**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: **[10 × 3]**

(i) If , find the values of such that

(ii) Find the eccentricity and the coordinates of foci of the hyperbola

(iii) Evaluate:

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

(v) Evaluate:

(vi) Using the properties of definite integrals, evaluate:

(vii) For the given lines of regression, and , find:

(a) regression coefficient and

(b) coefficient of correlation

(viii) Express the complex number in the form of . Hence, find the modulus and argument of the complex number.

(ix) A bag contains balls numbered from to . One ball is drawn at random from the bag. What is the probability that the ball drawn is marked with a number with is multiple of or ?

(x) Solve the differential equation:

Answer:

(i) Given,

Now,

and

(ii)

eccentricity

Foci Vertices

Latus rectum

(iii)

(iv)

v)

(vi) … … … … … i)

… … … … … ii)

Adding i) and ii) we get

(vii) (a) The two regression lines are

… … … … … i)

… … … … … ii)

From equation i)

(Regression of on )

From equation ii)

(Regression of on )

(b) . Therefore our assumption is correct.

Hence i) is the regression line of on and ii) is the regression line of on .

(viii)

(ix) Number of balls

Number of balls marked with multiple of

Number of balls marked with multiple of

Number of ball multiple of both and

The required marked with probability of

(x)

Question 2:

(a) Using properties of determinants, prove that:

** [5]**

(b) Solve the following system of linear equations using matrix method:

** [5]**

Answer:

(a)

Applying

Applying

Applying

Expanding along , we get

RHS. Hence proved.

(b) Given system of equation is,

exists

Therefore The system has the unique solution

Therefore

Now,

Question 3:

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

(b) represent switches in ‘on’ position and represent switches in off position. Construct a switching circuit representing the polynomial . Using Boolean algebra, simply the polynomial expression and construct the simplified circuit. **[5]**

Answer:

(a) Given

Squaring both sides we get

(b) The given Boolean expression is

The switching circuit is

Now

Question 4:

(a) Verify Rolle’s Theorem for the function on

(b) Find the equation of the parabola with latus rectum joining points and .

Answer:

(a) Given function is, on

We know, and exponential functions are always continuous

Therefore given function is continuous in

Differentiating w.r.t to , we get

Which exists for all in

and

Therefore, the given function satisfies all three conditions of Rolle’s theorem. For maxima or minima,

Since lies between so Rolle’s theorem is verified.

Question 5:

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

(b) A wire of length m is cut into two pieces. One piece of the with is bent in the shape of a square and the other int he shape of a circle. What should be the length of each piece so that the combines area of the two is minimum? ** [5]**

Answer:

(a) Given

Differentiating both sides w.r.t , we get

Hence proved.

(b) Given length of the wire

Let the length of the square

Therefore the length of circle

Let the side of the square

Therefore perimeter

Let the radius of the circle

Therefore circumference

Combined area

Differentiating w.r.t we get

For maxima and minima,

Therefore

Since

is minimum at

Therefore the length of the square wire

Therefore length of circle wire

Question 6:

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

(b) Sketch the graph of the curves and and find the area enclosed between them. ** [5]**

Answer:

(a)

(b) Given curves are

The vertex L of this parabola is It intercepts the y-axis at and .

The points of intersection of these two parabolas are given by the equation

Then

Therefore the points of intesection are and . Let intercepts the x-axis at .

Therefore total area of area of

sq. units.

Question 7:

(a) A psychologist selected at random sample of students. He grouped them in 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 and the other by method 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 |

Method A | 24 | 29 | 19 | 14 | 30 | 19 | 27 | 30 | 20 | 28 | 11 |

Method B | 37 | 35 | 16 | 26 | 23 | 27 | 19 | 20 | 16 | 11 | 21 |

Calculate Spearman’s Rank correlation. ** [5]**

(b) The coefficient of correlation between the values denoted by and is . The mean of is and that of is . Their standard deviations are and respectively. Find:

(i) the two lines of regression

(ii) the expected value of , when is given .

