**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) Determine whether the binary operations on defined by is commutative. Also, find the value of .

ii) Prove that

iii) Without expanding at any stage, find the value of the determinant:

iv) If , find

v) Find if

vi) The edge of a variable cube is increasing at a rate of cm/sec. How fast is the volume of the cube increasing when the edge is cm long?

vii) Evaluate

viii) Form a differential equation of the family of the curves

ix) A bag contains white, red and black balls. If four balls are drawn one by one with replacements, what is the probability that none is white?

x) Let A and B be two events such that and . Find and and are independent events.

Answer:

i) For commutative

for all

Given

. Hence proved.

.

ii) Given

Let

And let

LHS

RHS. Hence proved.

iii) Given

*Note: Determinant of a Matrix with two Identical rows or columns is equal to 0. It is one of the property of determinants. If, we have any matrix with two identical rows or columns then its determinant is equal to zero.*

iv) Given

v)

Differentiating on both sides

vi) Let the edge be

Given cm/sec

We need to find at cm

We know

vii)

viii) Consider the given equation.

… … … … … (i)

On differentiating both sides w.r.t , we get

… … … … … (ii)

From equations (i) and (ii), we get

ix) Balls are drawn one by one with replacements and each try is independent

Probability of not drawing a White ball

Therefore the probability of not drawing white ball in four consecutive draws

x)

Question 2: If the function

be defined as and

be defined as .

Show that **[ 4 ]**

Answer:

If and

Similarly,

Hence proved.

Question 3:

a) If , then prove that

**OR**

b) Evaluate at ** [ 4 ]**

Answer:

a)

Squaring both sides

**OR**

b)

Let

Question 4: Using the properties of determinants, show that

** [ 4 ]**

Answer:

LHS

. Hence proved.

Question 5: Verify Rolle’s theorem for the function in the interval ** [ 4 ]**

Answer:

Step I: The function is continuous in

Step II: The function is derivable in

Step III:

Therefore there must be at least one point in at which

Hence verified.

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

Answer:

Differentiating w.r.t

Squaring both sides

Differentiating again

Question 7:

(a) The equation of a tangent at on the curve is . Find the value of and .

**OR**

(b) Using L’Hospital’s rule, evaluate ** [ 4 ]**

Answer:

(a)

Differentiating w.r.t

Now as , slope of tangent

Since is lying on the curve,

**OR**

(b)

Question 8:

(a) Evaluate:

** OR**

(b) Evaluate: ** [ 4 ]**

Answer:

(a)

Using the method of completing the square first

Therefore

Since

**OR**

(b)

Let

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

Answer:

Integration Factor

Now

Question 10: Three persons A, B and C shoot to hit a target. Their probability of hitting the target are , and respectively. Find the probability that

i) Exactly two persons hit the target

ii) At least one person hits the target ** [ 4 ]**

Answer:

Given:

i) exactly two persons hit the target

ii) none of the three persons hit the target

Therefore Probability of at least one hitting the target

Question 11: Solve the following system of linear equations using matrices: and **[6]**

Answer:

Given equations

and

where

exists means there is a unique solution

Question 12:

(a) Show that the radius of a closed right circular cylinder of a given surface area and maximum volume to is equal to half of its height.

**OR**

(b) Prove that the area of a right angled triangle of given hypotenuse is maximum when the triangle is isosceles. ** [6]**

Answer:

(a)

Differentiating w.r.t.

Now

Differentiating again w.r.t.

which is negative. Hence volume maximum at

Substituting

Now and

. Hence proved.

**OR**

(b)

We know

Differentiating w.r.t.

Differentiating again w.r.t. we get less than 0 at . Hence it’s maximum.

Therefore the triangle is an isosceles triangle.

Question 13:

(a) Evaluate:

**OR**

(b) Evaluate: **[6]**

Answer:

(a)

Let

Let

Where

**OR**

(b)

Now, if

Now, if

Now, if

Question 14: The probability that a bulb produced in a factory will fuse after days is . Find the probability that out of five such bulbs:

i) None will fuse after days of use

ii) Not more than one will fuse after days of use

iii) More than one will fuse after days of use

iv) At least one will fuse after days of use

Answer:

Let be the number of bulbs that will fuse after days of use in an experiment of trials.

i)

ii)

iii)

iv)

**SECTION B (20 Marks)**

Question 15: **[ 3 × 2]**

(a) Write a vector of magnitude 18 units in the direction of the vector .

