**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) If the matrix is symmetric, find the value of .

(ii) If touches the conic , find the value of .

(iii) Prove that

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

(v) Evaluate:

(vi) Evaluate:

(vii) By using the data and , find:

(a) The regression equation on

(b) What is the most likely value of when ?

(c) What is the coefficient of correlation between and ?

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

(ix) If prove that and

(x) Solve:

Answer:

(i) Since is symmetric, therefore

We can compare corresponding terms. We get

(ii)

Therefore

Equation of line

Since is hyperbola

(iii) LHS

Putting

LHS

RHS

Hence Proved.

(iv)

(v) Put

(vi) Let I = … … … … (i)

Also we can write

I =

… … … … (ii)

Adding (i) and (ii) we get

2I = +

(vii) Given and

(a) The regression equation of is

(b) When

(c) Given as positive, (coefficient of correlation) will be positive

(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.

Therefore the Probability of solving the problem

(ix)

Now

. Hence proved.

Also

. Hence proved.

(x)

Integrating both sides we get

(constant of integration)

Question 2:

(a) Using properties of determinants, prove that:

** [5]**

(b) Given that: and . Find .

Using this result, solve the following system of equation:

and **[5]**

Answer:

(a) LHS

Applying

Now applying

Expanding along

RHS. Hence proved.

(b)

Now write the given system of equation in the form of .

Therefore

Question 3:

(a) Solve the equation for :

**[5]**

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

Answer:

(a) Given

or

Hence

(b) To be solved

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:

(a) Given in

The function f(x) is continuous in as and are continuous in

Now,

which exists for all values in

Therefore is differentiable in

Therefore by Lagrange’s mean value theorem we have

which lies in the interval . Hence verified.

(b) To be solved

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:

(a) Given … … … … … (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

Surface Area

Volume

Therefore

Differentiating with respect to

… … … … … (i)

Putting

Differentiating (i) with respect to we get

When

Therefore the Surface Area is minimum when .

Question 6:

(a) Evaluate:

**[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:

(a)

Now Put

Therefore

Let

For and for

Therefore Therefore

(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 | 9 | 0 | 0 |

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

80 | 75 | 5 | 4 | 1 | 1 |

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

73 | 70 | 7 | 7 | 0 | 0 |

73 | 75 | 6 | 5 | 1 | 1 |

35 | 38 | 10 | 10 | 0 | 0 |

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

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

81 | 70 | 4 | 6 | 2 | 4 |

Spearman’s rank correlation coefficient =

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

Therefore

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

Given

(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

(b) Let Event is chosen

Event is chosen

Event is chosen

and Event ball drawn is white and red

The bags are chosen totally at random therefore

Now

Probability of drawing a white and red ball from

Probability of drawing a white and red ball from

Probability of drawing a white and red ball from

By law of total probability we get:

Question 9:

(a) Prove that locus of is circle and find its center and radius if is purely imaginary. **[5]**

(b) Solve: **[5]**

Answer:

(a) Let

Since is purely imaginary ,

, 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 and **[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

Therefore

Let the angles between be respectively.

Therefore

Similarly and

Therefore

(b) Let be the given points. Therefore,

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

i.e.

Question 11:

(a) Show that the lines and intersect. Find the coordinate of their point of intersection. ** [5]**

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

Answers:

(a) Let and

The direction ratios of the two lines are .

Since

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

Now, and 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 and

Verifying that and 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 and 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 turns up, a ball is drawn from . If face turns up, a ball is drawn from . If face turns up, a ball is drawn from . contains and balls, contains and white balls and contains and 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 . ** [5]**

(b) An urn contains balls of which 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 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 is chosen

Event is chosen

Event is chosen

and Event red ball is drawn

Therefore , ,

Now

Probability of drawing a red ball from

Probability of drawing a red ball from

Probability of drawing a red ball from

Applying Baye’s theorem

(b) Probability of drawing a red ball

Probability of drawing a green ball

Probability of drawing red balls , where

(i) Probability of drawing all red balls

(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 and its effective life is estimated to be 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 . The price of the new machine is estimated to be 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 per annum compounded annually. **[5]**

(b) A farmer has a supply of chemical fertilizer of which contains nitrogen and phosphoric acid and of which contains nitrogen and phosphoric acid. After soil test, it is found that at least of nitrogen and same quantity of phosphoric acid is required for a good crop. The fertilizer of costs per kg and the costs 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.

Therefore:

Similarly, which contains phosphoric acid and which contains phosphoric acid.

Therefore:

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

We also know

Therefore cost of kg of and kg of fertilizer

We need to minimize . We will solve it graphically.

We see that the coordinates of the vertices of the feasible region are . Now calculate Cost for each of the three coordinates:

For

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:

(a) Given

(i) Total Revenue function

Therefore Marginal Revenue function

(ii) Marginal Revenue when ,

(b) Face Value

Discounted Value

Rate

Discount

Therefore

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)

Also

… … … … … (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 | – | – |