Maharashtra State BoardHSC Commerce 12th Board Exam
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Mathematics and Statistics 2015-2016 HSC Commerce 12th Board Exam Question Paper Solution

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Mathematics and Statistics
Marks: 80Academic Year: 2015-2016
Date: July 2016
  • All questions are compulsory.
  • Figure to the right indicateJull marks. 
  • Answer to every question must be written on a new page. 
  • Graph paper is necessaryJor L.P.P. 

[12]1 | Attempt any SIX of the following :
[2]1.A

Evaluate : `int  "e"^(3"x")/("e"^(3"x") + 1)` dx

Concept: Properties of Definite Integrals
Chapter: [0.016] Definite Integration [0.07] Definite Integrals
[2]1.B

The price P for demand D is given as P = 183 + 120 D – 3D2.
Find D for which the price is increasing.

Concept: Increasing and Decreasing Functions
Chapter: [0.013999999999999999] Applications of Derivatives [0.05] Applications of Derivative
[2]1.C

Write the Truth Value of the Negation of the Following Statement :

The Sun sets in the East. 

Concept: Mathematical Logic - Algebra of Statements
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic

Write the truth value of the negation of the following statement : 

cos2 θ + sin2 θ = 1, for all θ ∈ R 

Concept: Mathematical Logic - Algebra of Statements
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[2]1.D

Simplify the following :

`{3 [(1,2,0),(0,-1,3)] - [(1,5,-2),(-3,-4,4)]} [(1),(2),(1)]`

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

Examine the continuity off at x = 1, if

f (x) = 5x - 3 , for 0 ≤ x ≤ 1

       = x2 + 1 , for 1 ≤ x ≤ 2

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

Find `"dy"/"dx"` , if `"y" = "x"^("e"^"x")`

Concept: Logarithmic Differentiation
Chapter: [0.04] Differentiation
[2]1.G

If A = `[(1,2,3),(2,"a",2),(5,7,3)]` is a singular matrix , find the value of 'a'.

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

Evaluate :  `int 1/sqrt("x"^2 - 4"x" + 2) "dx"`

Concept: Properties of Definite Integrals
Chapter: [0.016] Definite Integration [0.07] Definite Integrals
[14]2
[6]2.A |  Attempt any TWO of the following :
[3]2.A.i

Show that the following statement pattern in contingency : 

(~p v q) → [p ∧ (q v ~ q)] 

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

If f(x) = `("e"^(2"x") - 1)/"ax"`   , for x < 0 , a ≠ 0

         = 1                                    for x = 0

       = `("log" (1 + 7"x"))/"bx"` , for x > 0 , b ≠ 0

is continuous at x = 0, then find a and b.

Concept: Applications of Definite Integrals
Chapter: [0.07] Definite Integrals
[3]2.A.iii

xy = ex-y, then show that  `"dy"/"dx" = ("log  x")/("1 + log x")^2`

Concept: Logarithmic Differentiation
Chapter: [0.04] Differentiation
[8]2.B | Attempt any TWO of the following:
[4]2.B.i

Evaluate : `int_0^1 "x" . "tan"^-1  "x"  "dx"`

Concept: Indefinite Integration - Methods of Integration
Chapter: [0.06] Indefinite Integration
[4]2.B.ii

If A = `[(1,-1,2),(3,0,-2),(1,0,3)]` ,

verify that A (adj A) = (adj A) A = |A| . I

Concept: Algebra of Matrices
Chapter: [0.012] Matrices [0.02] Matrices
[4]2.B.iii

A manufacturer can sell x items at a price of ₹ (280 - x) each .The cost of producing items is ₹ (x2 + 40x + 35) Find the number of items to be sold so that the manufacturer can make maximum profit.

Concept: Maxima and Minima
Chapter: [0.05] Applications of Derivative
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[14]3
[6]3.A | Attempt any TWO of the following:
[3]3.A.i

Find k, if the function f is continuous at x = 0, where

`f(x)=[(e^x - 1)(sinx)]/x^2`,      for x ≠ 0

     = k                             ,        for x = 0

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

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

Concept: Logarithmic Differentiation
Chapter: [0.04] Differentiation
[3]3.A.iii

Using the Venn diagram, examine the logical equivalence of the following statements: 

(a) Some politicians are actors.

(b) There are politicians who are actors.

