demography
Chapter: [0.11] Demography
[2]4.A.viii

Find the sequence that minimizes the total elapsed time (in hours) required to
complete the following jobs on the machine M1 , M2 and M3 in the order M1M2M3 :

 Machine          Jobs A B C D M1 5 6 9 5 M2 2 4 5 3 M3 3 5 6 7
Concept: Solution of Equations
Chapter: [0.02] Matrices
[14]5
[6]5.A | Attempt any two
[3]5.A.i

Find accumulated value after 1 year of annuity immediate in which Rs 20,000 is invested every quarter, at .16% p.a. compounded quarterly.
[Given : (1.04)4 = 1.1699]

Concept: Commission, Brokerage and Discount
Chapter: [0.09] Commission, Brokerage and Discount
[3]5.A.ii

If a random variable X follows Poisson distribution sucli that P(X = 1) = P(X = 2), find:
(a) the mean and
(b) P(X = 0). [Given: e-2 = 0.1353)

Concept: Probability Distribution - Distribution Function of a Continuous Random Variable
Chapter: [0.14] Random Variable and Probability Distribution
[3]5.A.iii

The probability of defective bolts in a workshop is 40%. Find the mean and variance of defective bolts out of 10 bolts.

Concept: Probability Distribution of a Discrete Random Variable
Chapter: [0.14] Random Variable and Probability Distribution
[8]5.B | Attempt any two
[4]5.B.i

Given the following table which relates to the number of animals of a certain species at age x . Complete the life table with dx, qx, Lx and Tx.

 x 0 1 2 3 4 5 lx 1000 850 760 360 25 0
Concept: Probability Distribution of a Discrete Random Variable
Chapter: [0.14] Random Variable and Probability Distribution
[4]5.B.ii

Co-efficient of correlation between the variables X and Y is 0.3 and their covariance is 12. The variance of X is 9, find the standard deviation of Y.

Concept: Probability Distribution - Distribution Function of a Continuous Random Variable
Chapter: [0.14] Random Variable and Probability Distribution
[4]5.B.iii

Compute correlation coefficient for the following data :

 X 9 7 6 8 5 Y 19 17 16 18 15
Concept: Probability Distribution - Distribution Function of a Continuous Random Variable
Chapter: [0.14] Random Variable and Probability Distribution
[14]6
[6]6.A
[3]6.A.i

Surabhi, Simona and Shruti started a business in a partnership by investing ₹ 60,000, ₹ 40,000 and ₹ 75,000 respectively. At the end of the year they found that they have incurred a loss of ₹ 24,500. Find that how much loss each one had to bear?

Concept: Ratio, Proportion and Partnership
Chapter: [0.08] Ratio, Proportion and Partnership
[3]6.A.ii

Three cars were sold through an agent for Rs 1,60,000, Rs1,48,000 and Rs 1,50,000 respectively. The rates of commission were 17 .5% on the first, and 12.5% on the second. If on the whole, the agent received 14% commission on the total sales, find the rate of commission paid on the third car.

Concept: Commission, Brokerage and Discount
Chapter: [0.09] Commission, Brokerage and Discount
[3]6.A.iii

The equations given of the two regression lines are 2x + 3y - 6 = 0 and 5x + 7y - 12 = 0.

Find:

(a) Correlation coefficient

(b) sigma_x/sigma_y

Concept: Lines of Regression of X on Y and Y on X Or Equation of Line of Regression
Chapter: [0.023] Linear Regression [0.13] Regression Analysis Introduction
[8]6.B | Attempt any two
[4]6.B.i

If Σx1 = 56 Σy1 = 56, Σx_1^2 = 478,
Σy_1^2` = 476, Σx1y1 = 469 and n = 7, Find
(a) the regression equation of y on x.
(b) y, if x = 12.

Concept: Lines of Regression of X on Y and Y on X Or Equation of Line of Regression
Chapter: [0.023] Linear Regression [0.13] Regression Analysis Introduction
[4]6.B.ii

Solve the following minimal assignment problem :

 Machines A B C D E M1 27 18 ∞ 20 21 M2 31 24 21 12 17 M3 20 17 20 ∞ 16 M4 21 28 20 16 27
Concept: Assignment Problem
Chapter: [0.027000000000000003] Assignment Problem and Sequencing [0.15] Management Mathematics
[4]6.B.iii

Solve the following L.P.P.:

Maximize Z = 4x + 5y
subject to       2x + y ≥ 4
x + y ≤ 5,
0 ≤ x ≤ 3,
0 ≤ y ≤ 3

Concept: Linear Programming Problem in Management Mathematics
Chapter: [0.15] Management Mathematics

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