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) If and
, and
is the set of real numbers, then find
and
.
ii) Solve:
iii) Using determinants, find the values of , if the area of triangle with vertices
and
is
square units.
iv) Show that is symmetric matrix, if
v)
is not defined at
. What value should be assigned to
for continuity of
at
?
vi) Prove that the function is increasing on
.
vii) Evaluate:
viii) Using L’Hospital’s Rule, evaluate:
ix) Two balls are drawn from an urn containing white,
red and
black balls, one by one without replacement. What is the probability that at least one ball is red?
x) If events and
are independent, such that
and
, find
.
Answer:
i) Given and
Also
ii)
iii)
or
or
iv)
is symmetric matrix
v) To be continuous at
vi)
is increasing for
vii)
viii)
ix) at least one ball is Red
no ball is Red
no Red ball in first draw
no Red ball in second draw
x) Since A and B are independent,
Question 2: If and
show that the function
is one
one onto. Hence, find
. [ 4 ]
Answer:
Given
Therefore
and
is one
one function.
Let
Therefore Co-domain Range
is onto function
Replacing by
and
by
.
Question 3:
a) Solve for
OR
b) If , show that
[ 4 ]
Answer:
a) Simplify the expression
We know, that
Therefore
OR
b) Given
Squaring both sides
Hence Proved
Question 4: Using properties of determinants prove that:
[ 4 ]
Answer:
LHS
We know
RHS
Hence Proved
Question 5:
a) Show that the function is continuous, but not differentiable at
.
OR
b) Verify the Lagrange’s mean value theorem for the function:
in the interval
[ 4 ]
Answer:
a) for
and for
L.H.L.
R.H.L.
is continuous at
R.H.D
L.H.D
Since L.H.D R.H.D,
does not exist.
is continuous at
but is non differentiable at
.
OR
b) Given
and
is continuous for
is differentiable for
Therefore Lagrange’s mean value theorem is applicable.
We know,
Hence Lagrange’s mean value theorem is verified.
Question 6: If and
, prove that
[ 4 ]
Answer:
Given and
Therefore
Question 7: A m long ladder is leaning against a wall, touching the wall at a certain height from the ground level. The bottom of the ladder is pulled away from the wall, along the ground, at the rate of
m/s. How fast is the height on the wall decreasing when the foot of the ladder is
m away from the wall? [ 4 ]
Answer:
… … … … … i)
The bottom of the ladder is begin pulled, so these distances are changing with time,
… … … … … ii)
… … … … … iii)
If m, then
m from i) (Pythagoras theorem)
From equation iii) we get
sign indicates the height is decreasing
Question 8:
a) Evaluate:
OR
b) Evaluate:
[ 4 ]
Answer:
a)
Let
OR
b) Given
Therefore
for all
for all
Question 9: Solve the differential equation:
[ 4 ]
Answer:
Given
Let
Question 10: Bag contains
white balls and
black balls, while Bag
contains
white balls and
black balls. Two balls are drawn from Bag
and placed in Bag
. Then, what is the probability of drawing a white ball from Bag
? [ 4 ]
Answer:
Bag contains
white balls and
black balls
Bag contains
white balls and
black balls
Case I : Let Both white balls are transferred from Bag to Bag
and then a white ball is drawn from Bag
.
Required probability
Case II: Let both black balls are transferred from Bag to Bag
and then a white ball is drawn from Bag
.
Required probability
Case III: 1 white and 1 black ball is transferred and then a white ball is drawn from Bag .
Required probability
Total probability
Question 11: Solve the following system of linear equations using matrix method: [ 6 ]
Answer:
Let
… … … … … i)
… … … … … ii)
… … … … … iii)
and
Question 12:
(a) The volume of a closed rectangular metal box with a square base is . The cost of polishing the outer surface of the box is Rs.
. Find the dimensions of the box for the minimum cost of polishing it.
OR
(b) Find the point on the straight line , which is closest to the origin. [ 6 ]
Answer:
a) Let the base of the box be and height be
.
Therefore Volume
Hence
… … … … … i)
Total surface area
Therefore the cost function
Rs. … … ii)
Differentiating w.r.t. we get,
… … … … … iii)
Let
Differentiating equation (iii) w.r.t. we get,
at
Also
cm
Therefore The cost for polishing the surface area is minimum when length of base is cm and height of box is
cm.
OR
b) The equation of line is given as
Therefore The point of the line can be taken as
Distance from origin
Let is minimum when
is minimum
… … … … … i)
Differentiating equation (i) w.r.t. we get,
… … … … … ii)
Let
Differentiating equation (ii) w.r.t. we get,
is minimum at
The closest point on the line
with origin is
Question 13: Evaluate:
[ 6 ]
Answer:
… … … … … i)
Using
… … … … … ii)
Adding i) and ii) we get
Question 14:
a) Given three identical Boxes and
, Box
contains
gold and
silver coin, Box
contains
gold and
silver coins and Box
contains
silver coins. A person chooses a Box at random and takes out a coin. If the coin drawn is of silver, find the probability that it has been drawn from the Box which has the remaining two coins also of silver.
