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 × 3]
(i) If , where and , find the value of .
(ii) Find the value(s) of so that the line may touch the hyperbola .
(iii) Prove that:
(iv) Using L’Hospital’s Rule, evaluate:
(v) Evaluate:
(vi) Evaluate:
(vii) Two regression lines are represented by and . Find the line of regression of .
(viii) If are the cube roots of unity, evaluate:
(ix) Solve the differential equation:
(x) If two balls are drawn from a bag containing three red and four blue balls, find the probability that:
(a) They are the same color
(b) They are of different colors
Answer:
(i) Given,
Therefore
However, does not satisfy all the equations. Hence
(ii)
Therefore
Condition for tangent
(iii) LHS
RHS. Hence Proved.
(iv)
Using L’Hospital’s Rule
Again applying L’Hospital’s Rule
Once again applying L’Hospital’s Rule,
(v) Given,
Put
Therefore
(vi)
Using property,
Therefore
(vii) Let us assume that be the regression line of on
Then
Therefore
Hence our assumption is true i.e. regression line on is
(viii)
(ix)
Integrating both sides
(x) Total number of ways of drawing balls
(a) (Balls are of same color)
(b) (Balls are of different color)
Question 2:
(a) Using properties of determinants, prove that:
[5]
(b) Find , where
Hence, solve the system of linear equations:
[5]
Answer:
(a)
Applying , we get
Applying and , we get
Expanding along , we get
RHS. Hence proved.
(b)
exists
Therefore
Given system of equation is
This can be written as
Where and
Since , the given system of equations have a unique solution
and which is the solution to the system of the equations.
Question 3:
(a) Solve for [5]
(b) Construct a circuit diagram for the following Boolean function:
Using laws of Boolean Algebra, simplify the function and draw the simplified circuit. [5]
Answer:
(a)
Let
Therefore we get
Hence the solution of the given equations are
(b)
Question 4:
(a) Verify Lagrange’s Mean Value Theorem for the function in the interval [5]
(b) From the following information, find the equation of the Hyperbola and the equation of its Transverse Axis:
Focus: , Directrix: , [5]
Answer:
(a) Given,
Since is continuous on and exists in
is continuous in
Differentiating the given function w.r.t
which exists for all
Thus, both the conditions of Lagrange’s mean value theorem is satisfied therefore at least one c exists in .
Such that
Squaring both sides we get
Clearly, lies in the interval
Hence, Lagrange’s mean value theorem is verified and
(b) Let be the point on the conic, then
i.e.
i.e. is the required hyperboloa.
Transverse axis passes through and is perpendicular to Directrix,
where
Therefore
Question 5:
(a) If , show that [5]
(b) Find the maximum volume of the cylinder which can be inscribed in a sphere of radius cm. (Leave the answer in terms of ) [5]
Answer:
(a) Given
Now,
Differentiating once again w.r.t
(b) Let the height of the cylinder and the radius of the cylinder
Given Radius of the sphere
If is the volume of the cylinder, then
Let be the center of the sphere and
For ,
Therefore
Therefore
For maxima and minima,
and
is maximum when , putting
Question 6:
(a) Evaluate: [5]
(b) Find the area bounded by the curve and the line [5]
Answer:
(a) Given,
Put
Therefore
Now putting the value of t back in the expression
(b) Given: and line of intersection
which represents a downward parabola with vertex at
Line of intersection
Putting in the above equation
Therefore the point of intersections are and
Therefore the area enclosed between the curve and the line is
sq. units.
