QUESTIONS- MATHEMATICS - 2012-2013
Time: 3 hrs.
questions are compulsory.
(ii) The question paper consists 3 section --------A,B & C.
Section A comprises ten questions of one mark
comprises twelve questions of four marks each.
comprises seven questions of six marks each.
1. What is
the principal value of
the value of
3. A bird
is located at A (3,2,8) in space. She wants to move to the plane given by
3x + 2y + 6z+ 16 = 0 in shortest time.
Find the distance she covered.
4. What is
the degree of the following differential equation?
5x - - 6y
= log x
5. If A is
a square matrix of order 3 such that =
6. Find the values of x and y if: 2 +
contentment obtained after eating x units of new dish at a trial
function is given by the function: C(x) = . If the marginal contentment is
defined as rate of change of C(x) with respect to the number of units consumed
at an instant, then find the marginal contentment when four units of dish are
what value of are the vectors and
to each other.
λ when the projection of and is 4 units.
11. If, find
If Differentiate, w.r.t. .
12. In a backward state, there are 729
families having six children each. If probability of survival of a girl is and that of boy is , find the number of families having 2
girls and 4 boys. Do you believe that a female child is neglected in backward
areas? What steps will you take to restore respect of a female child in
13.If y = , the show that
14. Evaluate dx as a limit of sums.
15.If any three vectors are coplanar, prove that the vectors
16.Find the relationship between “a” and “b” so that
the function ‘f’ defined by:
= is continuous at x= 3.
If show that
17. Let * be the binary
operation on N defined by a * b = H.C.F. of a and b.
Is * commutative? Is * associative? Does there exist identity for this binary
operation on N?
f: given by f(x) = . Show that f is
invertible with the inverse (, where is
the set of all non- negative real numbers.
19. Prove the following:
20. The volume of the parallelepiped whose
edges are -12
units. Find the value of
21. Prove, using properties of determinants:
that: = -
23. Find the area of the region bounded by the parabola y
= x2 and y =
24.Using matrices, solve the following system of
X + 2y + z = 7, x + 3z = 11, 2x –
3y = 1
25.A square tank of capacity 250 cubic metres has
to be dug out. The cost of land is RS. 50 per square metres. The cost of
digging increases with the depth and for the whole tank, it is RS.(400 x ),
where h metres is the depth of the tank. What should be the dimensions of the
tank so that the cost is minimum?
the semi vertical angle of the right circular cone of given total surface area
and maximum volume is
26.Solve the following differential equation:
given that y = 0, when x=
27.Find the vector equation of a line passing
through the point with position vector
to the plane , Also find the point of intersection of this line and the
28.There are three coins. One is a two headed
coin (having heads on the both faces), another is a biased coin that comes up
heads 75% of the times and the third is an unbiased coin. One of three coins is
chosen at random and tossed , and it shows head. W hat is the probability that
it was the two-headed coin?
has two machines A and B .Past records shows that machine A produced 60% of the
items of output and machine B produced 40% of the items. Further, 2 % of the
items produced by machine A and 1% produced by machine B were defective. All
the items are put into one stockpile and then one item is chosen at random from
this and is found to be defective. What is the probability that it was produced
by machine B ?
29. A cottage industry manufactures pedestal
lamps and wooden shades, each requiring the use of a grinding/cutting machine
and a sprayer. It takes 2 hours on grinding/cutting machine and 3 hours on the
sprayer to manufacture a pedestal lamp. It takes 1 hour on the grinding/cutting
machine and 2 hours on the sprayer to manufacture a shade. On any day, the
sprayer is available for at the most 20 hours and the grinding/cutting machine
for at the most 12 hours. The profit from the sale of a lamp is Rs 5 and that
from a shade is Rs 3. Assuming that the manufacturer can sell all the lamps and
shades that he produces, how should he schedule his daily production in order
to maximize his profit?
The Principal value of = =
We have = = = =
She must fly perpendicular to the given plane.
Shortest distance of A (3,2,8) from the
plane is p = = = 11 units
The degree of the given differential equation is 1.
5. = 256
X = 2 and y = -8
Given C(x) =
Marginal contentment (MC) = =
MC at x= 4 =
The vector are perpendicular to each
other if = 0
2 + λ – 6 = 0
The projection of = =
But the projection of is given as 4 units.
λ = 5
Taking logarithm of both sides, we have
y = x
Differetiating both sides w.r.t. x, we
Let N be the number of families having six
children. Let probability of survival of a girl child be given
and that of boy q = . Let X be
the number of girls in the Family.
Then X = 0,1,2,3,4,5,6
Probabilities of 0,1,2,3,4,5,6, girls in the
family are given by the expansion
Where p(r) =
Probability of 2 girls and 4 boys in the
family of 6
Number of families having 2 girls and 4 boys
in the backward state = N X P(2) = 729 X
Yes, the female child is neglected in backward
areas. Moral education to society and incentives for the female child just like
free education etc.
13. Given y = ..........................(1)
Differentiating both sides w.r.t.x
.................. using (1)
both sides of (2) w.r.t. x, we get
+ 2x + y(2x) +
It is known that,
From equations (2) and (3), we obtain
15. Since the three vectors
are coplanar, therefore = 0 ........(1)
= , by
= , by
2 = 0 (using 1)
⇒ = 0
⇒ are coplanar.
