Mathematics and Statistics Set 1 2017-2018 HSC Commerce (Marketing and Salesmanship) 12th Board Exam Question Paper Solution

Mathematics and Statistics [Set 1]
Date: July 2018
• All questions are compulsory.
• Figures to the right lndicatefull marks.
• Graph paper Is necessary for L.P.P.
• Write answer of every question on separate page.

Section - I
[12] 1 | Attempt any SIX
[2] 1.A

If p : It is a day time

q : It is warm

Give the verbal statement for the following symbolic statement :

(a) p ^ - q  (b) P → q

Concept: Mathematical Logic - Statements - Introduction in Logic
Chapter: [0.01] Mathematical Logic
[2] 1.B

Express the truth of each of the following statements using Venn diagrams:

(a) No circles are polygons

(b) Some quadratic equations have equal roots

Concept: Venn Diagrams
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[2] 1.C

Find the values of x and y if

2 [(x,5),(7,y-3)] [(3,-4),(1,2)] = [(7,6),(15,14)]

Concept: Algebra of Matrices
Chapter: [0.012] Matrices [0.02] Matrices
[2] 1.D

Find "dy"/"dx" ; if x = sin3θ , y = cos3θ

Concept: Derivatives of Implicit Functions
Chapter: [0.013000000000000001] Differentiation [0.04] Differentiation
[2] 1.E

Find "dy"/"dx" ; if y = cos-1 ("2x" sqrt (1 - "x"^2))

Concept: Derivatives of Implicit Functions
Chapter: [0.013000000000000001] Differentiation [0.04] Differentiation
[2] 1.F

Evaluate: ∫ x . log x dx

Concept: Mathematical Logic - Logical Connectives
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[2] 1.G

The cost C of producing x articles is given as C = x3-16x2 + 47x.  For what values of x, with the average cost is decreasing'?

Concept: Derivatives of Functions in Parametric Forms
Chapter: [0.04] Differentiation
[2] 1.H

Evaluate : int _0^(pi/4) 1/(1 + "x"^2)  "dx"

Concept: Applications of Definite Integrals
Chapter: [0.07] Definite Integrals
[14] 2
[6] 2.A | Attempt any TWO
[3] 2.A.i

Solve the following equations by reduction method:

x+ y+z = 6,

3x-y+3z = 10

5x+ y-4z = 3

Concept: Algebra of Matrices
Chapter: [0.012] Matrices [0.02] Matrices
[3] 2.A.ii

Evaluate : int (2"x" + 1)/(("x" + 1)("x"-2))  "dx"

Concept: Indefinite Integration - Integration by Parts
Chapter: [0.06] Indefinite Integration
[3] 2.A.iii

Evaluate int _0^l "x" (1 - "x")^(3/2)  "dx"

Concept: Applications of Definite Integrals
Chapter: [0.07] Definite Integrals
[8] 2.B |  Attempt any TWO
[4] 2.B.i

Using the rules of negation, write the negatlon of the following:

(a) p ∧ (q → r)

(b)  ~P ∨ ~q

Concept: Mathematical Logic - Algebra of Statements
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[4] 2.B.ii

If the function f is continuous at x = 2 and x = 4 then find the values of a and b.

Where f(x) = x2 + ax + b, x < 2

= 3x + 2, 2 ≤ x ≤ 4

=  2ax + 5b, 4 < x

Concept: Indefinite Integration - Integration by Parts
Chapter: [0.06] Indefinite Integration
[4] 2.B.iii

A manufacturing company produces x items at the total cost of Rs (180+4x). The demand function of this product is P=(240 - x). Find x for which profit is increasing.

Concept: Application of Derivatives to Economics
Chapter: [0.013999999999999999] Applications of Derivatives
[14] 3
[6] 3.A | Attempt any TWO
[3] 3.A.i

If A = [(1,2),(3,-1)] , "B" = [(7,1),(2,5)]

Verify that |AB| = |A|.|B|

Concept: Algebra of Matrices
Chapter: [0.012] Matrices [0.02] Matrices
[3] 3.A.ii

Evaluate : int 1/("x" [("log x")^2 + 4])  "dx"

Concept: Properties of Definite Integrals
Chapter: [0.016] Definite Integration [0.07] Definite Integrals
[3] 3.A.iii

Find the volume of solid generated by rotating the area bounded by x2+y2 =36 and the lines x = 0, x = 3 about X -axis.

Concept: Indefinite Integration - Methods of Integration
Chapter: [0.06] Indefinite Integration
[8] 3.B | Attempt any TWO
[4] 3.B.i

If f is continuous at x = 0, then find f (0).

Where f(x) = (3^"sin x" - 1)^2/("x" . "log" ("x" + 1)) , "x" ≠ 0

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

If x7 . y9 = (x + y)16 then show that "dy"/"dx" = "y"/"x"

Concept: Second Order Derivative
Chapter: [0.013000000000000001] Differentiation [0.04] Differentiation
[4] 3.B.iii

In a firm the cost function for output x is given as C = "x"^3/3 - 20"x"^2 + 70 "x".  Find the 3 output for which marginal cost  (Cm) is minimum.

Concept: Maxima and Minima
Chapter: [0.05] Applications of Derivative
Section II
[12] 4 | Attempt any SIX
[2] 4.A

The price of a T.V. set is Rs 17,000. An agent charges at 3% and earns Rs. 25,500. Find the number of T.V. sets sold by him.

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

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

 Age groups (In year) Population (in '000) Number of Dearths 0 - 10 11 275 10 - 20 12 180 20 - 60 9 81 60 and above 2 32
Concept: Life Tables
Chapter: [0.11] Demography
[2] 4.C

The regression equation of y on x is given by 3x + 2y - 26 = O. Find byx.

