**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)**

(ii) Find the value(s) of so that the line may touch the hyperbola

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

(vii) Two regression lines are represented by Find the line of regression of

(ix) Solve the differential equation:

(x) If two balls are drawn from a bag containing three red and four blue balls, find the probability that:

(a) They are the same color

(b) They are of different colors

Answer:

(i)

Therefore

However, does not satisfy all the equations. Hence

(ii)

Condition for tangent

Using L’Hospital’s Rule

Again applying L’Hospital’s Rule

Once again applying L’Hospital’s Rule,

Put

Therefore

(vii) Let us assume that be the regression line of on

Hence our assumption is true i.e. regression line on is

(viii)

Integrating both sides

(x) Total number of ways of drawing balls

Question 2:

(a) Using properties of determinants, prove that:

Hence, solve the system of linear equations:

Answer:

Applying , we get

Applying , we get

Expanding along , we get

RHS. Hence proved.

exists

Given system of equation is

This can be written as

Since , the given system of equations have a unique solution

Question 3:

(a) Solve for

(b) Construct a circuit diagram for the following Boolean function:

Using laws of Boolean Algebra, simplify the function and draw the simplified circuit. [5]

Answer:

Let

Therefore we get

Question 4:

(a) Verify Lagrange’s Mean Value Theorem for the function in the interval

(b) From the following information, find the equation of the Hyperbola and the equation of its Transverse Axis:

Answer:

(a) Given,

Since is continuous on and exists in

is continuous in

Differentiating the given function w.r.t

which exists for all

Thus, both the conditions of Lagrange’s mean value theorem is satisfied therefore at least one c exists in .

Squaring both sides we get

(b) Let be the point on the conic, then

is the required hyperboloa.

Transverse axis passes through and is perpendicular to Directrix,

where

Therefore

Question 5:

(b) Find the maximum volume of the cylinder which can be inscribed in a sphere of radius cm. (Leave the answer in terms of ) [5]

Answer:

(a) Given

Differentiating once again w.r.t

(b) Let the height of the cylinder and the radius of the cylinder

Given Radius of the sphere

If is the volume of the cylinder, then

Let be the center of the sphere and

For ,

For maxima and minima,

is maximum when , putting

Question 6:

(b) Find the area bounded by the curve and the line

Answer:

Put

Now putting the value of t back in the expression

(b) Given: and line of intersection

which represents a downward parabola with vertex at

Line of intersection

Putting in the above equation

Therefore the point of intersections are

Therefore the area enclosed between the curve

Question 7:

16 | 18 | 21 | 20 | 22 | 26 | 27 | 15 | |

22 | 25 | 24 | 26 | 25 | 30 | 33 | 14 |

(b) The following table shows the mean and standard deviation of the marks of Mathematics and Physics scored by students in a school:

Mathematics | Physics | |

Mean | 84 | 81 |

Standard Deviation | 7 | 4 |

The correlation co-efficient between the given marks is . Estimate the likely marks in Physics in the marks in Mathematics at . **[5]**

Answer:

(a)

16 | 22 | -4 | -3 | 16 | 9 | 12 |

18 | 25 | -2 | 0 | 4 | 0 | 0 |

21 | 24 | -1 | -1 | 1 | 1 | -1 |

20 | 26 | 0 | 1 | 0 | 1 | 0 |

22 | 25 | 2 | 0 | 4 | 0 | 0 |

26 | 30 | 6 | 5 | 36 | 25 | 30 |

27 | 33 | 7 | 8 | 49 | 64 | 56 |

15 | 14 | -5 | -11 | 25 | 121 | 55 |

5 | -1 | 135 | 221 | 152 |

(b) Mean marks in Mathematics

Mean marks in Physics

Therefore regression equation of on

Putting

Hence the likely marks in Physics are

Question 8:

(a) contains three red and four white balls. contains two red and three white balls. If one ball is drawn from and two balls are drawn from , find the probability that:

(i) One ball is red and two balls are white

(ii) All the three balls are of the same color **[5]**

(b) Three persons, Aman, Bipin and Mohan attempt a Mathematics problems independently. The odds in favor of Aman and Mohan solving the problem are and respectively and the odds against Bipin solving the problem are . Find:

(i) The probability that all the three will solve the problem

(ii) the probability that the problem will be solved. **[5]**

Answer:

(a) Possible selection are as follows:

i) 1 Red ball from Bag A, 2 white ball from Bag B

1 white ball from bag A, 1 white ball from bag B, 1 red ball from Bag B

Therefore P(one ball is red and two balls are white)

ii) Possible selections are as follows:

1 red ball from Bag A, 2 red balls from Bag B

1 white ball from Bag A and 2 white balls from bag B

Therefore P (all the three balls are of the same color)

: Event Aman solves the problem

: Event Bipin solves the problem

: Event Mohan solves the problem

Aman | Bipin | Mohan |

ii) Probability that the problem is not solved = probability that all three fail to solve the problem

Question 9:

(a) Find the locus of the complex number , satisfying the relation

Illustrate the locus on the Argand plane. ** [5]**

(b) Solve the following differential equation:

Answer:

(a) Let

Therefore

It is a linear differential equation in .

I.F.

b

Therefore, general solution is

Given that and

Substituting the value of we get

Question 10:

(a) If and are unit vectors and is the angle between them,

(b) If the value of for which the four points with position vectors ; ; are coplanar. [5]

Answer:

a)

b) Let be the given points whose position vectors are ; ; and

Since the points A, B, C and D are coplanar, vectors and coplanar.

