**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.

(ii) If

(a) The regression equation

(b) What is the most likely value of ?

(c) What is the coefficient of correlation between ?

(viii) A problem is given to three students whose chances of solving it are , and respectively. Find the probability that the problem is solved.

Answer:

We can compare corresponding terms. We get

(ii)

Hence Proved.

… … … … (i)

Also we can write

… … … … (ii)

Adding (i) and (ii) we get

+

(a) The regression equation of is

(b) When

(c) Given

(viii) If are three events representing that students can solve the problem. Therefore

The problem would be solved if anyone of them solves the problem. Also remember, these are mutually exclusive events.

. Hence proved.

Integrating both sides we get

Question 2:

(a) Using properties of determinants, prove that:

Using this result, solve the following system of equation:

Answer:

.

Question 3:

(a) Solve the equation for :

(b) If and are elements of Boolean Algebra, simplify the expression . Draw the simplified circuit. [5]

Answer:

(b)

Question 4:

(a) Verify Lagrange’s mean value theorem for the function: and find the value of in the interval [5]

(b) Find the coordinates of center, foci and equation of directrix of hyperbola: [5]

Answer:

The function is continuous in and are continuous in

Therefore by Lagrange’s mean value theorem we have

(b) Here equation of the given hyperbola is

Therefore the coordinates of the center are

The equation of directrices are

Question 5:

(a) If , show that:

[5]

(b) Show that the surface are of a closed cuboid with square base and a given volume is minimum when it is a cube. [5]

Answer:

… … … … … (i)

Differentiating both sides, we get,

… … … … … (ii)

Again differentiating both sides

Using (i) and (ii) we get

Hence proved.

(b) Let be the height and be the side of the square base of the cuboid. Therefore

… … … … … (i)

Question 6:

[5]

(b) Draw a rough sketch of the curve and find the area of the region enclosed by the curve and the line . [5]

Answer:

For

(b) Equation of the curve given is and the equation of line is

Solving

When and when

Hence the two point of intersection are and

Therefore the required enclosed area is

Question 7:

(a) Calculate the Spearman’s rank correlation coefficient for the following data and interpret the results: [5]

X | 35 | 54 | 80 | 95 | 73 | 73 | 35 | 91 | 83 | 81 |

Y | 40 | 60 | 75 | 90 | 70 | 75 | 38 | 95 | 75 | 70 |

(b) Find the line of best fit for the following data, treating as a dependent variable (Regression equation on ):

X | 14 | 12 | 13 | 14 | 16 | 10 | 13 | 12 |

Y | 14 | 23 | 17 | 24 | 18 | 25 | 23 | 24 |

Hence, estimate the value of when . [5]

Answer:

(a)

Rank | Rank | ||||

35 | 40 | 9.5 | 9 | 0.5 | 0.25 |

54 | 60 | 8 | 8 | 0 | 0 |

80 | 75 | 5 | 4.5 | 0.5 | 0.25 |

95 | 90 | 1 | 2 | 1 | 1 |

73 | 70 | 6.5 | 6.5 | 0 | 0 |

73 | 75 | 6.5 | 4.5 | 2 | 4 |

35 | 38 | 9.5 | 10 | -0.5 | 0.25 |

91 | 95 | 2 | 1 | 1 | 1 |

83 | 75 | 3 | 3 | 0 | 0 |

81 | 70 | 4 | 6.5 | -2.5 | 6.25 |

This indicates a strong positive relationship between . That is, the higher is , the higher is .

(b)

Now,

The regression equation of is

Value of when

Question 8:

(a) In a class of students, opted for Mathematics, opted for Biology and opted for both mathematics and Biology. If one of these students is selected at random, find the probability that:

(i) The student opted for Mathematics or Biology

(ii) The student has opted neither for Mathematics nor Biology

(iii) The student has opted Mathematics but not Biology [5]

(b) Bag contains white, blue and red balls. Bag contains white, blue and red balls. Bag contains white, blue and red balls. One bag is selected at random and then two balls are drawn from the selected bag. Find the probability that the balls drawn are white and red. [5]

Answer:

(a) Let Student who opted for Mathematics and Students who opted for Biology

(i) The student opted for Mathematics or Biology

(ii) The student has opted neither for Mathematics nor Biology

(iii) The student has opted Mathematics but not Biology

ball drawn is white and red

Now

By law of total probability we get:

Question 9:

Answer:

, which is a circle of radius unit .

is a complex number such that lie on a circle centered at origin and radius 1 unit.

(b)

Integrating on both sides,

Question 10:

(a) If are three mutually perpendicular vectors of equal magnitude, prove that is equally inclined with vectors [5]

(b) Find the value of for which the four points with position vectors , are co-planar. [5]

Answer:

(a) Since are three mutually perpendicular vectors of equal magnitude, therefore

and

Let

Let the angles between be respectively.

(b) Let be the given points. Therefore,

The points would be co-planar, if are co-planar

i.e.

Question 11:

(b) Find the equation of the plane passing through the point and perpendicular to the line joining the points . [5]

Answers:

The direction ratios of the two lines are

Since

Therefore the lines are not parallel. Hence they will intersect somewhere.