(iii) the expected value of , when is given as . ** [5]**

Answer:

(a)

Pairs | A | B | Rank A | Rank B | D | D² |

1 | 24 | 37 | 6 | 1 | 5 | 25 |

2 | 29 | 35 | 3 | 2 | 1 | 1 |

3 | 19 | 16 | 8.5 | 9.5 | 1 | 1 |

4 | 14 | 26 | 10 | 4 | 6 | 36 |

5 | 30 | 23 | 1.5 | 5 | -3.5 | 12.25 |

6 | 19 | 27 | 8.5 | 3 | 5.5 | 30.25 |

7 | 27 | 19 | 5 | 8 | -3 | 9 |

8 | 30 | 20 | 1.5 | 7 | -5.5 | 30.25 |

9 | 20 | 16 | 7 | 9.5 | -2.5 | 6.25 |

10 | 28 | 11 | 4 | 11 | -7 | 49 |

11 | 11 | 21 | 11 | 6 | 5 | 25 |

, shows negative correlation.

(b) Given and

i) Regression equation of on is:

Regression equation of on is:

ii) When

iii) When

Question 8:

(a) In a college, students pass in Physics, pass in Mathematics and 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. ** [5]**

(b) A bag contains white and black balls and another bag contains white and 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 white and black balls? ** [5]**

Answer:

(a) Let students pass in both Mathematics and Physics

Students who pass in Physics

Students who pass in Mathematics

Students who fail in Both

i) (Pass in Physics and Mathematics)

ii)

iii)

(b) Let : the event of getting 1 black ball from Bag I

: the event of getting 1 black and 1 white ball from Bag II

: the event of getting 1 white ball from Bag I

: the event of getting 2 black balls from Bag II

Then,

Therefore required probability

Question 9:

(a) Using De Moivre’s theorem, find the least positive integer n such that is a positive integer. ** [5]**

(b) Solve the following differential equation: ** [5]**

Answer:

(a) We have

Which is positive integer if

Therefore the least positive value of

(b)

Put

Integrating both sides

**Section – B (20 Marks)**

Question 10:

(a) In a triangle , using vectors, prove that: . ** [5]**

(b) Prove that:

** [5]**

Answer:

(a)

Squaring both sides

(b) LHS

RHS

Question 11:

(a) Find the equation of a line passing through the points and . Also find the point is collinear with the points and , then find the value of . ** [5]**

(b) Find the equation of the plan passing through the points and and perpendicular to the plane ** [5]**

Answer:

(a) Equation of the line passing through point and is

Since Point lies on it

(b) The required plan is perpendicular to the given plane

Therefore required plan is parallel to the line which is perpendicular to the given plan. Direction ratio of line

Hence the required plane is

Question 12:

(a) In a bolt factory, three machines manufacturer of the total production respectively. Of their respective outputs, 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 . ** [5]**

(b) On 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. ** [5]**

Answer:

(a) Let the event that the bolt is produced by machine

the event that the bolt is produced by machine

the event that the bolt is produced by machine

and are mutually exclusive and exhaustive events

We have

Let be the event that bolt chosen is found to be defective

Therefore

(defective bolt produced by machine )

(b) The probability of one phone number out of five is busy is

Then probability of phone not busy is

Given

Therefore required probability (at least three phones are busy)

(Probability maximum two phones busy)

**Section – C (20 Marks)**

Question 13:

(a) A person borrows on the condition that he will repay the money with compound interest at per annum in equal annual installments, the first one being payable t the end of the first year. Find the value of each installment. ** [5]**

(b) A company manufactures two types of toys and . A toy of type requires minutes for cutting and minutes for assembling. A $ toy of type requires minutes of cutting and minutes of assembling. There are three hours available for cutting and hours for assembling in a day. The profit is each on a toy of type and each on a toy of type . How many toys of each type should a company manufacture in a day to maximize the profit? Use linear programming to find the solution. ** [5]**

