**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 × 2 ]

is the set of real numbers, then find

iii) Using determinants, find the values of , if the area of triangle with vertices is square units.

ix) Two balls are drawn from an urn containing 3 white, 5 red and 2 black balls, one by one without replacement. What is the probability that at least one ball is red?

Answer:

i)

Also

ii)

v) To be continuous at

x) Since A and B are independent,

Question 2:

Answer:

Therefore Co-domain = Range

Replacing by by

Question 3:

**OR**

Answer:

Therefore

**OR**

Squaring both sides

Hence Proved

Question 4: Using properties of determinants prove that:

[ 4 ]

Answer:

RHS

Hence Proved

Question 5:

a) Show that the function is continuous, but not differentiable at

OR

b) Verify the Lagrange’s mean value theorem for the function:

Answer:

a)

is continuous at

R.H.D, does not exist.

is continuous at but is non differentiable at

**OR**

Therefore Lagrange’s mean value theorem is applicable.

We know,

Hence Lagrange’s mean value theorem is verified.

Answer:

Question 7: A 13 m long ladder is leaning against a wall, touching the wall at a certain height from the ground level. The bottom of the ladder is pulled away from the wall, along the ground, at the rate of 2 m/s. How fast is the height on the wall decreasing when the foot of the ladder is 5 m away from the wall? [4]

Answer:

… … … … … i)

The bottom of the ladder is begin pulled, so these distances are changing with time,

… … … … … ii)

… … … … … iii)

m, then m from i) (Pythagoras theorem)

sign indicates the height is decreasing

Question 8:

OR

Answer:

Let

**OR**

Answer:

Question 10: Bag A contains 4 white balls and 3 black balls, while Bag B contains 3 white balls and 5 black balls. Two balls are drawn from Bag A and placed in Bag B. Then, what is the probability of drawing a white ball from Bag B? [ 4 ]

Answer:

Bag contains white balls and black balls

Bag contains white balls and black balls

Case I : Let Both white balls are transferred from Bag to Bag and then a white ball is drawn from Bag

Case II: Let both black balls are transferred from Bag to Bag and then a white ball is drawn from Bag

Case III: 1 white and 1 black ball is transferred and then a white ball is drawn from Bag

Question 11: Solve the following system of linear equations using matrix method: [ 6 ]

Answer:

… … … … … i)

… … … … … ii)

… … … … … iii)

Question 12:

(a) The volume of a closed rectangular metal box with a square base is The cost of polishing the outer surface of the box is Rs. Find the dimensions of the box for the minimum cost of polishing it.

OR

(b) Find the point on the straight line , which is closest to the origin. [ 6 ]

Answer:

a) Let the base of the box be and height be

Differentiating equation (iii) w.r.t. we get,

at

Therefore The cost for polishing the surface area is minimum when length of base is 16 cm and height of box is 16 cm.

**OR**

b) The equation of line is given as

Differentiating equation (i) w.r.t. we get,

Differentiating equation (ii) w.r.t. we get,

Answer:

… … … … … i)

… … … … … ii)

Adding i) and ii) we get

Question 14:

a) Given three identical Boxes A, B and C , Box A contains 2 gold and 1 silver coin, Box B contains 1 gold and 2 silver coins and Box C contains 3 silver coins. A person chooses a Box at random and takes out a coin. If the coin drawn is of silver, find the probability that it has been drawn from the Box which has the remaining two coins also of silver.

**OR**

b) Determine the binomial distribution where mean is 9 and standard deviation is Also, find the probability of obtaining at most 1 success. [ 6 ]

Answer:

a) Given Box A contains 2 gold and 1 silver coin.

Box B contains 1 gold and 2 silver coins.

Box C contains 3 silver coins.

Using Baye’s theorem,

OR

**SECTION B (20 Marks)**

Question 15: [ 3 × 2 ]

a) are perpendicular vectors, Find the value of

b) Find the length of the perpendicular from origin to the plane

c) Find the angle between the two lines .

Answer:

a) are perpendicular vectors,

b)

Perpendicular length from origin to plane

Therefore the angle between the two lines

Question 16:

a) , prove that are perpendicular.

OR

b) are non-collinear vectors, find the value of such that the vectors are collinear. [ 4 ]

Answer:

a)

OR

b) are collinear.

On comparing coefficient of

Also

Question 17:

(a) Find the equation of the plane passing through the intersection of the planes such that the intercepts made by the resulting plane on the x-axis and the z-axis are equal.

OR

b) Find the equation of the lines passing through the point and perpendicular to the lines

Answer:

a) Equation of intersecting plane is

… … … … … i)

For intercept made on x-axis, Put ,

For intercept made on z-axis Put

Given Intercept made on x-axis intercept made on z-axis

Substituting in i) we get

OR

b) Lines are passing through the point

Also given, that they are perpendicular to

Denominators of the required lines are

Question 18: Draw a rough sketch and find the area bounded by the curve and [ 6 ]

Answer:

Equation of parabola is … i)

Equation of straight line is

Substituting it in i) we get

When and when

Hence we have points and

Area under line

Area under curve

square units

Area bounded by the curve

**SECTION C (20 Marks)**

Question 19: [ 3 × 2 ]

(a) A company produces a commodity with Rs. 24,000 as fixed cost. The variable cost estimated to be 25% of the total revenue received on selling the product, is at the rate of Rs. 8 per unit. Find the break-even point.

Find the number of units produced for which M.C. = A.C. (M.C.= Marginal Cost and A.C. = Average Cost.)

c) and correlation coefficient , find regression equation of on .

Answer:

a) Revenues Variable cost Fixed cost

Let the total revenue

Therefore total revenue

Therefore units.

c) Regression line on on is

Question 20:

a) The following results were obtained with respect to two variables and : and

(i) Find the regression coefficient .

(ii) Find the regression equation of on .

OR

b) Find the equation of the regression line of y on , if the observations are as follows: Also, find the estimated value of when . [ 4 ]

Answer:

a) Given and

OR

b)

1 | 4 | 4 | 1 |

2 | 8 | 16 | 4 |

3 | 2 | 6 | 9 |

4 | 12 | 48 | 16 |

5 | 10 | 50 | 25 |

6 | 14 | 84 | 36 |

7 | 16 | 112 | 49 |

8 | 6 | 48 | 64 |

9 | 18 | 162 | 81 |

45 | 90 | 530 | 285 |

When

Question 21:

a) The cost function of a product is given by where is the number of units produced. How many units should be produced to minimize the marginal cost?

OR

b) The marginal cost function of units of a product is given by . The cost of producing one unit is Rs. 7. Find the total cost function and average cost function. [ 4 ]

Answer:

is minimum

**OR**

b) Let total cost function

Also, the average cost function is given by

Question 22: A carpenter has 90, 80 and 50 running feet respectively of teak wood, plywood and rosewood which is used to produce product A and product B. Each unit of product A requires 2, 1 and 1 running feet and each unit of product B requires 1, 2 and 1 running feet of teak wood, plywood and rosewood respectively. If product A is sold for Rs. 48 per unit and product B is sold for Rs. 40 per unit, how many units of product A and product B should be produced and sold by the carpenter, in order to obtain the maximum gross income?

Formulate the above as a Linear Programming Problem and solve it, indicating clearly the feasible region in the graph. [ 6 ]

Answer:

Product | A | B | |

Teakwood | |||

Plywood | |||

Rosewood |

Selling Price for Product Rs./unit

Selling Price for Product Rs./unit

From equation iii) and ii)

From equation i) and iii)

We know

is maximum at

Hence the optimal solution is and units (Product units, Product units)