# Mathematics 2013-2014 ISC (Commerce) Class 12 Question Paper Solution

Mathematics
Date: March 2014
• Question 1 is Compulsory
• Attempt  Five Question From Question 2 to Question 9
• Attempt Any two Question From Question 10 to Question 15

Section A
 1 | All questions are compulsory.
 1.i

If A = [[3,1] , [7,5]], find the values of x and y such that A2 + xI2 = yA.

Concept: Equality of Matrices
Chapter: [0.021] Matrices and Determinants
 1.ii

Find the eccentricity and the coordinates of foci of the hyperbola 25x2 + 9y2 = 225.

Concept: Application on Coordinate Geometry
Chapter: [0.034] Differential Equations
 1.iii

Evaluate: tan [ 2 tan^-1  (1)/(2) – cot^-1 3]

Concept: Basic Concepts of Inverse Trigonometric Functions
Chapter: [0.01] Relations and Functions (Section A)
 1.iv

Using L’Hospital’s Rule, evaluate: limx → 0 ( 1 + sin x )cot x

Concept: L' Hospital'S Theorem
Chapter: [0.031] Continuity, Differentiability and Differentiation
 1.v

Evaluate: int_  e^x ((2+sin2x))/cos^2 x dx

Concept: Evaluation of Definite Integrals by Substitution
Chapter: [0.033] Integrals
 1.vi

Using properties of definite integrals, evaluate

int_0^(π/2)  sqrt(sin x )/ (sqrtsin x + sqrtcos x)dx

Concept: Properties of Definite Integrals
Chapter: [0.033] Integrals
 1.vii

For the given lines of regression, 3x – 2y = 5 and x – 4y = 7, find:
(a) regression coefficients byx and bxy
(b) coefficient of correlation r (x, y)

Concept: Lines of Regression of X on Y and Y on X Or Equation of Line of Regression
Chapter: [0.09] Linear Regression (Section C)
 1.viii

Express the complex number (1+sqrt3i)^2/(sqrt3 -i in the form of a + ib. Hence, find the modulus and argument of the complex number.

Concept: Applications of Integrations - Application of Integrals - Modulus Function
Chapter: [0.07] Application of Integrals (Section B)
 1.ix

A bag contains 20 balls numbered from 1 to 20. One ball is drawn at random from the bag. What is the probability that the ball drawn is marked with a number which is multiple of 3 or 4?

Concept: Mean of a Random Variable
Chapter: [0.04] Probability (Section A)
 1.x

Solve the differential equation: (x + 1) dy – 2xy dx = 0

Concept: Solutions of Linear Differential Equation
Chapter: [0.034] Differential Equations
 2
 2.i

Using properties of determinants, prove that:

|[a^2 + 1, ab, ac], [ba, b^2 + 1, bc ], [ca, cb, c^2+1]| = a^2 + b^2 + c^2 + 1

Concept: Properties of Determinants
Chapter: [0.021] Matrices and Determinants
 2.ii

Using matrix method, solve the following system of equations:
x – 2y = 10, 2x + y + 3z = 8 and -2y + z = 7

Concept: Inverse of Matrix - Inverse of a Square Matrix by the Adjoint Method
Chapter: [0.021] Matrices and Determinants
 3
 3.i

If cos-1 x + cos -1 y + cos -1 z = π , prove that x2 + y2 + z2 + 2xyz = 1.

Concept: Properties of Inverse Trigonometric Functions
Chapter: [0.01] Relations and Functions (Section A)
 3.ii

P, Q, and R represent switches in on position and P’, Q’ and R’ represent switches in off position. Construct a switching circuit representing the polynomial PR + Q (Q’ + R) (P + QR). Using Boolean Algebra, simplify the polynomial expression and construct the simplified circuit.

Concept: Applications of Integrations - Application of Integrals - Polynomial Functions
Chapter: [0.07] Application of Integrals (Section B)
 4
 4.i

Verify Rolle’s Theorem for the function f(x) = ex (sin x – cos x) on [ (π)/(4), (5π)/(4)].