Answer:

(a) Let the installment

Money borrowed Time period Rate of Interest

Rs

(b)

Toy A | Toy B | Time in a Day | |

Cutting Time | 5 minutes | 8 minutes | 180 minutes |

Assembling Time | 10 minutes | 8 minutes | 240 minutes |

Profit | 50 | 60 | |

Assumed Quantity | x | y |

Profit function

Now plot the line

A | B | |

x | 0 | 36 |

y | 22.5 | 0 |

Now plot the line

C | D | |

x | 0 | 24 |

y | 30 | 0 |

Corner Point | Objective Function: z = 50x + 60y |

O (0,0) | z = 50 × 0 + 60 × 0 = 0 |

D (24, 0) | z = 50 × 24 + 60 × 0 = 1200 |

E (12, 15) | z = 50 × 12 + 60 × 15 = 1500 |

A (0, 22.5) | z = 50 × 0 + 60 × 22.5 = 1350 |

Hence the maximum profit is at

Question 14:

(a) A firm has the cost function and demand function .

(i) Write the total revenue function in terms of

(ii) Formulate the total profit function in terms of

(iii) Find the profit maximizing level of output. ** [5]**

(b) A bill of is drawn on 13th April 2013. It was discounted on 4th July 2013 at per annum. If the banker’s gain on the transaction is , find the nominal date of the maturity of the bill. ** [5]**

Answer:

(a) Cost function

Demand function

i) Revenue function

ii) Profit function, Revenue function – Cost function

iii) For profit function to be maximum

Differentiating w.r.t

Now,

Again differentiating equation iii) w.r.t we get

At (minimum value)

at (maximum value)

Hence, the profit maximizing level of output at

(b) Let the unexpired period of the bill at the time of discounting be years

where A is the face value of the bill.

Here

Therefore

days

Therefore , the legal due date of maturity is 73 days after 4th July which comes to 15th September .

Therefore the legal due date is 15th September and the nominal due date is 12th September.

Question 15:

(a) The price of six different commodities for year 2009 and 2011 are as follows:

Commodities | A | B | C | D | E | F |

Price in 2009 (Rs.) | 35 | 80 | 25 | 30 | 80 | |

Price in 2011 (Rs.) | 50 | 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 if the total price in 2009 is . ** [5]**

(b) 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-Mar | 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 on original figures on one graph using the same axes for both. ** [5]**

Answer:

(a)

Commodities | Price in 2009 (Rs.) | Price in 2011 (Rs) |

A | 35 | 50 |

B | 80 | y |

C | 25 | 45 |

D | 30 | 70 |

E | 80 | 120 |

F | x | 105 |

Given

Price Index

(b) Calculations for trends by 4 quarterly moving average:

Year | Quarter | Values | 4- Quarterly moving total | 4-Quarterly moving average | 4-Quarterly moving average centered |

2010 | 1 | 70 | |||

2 | 60 | ||||

247 | 247/4=61.75 | ||||

3 | 45 | 125.75/2=62.875 | |||

256 | 256/4=64 | ||||

4 | 72 | 127/2=63.5 | |||

252 | 252/4/=63 | ||||

2011 | 1 | 79 | 126.25/2=63.125 | ||

253 | 253/4=63.25 | ||||

2 | 56 | 129.50/2=64.750 | |||

265 | 265/4=66.25 | ||||

3 | 46 | 135.25/2=67.625 | |||

276 | 276/4=69 | ||||

4 | 84 | 140/2=70 | |||

284 | 284/4=71 | ||||

2012 | 1 | 90 | 141.75/2=70.875 | ||

283 | 283/4=70.75 | ||||

2 | 64 | 141/2=70.50 | |||

281 | 281/4=70.25 | ||||

3 | 45 | ||||

4 | 82 |