**MATHEMATICS**

*(Maximum Marks: 100)*

*(Time Allowed: Three Hours)*

*(Candidates are allowed additional 15 minutes for only reading the paper. *

*They must NOT start writing during this time)*

*The Question Paper consists of three sections A, B and C. *

*Candidates are required to attempt all questions from Section A and all question EITHER from Section B OR Section C*

**Section A: **Internal choice has been provided in three questions of four marks each and two questions of six marks each.

**Section B:** Internal choice has been provided in two question of four marks each.

**Section C:** Internal choice has been provided in two question of four marks each.

*All working, including rough work, should be done on the same sheet as, and adjacent to, the rest of the answer. *

*The intended marks for questions or parts of questions are given in brackets [ ].*

**Mathematical tables and graphs papers are provided.**

**Section – A (80 Marks)**

Question 1:

(i) Find the value of if and

(ii) Find the equation of an ellipse whose latus rectum is and eccentricity is

(iii) Solve:

(iv) Using L’Hospital’s rule, evaluate:

(v) Evaluate:

(vi) Evaluate: , where

(vii) The two lines of regressions are and . Find the correlation co-efficient between .

(viii) 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?

(ix) If are the cube roots of unity, prove that

(x) Solve the differential equation:

Answer:

(i)

Similarly,

(ii)

Equation of ellipse:

(iii)

(iv)

Apply L’Hospital’s rule

Apply L’Hospital’s rule

Apply L’Hospital’s rule

(v)

(vi) , where

Therefore

(vii) Let the line of regression of on be

Let the line of regression of on be

Therefore

Hence since both and are negative.

(viii) Probability P(E) = Probability of drawing a spade + Probability of drawing an ace – Probability of drawing ace of spade

(ix)

Multiplying numerator and denominator by

We know

(x)

Let

Question 2:

(a) Using properties of determinants, prove that:

** [5]**

(b) Given two matrices

and . Find .

Using this result, solve the following system of equation:

**[5]**

Answer:

(a) LHS =

RHS. Hence proved.

(b)

Now,

Hence

Question 3:

(a) Solve the equation for :

**[5]**

(b) represent switches in ‘on’ position and represent them in ‘off’ position. Construct a switching circuit representing the polynomial . Using Boolean Algebra, prove that the given polynomial can be simplified to . Construct an equivalent switching circuit. ** [5]**

Answer:

(a)

(b)

Since

Question 4:

(a) Verify Lagrange’s Mean Value Theorem for the following function:

** [5]**

(b) Find the equation of the hyperbola whose foci are and passing through the point . **[5]**

Answer:

(a) is continuous in

exists in

All the conditions of Lagrange’s Mean Value theorem satisfied there exists in such that

(not possible)

or which lies between and .

Hence Lagrange’s Mean Value theorem is verified.

(b) Foci

Let the equation be

not possible as cannot be negative or

Therefore and

Hence the equation is

Question 5:

(a) If , prove that ** [5]**

(b) Show that the rectangle of maximum perimeter which can be inscribed in a circle of radius is a square of side ** [5]**

Answer:

(a)

Differentiating with respect to

Differentiating again with respect to

(b) Please refer to the adjoining figure

and

Differentiating with respect to

Differentiating with respect to

Hence the perimeter is maximum when

Therefore

Therefore is a square with side of

Question 6:

(a) Evaluate: **[5]**

(b) Find the smaller are enclosed by the circle and the line . **[5]**

Answer:

(a)

Now put

Now

Thus

Therefore

(b) We have to calculate

First Integral:

Let

Now , Therefore

Now substituting back

Second Integral:

Hence

Question 7:

(a) Given that the observations are: . Find the two lines of regression and estimate the value of when . ** [5]**

(b) In a contest the competitors are awarded marks out of by two judges. The scores of the competitors are given below. Calculate Spearman’s rank correlation. **[5]**

Competitors | ||||||||||

Judge A | ||||||||||

Judge B |

Answer:

(a)

Line of regression of

Line of regression of

When

(b)

Judge A | Judge B | Rank | Rank | ||

Question 8:

(a) An urn contains white and 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 color. The process is repeated. Find the probability that the third ball drawn is black. ** [5]**

(b) Three person shoot to hit a target. If hits the target times out of trials, hits it times in trials and hits is times in trials, find the probability that:

(i) Exactly two person hit the target.

(ii) At least two person hit the target.

(iii) Non hit the target. ** [5]**

Answer:

(a) Please note: The number of balls in the urn change based on what color is drawn.

Every time you get a Black ball, not only the ball is put back in the urn, another black ball is put in the urn. So the number of balls increase every time you pull a black ball.

In case you draw a white ball, it is not put back in the urn. So the number of balls decrease when you pull white ball.