16. L = = 3a
R L = = 3b
Also, f(3) = a(3) + 1 =3a+1
Since f is continuous at x = 3,
3a + 1 = 3b + 3 = 3a+1
3a – 3b – 2 = 0
Hence, the relation between ‘a’ and ‘b’ is 3a-3b-2 = 0
Taking log of both sided of (1) , we have
y = ..................(2)
Differentiating (2) w.r.t.x
⇒ = =
binary operation * on N is defined as:
a * b
= H.C.F. of a and b
It is known that:
H.C.F. of a and b = H.C.F. of b and a
" a, b
\a * b = b * a
Thus, the operation * is commutative.
For a, b, c Î N, we have:
(a * b)* c = (H.C.F. of a and
b) * c = H.C.F. of a, b, and c
a *(b *
c)= a *(H.C.F. of b and c) = H.C.F. of a, b,
\(a * b)
* c = a * (b * c)
Thus, the operation * is associative.
Now, an element e Î N will be
the identity for the operation * if a * e = a = e* a
a Î N.
But this relation is not true for any a Î N.
Thus, the operation * does not have any identity in N.
f: R+ → [4, ∞) is given as f(x) = x2
Let f(x) = f(y).
\ f is a
For y Î [4, _), let y = x2 + 4.
Therefore, for any y Î R, there
exists such that
\ f is onto.
Thus, f is one-one and onto and therefore, f−1
Let us define g: [4, ∞) → R+ by,
Hence, f is invertible and the inverse of f
is given by
18. Put , then x =
dx. = =
= t dt =
= -t =
Let I =
By solving above A = , B = ,C =
Substituting the values of A,B and C, we get
I = =
⇒ = + C
Let -12 =
We know that the volume of a parallelepiped whose three
adjacent edges are is
Now, = =
528 - 6λ
So, the volume of the parallelepiped = =
But the given volume of the parallelepiped is 546 cubic
⇒ 528 - 6λ = 546
λ =-3 .
Applying C1 → C1 + C2 + C3, we have:
Applying R2 → R2 − R1 and R3 → R3 − R1, we have:
Expanding along R3, we have:
Hence, the given result is proved.
22. We have
Put x = a , then dx = 2a d
And a +x = a +
a = a
when x = a, then a = a
Integrating by parts by taking as first function and as the second function
= a = a
Put x = , then dx =
When x= 0, then = 0 and when x = 1, then =
Let I = d
= d = d
= d - d
= d - I
I + I =
area bounded by the parabola, x2 = y,and the line,, can be
The given area is symmetrical about y-axis.
\ Area OACO = Area
The point of intersection of parabola, x2
= y, and line, y = x, is A (1, 1).
Area of OACO = Area ΔOAB – Area OBACO
Þ Area of OACO = Area
of ”OAB _ Area of OBACO
Therefore, required area = units
24. The given system of equations can be written in
matrix form as AX = B
Where A =
X = 2, y = 1, z =
26. The given
differential equation is, ..........(1)
This is a linear
differential equation of the form
, where P =
2 and Q =
I.F. = =
So , y =
⇒ y =
It is given that y
= 0, when x =
⇒ 0 = 2 + C
⇒ C = -2
Substituting this value of C in (2), we get
⇒ y =
As the required
27.The vector equation of line passing through the
Point and is in the direction of is
Since the line (1) is perpendicular to the
There fore , it is parallel to the normal
= in (1), we get
required vector equation of a line.
Rewriting (2) as .......... (3)
The equation of the given plane is
Since the point of intersection of the
line (3) and the plane (4) lies on the plane as well as on the line. Therefore
lies on the plane (4)
Substituting this value of λ in (3) , we
Hence, the required point of intersection
of the given line and the given plane is
28. Let E1, E2, and E3 be
the respective events of choosing a two headed coin, a biased coin, and an
Let A be the event that the coin shows heads.
A two-headed coin will always show heads.
Probability of heads coming up, given that it is a biased
Since the third coin is unbiased, the probability that it
shows heads is always.
The probability that the coin is two-headed, given that
it shows heads, is given by
By using Bayes’ theorem, we obtain
Let E1 and E2 be the respective
events of items produced by machines A and B. Let X be the event that the
produced item was found to be defective.
\ Probability of
items produced by machine A, P (E1)
Probability of items produced by machine B, P (E2)
Probability that machine A produced defective items, P
Probability that machine B produced defective items, P
The probability that the randomly selected item was from
machine B, given that it is defective, is given by P (E2|X).
By using Bayes’ theorem, we obtain
the cottage industry manufacture x pedestal lamps and y wooden
0 and y ≥ 0
The given information can be compiled in a table as
The profit on a lamp is Rs 5 and on the shades is Rs 3.
Therefore, the constraints are
Total profit, Z = 5x + 3y
The mathematical formulation of the given problem is
Maximize Z = 5x + 3y … (1)
subject to the constraints,
≥ 0 … (4)
The feasible region determined by the system of
constraints is as follows.
The corner points are A (6, 0), B (4, 4), and C (0, 10).
The values of Z at these corner points are as follows
Z = 5x + 3y
The maximum value of Z is 32 at (4, 4).
Thus, the manufacturer should produce 4 pedestal lamps
and 4 wooden shades to maximize his profits.