Concept: Mean Value Theorem
Chapter: [0.031] Continuity, Differentiability and Differentiation
 4.ii

Find the equation of the parabola with latus-rectum joining points (4, 6) and (4, -2).

Concept: Area of the Region Bounded by a Curve and a Line
Chapter: [0.07] Application of Integrals (Section B)
 5
 5.i

If y = (x sin^-1 x)/sqrt(1 -x^2), prove that: (1 - x^2)dy/dx = x + y/x

Concept: Properties of Inverse Trigonometric Functions
Chapter: [0.01] Relations and Functions (Section A)
 5.ii

A wire of length 50 m is cut into two pieces. One piece of the wire is bent in the shape of a square and the other in the shape of a circle. What should be the length of each piece so that the combined area of the two is minimum?

Concept: Maximum and Minimum Values of a Function in a Closed Interval
Chapter: [0.032] Applications of Derivatives
 6
 6.i

Evaluate: int_  (x + sin x)/(1 + cos x )  dx

Concept: Evaluation of Simple Integrals of the Following Types and Problems
Chapter: [0.033] Integrals
 6.ii

Sketch the graphs of the curves y2 = x and y2 = 4 – 3x and find the area enclosed between them.

Concept: Area of the Region Bounded by a Curve and a Line
Chapter: [0.07] Application of Integrals (Section B)
 7
 7.i

A psychologist selected a random sample of 22 students. He grouped them in 11 pairs so that the students in each pair have nearly equal scores in an intelligence test. In each pair, one student was taught by method A and the other by method B and examined after the course. The marks obtained by them after the course are as follows:

 Pairs 1 2 3 4 5 6 7 8 9 10 11 Methods A 24 29 19 14 30 19 27 30 20 28 11 Methods B 37 35 16 26 23 27 19 20 16 11 21

Calculate Spearman’s Rank correlation.

Concept: Regression Coefficient of X on Y and Y on X
Chapter: [0.09] Linear Regression (Section C)
 7.ii

The coefficient of correlation between the values denoted by X and Y is 0.5. The mean of X is 3 and that of Y is 5. Their standard deviations are 5 and 4 respectively.
Find:
(i) the two lines of regression.
(ii) the expected value of Y, when X is given 14.
(iii) the expected value of X, when Y is given 9.

Concept: Regression Coefficient of X on Y and Y on X
Chapter: [0.09] Linear Regression (Section C)
 8
 8.i

In a college, 70% of students pass in Physics, 75% pass in Mathematics and 10% of students fail in both. One student is chosen at random. What is the probability that:
(i) He passes in Physics and Mathematics?
(ii) He passes in Mathematics given that he passes in Physics.
(iii) He passes in Physics given that he passes in Mathematics.

Concept: Conditional Probability
Chapter: [0.04] Probability (Section A)
 8.ii

A bag contains 5 white and 4 black balls and another bag contains 7 white and 9 black balls. A ball is drawn from the first bag and two balls are drawn from the second bag. What is the probability of drawing one white and two black balls?

Concept: Multiplication Theorem on Probability
Chapter: [0.04] Probability (Section A)
 9
 9.i

Using De Moivre’s theorem, find the least positive integer n such that ((2i)/(1+i))^n  is a positive integer.

Concept: Introduction of Integrals
Chapter: [0.033] Integrals
 9.ii

Solve the following differential equation: (3xy + y2) dx + (x2 + xy) dy = 0

Concept: Logarithmic Differentiation
Chapter: [0.031] Continuity, Differentiability and Differentiation
Section – B
 10
 10.i

In a triangle ABC, using vectors, prove that c2 = a2 + b2 – 2ab cos c.

Concept: Operations - Sum and Difference of Vectors
Chapter: [0.05] Vectors (Section B)
 10.ii

Prove that: veca . (vecb + vecc) × (veca + 2vecb + vec3c) = [veca vecb vecc]

Concept: Operations - Sum and Difference of Vectors
Chapter: [0.05] Vectors (Section B)
 11
 11.i

Find the equation of a line passing through the points P (-1, 3, 2) and Q (-4, 2, -2). Also, if the point R (5, 5, λ) is collinear with the points P and Q, then find the value of λ.