Concept: Regression Coefficient of X on Y and Y on X
Chapter: [0.13] Regression Analysis Introduction
[2] 4.D

Verify whether the following function can be regarded as p.m.f. of the random variable X:

$P(X) = \begin{cases} \frac{x - 1}3 & \quad \text{x } = \text{ 1,2,3}\\ 0 & \quad \text{} , \text{otherwise} \end{cases}$

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

If X has a binomial distribution with n = 20 , p = 1/10 , find E (X) and V (X).

Concept: Binomial Theorem and Binomial Distribution
Chapter: [0.14] Random Variable and Probability Distribution
[2] 4.F

Bring out the inconsistency; if any: byx + bxy = 1.30. and r = 0.75

Concept: Regression Coefficient of X on Y and Y on X
Chapter: [0.13] Regression Analysis Introduction
[2] 4.G

A train travelled between two stations. The distance and time were recorded as below:

 Distance (Km) 80 120 160 200 240 Time (Hr) 2 3 4 5 6

Draw scatter diagram and identify the type of correlation.

Concept: Statistics - Karl Pearson’s Coefficient of Correlation
Chapter: [0.12] Bivariate Data and Correlation
[2] 4.H

If r = 0.5, σx = 1 and σy = 4, then find Cov.(X,Y).

Concept: Statistics - Karl Pearson’s Coefficient of Correlation
Chapter: [0.12] Bivariate Data and Correlation
[14] 5
[6] 5.A | Attempt any TWO
[3] 5.A.i

Calculate "e"_0^circ ,"e"_1^circ , "e"_2^circ from the following:

 Age x 0 1 2 lx 1000 880 876 Tx - - 3323
Concept: Random Variables and Its Probability Distributions
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
[3] 5.A.ii

Calculate the CDR for District A and B and compare them:

 Age group (in years) District A District B No.of.persons(in '000) No.of.deaths No.of.persons(in '000) No.of. deaths 0 - 15 1 20 2 50 15 - 60 3 30 7 70 60 and above 2 40 1 25
Concept: Linear Programming Problem in Management Mathematics
Chapter: [0.15] Management Mathematics
[3] 5.A.iii

The equation of the line of regression of Y on X is 3x + 2y = 26 and X on Y is 6x + y = 31. Find Var. (X) if Var. (Y) = 36.

Concept: Regression Propertise
Chapter: [0.13] Regression Analysis Introduction
[8] 5.B | Attempt any TWO
[4] 5.B.i

Calculate Spearman's Rank Correlation Coefficient between the following marks given by 'two' judges (A) and (B) to •eight' contestants in the elocution competition:

 Contestants 1 2 3 4 5 6 7 8 Marks by A 81 72 60 33 29 11 56 42 Marks byB 75 56 42 15 30 20 60 80
Concept: Rank Correlation
Chapter: [0.12] Bivariate Data and Correlation
[4] 5.B.ii

Solve the following assignment problem to minimize the cost:

 Persons Jobs I II III A 7 3 5 B 2 7 4 C 6 5 3 D 3 4 7
Concept: Maxima and Minima
Chapter: [0.05] Applications of Derivative
[4] 5.B.iii

Find the sequence of the following five jobs to be processed on three machines M1 M2, , M3 that will minimize the total elapsed time to complete the tasks . Each job is to be processed in the order M1 - M2 - M3 :

 Jobs 1 2 3 4 5 Machine M1 5 11 5 7 6 Machine M2 1 4 2 5 3 Machine M3 1 5 2 3 4
Concept: Sequencing in Management Mathematics
Chapter: [0.15] Management Mathematics
[14] 6
[6] 6.A | Attempt any TWO
[3] 6.A.i

Minimize: Z = 2x + y

Subject to: x + y ≤ 5

x + 2y ≤  8

4x + 3y ≥ 12

x ≥  o,  y ≥ o

Solve graphically.

Concept: Linear Programming Problem in Management Mathematics
Chapter: [0.15] Management Mathematics
[3] 6.A.ii

Find the graphical solution for the following system of linear equations :

3x + 4y ≥ 12 , 4x + 7y ≤ 28 , y ≥ 1 , x ≥ 0 , y ≥ 0 ,

Hence find the co-ordinates of comer points of the common region.

Concept: Linear Programming Problem in Management Mathematics
Chapter: [0.15] Management Mathematics
[3] 6.A.iii

A wholesaler allows 25% trade discount and 5% cash discount.  Find the list price of an article, if it was sold for the net amount of Rs 1,140.

Concept: Commission, Brokerage and Discount
Chapter: [0.09] Commission, Brokerage and Discount
[8] 6.B | Attempt TWO of the following :
[4] 6.B.i

Find the accumulated value after 3 years of an immediate annuity of Rs 2,000 p.a. with interest compounded at 10% p.a. [Given (1.1)3 = 1.331]

Concept: Insurance and Annuity
Chapter: [0.1] Insurance and Annuity
[4] 6.B.ii

If the difference between true discount and banker's discount on a sum due 4 months hence is Rs 20,  find true discount, banker's discount and amount of the bill, the rate of simple interest charged being 5% p.a .

Concept: Commission, Brokerage and Discount
Chapter: [0.09] Commission, Brokerage and Discount
[4] 6.B.iii

If a random variable X follows Poisson distribution such that P(X = l) =P(X = 2), then find P(X ≥ 1).  [Use e-2 = 0.1353]

Concept: Poisson Distribution
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution

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