# Mathematics and Statistics 2016-2017 HSC Commerce (English Medium) 12th Standard Board Exam Question Paper Solution

Mathematics and Statistics
Date & Time: 6th March 2017, 11:00 am
Duration: 3h

(I) All questions are compulsory.
(2) Answer lo every question must be written on a new page .
(3) Log table will be provided on demand .
(4) Write answers of section - I and Section - II in two separate answer books .

Section - I
 1 | Attempt any SIX of the following :
 1.A

Find x , y , z , w if [("x+y","x-y"),("y+z+w","2w-z")] = [(2,-1),(9,5)]

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

Express the truth of the following statements with the help of Venn diagram:
(a) No circles are polygon
(b) If a quadrilateral is rhombus , then it is a parallelogram .

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

Find the points of discontinuity , if any for the function : f(x) = (x^2 - 9)/(sinx - 9)

Concept: Continuous Function of Point
Chapter: [0.03] Continuity
 1.D

Write negation of the following statements :
(a) The number 6 is an even number or the number 25 is a perfect square.
(b) If x ∈ A ∩ B , then x ∈ A and x ∈ B.

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

Evaluate : ∫ cos^2x dx

Concept: Indefinite Integration - Rules of Integration
Chapter: [0.06] Indefinite Integration
 1.F

Find (d^2y)/(dx^2) , if y = log x

Concept: Logarithmic Differentiation
Chapter: [0.04] Differentiation
 1.G

Evaluate : ∫(e^x + 1)/(e^x + x) dx

Concept: Indefinite Integration - Rules of Integration
Chapter: [0.06] Indefinite Integration
 1.H

Find (dy)/(dx) , "If"   x^3 + y^2 + xy = 10

Concept: Derivatives of Implicit Functions
Chapter: [0.013000000000000001] Differentiation [0.04] Differentiation
 2
 2.A | Attempt any Two of the following:
 2.A.i

Find the inverse of the matrix [(1      2     3),(1    1     5),(2    4     7)] by adjoint method

Concept: Inverse of Matrix
Chapter: [0.012] Matrices [0.02] Matrices
 2.A.ii

If f(x) = (e^(2x) - 1)/(ax) .                for x < 0 , a ≠ 0
= 1.                             for x = 0
= (log(1 + 7x))/(bx).        for x > 0 , b ≠ 0
is continuous at x = 0 . then find a and b

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

Demand function x, for a certain commodity is given as x = 200 - 4p where p is the unit price. Find :
(a) elasticity of demand as function of p.
(b) elasticity of demand when p = 10 , interpret your result.

Concept: Random Variables and Its Probability Distributions
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
 2.B | Attempt any Two of the following:
 2.B.i

Using the truth table verify that p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r).

Concept: Random Variables and Its Probability Distributions
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
 2.B.ii

If the demand function is D = 150 - p2 - 3p, find marginal revenue, average revenue and elasticity of demand for price p = 3.

Concept: Random Variables and Its Probability Distributions
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
 2.B.iii

Evaluate : ∫_0^(pi/2) (sinx.cosx)/(1 + sin^4x).dx

Concept: Indefinite Integration - Methods of Integration
Chapter: [0.06] Indefinite Integration
 3
 3.A | Attempt any Two of the following :
 3.A.i

Solve the following equations by reduclion method

x+3y+3z= 16 ,  x+4y+4z=21 , x+3y+4z = 19

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

If the function f (x) = (15^x - 3^x - 5^x + 1)/(x tanx),  x ≠ 0 is continuous at x = 0 , then find f(0).

Concept: Continuous Function of Point
Chapter: [0.03] Continuity
 3.A.iii

Examine the function f(x) = x + 25/x  for maxima and minima

Concept: Maxima and Minima
Chapter: [0.05] Applications of Derivative
 3.B | Attempt any Two of the following
 3.B.i

Find the volume of a solid obtained by the complete revolution of the ellipse x^2/36 + y^2/25 = 1 about X-axis.

Concept: Applications of Definite Integrals
Chapter: [0.07] Definite Integrals
 3.B.ii

If x^3y^5 = (x + y)^8 , then show that (dy)/(dx) = y/x

Concept: Second Order Derivative
Chapter: [0.013000000000000001] Differentiation [0.04] Differentiation

Evaluate : ∫((1 + logx))/(x(2 + logx)(3 + logx)dx

Concept: Indefinite Integration - Integration by Parts
Chapter: [0.06] Indefinite Integration
Section - II
 4 | Attempt any Six of the following.
 4.A
 4.A.i

The ratio of number of boys and girls in a school is 3 : 2. If 20% of the boys and 30% of the girls are scholarship holders, find the percentage of students who are not scholarship holders.

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

Calculate crude death rates (CDR) for district A.

 Age group (in years) Number or persons (in thousands) Number of deaths 0 - 15 1 20 15 - 60 3 30 60 and above 2 40
Concept: Probability Distribution of a Discrete Random Variable
Chapter: [0.14] Random Variable and Probability Distribution
 4.A.iii

What is the sum due of ₹5,000, due 4 months, hence at 12.5% p.a. simple interest?