(b) Find the angle between the two lines

and

(c) Find the equation of the plane passing through the point and perpendicular to the line joining the points and .

Answer:

(a)

(b)

and

Angle between and is given by

or

(c) Given point ). d.rs of line joining the points and is i.e. or

Equation of plant passing through is

Question 16: **[ 4]**

(a) Prove that

**OR**

(b) Using vectors, find the area of the triangle whose vertices are: and .

Answer:

(a) We have

RHS. Hence proved.

**OR**

(b) Given and .

Area of

Area of sq. units.

Question 17: ** [ 4 ]**

(a) Find the image of the point on the plant

**OR**

(b) Determine the equation of the line passing through the point and perpendicular to the lines and

Answer:

(a) The directional ratios of normal to plane are

Equation of

Coordinate of a general point on line is

This point lies on the plane

Therefore

coordinate of are

**OR**

(b) Equation of line passing through the point is

where are direction ratios of the line.

Line is to lines and

The directional ratios of like can be found by solving equation

Therefore the equation of line L is

Question 18: Draw a rough sketch of the curves and and find the area enclosed between them.** **** [ 6 ]**

Answer:

… … … … … i) (equation of a parabola)

… … … … … ii) (equation of a parabola)

Points of intersection:

Therefore

Hence the two points of intersection are and

The vertex of this parabola is . It cuts the y axis at and

Let cut the x-axis at

Total area area of

sq. units

**SECTION C (20 Marks)**

Question 19: ** [3 × 2]**

(a) The selling price of a commodity is fixed at Rs. and its cost function is

i) Determine it’s profit margin

ii) Find the break even point

(b) The Revenue function is given by . Find

i) The demand function

ii) Marginal revenue function

(c) For the lines of regression and , find the value of and the mean of

Answer:

(a) and

i)

ii)At break even point

(b)

i)

ii)

(c) … … … … … i)

… … … … … ii)

Solving i) and ii) we get and

and

Question 20: ** [4]**

(a) The correlation coefficient between and is . If the variance of and the variance of is , mean of is and the mean of , find:

i) the equations of two regression lines

ii) the expected value of when

**OR**

(b) Find the regression coefficient and the correlation coefficient for the following data:

Answer:

(a) Given

i) Line of regression of on

Line of regression of on

ii) At

**OR**

(b)

2 | 4 | 8 | 64 | 16 |

6 | 36 | 8 | 64 | 48 |

4 | 14 | 5 | 25 | 20 |

7 | 49 | 6 | 36 | 42 |

5 | 25 | 2 | 4 | 10 |

Since has the same sign as regressions coefficient

Question 21: ** [4]**

(a) The marginal cost of the production of the commodity is , it is know that fixed cost are Rs. , find;

i) The total cost

ii) The cost of increasing output from to units

**OR**

(b) The total cost function of the firm is given by where the selling price per unit is given as Rs. . Find for what value of will the profit be maximum.

Answer:

(a)

i)

Integrating both sides

When

cost function

ii) Cost of increasing the output from to units

cost of increasing output Rs.

**OR**

(b)

Put

[ Therefore is minimum at ]

[ Therefore is maximum at ]

Question 22: A company uses three machines to manufacture two types of shirts, half sleeves shirts and full sleeves shirt. The number of hours required per week on machines and for one shirt of each type is given in the following table:

Half Sleeves Shirt | 1 | 2 | 8/5 |

Full Sleeves Shirt | 2 | 1 | 8/5 |

None of the machines can be in operations for more than hours per week. The profit on each half sleeves shirt is Rs. and the profit on each full sleeves shirt is Rs. .

How many of each type of shirts should be made per week to maximize the company’s profit. ** [6]**

Answer:

Let the number of half sleeves shirts produced

Let the number of full sleeves shirts produced

Maximize

Constraints:

Now draw the three lines:

0 | 40 | 0 | 20 | 0 | 25 | |||||

20 | 0 | 40 | 0 | 25 | 0 |

A( 0, 20) | 30 |

B(10, 15) | 32.5 |

C( 15, 10) | 30 |

D( 20, 0) | 20 |

Therefore the maximum value is at . Hence the company should make Half Sleeves Shirts and Full Sleeves Shirts to maximize profit.