Question 11:

(a) Find the equation of a line passing through the point and perpendicular to the lines:

(b) Find the equation of planes parallel to the plane and which are at a distance of five units from the point

Answer:

a) Any line passing through the point is

then

… … … … … i)

and

… … … … … ii)

Subtracting ii) from i) we get

… … … … … iii)

and hence from i) we get

… … … … … iv)

From iv) and iii) we get

Putting these values in the equation of the line

b) The given plane is … … … … … i)

Equation of any plane parallel to above equation is

… … … … … ii)

Now ii) is at a distance of units from the point of

Substituting these values in ii) the equations of the required planes are

Question 12:

(a) If the sum and the product of the mean and variance of a Binomial Distribution are respectively, find the probability distribution and the probability of at least one success. [5]

(b) For , the chances of being selected as the manager of firm are respectively. The probabilities for them to introduce a radical chance in the marketing strategy are respectively. If a chance takes place; find the probability that it is due to the appointment of B. [5]

Answer:

a) Given,

… … … … … i)

… … … … … ii)

Dividing the square of i) by ii) we get

Probability of getting at least 1 success

b) Let and and be the events as defined below

is selected as a manager

is selected as a manager

is selected as a manager

and radical change occurs in marketing strategy

Given, , ,

We want to find the probability that the radical change in marketing strategy occurred due to the appointment of B i.e

Question 13:

(a) If Mr. Nirav deposits at the beginning of each month in an account that pays an interest of per annum compounded monthly, how many months will be required for the deposit to at least ? [5]

(b) A mill owner buys two types of machines for his mill. occupies sqm of area and requires men to operate it.; while occupies sqm of area and requires men to operate it. The owner has sqm of area available and men to operate the machines. If produces units and produces units daily, how many machines of each type should be buy to maximize the daily output? Use linear programming to find the solution. [5]

Answer:

a)

months

b) Data given

Machine A | Machine B | Machine C | |

Area Needed (Sq. m) | 1000 | 1200 | 7600 |

Labor Force | 12 | 8 | 72 |

Daily Output (units) | 50 | 40 | – |

Let and be the number of Machines A and Machine B respectively.

Constraints

Total output

So minimize

Subject to And

6 | 0 | |

0 | 9 |

7.2 | 0 | |

0 | (19/3) |

The vertices of the feasible region OAPD are and

Corner Points | Object Function |

Thus we see that is maximum at . Therefore number of Machine A and number of Machine .

Question 14:

(a) A bill of was drawn on 1st April 2011 at 4 months and discounted for at a bank. If the rate of interest was per annum, on what date was the bill discounted? [5]

(b) A company produces a commodity with fixed cost. The variable cost is estimated to be of the total revenue recovered on selling the product at a rate of per unit. FInd the following:

(i) Cost function

(ii) Revenue function

(iii) Break even point [5]

Answer:

Let be the expired period in years

Legal due date of the bill

1st April 2011 + 4 months + 3 days of grace = 4th August 2011

The bill was cashed 73 days before 4th August (4 days in August, 31 days in July, 30 days in June, 8 days in May) which comes to 23rd May 2011 as the date

b) Supposed that number of units are produced and sold.

Given fixed cost Rs.

Revenue

i) As each unit’s variable cost of units is of revenue,

Therefore Variable cost of units

Therefore total cost of producing units

ii) Price of one unit Rs.

Therefore revenue on selling units

ii) At break even values

Hence the break even point is units produced.

Question 15:

(a) The price index for the following data for the year 2011 taking 2001 as the base year was 127. The simple average of price relative method was used. Find the value of . [5]

Items | A | B | C | D | E | F |

Price (Rs. Per unit) in year 2001 | 80 | 70 | 50 | 20 | 18 | 25 |

Price (Rs. Per unit) in year 2011 | 100 | 87.50 | 61 | 22 | 32.50 |

(b) The profit of a paper bag manufacturing company (in lakhs / millions of Rs.) during each month pf a year are:

Month | Jan. | Feb. | Mar. | Apr. | May | Jun. | Jul. | Aug. | Sept. | Oct. | Nov. | Dec. |

Profit | 1.2 | 0.8 | 1.4 | 1.6 | 2.0 | 2.4 | 3.6 | 4.8 | 3.4 | 1.8 | 0.8 | 1.2 |

Plot the given data on a graph sheet. Calculate the four monthly moving averages and plot these on the same graph sheet. [5]

Answer:

a) Using simple averages of price relatives, the price index of 2011 taking 2001 as the base year was . From the following data we find

80 | 70 | 50 | 20 | 18 | 25 | |

100 | 87.50 | 61 | 22 | 32.5 | ||

125 | 125 | 122 | 110 | 130 |

b)

Month | Profit | 4 monthly totals | 4 monthly average | 4 monthly centered moving average |

Jan | 1.2 | |||

Feb | 0.8 | |||

5 | 1.25 | |||

Mar | 1.4 | 1.35 | ||

5.8 | 1.45 | |||

Apr | 1.6 | 1.65 | ||

7.4 | 1.85 | |||

May | 2.0 | 2.125 | ||

9.6 | 2.4 | |||

Jun | 2.4 | 2.8 | ||

12.8 | 3.2 | |||

July | 3.6 | 3.375 | ||

14.2 | 3.55 | |||

Aug | 4.8 | 3.475 | ||

13.6 | 3.4 | |||

Sep | 3.4 | 3.05 | ||

10.8 | 2.7 | |||

Oct | 1.8 | 2.25 | ||

7.2 | 1.8 | |||

Nov | 0.8 | |||

Dec | 1.2 | |||