**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 ]**

i) If and , and is the set of real numbers, then find and .

ii) Solve:

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

iv) Show that is symmetric matrix, if

v) is not defined at . What value should be assigned to for continuity of at ?

vi) Prove that the function is increasing on .

vii) Evaluate:

viii) Using L’Hospital’s Rule, evaluate:

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

x) If events and are independent, such that and , find .

Answer:

i) Given and

Also

ii)

iii)

or

or

iv)

is symmetric matrix

v) To be continuous at

vi)

is increasing for

vii)

viii)

ix) at least one ball is Red no ball is Red

no Red ball in first draw no Red ball in second draw

x) Since A and B are independent,

Question 2: If and show that the function is one one onto. Hence, find . **[ 4 ]**

Answer:

Given

Therefore and

is one one function.

Let

Therefore Co-domain Range

is onto function

Replacing by and by .

Question 3:

a) Solve for

**OR**

b) If , show that **[ 4 ]**

Answer:

a) Simplify the expression

We know, that

Therefore

**OR**

b) Given

Squaring both sides

Hence Proved

Question 4: Using properties of determinants prove that:

** [ 4 ]**

Answer:

LHS

We know

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: in the interval **[ 4 ]**

Answer:

a) for

and for

L.H.L.

R.H.L.

is continuous at

R.H.D

L.H.D

Since L.H.D R.H.D, does not exist.

is continuous at but is non differentiable at .

**OR**

b) Given and

is continuous for

is differentiable for

Therefore Lagrange’s mean value theorem is applicable.

We know,

Hence Lagrange’s mean value theorem is verified.

Question 6: If and , prove that ** [ 4 ]**

Answer:

Given and

Therefore

Question 7: A 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 m/s. How fast is the height on the wall decreasing when the foot of the ladder is 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)

If m, then m from i) (Pythagoras theorem)

From equation iii) we get

sign indicates the height is decreasing

Question 8:

a) Evaluate:

**OR**

b) Evaluate: **[ 4 ]**

Answer:

a)

Let

**OR**

b) Given

Therefore

for all

for all

Question 9: Solve the differential equation: ** [ 4 ]**

Answer:

Given

Let

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

Required probability

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

Required probability

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

Required probability

Total probability

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

Answer:

Let

… … … … … i)

… … … … … ii)

… … … … … iii)

and

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 .

Therefore Volume

Hence … … … … … i)

Total surface area

Therefore the cost function Rs. … … ii)

Differentiating w.r.t. we get,

… … … … … iii)

Let

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

at

Also cm

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

**OR**

b) The equation of line is given as

Therefore The point of the line can be taken as

Distance from origin

Let is minimum when is minimum

… … … … … i)

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

… … … … … ii)

Let

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

is minimum at

The closest point on the line with origin is

Question 13: Evaluate: ** [ 6 ]**

Answer:

… … … … … i)

Using

… … … … … ii)

Adding i) and ii) we get

Question 14:

a) Given three identical Boxes and , Box contains gold and silver coin, Box contains gold and silver coins and Box contains 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 contains gold and silver coin.

Box contains gold and silver coins.

Box contains silver coins.

Probability of choosing a bag is

Using Baye’s theorem,

**OR**

b) Given

Since

**SECTION B (20 Marks)**

Question 15: ** [ 3 × 2 ]**

a) If and are perpendicular vectors, and . Find the value of .

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

c) Find the angle between the two lines and .

Answer:

a) Given and are perpendicular vectors, and

b) Given

Perpendicular length from origin to plane

units

c) Given

and

Therefore the angle between the two lines and is

Question 16:

a) If , prove that and are perpendicular.

**OR**

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

Answer:

a) Given,

**OR**

b) Given and are collinear.

On comparing coefficient of and

Also

Question 17:

(a) Find the equation of the plane passing through the intersection of the planes and 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

and ** [ 4 ]**

Answer:

a) Equation of intersecting plane is

… … … … … i)

For intercept made on x-axis, Put , and

For intercept made on z-axis Put and

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 and

Denominators of the required lines are or

Therefore equation of line is

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

or

When and when

Hence we have points and

Area under line

square units

Area under curve

square units

Area bounded by the curve and

square units

**SECTION C (20 Marks)**

Question 19: ** [ 3 × 2 ]**

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

b) The total cost function for a production is given by .

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 break even point (in units) units

b) Given … … … … … ii)

… … … … … ii)

… … … … … iii)

Given that

Therefore units.

c) Regression line on on is

We know,

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

i) Regression coefficient

ii) Regression equation of on

**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 x units of a product is given by . The cost of producing one unit is Rs. . Find the total cost function and average cost function.** [ 4 ]**

Answer:

a) Given

is minimum

For minimum,

**OR**

b) Let total cost function

Therefore Marginal Cost function

If

Also, the average cost function is given by

Question 22: A carpenter has and running feet respectively of teak wood, plywood and rosewood which is used to produce product and product . Each unit of product requires and running feet and each unit of product requires and running feet of teak wood, plywood and rosewood respectively. If product is sold for Rs. per unit and product is sold for Rs. per unit, how many units of product and product 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

Therefore the objective function is

… … … … … i)

… … … … … ii)

… … … … … iii)

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)