Question 7:
(a) Find the Karl Pearson’s coefficient of correlation between x and y for the following data. [5]
16  18  21  20  22  26  27  15  
22  25  24  26  25  30  33  14 
(b) The following table shows the mean and standard deviation of the marks of Mathematics and Physics scored by students in a school:
Mathematics  Physics  
Mean  84  81 
Standard Deviation  7  4 
The correlation coefficient between the given marks is . Estimate the likely marks in Physics in the marks in Mathematics at . [5]
Answer:
(a)
16  22  4  3  16  9  12 
18  25  2  0  4  0  0 
21  24  1  1  1  1  1 
20  26  0  1  0  1  0 
22  25  2  0  4  0  0 
26  30  6  5  36  25  30 
27  33  7  8  49  64  56 
15  14  5  11  25  121  55 
5  1  135  221  152 
(b) Mean marks in Mathematics
Mean marks in Physics
Therefore
Therefore regression equation of on
Putting
Hence the likely marks in Physics are
Question 8:
(a) contains three red and four white balls. contains two red and three white balls. If one ball is drawn from and two balls are drawn from , find the probability that:
(i) One ball is red and two balls are white
(ii) All the three balls are of the same color [5]
(b) Three persons, Aman, Bipin and Mohan attempt a Mathematics problems independently. The odds in favor of Aman and Mohan solving the problem are and respectively and the odds against Bipin solving the problem are . Find:
(i) The probability that all the three will solve the problem
(ii) the probability that the problem will be solved. [5]
Answer:
(a) Possible selection are as follows:
i) 1 Red ball from Bag A, 2 white ball from Bag B
1 white ball from bag A, 1 white ball from bag B, 1 red ball from Bag B
Therefore P(one ball is red and two balls are white)
ii) Possible selections are as follows:
1 red ball from Bag A, 2 red balls from Bag B
1 white ball from Bag A and 2 white balls from bag B
Therefore P (all the three balls are of the same color)
(b) Odd against
Odds in favor
: Event Aman solves the problem
: Event Bipin solves the problem
: Event Mohan solves the problem
Aman  Bipin  Mohan 



i) Probability that all three will solve the problem
ii) Probability that the problem is not solved = probability that all three fail to solve the problem
Therefore the probability that the problem will be solved
Question 9:
(a) Find the locus of the complex number , satisfying the relation and . Illustrate the locus on the Argand plane. [5]
(b) Solve the following differential equation:
, given that . [5]
Answer:
(a) Let
Therefore
When and when
Therefore the locus will be and
(b) Given,
It is a linear differential equation in .
I.F.
Therefore, general solution is
Given that and
Substituting the value of we get
Question 10:
(a) If and are unit vectors and is the angle between them, then show that . [5]
(b) If the value of for which the four points with position vectors ; ; and are coplanar. [5]
Answer:
a)
Now,
b) Let be the given points whose position vectors are ; ; and
Since the points A, B, C and D are coplanar, vectors and coplanar.
Question 11:
(a) Find the equation of a line passing through the point and perpendicular to the lines: and [5]
(b) Find the equation of planes parallel to the plane and which are at a distance of five units from the point . [5]
Answer:
a) Any line passing through the point is
where and are its direction vectors
It is perpendicular to the lines
then
… … … … … i)
and
… … … … … ii)
Subtracting ii) from i) we get
… … … … … iii)
and hence from i) we get
… … … … … iv)
From iv) and iii) we get
Putting these values in the equation of the line
b) The given plane is … … … … … i)
Equation of any plane parallel to above equation is
… … … … … ii)
Now ii) is at a distance of units from the point of
or
or
Substituting these values in ii) the equations of the required planes are
and
or and
Question 12:
(a) If the sum and the product of the mean and variance of a Binomial Distribution are respectively, find the probability distribution and the probability of at least one success. [5]
(b) For , the chances of being selected as the manager of firm are respectively. The probabilities for them to introduce a radical chance in the marketing strategy are respectively. If a chance takes place; find the probability that it is due to the appointment of B. [5]
Answer:
a) Given,
… … … … … i)
… … … … … ii)
Dividing the square of i) by ii) we get
Therefore
But
Therefore
From i),
Therefore
Hence, the binomial distribution is
Probability of getting at least 1 success
b) Let and and be the events as defined below
is selected as a manager
is selected as a manager
is selected as a manager
and radical change occurs in marketing strategy
Given, , ,
We want to find the probability that the radical change in marketing strategy occurred due to the appointment of B i.e
Question 13:
(a) If Mr. Nirav deposits at the beginning of each month in an account that pays an interest of per annum compounded monthly, how many months will be required for the deposit to at least ? [5]
(b) A mill owner buys two types of machines for his mill. occupies sqm of area and requires men to operate it.; while occupies sqm of area and requires men to operate it. The owner has sqm of area available and men to operate the machines. If produces units and produces units daily, how many machines of each type should be buy to maximize the daily output? Use linear programming to find the solution. [5]
Answer:
a)
months
b) Data given
Machine A  Machine B  Machine C  
Area Needed (Sq. m)  1000  1200  7600 
Labor Force  12  8  72 
Daily Output (units)  50  40  – 
Let and be the number of Machines A and Machine B respectively.
