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 value of if
and
(ii) Find the equation of an ellipse whose latus rectum is and eccentricity is
(iii) Solve:
(iv) Using L’Hospital’s rule, evaluate:
(v) Evaluate:
(vi) Evaluate: , where
(vii) The two lines of regressions are and
. Find the correlation co-efficient between
.
(viii) A card is drawn from a well shuffled pack of playing cards. What is the probability that it is either a spade or an ace or both?
(ix) If are the cube roots of unity, prove that
(x) Solve the differential equation:
Answer:
(i)
Similarly,
(ii)
Equation of ellipse:
(iii)
(iv)
Apply L’Hospital’s rule
Apply L’Hospital’s rule
Apply L’Hospital’s rule
(v)
(vi) , where
Therefore
(vii) Let the line of regression of on
be
Let the line of regression of on
be
Therefore
Hence
since both
and
are negative.
(viii) Probability P(E) = Probability of drawing a spade + Probability of drawing an ace – Probability of drawing ace of spade
(ix)
Multiplying numerator and denominator by
We know
(x)
Let
Question 2:
(a) Using properties of determinants, prove that:
[5]
(b) Given two matrices
and
. Find
.
Using this result, solve the following system of equation:
[5]
Answer:
(a) LHS =
RHS. Hence proved.
(b)
Now,
Hence
Question 3:
(a) Solve the equation for :
[5]
(b) represent switches in ‘on’ position and
represent them in ‘off’ position. Construct a switching circuit representing the polynomial
. Using Boolean Algebra, prove that the given polynomial can be simplified to
. Construct an equivalent switching circuit. [5]
Answer:
(a)
(b)
Since
Question 4:
(a) Verify Lagrange’s Mean Value Theorem for the following function:
[5]
(b) Find the equation of the hyperbola whose foci are and passing through the point
. [5]
Answer:
(a) is continuous in
exists in
All the conditions of Lagrange’s Mean Value theorem satisfied there exists in
such that
(not possible)
or
which lies between
and
.
Hence Lagrange’s Mean Value theorem is verified.
(b) Foci
Let the equation be
not possible as
cannot be negative
or
Therefore and
Hence the equation is
Question 5:
(a) If , prove that
[5]
(b) Show that the rectangle of maximum perimeter which can be inscribed in a circle of radius is a square of side
[5]
Answer:
(a)
Differentiating with respect to
Differentiating again with respect to
(b) Please refer to the adjoining figure
and
Differentiating with respect to
Differentiating with respect to
Hence the perimeter is maximum when
Therefore
Therefore is a square with side of
Question 6:
(a) Evaluate:
[5]
(b) Find the smaller are enclosed by the circle and the line
. [5]
Answer:
(a)
Now put
Now
Thus
Therefore
(b) We have to calculate
First Integral:
Let
Now
, Therefore
Now substituting back
Second Integral:
Hence
Question 7:
(a) Given that the observations are: . Find the two lines of regression and estimate the value of
when
. [5]
(b) In a contest the competitors are awarded marks out of by two judges. The scores of the
competitors are given below. Calculate Spearman’s rank correlation. [5]
Competitors | ||||||||||
Judge A | ||||||||||
Judge B |
Answer:
(a)
Line of regression of
Line of regression of
When
(b)
Judge A | Judge B | Rank |
Rank |
||
Question 8:
(a) An urn contains white and
black balls. A ball is drawn at random. If it is white, it is not replaced into the urn. Otherwise, it is replaced with another ball of the same color. The process is repeated. Find the probability that the third ball drawn is black. [5]
(b) Three person shoot to hit a target. If
hits the target
times out of
trials,
hits it
times in
trials and
hits is
times in
trials, find the probability that:
(i) Exactly two person hit the target.
(ii) At least two person hit the target.
(iii) Non hit the target. [5]
Answer:
(a) Please note: The number of balls in the urn change based on what color is drawn.
Every time you get a Black ball, not only the ball is put back in the urn, another black ball is put in the urn. So the number of balls increase every time you pull a black ball.
In case you draw a white ball, it is not put back in the urn. So the number of balls decrease when you pull white ball.
Therefore the probability that the third ball is black is:
(b) Given:
and
Therefore
and
(i) Probability of at least two hitting the target
(ii) At least two people hit the target
(iii) None hit the target
Question 9:
(a) If
and
, find the locus of
and illustrate it in Argand Plane. [5]
(b) Solve the differential equation:
[5]
Answer:
(a)
Given
… … … … … (i)
Now we know that the equation of a circle with center and radius
is of the form
Therefore (i) can be written as
Therefore center is
and radius is
.