Therefore the probability that the third ball is black is:

(b) Given: and

Therefore and

(i) Probability of at least two hitting the target

(ii) At least two people hit the target

(iii) None hit the target

Question 9:

(a) If and , find the locus of and illustrate it in Argand Plane.** [5]**

(b) Solve the differential equation:

[5]

Answer:

(a)

Given

… … … … … (i)

Now we know that the equation of a circle with center and radius is of the form

Therefore (i) can be written as

Therefore center is and radius is .

(b)

Substitute

When and we get

Hence

**Section – B (20 Marks)**

Question 10:

**(a)** Using vectors, prove that an angle in a semicircle is a right angle. ** [5]**

(b) Find the volume of a parallelopiped whose edges are represented by the vectors: **[5]**

Answer:

(a) In the adjoining diagram, we have a circle with center (origin) and the vertices of the triangle are and is the diameter.

Let

Therefore

(since is the radius)

Therefore is a right angle.

(b) The volume of a parallelopiped:

Question 11:

(a) Find the equation of the plane passing through the intersection of the planes and and the point . ** [5]**

(b) Find the shortest distance between the lines and ** [5]**

Answer:

(a) Given planes and and the point

The equation of a line passing through the intersection of the plans is given as

Hence the equation of line will be

(b)

Hence the shortest distance units

Question 12:

(a) contains white and black balls. contains white and black balls and contains white and black balls. A dice having red, yellow and green face, is thrown to select a box. If red face turns up we pick , if yellow face turns up we pick otherwise we pick . The we draw a ball from a selected box. If the ball drawn is white, what is the probability that the dice had turned up with a Red face. **[5]**

(b) 5 dices are thrown simultaneously. If the occurrence of an odd number in a single die is considered a success, find the probability of maximum three successes. ** [5]**

Answer:

(a) Given: Probability of a red face when the dice is rolled :

Probability of a yelloq face when the dice is rolled :

Probability of a green face when the dice is rolled :

Let D be the probability of drawing a White ball. Therefore

Probability of drawing a white ball from Bag I:

Probability of drawing a white ball from Bag II:

Probability of drawing a white ball from Bag III:

Therefore the probability of white ball drawn when red face show up:

(b) Number of dice

P(odd number)

P(even number)

**Section – C (20 Marks)**

Question 13:

(a) Mr. Nirav borrowed from the bank for years. The rate of interest is per annum compounded monthly. Find the payment he makes if he pays back at the beginning of each month. ** [5]**

(b) A dietitian wishes to mix two kinds of food and in such a way that the mixture contains at least units of vitamin , units of vitamin and units of vitamin . The vitamin contents of one kg of food is given below:

Food | Vitamin A | Vitamin B | Vitamin C |

X | 1 unit | 2 units | 3 units |

Y | 2 units | 2 units | 1 unit |

One kg of food costs and one kg of food costs . Using linear programming, find the least cost of the total mixture which will contain the required vitamins. ** [5]**

Answer:

(a) Principal

Annual rate of interest Monthly rate of interest

Now

Hence monthly installment

(b) Let us say we have kgs of food and kgs of food .

Therefore we have the following inequations:

Now lets plot these lines. We got the following graph

We have to minimize

Hence the four point of intersection from the graph are:

Hence, the and should be bought.

Question 14:

(a) A bill of was drawn on 8th March 2013, at months. It was discounted on 18th May, 2013 and the holder of the bill received . What is the rate of interest charged by the bank? ** [5]**

(b) The average cost function, for a commodity is given by in terms of output . Find:

(i) The total cost, and marginal cost, as a function of

(ii) The outputs for which increases. ** [5] **

Answer:

(a) Face value of the bill

Discounted value of the bill

Banker’s discount

Nominal date due is 8th October

Legal due date of the bill is 11th October (3 days of grace period)

Number of unexpired days from 8th May to 11th October is 146 days

Banker’s discount

(b)

Cost function

Marginal cost

For to be increasing,

Case 1:

Case 2:

But cannot be negative. Therefore

Hence the average cost function increase if the output

Question 15:

(a) Calculate the index number for the year 2014, with 2010 as the base year by the weighted aggregate method from the following data: ** [5]**

Commodity | Price in Rs. | Weight | |

2010 | 2014 | ||

A | |||

B | |||

C | |||

D |

(b) The quarterly profits of a small scale industry (in thousands of Rupees) is as follows: Calculate the quarterly moving averages. Display these and the original figures graphically on the same graph sheet. ** [5]**

Year | Quarter 1 | Quarter 2 | Quarter 3 | Quarter 4 |

2012 | ||||

2013 | ||||

2014 |

Answer:

(a)

Commodity | ||||||

Index number for the year 2014 with respect to 2010 as the base year

(b)

Year | Quarter | Quarterly Profits | 4 years moving total | 4 year average |