(c) There are politicians who are not actors.

Concept: Venn Diagrams
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[8]3.B | Attempt any TWO of the following :
[4]3.B.i

Find the volume of the solid generated by the complete revolution of the ellipse `"x"^2/36 + "y"^2/25 = 1` about Y-axis.

Concept: Indefinite Integration - Methods of Integration
Chapter: [0.06] Indefinite Integration
[4]3.B.ii

Evaluate  : `int "x"^2/("x"^4 + 5"x"^2 + 6) "dx"`

Concept: Properties of Definite Integrals
Chapter: [0.016] Definite Integration [0.07] Definite Integrals
[4]3.B.iii

The total cost of manufacturing x articles is C = (47x + 300x2 - x4).  Find x, for which average cost is increasing.

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

The total cost of manufacturing x articles C = 47x + 300x2 – x4 . Find x, for which average cost is decreasing.

Concept: Application of Derivatives to Economics
Chapter: [0.013999999999999999] Applications of Derivatives
Section-II
[12]4 |  Attempt any SIX of the following:
[2]4.A

Let X = amount of time for which a book is taken out of a college library by a randomly selected student and suppose X has p.d.f. 

`f(x)={(0.5x",",0≤x≤2,,),(0",","otherwise",,):}`

Calculate (a) P (X ≤ 1) (b) P (0.5 ≤ X ≤ 1.5)   

Concept: Applications of Definite Integrals
Chapter: [0.07] Definite Integrals
[2]4.B

If the correlation coefficient between X and Y is 0.8, what is the correlation coefficient between:

`"X"/2` and Y

Concept: Statistics - Bivariate Frequency Distribution
Chapter: [0.12] Bivariate Data and Correlation

If the correlation coefficient between X and Y is 0.8, what is the correlation coefficient between:

`"X - 5"/7` and  `"Y - 3"/8`

Concept: Statistics - Bivariate Frequency Distribution
Chapter: [0.12] Bivariate Data and Correlation
[2]4.C

Raghu, Madhu and Ramu 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
[2]4.D

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
[2]4.E

For an immediate annuity paid for 3 years with interest compounded at 10% p.a.  its present value is ₹ 10,000. What is its accumulated value after 3 years. 

[Given: (1.1)3 = 1.331)].

Concept: Insurance and Annuity
Chapter: [0.1] Insurance and Annuity
[2]4.F

Calculate CDR for district A and B. 

Age group (in years)  District A  District B
Population No.of.Deaths Population No.of.Deaths
0 - 10 1000 18 3000 70
10 - 55 3000 32 7000 50
Above 55 2000 41 1000 24
Concept: Life Tables
Chapter: [0.11] Demography
[2]4.G

Compute Age - Specific Death rate tor the following data 

Age group (in years)  Population (in thousands No. of deaths
Below 5 15 360
5 - 30 20 400
Above 30 10 280
Concept: Life Tables
Chapter: [0.11] Demography
[2]4.H

The ratio of prices of two houses was 2 : 3. Two years later when price of first house has increased by 30% and that of the second by ₹  90,000 the ratio of prices becomes 5 : 7. Find the original prices of two houses. 

Concept: Ratio, Proportion and Partnership
Chapter: [0.08] Ratio, Proportion and Partnership
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[14]5
[6]5.A | Attempt any TWO of the following:
[3]5.A.i

The income of an agent remains unchanged though the rate of commission is increased from 5% to 6.25%. Find the percentage reduction in the value of business.

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

A new treatment for baldness is known to be effective in 70% of the cases treated. Four bald members from different families are treated. Find the probability that exactly two members are successfully treated.

Concept: Binomial Theorem and Binomial Distribution
Chapter: [0.14] Random Variable and Probability Distribution

A new treatment for baldness is known to be effective in 70% of the cases treated. Four bald members from different families are treated. Find the probability that at least  one member is  successfully treated.

