**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.

Answer:

(i) Given

(ii) Since is symmetric, therefore

We can compare corresponding terms. We get

(iii)

(iv)

Applying and we get

since

(v) Given is continuous at

Therefore

Therefore

Since

Therefore

(vi) Volume of a cube

Therefore

Hence change in volume

Hence change in volume decrease by

(vii)

(viii) Family of concentric circles is

Therefore Differential w.r.t.

Therefore

(ix)

(x) Let win the race be win the race be

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

Answer:

Let

Squaring both sides

Therefore

Now,

Therefore

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

Answer:

Question 4: Use properties of determinants to solve for :

and ** [4]**

Answer:

Given and

we get

we get

or

But , therefore

Question 5:** [4]**

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

**OR**

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

Answer:

(a) Continuity at

Therefore

Therefore is continuous at

Now differentiate at

Therefore

Hence is not differentiable at

(b)

(i) is continuous on because are continuous function on its domain.

(ii) and is differentiable on

(iii)

(iv) Let be number such that

Therefore

Therefore

Therefore

Therefore

Therefore

Therefore Rolle’s theorem verified

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

Answer:

Therefore

differentiating both sides w.r.t.

Again differentiating both sides w.r.t.

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

Answer:

Put

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.

Answer:

(a) … … … … … (i)

Given that lines is parallel to

Therefore

Put in equation (i)

When then

Therefore Point

When , then

Therefore Point

Therefore Points and

(b) Let be the radius, be the height and be the volume of the funnel at any time .

… … … … … (i)

Let be the slant height of the funnel

Given : Semi-vertical angle in the :

… … … … … (ii)

therefore the equation (i) can be rewritten as:

Differentiate w.r.t. :

Since it is given that rate of change (decrease) of volume of water w.r.t. is

Thereofore

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.

Answer:

(a)

Compare with

Therefore solution of the linear differential equation

(b) (Since increase in population speeds up with increase in population) and let be the population at anytime .

Therefore (where r is proportionality constant)

Therefore

integrating both sides

, (where c is the integration constant)

Therefore

where

Here is the rate of increase and is the initial population let then

Given to find the time taken to attain times population, so

Therefore

Taking log on both sides

Question 10: **[6]**

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

**OR**

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

Answer:

(a) Given, the three equations:

We can write this in the form of , i.e. as follows:

We know,

Hence it is a non – singular matrix. Therefore A^{-1} exists. Let us find the (adj A) by finding the minors and co-factors

We know , then

Matrix multiplication can be done by multiplying the rows of matrix with the column of matrix .

Hence , and

(b) Let

Therefore exists.

Interchanging and

and

Therefore

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]**

Answer:

speaks truth

speaks truth

They contradict each other

of cases they likely to contradict each other

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

Answer:

Let be a cone of greatest volume inscribed in a sphere of radius . It is obvious that for maximum volume the axis of the cone must be along a diameter of the sphere. Let be the axis of the cone and be the center of the sphere such that .

Then, height of cone. Applying Pythagoras theorem,

Let be the volume of the cone, then

… … … … … (i)

Now,

Thus, is maximum when . Putting in (i), we obtain

Height of the cone

Question 13: ** [6]**

(a) Evaluate:

**OR**

(b) Evaluate:

Answer:

(a) Let

Therefore

and

(b) … … … … … (i)

Using

Therefore

… … … … … (ii)

Adding equation (i) and (ii)

Put and

When

When

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]**

Answer:

In items defective and non-defective. Let is the probability of defective items

Let number of defective items. Therefore

Therefore

(i) Mean

(ii) Variance

**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.

Answer:

(a) Projection on on is

Given and

Therefore

Therefore

(b) Cartesian equation of a line is

i.e.

Dividing by throughout we get

Therefore D.r.s of the above line is . Now, equation of a line passing through point and parallel to the above line whose d.r.s. is is

(c) Equation of 1st plane is:

i.e.

… … … … … (i)

Equation of 2nd plane is:

i.e.

… … … … … (ii)

Now, equation of a plane passing through intersection of given planes is:

Since plane is 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

Answer:

(a) Let be a triangle and let be the position vectors of its vertices respectively. Let be the perpendicular from on . Then,

Area of

Also, Area of

Area of

Therefore

(b) Given:

are coplanar if

i.e.

Therefore

Therefore are coplanar.

Therefore Points and are coplanar

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 .

Answer:

(a) Given equation is

Comparing with

We get

Given

Also, meets

Therefore and are their point of intersection.

Required are

sq. units.

(b) , if

, if

For
when when Points are and |
For
when when Points are and |

Therefore the required area:

sq. units

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

Answer:

Let be root of point in the plane ** **can of is

substitute and in plane

Therefore

Therefore by mid point formula

**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.

Answer:

(a)

Marginal cost function

Average cost function

(b) Let be of on .

Therefore

Similarly, let be of on

Therefore

Hence correlation

(c) Given

Now

Therefore for will keep 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

Answer:

(a)

1 | 7 | 0 | 2 | 0 | 0 | 4 |

2 | 6 | 1 | 1 | 1 | 1 | 1 |

3 | 5 | 2 | 0 | 0 | 4 | 0 |

4 | 4 | 3 | -1 | -3 | 9 | 1 |

5 | 3 | 4 | -2 | -8 | 16 | 4 |

Hence there is high negative correlation between and .

Therefore equation of line:

When we get

(b) To Be Solved

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.

Answer:

(a) Total Cost

Price

If units are produced, then

For maximizing profits

units.

units

(b)

The cost involved to increase production from units to units:

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]**

Answer:

Let quantity of teaching aid and Quantity of teaching aid

Each type of requires hours for fabricating and hour for finishing.

Therefore … … … … … (i)

Each type of requires hours for fabricating and hours for finishing.

Therefore … … … … … (ii)

We know the company makes a profit of on each piece of type and on each piece of type .

We also know

Therefore Profit … … … … … (iii)

We need to maximize . We will solve it graphically.

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

For

For

For

Therefore, the quantity of teaching aid and Quantity of teaching aid