Constraints
Total output
So minimize
Subject to And
Now for
$latex x $  6  0 
0  9 
Now for
7.2  0  
0  (19/3) 
The vertices of the feasible region OAPD are and
Corner Points  Object Function 
Thus we see that is maximum at . Therefore number of Machine A and number of Machine .
Question 14:
(a) A bill of was drawn on 1st April 2011 at 4 months and discounted for at a bank. If the rate of interest was per annum, on what date was the bill discounted? [5]
(b) A company produces a commodity with fixed cost. The variable cost is estimated to be of the total revenue recovered on selling the product at a rate of per unit. FInd the following:
(i) Cost function
(ii) Revenue function
(iii) Break even point [5]
Answer:
a) Banker’s discount
Let be the expired period in years
days
Legal due date of the bill
1st April 2011 + 4 months + 3 days of grace = 4th August 2011
The bill was cashed 73 days before 4th August (4 days in August, 31 days in July, 30 days in June, 8 days in May) which comes to 23rd May 2011 as the date
b) Supposed that number of units are produced and sold.
Given fixed cost Rs.
Revenue
i) As each unit’s variable cost of units is of revenue,
Therefore Variable cost of units
Therefore total cost of producing units
ii) Price of one unit Rs.
Therefore revenue on selling units
ii) At break even values
Hence the break even point is units produced.
Question 15:
(a) The price index for the following data for the year 2011 taking 2001 as the base year was 127. The simple average of price relative method was used. Find the value of . [5]
Items  A  B  C  D  E  F 
Price (Rs. Per unit) in year 2001  80  70  50  20  18  25 
Price (Rs. Per unit) in year 2011  100  87.50  61  22  32.50 
(b) The profit of a paper bag manufacturing company (in lakhs / millions of Rs.) during each month pf a year are:
Month  Jan.  Feb.  Mar.  Apr.  May  Jun.  Jul.  Aug.  Sept.  Oct.  Nov.  Dec. 
Profit  1.2  0.8  1.4  1.6  2.0  2.4  3.6  4.8  3.4  1.8  0.8  1.2 
Plot the given data on a graph sheet. Calculate the four monthly moving averages and plot these on the same graph sheet. [5]
Answer:
a) Using simple averages of price relatives, the price index of 2011 taking 2001 as the base year was . From the following data we find
80  70  50  20  18  25  
100  87.50  61  22  32.5  
125  125  122  110  130 
b)
Month  Profit  4 monthly totals  4 monthly average  4 monthly centered moving average 
Jan  1.2  
Feb  0.8  
5  1.25  
Mar  1.4  1.35  
5.8  1.45  
Apr  1.6  1.65  
7.4  1.85  
May  2.0  2.125  
9.6  2.4  
Jun  2.4  2.8  
12.8  3.2  
July  3.6  3.375  
14.2  3.55  
Aug  4.8  3.475  
13.6  3.4  
Sep  3.4  3.05  
10.8  2.7  
Oct  1.8  2.25  
7.2  1.8  
Nov  0.8  
Dec  1.2  