Concept: Binomial Theorem and Binomial Distribution
Chapter: [0.14] Random Variable and Probability Distribution
[3]5.A.iii

Find the graphical solution for following system of linear inequations

`"x"_1/60 + "x"_2/90 <= 1 ; 5x_1 + 8x_2 ≤ 600 , x_1 ≥ 0 , x_2 ≥ 0`

Concept: Linear Programming Problem in Management Mathematics
Chapter: [0.15] Management Mathematics
[8]5.B | Attempt any TWO of the following:
[4]5.B.i

Bring out the inconsistency, if any in the following : 

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

Bring out the inconsistency, if any in the following : 

bYX  = bXY = 1.50 and r = -0.9

Concept: Regression Coefficient of X on Y and Y on X
Chapter: [0.13] Regression Analysis Introduction

Bring out the inconsistency, if any in the following : 

bYX  = 1.9 and bXY = -0.25

Concept: Regression Coefficient of X on Y and Y on X
Chapter: [0.13] Regression Analysis Introduction

Bring out the inconsistency, if any in the following : 

bYX = 2.6 and bXY  = `1/2.6`

Concept: Regression Coefficient of X on Y and Y on X
Chapter: [0.13] Regression Analysis Introduction
[4]5.B.ii

A pharmaceutical company has four branches, one ea.ch at city A, B, C, D. A branch manager is to be appointed one at each city, out of four candidates P, Q, R and S. The monthly business deyending upon the city and the effectiveness of the branch manager in that city is given below :

  City
  A B C D
  Monthly business (₹ lakh) 
P 10 10 8 8
Q 12 15 10 9
R 11 16 12 7
S 15 13 15 11

Which manager should be appointed at which city so as to get the maximum total monthly business·? 

Concept: Solution of Equations
Chapter: [0.02] Matrices
[4]5.B.iii

Fill in the blanks which are marked with a question mark in the following marked from the life table : 

x lx dx qx px Lx Tx
20 88230 ? ? ? ? ?
21 79473         320552
Concept: Life Tables
Chapter: [0.11] Demography
[14]6
[6]6.A | Attempt any Two of the following :
[3]6.A.i

For a bivariate data,

`bar x = 53 , bar y = 28 , "b"_"xy" = -1.5 and "b"_"xy"=- 0.2` Find 

Correlation coefficient between X and Y 

Concept: Regression Coefficient of X on Y and Y on X
Chapter: [0.13] Regression Analysis Introduction

For a bivariate data, 

`bar x = 53 , bar y = 28 , "b"_"xy" = - 0.2` , `"b"_"yx" = -1.5` Find 

 Estimate of Y , When X = 50. 

Concept: Regression Coefficient of X on Y and Y on X
Chapter: [0.13] Regression Analysis Introduction

For a bivariate data, 

`bar x = 53 , bar y = 28 , "b"_"yx"=-1.5 and "b"_"xy"=- 0.2` Find Estimate of X for y = 25. 

Concept: Regression Coefficient of X on Y and Y on X
Chapter: [0.13] Regression Analysis Introduction
[3]6.A.ii

If X has a Poisson distribution with variance 2, find P (X = 4) 

[Use e-2 = 0.1353] 

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

If X has a Poisson distribution with variance 2, find P(X ≤ 4) 

[Use e-2 = 0.1353] 

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

If X has a Poisson distribution with variance 2, find 

Mean of X [Use e-2 = 0.1353] 

Concept: Poisson Distribution
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
[3]6.A.iii

There are 5 jobs each of which is to be processed through three machines A, B and C in the order ABC. Processing times in hours are as shown in the following table : 

Job 1 2 3 4 5
A 3 8 7 5 4
B 4 5 1 2 3
C 7 9 5 6 10

Determine the optimum sequence for the five jobs and the minimum elapsed time. 

Concept: Sequencing in Management Mathematics
Chapter: [0.15] Management Mathematics
[8]6.B | Attempt any TWO of the following :
[4]6.B.i

A bill of Rs.7,500 was discounted for Rs. 7,290 at a bank on 28th October 2006. If the rate of interest was 14% p.a., what is the legal due date ?

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

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.
Also prepare marginal distributions :

(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: Probability Distribution - Distribution Function of a Continuous Random Variable
Chapter: [0.14] Random Variable and Probability Distribution
[4]6.B.iii

 Minimize : Z = 3x1 + 2x2

Subject to constraints 

5x1 + x2 ≥ 10

2x1 + 2x2 ≥ 12

x1 + 4x2 ≥ 12

x1 , x2 ≥ 0

Concept: Linear Programming Problem in Management Mathematics
Chapter: [0.15] Management Mathematics
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Maharashtra State Board previous year question papers 12th Board Exam Mathematics and Statistics with solutions 2015 - 2016

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