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

The following data gives the marks of 20 students in mathematics (X) and statistics (Y) each out of 10, expressed as (x, y). construct ungrouped frequency distribution considering single number as a class :
(2, 7) (3, 8) (4, 9) (2, 8) (2, 8) (5, 6) (5 , 7) (4, 9) (3, 8) (4, 8) (2, 9) (3, 8) (4, 8) (5, 6) (4, 7) (4, 7) (4, 6 ) (5, 6) (5, 7 ) (4, 6 )

Concept: Random Variables and Its Probability Distributions
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
 4.A.v

A wholesaler allows 25% trade discount and 5% cash discount, what will be the net price of an article marked at ₹1,600?

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

Verify the following function, which can be regarded as p.m.f. for the given values of X :

 X = x -1 0 1 P(x) -0.2 1 0.2
Concept: Random Variables and Its Probability Distributions
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
 4.A.vii

Solve the following minimal assignment problem :

 Machines Jobs I II III M1 1 4 5 M2 4 2 7 M3 7 8 3
Concept: Introduction of Matrices
Chapter: [0.02] Matrices
 4.A.viii

If X has Poisson distribution with parameter m = 1, find P[X ≤ 1]  [Use e^-1 = 0.367879]

Concept: Poisson Distribution
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
 5 | Attempt any Two of the following:
 5.A
 5.A.i

Find the present value of annuity immediate of ₹ 18000 p.a. for 3 years at 9% p.a. compounded annually. [(Given : (1.09)^-3 = 0.7722]

Concept: Insurance and Annuity
Chapter: [0.1] Insurance and Annuity
 5.A.ii

Complete the following life table :

 x lx dx qx px Lx 4 9100 60 ? ? ? 5 ? 45 ? ?
Concept: Life Tables
Chapter: [0.11] Demography
 5.A.iii

Given that r = 0.4 , Σ(x - barx)(y - bary) = 108 , σ_y = 3 and Σ(x - barx)^2 = 900 . Find the number of pairs of observations.

Concept: Statistics (Entrance Exam) - Karl Pearson’s Coefficient of Correlation
Chapter: [0.12] Bivariate Data and Correlation
 5.B | Attempt any Two of the following:
 5.B.i

Find mean and standard deviation of the continuous random variable X whose p.d.f. is given by f(x) = 6x(1 - x);= (0);      0 < x < 1(otherwise)

Concept: Random Variables and Its Probability Distributions
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
 5.B.ii

Solve the following L.P.P. graphically Minimize Z = 3x + 5y Subject to 2x + 3y ≥ 12
-x + y ≤ 3
x ≤ 4
y ≥ 3

Concept: Linear Programming Problem in Management Mathematics
Chapter: [0.15] Management Mathematics
 5.B.iii

We have seven jobs each of which has to go through two machines M1 and M2 in the order  M1 - M2 . Processing times (in hours) are given as :

 Jobs A B C D E F G Machine M1 3 12 15 6 10 11 9 Machine M2 8 10 10 6 12 1 3

Determine a sequence of these jobs that Will minimize the total elapsed time 'T', and idle time for each machine .

Concept: Sequencing in Management Mathematics
Chapter: [0.15] Management Mathematics
 6 |
 6.A | Attempt any Two of the following:
 6.A.i

Compute the age specific death rate for the following data :

 Age group (years) Population (in thousands) Number of deaths Below 5 15 360 5-30 20 400 Above 30 10 280
Concept: Random Variables and Its Probability Distributions
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
 6.A.ii

If the rank correlation coefficient is 0.6 and the sum of squares of differences of ranks is 66, then find the number of pairs of observations.

Concept: Rank Correlation
Chapter: [0.12] Bivariate Data and Correlation
 6.A.iii

The equations of the two regression lines are 2x + 3g - 6 = 0 and 5x + 7g - 12 = 0
Find: (a) Correlation coefficient.
(b) sigma_x/sigma_y

Concept: Statistics (Entrance Exam) - Karl Pearson’s Coefficient of Correlation
Chapter: [0.12] Bivariate Data and Correlation
 6.B | Attempt any Two of the following
 6.B.i

John and Mathew started a business with their capitals in the ratio 8 : 5. After 8 months, john added 25% of his earlier capital as further investment. At the same time, Mathew withdrew 20% of bis earlier capital. At the end of the year, they earned ₹ 52000 as profit. How should they divide the profit between them?

Concept: Random Variables and Its Probability Distributions
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
 6.B.ii

A departmental store gives trafnfng to the salesmen in service followed by a test. It is experienced that the performance regarding sales of any salesman is linearly related to the scores secured by him. The following data gives the test scores and sales made by nine (9) salesmen during a fixed period.

 Test scores (X) 16 22 28 24 29 25 16 23 24 Sales (Y) (₹ in hundreds) 35 42 57 40 54 51 34 47 45

(a) Obtain the line of regression of Y on X.
(b) Estimate Y when X = 17.

Concept: Random Variables and Its Probability Distributions
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
 6.B.iii

Three different aeroplanes are to be assigned to carry three cargo consignments with a view to maximize profit. The profit matrix (in lakhs of ₹) is as follows :

 Aeroplanes Cargo consignments C1 C2 C3 A1 1 4 5 A2 2 3 3 A3 3 1 2

How should the cargo consignments be assigned to the aeroplanes to maximize the profit?

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

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