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:
(i) Find the matrix for which:
(ii) Solve for , if :
(iii) Prove that the line touches the conic . Also, find the point contact.
(iv) Using L’Hospitals rule, evaluate:
(v) Evaluate:
(vi) Using properties of definite integrals, evaluate:
(vii) The two lines of regression are and and the variance of . Find the variance of y and the coefficient of correlation.
(viii) Express in the form of . Find its modulus and argument.
(ix) A pair of dice is thrown. What is the probability of getting an even number on the first die or a total of ?
(x) Solve the differential equation:
Answer:
(i)
(ii) We know,
Squaring on both sides
(iii) Line
The condition
. Therefore the lines touches the parabola
and
Therefore the point of contact is
(iv)
Since
form, applying L’Hospital’s Rule
form, applying L’Hospital’s Rule
(v)
In the first integral, put
(vi) Let
Now, . Using the property
in the first integral
(vii) Two lines of regression are:
(regression line of )
Therefore
(regression line of )
Therefore
Therefore
Since both and are negative, is also negative. Hence Correlation Coefficient
Variance of
Therefore
Therefore Variance of
(viii)
Argument of
(ix) Total number of possible outcomes
Even : Getting a total of
Favorable outcomes
Therefore
Hence
Event : Getting even number on first die
Favorable outcomes
Therefore
Hence
Therefore
Hence
(x)
Therefore
Integrating
Question 2:
(a) Using properties of determinants, prove that:
[5]
(b) Solve the following system of linear equations using matrix method:
[5]
Answer:
(a) Let
Operating:
Therefore
Now applying and
Therefore
Expanding along , we will get
. Hence proved.
(b) We have to write the given equations in the form
This shows that A is a nonsingular matrix. Therefore the system has a unique solution defined by
Hence
Question 3:
(a) If , prove that: [5]
(b) Write the Boolean function corresponding to the switching circuit given below:
represent switches in ‘on’position and represent them in ‘off’ position. Using Boolean algebra, simplify the function and construct an equivalent circuit. [5]
Answer:
(a) Given
Therefore . But cannot be negative, hence or . Hence proved.
(b)
Question 4:
(a) Verify the conditions of Rolle’s Theorem for the following function:
Find a point in the interval, where the tangent to the curve is parallel to . [5]
(b) Find the equation of a standard ellipse, taking its axes as the coordinate axes, whose minor axis is equal to the distance between the foci and whose lengths of latus rectum is . Also find its eccentricity. [5]
Answer:
(a) Since , therefore the function is continuous in
which exists for all . Therefore the function is differentiable in
Therefore
Thus satisfies all the conditions of Rolle’s Theorem.
Therefore there must exists at least one value of say in the open interval such that
Hence Rolle’s Theorem is verified.
When
So, there exist a point on the given curve where the tangent to the curve is parallel to
(b) Let the equation of ellipse be
Given … … … … … (i)
and … … … … … (ii)
We also know that … … … … … (iii)
Substituting (i) and (ii) into (iii)
Therefore the equation of the ellipse:
Also
Question 5:
(a) If , prove that:
[5]
(b) A rectangle is inscribed in a semicircle of radius with one of its sides on the diameter of the semicircle. Find the dimensions of the rectangle to get maximum area. Also, find the maximum are. [5]
Answer:
(a) Given:
Differentiating both sides with respect to
Differentiating both sides with respect to
Hence proved.
(b) Please refer to the figure on the right.
Consider the rectangle
Therefore the area of the rectangle
Differentiating both sides with respect to
For maximizing the area of
Hence the area is maximum when
and
Hence sides are
Therefore the area of units.
Question 6:
(a) Evaluate: [5]
(b) Find the area of the region bounded by the curve and [5]
Answer:
(a) Let
Now let . Differentiating both sides with respect to
(b) Given: . This represents a parabola with vertex at (3, 9) and it opens downwards
Similarly for . This represents a parabola with vertex and it opens upwards.
Both the curves passes through origin . Solving the two equation:
Hence the integration point is
Therefore the enclosed area is:
Question 7:
(a) Calculate Carl Pearson’s coefficient of correlation between for the following data and interpret the result:
[5]
(b) The marks obtained by 10 students in English and Mathematics are given below:
Marks in English  20  13  18  21  11  12  17  14  19  15 
Marks in Mathematics  17  12  23  25  14  8  19  21  22  19 
Estimate the probable score for Mathematics of the marks obtained in English are 24. [5]
Answer:
(a)
1  6  6  1  36 
2  5  10  4  25 
3  7  21  9  49 
4  9  36  16  81 
5  8  40  25  64 
6  10  60  36  100 
7  11  77  49  121 
8  13  104  64  169 
9  12  108  81  144 
Hence there is high correlation between and .
(b)
(English)  (Math)  
20  17  4  1  4  16  1 
13  12  3  6  18  9  36 
18  23  2  5  10  4  25 
21  25  5  7  35  25  49 
11  14  5  4  20  25  16 
12  8  4  10  40  16  100 
17  19  1  1  1  1  1 
14  21  2  3  6  4  9 
19  22  3  4  12  9  16 
15  19  1  1  1  1  1 
Therefore equation of line:
When we get
marks
Question 8:
(a) A committee of person has to be chosen from boys and girls, consisting at least of one girl. Find the probability that the committee consists of more girls than boys. [5]
(b) An urn contains white and black balls while another urn contains white and black balls. Two balls are drawn from the first urn and put into the second urn and then a ball is drawn from the second urn. Find the probability that the ball drawn from the second urn is a white ball. [5]
Answer:
(a) The various possibilities of forming a committee:
Boys  Girls  Committee Size 
4  0  4 
3  1  4 
2  2  4 
1  3  4 
0  4  4 
All possible combinations where a committee can be formed with at least one girl:
Number of committees where the number of girls are more than boys
Hence the
(b) When one transfers 2 balls from first urn to another, three possible scenarios can happen:
 Both the balls transferred can be white.
