**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) The binary operation is defined as Find .

(ii) If and is symmetric matrix, show that

(iii) Solve :

(iv) Without expanding at any stage, find the value of:

(v) Find the value of constant so that the function defined as:

is continuous at .

(vi) Find the approximate change in the volume of a cube of side meters caused by decreasing the side by .

(vii) Evaluate :

(viii) Find the differential equation of the family of concentric circles

(ix) If and are events such that and , then find:

(a)

(b)

(x) In a race, the probabilities of A and B winning the race are and respectively. Find the probability of neither of them winning the race.

Question 2: If the function is invertible then find its inverse. Hence prove that . **[4]**

Question 3: If , prove that . ** [4]**

Question 4: Use properties of determinants to solve for :

and ** [4]**

Question 5:** [4]**

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

**OR**

(b) Verify Rolle’s theorem for the following function:

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

Question 7: Evaluate: **[4]**

Question 8: **[4]**

(a) Find the points on the curve at which the equation of the tangent is parallel to the x-axis.

**OR**

(b) Water is dripping out from a conical funnel of semi-vertical angle at the uniform rate of in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is 4 cm, find the rate of decrease of the slant height of the water.

Question 9:** [4]**

(a) Solve:

**OR**

(b) The population of a town grows at the rate of per year. Using differential equation, find how long will it take for the population to grow times.

Question 10: **[6]**

(a) Using matrices, solve the following system of equations :

**OR**

(b) Using elementary transformation, find the inverse of the matrix :

Question 11: speaks truth in of the cases, while is of the cases. In what percent of cases are they likely to contradict each other in stating the same fact ? **[4]**

Question 12: A cone is inscribed in a sphere of radius . If the volume of the cone is maximum, find its height. **[6]**

Question 13: **[6]**

(a) Evaluate:

**OR**

(b) Evaluate:

Question 14: From a lot of items containing defective items, a sample of items are drawn at random. Let the random variable denote the number of defective items in the sample. If the sample is drawn without replacement, find :

(a) The probability distribution of

(b) Mean of

(c) Variance of **[6]**

**SECTION B (20 Marks)**

Question 15: **[3 × 2]**

(a) Find if the scalar projection of and is units.

(b) The Cartesian equation of line is : . Find the vector equation of a line passing through and parallel to the given line.

(c) Find the equation of the plane through the intersection of the planes and and passing through the origin.

Question 16: **[4]**

(a) If are three non- collinear points with position vectors respectively, then show that the length of the perpendicular from is

**OR**

(b) Show that the four points A,B, C and D with position vectors and

Question 17: **[4]**

(a) Draw a rough sketch of the curve and find the area of the region bounded by curve and the line .

**OR**

(b) Sketch the graph of . Using integration, find the area of the region bounded by the curve and and .

Question 18: Find the image of a point having position vector : in the plane **[6]**

**SECTION C (20 Marks)**

Question 19: **[3 × 2]**

(a) Given the total cost function of units of a commodity as

Find: (i) Marginal cost function (ii) Average cost function

(b) Find the coefficient of correlation from the regression lines: and .

(c) The average cost function associated with producing and marketing units of an item is given by . Find the range of values of the output , from which is increasing.

Question 20:** [4]**

(a) Find the line of regression of on from the following table.

1 | 2 | 3 | 4 | 5 | |

7 | 6 | 5 | 4 | 3 |

Hence estimate the value of when .

**OR **

(b) From the given data:

Variable | ||

Mean | 6 | 8 |

Standard Deviation | 4 | 6 |

And the correlation coefficient: . Find

(i) Regression coefficients and

(ii) Regression line on

(iii) Most likely value of when

Question 21:** [4]**

(a) A product can be manufactured at a total cost , where is the number of units produced. The price at which each unit can be sold is given by . Determine the production level at which the profit is maximum.What is the price per unit and profit at the level of production.

**OR**

(b) A manufacturer’s marginal cost function is . Find the cost involved to increase production from units to units.

Question 22: A manufacturing company produces two type of teaching aids and of Mathematics for Class X. Each type of requires hours for fabricating and hour for finishing. Each type of requires hours for fabricating and hours for finishing. For fabricating and finishing, the maximum labor hours available for a week are and respectively. The company makes a profit of on each piece of type and on each piece of type . How many pieces of type and type should be manufactured per week to get a maximum profit? Formulate this as a linear programming problem and solve it. Identify the feasible region from the rough sketch. **[6]**