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Question Paper Solutions - Mathematics 2014 - 2015-I.S.C.-Class 12 CISCE (Council for the Indian School Certificate Examinations)

Question 1 is Compulsory

Attempt  Five Question From Question 2 to Question 9

Attempt Any two Question From Question 10 to Question 15


Marks: 100
[10]1
[3]1.1

Find the value of k if M  = `[(1,2),(2,3)]` and `M^2 - km - I_2 = 0`

Chapter: [2.01] Matrices
Concept: Order of a Matrix
[3]1.2

Find the equation of an ellipse whose latus rectum is 8 and eccentricity is `1/3`

Chapter: [7] Application of Integrals (Section B)
Concept: Area Under Simple Curves
[3]1.3

Solve `cos^(-1)(sin cos^(-1)x) = pi/2`

Chapter: [3.01] Continuity, Differentiability and Differentiation
Concept: Derivatives of Inverse Trigonometric Functions
[3]1.4

Using L'Hospital's rule, evaluate : `lim_(x->0) (x - sinx)/(x^2 sinx)`

Chapter: [3.01] Continuity, Differentiability and Differentiation
Concept: L' Hospital'S Theorem
[3]1.5

Evaluate: `int (2y^2)/(y^2 + 4)dx`

Chapter: [3.03] Integrals
Concept: Methods of Integration - Integration by Substitution
[3]1.6

Evaluate: `int_0^3 f(x)dx` where f(x) = `{(cos 2x, 0<= x <= pi/2),(3, pi/2 <= x <= 3) :}`

Chapter: [3.03] Integrals
Concept: Methods of Integration - Integration by Substitution
[3]1.7

The two lines of regressions are 4x + 2y- 3 = 0 and 3x + 6y + 5 =0. Find the correlation co-efficient between x and y.

Chapter: [9] Linear Regression (Section C)
Concept: Regression Coefficient of X on Y and Y on X
[3]1.8

A card is drawn from a well-shuffled pack of playing cards. What is the probability that it is either a spade or an ace or both? 

Chapter: [6.01] Probability
Concept: Conditional Probability
[3]1.9

If 1, `omega` and `omega^2` are the cube roots of unity, prove `(a + b omega + c omega^2)/(c + s omega +  b omega^2) =  omega^2`

Chapter: [3.04] Differential Equations
Concept: Basic Concepts of Differential Equation
[3]1.10

Solve the differential equation `sin^(-1) (dy/dx) = x + y`

Chapter: [3.04] Differential Equations
Concept: Solutions of Linear Differential Equation
[10]2
[5]2.1

Using properties of determinants, prove that:

`|(1+a^2-b^2, 2ab, -2b),(2ab, 1-a^2+b^2, 2a),(2b, -2a, 1-a^2-b^2)| = (1 + a^2 + b^2)^3`

Chapter: [2.01] Matrices and Determinants
Concept: Properties of Determinants
[5]2.2

Given two matrices A and B 

`A = [(1,-2,3),(1,4,1),(1,-3, 2)]  and B  = [(11,-5,-14),(-1, -1,2),(-7,1,6)]`

find AB and use this result to solve the following system of equations:

x - 2y + 3z = 6, x + 4x + z = 12, x - 3y + 2z = 1

Chapter: [2.01] Matrices and Determinants
Concept: Types of Matrices
[10]3
[5]3.1

Solve the equation for x: `sin^(-1)  5/x + sin^(-1)  12/x = pi/2, x != 0`

Chapter: [3.04] Differential Equations
Concept: Basic Concepts of Differential Equation
[5]3.2

A, Band C represent switches in 'on' position and A', B' and C' represent them in off position. Construct a switching circuit representing the polynomial ABC + AB'C· + A'B'C. Using Boolean Algebra, prove that the .given 'polynomial can be simplified to C(A + B'). Construct an equivalent switching circuit

Chapter: [3.03] Integrals
Concept: Anti-derivatives of Polynomials and Functions
[10]4
[5]4.1