 Both the balls transferred can be black.
 One of the ball is white, while the other is black.
P (transferring 2 white balls & drawing 1 white ball)
P (transferring 2 black balls & drawing 1 white ball)
P (transferring 2 white balls & drawing 1 white ball)
Hence the probability is
Question 9:
(a) Find the locus of a complex number, , satisfying .
Illustrate the locus of in the Argand plane. [5]
(b) Solve the following differential equation:
, when [5]
Answer:
(a) ,
Squaring
The locus is the region outside the circle with center and radius
(b)
Put
Therefore
Integrating both sides we get
substituting
For and . Substituting back we get
Section – B (20 Marks)
Question 10:
(a) For any three vectors , show that are coplanar. [5]
(b) For a unit vector perpendicular to each of the vectors and where and [5]
Answer:
(a)
are coplanar
(b) Given and
Therefore
Unit vector perpendicular to and is given by
In our case
Now,
Therefore or
Question 11:
(a) Find the image of the point in the line . Also find the length of the perpendicular from point to the line. [5]
(b) Find the Cartesian equation of the plane, passing through the line of intersection of the planes: and and intersecting at . [5]
Answer:
(a) Let be the point and AB be the line given by equation
Draw the diagram as shown. is the reflection of on .
Any point of will be given by
Then DR’s of is
and DC’s of are proportional to
Therefore
Therefore
Let the coordinate of be
Since is the mid point of ,
Therefore the coordinates of
Hence the distance of M from units
(b) A plane passing through the intersection of the given plane is
Point of intersection on is . At , and will be .
Therefore
Therefore the required plane is
. Equation of the plane.
Question 12:
(a) In an automobile factory, certain parts are to be fixed into the chassis in a section before it moves into another section. One a given day, one of the three persons carries out this task. has chance, has change and has chance of doing the task. The probability that will take more than the allotted time is and respectively. If it is found that the time taken is more than the allotted time, what is the probability that A has done the task? [5]
(b) The difference between mean and variance of a binomial distribution is and the difference of their square is . Find the distribution. [5]
Answer:
(a) Let denote the events of carrying out the task by respectively. Let H denote the event of taking more time.
Then,
Then, using Bayes Theorem
(b) Let the binomial distribution be
mean and variance
Given: … … … … … (i)
… … … … … (ii)
Dividing (ii) by square of (i) we get
Therefore
Substituting in (i) we get
Hence the binomial distribution is
Section – C (20 Marks)
Question 13:
(a) A man borrows at per annum, compounded semiannually and agrees to pay it in equal semiannual installments. Find the value of each installment, if the first payment is due at the end of two years. [5]
(b) A company manufacturers two types of products and . Each unit of requires grams of nickel and gram of chromium, while each unit of requires gram of nickel and grams of chromium. The firm can produce grams of nickel and grams of chromium. The profit is on each product of type and on each unit of type . How many units of each type should the company manufacture so as to earn maximum profit. Use linear programming to find the solution. [5]
Answer:
(a) Principal amount
Rate of interest semiannually
Let the installment amount be
Therefore, if we were to calculate the net present value of all the installments, that should be equal to
Hence we have:
(b)
Type of Product 
Total Items 
Profit per Unit (Rs.)  
A  3  1  40 
B  1  2  50 
Let number of units of type
and Let number of units of type
So we have to maximize
given constraints: … … … … … (i)
and … … … … … (ii)
Also
For equation is and is
Similarly, for equation is ) and is
Now plot the two lines on the graph paper
Therefore the solution set of this system is the shaded region as shown in the graph above.
When
Therefore the maximum profit is . Hence the number of units of produced is and number of units of produced is .
Question 14:
(a) The demand function is where is the number of units demanded and is the price per unit. Find:
(i) The revenue function in terms of .
(ii) The price and the number of units demanded for which the revenue is maximum. [5]
(b) A bill of drawn on 10th September, 2010 at 6 months was discounted for at a bank. If the rate of interest was per annum, on what date was the bill discounted. [5]
Answer:
(a) Given
(i) Revenue function
(ii) For maximum revenue,
and which is
is maximum revenue, price per unit is
Therefore
Therefore Price is and no. of units is for which the revenue is maximum.
(b) Amount
Interest
Bill Discount
Now
Therefore Date of Expiry = 13 March 2011
and Date of Discounting is 30 December 2010
Question 15:
(a) The index number by the method of aggregates for the year 2010, taking 2000 as the base year, was found to be . If the sum of the price in year 2000 is , find the value of in the data given below. [5]
Commodity  A  B  C  D  E  F 
Price in Base year 2000 (Rs.)  50  30  70  116  20  
Price in the year 2010 (Rs.)  60  24  80  120  28 
(b) From the details given below, calculate the five yearly moving averages of the number of students who have studies in a school. Also, plot these and original data on the same graph paper. [5]
Year  1993  1994  1995  1996  1997  1998  1999  2000  2001  2002 
No. of Students  332  317  357  392  402  405  410  427  405  438 
Answer:
(a)
Commodity 
Price in Rs. 

2000  2011  
A  50  60 
B  24  
C  30  
D  70  80 
E  116  120 
F  20  28 
(b)
Year  Number  5 year moving total  5 year moving average 
1993  332  –  – 
1994  317  –  – 
1995  357  1800  360 
1996  392  1873  374.6 
1997  402  1966  393.2 
1998  405  2036  407.2 
1999  410  2049  409.8 
2000  427  2085  417 
2001  405  –  – 
2002  438  –  – 