Now, are general coordinates of points on the two lines respectively.

The lines will intersect if the points coincide.

Hence

Solve the first two equations we get

Verifying that satisfy the third equation: . Hence it satisfies the third equation.

Therefore the point of intersection is

If you were to take the other point you get the same coordinates.

(b) Coordinate of the point given.

Therefor the equation of line AB will be

Therefore the direction ratios of is .

Since the line is to the required plane, the direction ratios of the plane are proportional to . Hence the equation of the line passing through the point and normal is:

is the required equation of the plane.

Question 12:

(a) A fair die is rolled. If face 1 turns up, a ball is drawn from Bag A . If face 2 or 3 turns up, a ball is drawn from Bag B . If face 4 or 5 or 6 turns up, a ball is drawn from Bag C . Bag A contains 3 red and 2 white balls, Bag B contains 3 red and 4 white balls and Bag C contains 4 red and 5 white balls. The die is rolled, a Bag is picked and a ball is drawn. If the drawn ball is red, what is the probability that is it drawn from Bag B . [5]

(b) An urn contains 25 balls of which 10 balls are red and the remaining are green. A ball is drawn at random from the urn, the color is noted and the ball is replaced. If 6 balls are drawn in this way, find the probability that:

(i) All the balls are red

(ii) Not more than balls are green

(iii) Number of red balls and green balls are equal [5]

Answers:

(a) Let Event

Event

Event

and Event red ball is drawn

Now

Applying Baye’s theorem

(ii) Probability of drawing not more than 2 green balls i.e. Probability of drawing at least 4 red balls

(iii) The probability of drawing equal number of red and green balls

Question 13:

(a) A machine costs Rs. 60000 and its effective life is estimated to be 25 years. A sinking fund is to be created for replacing the machine at the end of its life when its scrap value is estimated as Rs. 5000 . The price of the new machine is estimated to be 100% more than the price of the present one. Find the amount that should be set aside at the end of each year, out of the profits, for the sinking fund it is accumulates at an interest of 6% per annum compounded annually. [5]

(b) A farmer has a supply of chemical fertilizer of Type A which contains 10% nitrogen and 6% phosphoric acid and of Type B which contains 5% nitrogen and 10% phosphoric acid. After soil test, it is found that at least 7 kg of nitrogen and same quantity of phosphoric acid is required for a good crop. The fertilizer of Type A costs Rs. 5.00 per kg and the Type B costs Rs. 8.00 per kg. Using linear programming, find how many kilograms of each fertilizer should be brought to meet the requirement and and for the cost to be minimum. Find the feasible region in the graph. [5]

Answers:

(a) Present cost of machine

Cost of Machine after years

The scrap value of the machine

Therefore we have to accumulate to replace the machine after years.

Rate of interest on the amount set aside

Let the amount set aside after the end of every year

Therefore

(b) Let quantity of fertilizer and Quantity of fertilizer

We know, which contains nitrogen and which contains nitrogen.

Similarly,

Given, The fertilizer of costs per kg Rs. $and the costs per kg.

We also know

Therefore cost of kg of kg of fertilizer

We need to minimize . We will solve it graphically.

Now calculate Cost for each of the three coordinates:

For

For

Hence cost will be minimum when and .

Question 14:

(a) The demand for a certain product is represented by the equation in Rupees where is the number of units and is the price per unit. Find:

(i) Marginal revenue function

(ii) The marginal revenue when units are sold. **[5]**

(b) The bill of payable months after date was discounted for on 30th June 2007. If the rate of interest was per annum, on what date was the bill drawn? **[5]**

Answers:

(i) Total Revenue function

(ii) Marginal Revenue when ,

(b) Face Value

Discounted Value

Rate

Discount

months

Therefore the date is months before 30th June which is 6th Feb 2007.

Question 15:

(a) The price relative and weights of a set of commodities are given below:

Commodity | A | B | C | D |

Price Relatives | 125 | 120 | 127 | 119 |

Weights |

If the sum of the weights is and weighted average of price relative index number is , find the numerical value of and . **[5]**

(b) Construct three yearly moving averages from the following data and show on a graph against the original data: **[5]**

Year | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 |

Annual Sales in Lakhs | 18 | 22 | 20 | 26 | 30 | 22 | 24 | 28 | 32 | 35 |

Answers:

(a) Given:

… … … … … (i)

… … … … … (ii)

Subtracting (ii) from (i) we get

Substituting in (i) we get

(b)

Year | Annual Sales | 3 year moving total | 3 year moving average |

2000 | 18 | – | – |

2001 | 22 | 60 | 20.00 |

2002 | 20 | 68 | 22.67 |

2003 | 26 | 76 | 25.33 |

2004 | 30 | 78 | 26.00 |

2005 | 22 | 76 | 25.33 |

2006 | 24 | 74 | 24.67 |

2007 | 28 | 84 | 28.00 |

2008 | 32 | 95 | 31.67 |

2009 | 35 | – | – |