Verify Lagrange's Mean Value Theorem for the following function:

`f(x ) = 2 sin x +  sin 2x " on " [0, pi]`

Chapter: [3.01] Continuity, Differentiability and Differentiation
Concept: Mean Value Theorem
[5]4.2

Find the equation of the hyperbola whose foci are `(0,+- sqrt10)` and passing through the point (2,3)

Chapter: [3.04] Differential Equations
Concept: Solve Population Based Problems on Application of Differential Equations
[10]5
[5]5.1

if `y = e^(mcos^(-1)x)`, prove that `(1 - x^2) (d^2y)/(dx^2) - x dy/dx = m^2y`

Chapter: [3.04] Differential Equations
Concept: Solve Population Based Problems on Application of Differential Equations
[5]5.2

Show that the rectangle of the maximum perimeter which can be inscribed in the circle of radius 10 cm is a square of side `10sqrt2` cm.

Chapter: [7] Application of Integrals (Section B)
Concept: Area Between Two Curves
[10]6
[5]6.1

Evaluate: `int (sec x)/(1 + cosec x) dx`

Chapter: [3.03] Integrals
Concept: Methods of Integration - Integration by Substitution
[5]6.2
[10]7
[5]7.1

Given that the observations are: (9, -4), (10, -3), (11, -1), (12, 0), (13, 1), (14, 3), (15, 5), (16, 8). Find the two lines of regression and estimate the value of y when x = 13·5.

Chapter: [9] Linear Regression (Section C)
Concept: Lines of Regression of X on Y and Y on X Or Equation of Line of Regression
[5]7.2

In a contest, the competitors are awarded marks out of 20 by two judges. The scores of the 10 competitors are given below. Calculate Spearman's rank correlation.

Competitors A B C D E F G H I J
Judge A 2 11 11 18 6 5 8 16 13 15
Judge B 6 11 16 9 14 20 4 3 13 17
Chapter: [9] Linear Regression (Section C)
Concept: Regression Coefficient of X on Y and Y on X
[10]8
[5]8.1

An urn contains 2 white and 2 black balls. A ball is drawn at random. If it is white, it is not replaced into the urn. Otherwise, it is replaced with another ball of the same colour. The process is repeated. Find the probability that the third ball is drawn is black.

Chapter: [6.01] Probability
Concept: Conditional Probability
[5]8.2

Three persons A, B and C shoot to hit a target. If A hits the target four times in five trials, B hits it three times in four trials and C hits it two times in three trials, find the probability that:

1) Exactly two persons hit the target.

2) At least two persons hit the target.

3) None hit the target.

Chapter: [19] Probability Distribution
Concept: Random Variables and Its Probability Distributions
[10]9
[5]9.1

if z = x + iy, `w = (2 -iz)/(2z - i)` and |w| = 1. Find the locus of z and illustrate it in the Argand Plane.

Chapter: [4.02] Three - Dimensional Geometry
Concept: Plane - Intercept Form of the Equation of a Plane
[5]9.2

Solve the differential equation:

`e^(x/y)(1-x/y) + (1 + e^(x/y)) dx/dy = 0` when x = 0, y = 1

Chapter: [3.04] Differential Equations
Concept: General and Particular Solutions of a Differential Equation
[10]10
[5]10.1

Using Vectors, prove that angle in a semicircle is a right angle

Chapter: [8] Vectors
Concept: Basic Concepts of Vector Algebra
[5]10.2

Find the volume of a parallelopiped whose edges are represented by the vectors:

`vec a = 2 hat i - 3 hat j - 4 hat k`, `vec b  = hat i + 2 hat j - hat k` and `vec c = 3 hat i +  hat j +  2 hatk`

Chapter: [4.01] Vectors
Concept: Scalar Triple Product of Vectors
[10]11
[5]11.1

Find the equation of the plane passing through the intersection of the planes: x + y + z + 1 = 0  and 2x -3y + 5z -2 = 0 and the point ( -1, 2, 1 ).