OR
b) Determine the binomial distribution where mean is 9 and standard deviation is . Also, find the probability of obtaining at most 1 success. [ 6 ]
Answer:
a) Given Box contains
gold and
silver coin.
Box contains
gold and
silver coins.
Box contains
silver coins.
Probability of choosing a bag is
Using Baye’s theorem,
OR
b) Given
Since
SECTION B (20 Marks)
Question 15: [ 3 × 2 ]
a) If and
are perpendicular vectors,
and
. Find the value of
.
b) Find the length of the perpendicular from origin to the plane
c) Find the angle between the two lines and
.
Answer:
a) Given and
are perpendicular vectors,
and
b) Given
Perpendicular length from origin to plane
units
c) Given
and
Therefore the angle between the two lines and
is
Question 16:
a) If , prove that
and
are perpendicular.
OR
b) If and
are non-collinear vectors, find the value of
such that the vectors
and
are collinear. [ 4 ]
Answer:
a) Given,
OR
b) Given and
are collinear.
On comparing coefficient of and
Also
Question 17:
(a) Find the equation of the plane passing through the intersection of the planes and
such that the intercepts made by the resulting plane on the x-axis and the z-axis are equal.
OR
b) Find the equation of the lines passing through the point and perpendicular to the lines
and
[ 4 ]
Answer:
a) Equation of intersecting plane is
… … … … … i)
For intercept made on x-axis, Put , and
For intercept made on z-axis Put and
Given Intercept made on x-axis intercept made on z-axis
Substituting in i) we get
OR
b) Lines are passing through the point
Also given, that they are perpendicular to
and
Denominators of the required lines are or
Therefore equation of line is
Question 18: Draw a rough sketch and find the area bounded by the curve and
. [ 6 ]
Answer:
Equation of parabola is … i)
Equation of straight line is
Substituting it in i) we get
or
When and when
Hence we have points and
Area under line
square units
Area under curve
square units
Area bounded by the curve and
square units
SECTION C (20 Marks)
Question 19: [ 3 × 2 ]
(a) A company produces a commodity with Rs. as fixed cost. The variable cost estimated to be
of the total revenue received on selling the product, is at the rate of Rs.
per unit. Find the break-even point.
b) The total cost function for a production is given by
.
Find the number of units produced for which M.C. = A.C. (M.C.= Marginal Cost and A.C. = Average Cost.)
c) and correlation coefficient
, find regression equation of
on
.
Answer:
a) Revenues Variable cost
Fixed cost
Let the total revenue
Therefore total revenue
Therefore break even point (in units)
units
b) Given
… … … … … ii)
… … … … … ii)
… … … … … iii)
Given that
Therefore units.
c) Regression line on on
is
We know,
Question 20:
a) The following results were obtained with respect to two variables and
:
and
(i) Find the regression coefficient .
(ii) Find the regression equation of on
.
OR
b) Find the equation of the regression line of y on , if the observations
are as follows:
Also, find the estimated value of
when
. [ 4 ]
Answer:
a) Given and
i) Regression coefficient
ii) Regression equation of on
OR
b)
1 | 4 | 4 | 1 |
2 | 8 | 16 | 4 |
3 | 2 | 6 | 9 |
4 | 12 | 48 | 16 |
5 | 10 | 50 | 25 |
6 | 14 | 84 | 36 |
7 | 16 | 112 | 49 |
8 | 6 | 48 | 64 |
9 | 18 | 162 | 81 |
45 | 90 | 530 | 285 |
When
Question 21:
a) The cost function of a product is given by
where
is the number of units produced. How many units should be produced to minimize the marginal cost?
OR
b) The marginal cost function of x units of a product is given by . The cost of producing one unit is Rs.
. Find the total cost function and average cost function. [ 4 ]
Answer:
a) Given
is minimum
For minimum,
OR
b) Let total cost function
Therefore Marginal Cost function
If
Also, the average cost function is given by
Question 22: A carpenter has and
running feet respectively of teak wood, plywood and rosewood which is used to produce product
and product
. Each unit of product
requires
and
running feet and each unit of product
requires
and
running feet of teak wood, plywood and rosewood respectively. If product
is sold for Rs.
per unit and product
is sold for Rs.
per unit, how many units of product
and product
should be produced and sold by the carpenter, in order to obtain the maximum gross income?
Formulate the above as a Linear Programming Problem and solve it, indicating clearly the feasible region in the graph. [ 6 ]
Answer:
Product | A | B | |
Teakwood | |||
Plywood | |||
Rosewood |
Selling Price for Product Rs./unit
Selling Price for Product Rs./unit
Therefore the objective function is
… … … … … i)
… … … … … ii)
… … … … … iii)
From equation iii) and ii)
From equation i) and iii)
We know
is maximum at
Hence the optimal solution is and
units (Product
units, Product
units)