(b)
Substitute
When and
we get
Hence
Section – B (20 Marks)
Question 10:
(a) Using vectors, prove that an angle in a semicircle is a right angle. [5]
(b) Find the volume of a parallelopiped whose edges are represented by the vectors: [5]
Answer:
(a) In the adjoining diagram, we have a circle with center (origin) and the vertices of the triangle are
and
is the diameter.
Let
Therefore
(since
is the radius)
Therefore is a right angle.
(b) The volume of a parallelopiped:
Question 11:
(a) Find the equation of the plane passing through the intersection of the planes and
and the point
. [5]
(b) Find the shortest distance between the lines and
[5]
Answer:
(a) Given planes and
and the point
The equation of a line passing through the intersection of the plans is given as
Hence the equation of line will be
(b)
Hence the shortest distance units
Question 12:
(a) contains
white and
black balls.
contains
white and
black balls and
contains
white and
black balls. A dice having
red,
yellow and
green face, is thrown to select a box. If red face turns up we pick
, if yellow face turns up we pick
otherwise we pick
. The we draw a ball from a selected box. If the ball drawn is white, what is the probability that the dice had turned up with a Red face. [5]
(b) 5 dices are thrown simultaneously. If the occurrence of an odd number in a single die is considered a success, find the probability of maximum three successes. [5]
Answer:
(a) Given: Probability of a red face when the dice is rolled :
Probability of a yelloq face when the dice is rolled :
Probability of a green face when the dice is rolled :
Let D be the probability of drawing a White ball. Therefore
Probability of drawing a white ball from Bag I:
Probability of drawing a white ball from Bag II:
Probability of drawing a white ball from Bag III:
Therefore the probability of white ball drawn when red face show up:
(b) Number of dice
P(odd number)
P(even number)
Section – C (20 Marks)
Question 13:
(a) Mr. Nirav borrowed from the bank for
years. The rate of interest is
per annum compounded monthly. Find the payment he makes if he pays back at the beginning of each month. [5]
(b) A dietitian wishes to mix two kinds of food and
in such a way that the mixture contains at least
units of vitamin
,
units of vitamin
and
units of vitamin
. The vitamin contents of one kg of food is given below:
Food | Vitamin A | Vitamin B | Vitamin C |
X | 1 unit | 2 units | 3 units |
Y | 2 units | 2 units | 1 unit |
One kg of food costs
and one kg of food
costs
. Using linear programming, find the least cost of the total mixture which will contain the required vitamins. [5]
Answer:
(a) Principal
Annual rate of interest Monthly rate of interest
Now
Hence monthly installment
(b) Let us say we have kgs of food
and
kgs of food
.
Therefore we have the following inequations:
Now lets plot these lines. We got the following graph
We have to minimize
Hence the four point of intersection from the graph are:
Hence, the and
should be bought.
Question 14:
(a) A bill of was drawn on 8th March 2013, at
months. It was discounted on 18th May, 2013 and the holder of the bill received
. What is the rate of interest charged by the bank? [5]
(b) The average cost function, for a commodity is given by
in terms of output
. Find:
(i) The total cost, and marginal cost,
as a function of
(ii) The outputs for which increases. [5]
Answer:
(a) Face value of the bill
Discounted value of the bill
Banker’s discount
Nominal date due is 8th October
Legal due date of the bill is 11th October (3 days of grace period)
Number of unexpired days from 8th May to 11th October is 146 days
Banker’s discount
(b)
Cost function
Marginal cost
For to be increasing,
Case 1:
Case 2:
But cannot be negative. Therefore
Hence the average cost function increase if the output
Question 15:
(a) Calculate the index number for the year 2014, with 2010 as the base year by the weighted aggregate method from the following data: [5]
Commodity | Price in Rs. | Weight | |
2010 | 2014 | ||
A | |||
B | |||
C | |||
D |
(b) The quarterly profits of a small scale industry (in thousands of Rupees) is as follows: Calculate the quarterly moving averages. Display these and the original figures graphically on the same graph sheet. [5]
Year | Quarter 1 | Quarter 2 | Quarter 3 | Quarter 4 |
2012 | ||||
2013 | ||||
2014 |
Answer:
(a)
Commodity |
|
|
||||
Index number for the year 2014 with respect to 2010 as the base year
(b)
Year | Quarter | Quarterly Profits | 4 years moving total | 4 year average |