 

Chapter: [6] Three - Dimensional Geometry (Section B)
Concept: Intersection of the Line and Plane
[5]11.2

Find the shortest distance between the lines `vec r = hat i + 2hat j + 3 hat k +  lambda(2 hat i +  3hatj +  4hatk)` and `vec r =  2hat i +  4 hat j + 5 hat k +  mu (4hat i + 6 hat j +  8 hat k)`

Chapter: [6] Three - Dimensional Geometry (Section B)
Concept: Shortest Distance Between Two Lines
[10]12
[5]12.1

Box I contains two white and three black balls. Box II contains four white and one black balls and box III contains three white ·and four black balls. A dice having three red, two yellow and one green face, is thrown to select the box. If red face turns up, we pick up the box I, if a yellow face turns up we pick up box II, otherwise, we pick up box III. Then, we draw a ball from the selected box. If the ball is drawn is white, what is the probability that the dice had turned up with a red face?

Chapter: [6.01] Probability
Concept: Conditional Probability
[5]12.2

Five dice are thrown simultaneously. If the occurrence of an odd number in a single dice is considered a success, find the probability of maximum three successes.

Chapter: [6.01] Probability
Concept: Conditional Probability
[10]13
[5]13.1

Mr. Nirav borrowed Rs 50,000 from the bank for 5 years. The rate of interest is 9% per annum compounded monthly. Find the payment he makes monthly if he pays back at the beginning of each month.

Chapter: [8] Application of Calculus (Section C)
Concept: Application of Calculus in Commerce and Economics in the Profit Function and Breakeven Point
[5]13.2

A dietician wishes to mix two kinds ·of food X· and Y in such a way that the  mixture contains at least 10 units of vitamin A, 12 units of vitamin B arid 8 units of vitamin C. The vitamin contents of one kg food is given below:

Food Vitamin A Vitamin.B Vitamin C
X 1 unit 2 unit 3 unit
Y 2 unit 2 unit 1 unit

Orie kg of food X costs Rs 24 and one kg of food Y costs Rs 36. Using Linear Programming, find the least cost of the total mixture. which will contain the required vitamins.

Chapter: [10] Linear Programming (Section C)
Concept: Graphical Method of Solving Linear Programming Problems
[10]14
[5]14.1

A bill for Rs 7,650 was drawn on 8th March 2013, at 7 months. It was discounted on 18th May 2013 and the holder of the bill received Rs 7,497. What is the rate of interest charged by the bank?

Chapter: [8] Application of Calculus (Section C)
Concept: Application of Calculus in Commerce and Economics in the Profit Function and Breakeven Point
[5]14.2

The average cost function, AC for a commodity is given by AC = `x + 5 + 36/x` in terms of output x. Find

1) The total cost, C and marginal cost, MC as a function of x.

2) The outputs for which AC increases

Chapter: [8] Application of Calculus (Section C)
Concept: Application of Calculus in Commerce and Economics in the Average Cost
[5]15
[5]15.1

Calculate the index number. for the year 2014, with 2010 as the base year by the weighted aggregate method from the following data:

Commodity Price in Rs Weight
2010 2014
A 2 4 8
B 5 6 10
C 4 5 14
D 2 2 19

 

Chapter: [8] Application of Calculus (Section C)
Concept: Application of Calculus in Commerce and Economics in the Profit Function and Breakeven Point
[5]15.2

The quarterly profits of a small scale industry (in thousands of rupees)· is as  follows:

Year

Quarter

1

Quarter

2

Quarter

3

Quarter

4

2012 39 47 20 56
2013 68 59 66 72
2014 88 60 60 67

Calculate four quarterly moving averages. Display these and the original figures graphically on the same graph sheet.

Chapter: [8] Application of Calculus (Section C)
Concept: Application of Calculus in Commerce and Economics in the Profit Function and Breakeven Point
S