Concept: Concept of Line - Equation of Line Passing Through Given Point and Parallel to Given Vector
Chapter: [0.06] Three - Dimensional Geometry (Section B)
 11.ii

Find the equation of the plane passing through the points (2, -3, 1) and (-1, 1, -7) and perpendicular to the plane x – 2y + 5z + 1 = 0.

Concept: Plane - Equation of Plane Passing Through the Intersection of Two Given Planes
Chapter: [0.06] Three - Dimensional Geometry (Section B)
 12
 12.i

In a bolt factory, three machines A, B, and C manufacture 25%, 35% and 40% of the total production respectively. Of their respective outputs, 5%, 4% and 2% are defective. A bolt is drawn at random from the total production and it is found to be defective. Find the probability that it was manufactured by machine C.

Concept: Introduction of Probability
Chapter: [0.04] Probability (Section A)
 12.ii

One dialing certain telephone numbers assume that on an average, one telephone number out of five is busy, Ten telephone numbers are randomly selected and dialed. Find the probability that at least three of them will be busy.

Concept: Introduction of Probability
Chapter: [0.04] Probability (Section A)
Section – C
 13
 13.i

A person borrows ₹ 68962 on the condition that he will repay the money with compound interest at 5% per annum in 4 equal annual installments, the first one being payable at the end of the first year. Find the value of each installment.

Concept: Application of Calculus in Commerce and Economics in the Cost Function
Chapter: [0.08] Application of Calculus (Section C)
 13.ii

A company manufactures two types of toys A and B. A toy of type A requires 5 minutes for cutting and 10 minutes for assembling. A toy of type B requires 8 minutes for cutting and 8 minutes for assembling. There are 3 hours available for cutting and 4 hours available for assembling the toys in a day. The profit is ₹ 50 each on a toy of type A and ₹ 60 each on a toy of type B. How many toys of each type should the company manufacture in a day to maximize the profit? Use linear programming to find the solution.

Concept: Introduction of Linear Programming
Chapter: [0.1] Linear Programming (Section C)
 14
 14.i

A firm has the cost function C = x^3/3 - 7x^2 + 111x + 50  and demand function x = 100 – p.
(i) Write the total revenue function in terms of x.
(ii) Formulate the total profit function P in terms of x.
(iii) Find the profit-maximizing level of output x.

Concept: Application of Calculus in Commerce and Economics in the Cost Function
Chapter: [0.08] Application of Calculus (Section C)
 14.ii

A bill of ₹ 5050 is drawn on 13th April 2013. It was discounted on 4th July 2013 at 5% per annum. If the banker’s gain on the transaction is ₹ 0.50, find the nominal date of the maturity of the bill.

Concept: Application of Calculus in Commerce and Economics in the Cost Function
Chapter: [0.08] Application of Calculus (Section C)
 15
 15.i

The price of six different commodities for years 2009 and year 2011 are as follows:

 Commodities A B C D E F Price in 2009 (₹) 35 80 25 30 80 x Price in 2011 (₹) 50 y 45 70 120 105

The Index number for the year 2011 taking 2009 as the base year for the above data was calculated to be 125. Find the values of x andy if the total price in 2009 is ₹ 360.

Concept: Basic Concepts of Differential Equation
Chapter: [0.034] Differential Equations
 15.ii

The number of road accidents in the city due to rash driving, over a period of 3 years, is given in the following table:

 Year Jan-March April-June July-Sept. Oct.-Dec. 2010 70 60 45 72 2011 79 56 46 84 2012 90 64 45 82

Calculate four quarterly moving averages and illustrate them and original figures on one graph using the same axes for both.

Concept: Simple Problems on Applications of Derivatives
Chapter: [0.032] Applications of Derivatives

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