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SET

HOT QUESTIONS- MATHEMATICS - 2012-2013

Class XII

Time: 3 hrs.

M.M .100

 

General Instruction:

(i) All questions are compulsory.

(ii) The question paper consists 3 section --------A,B & C.

Section A comprises ten questions of one mark each.

Section B comprises twelve questions of four marks each.

Section c comprises seven questions of six marks each.

SECTION A

1. What is the principal value of

2. Write 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 = 256. Find

6. Find the values of x and y if: 2 +

7. The 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 consumed.

8. For what value of are the vectors and perpendicular

to each other.

9. Find λ when the projection of and is 4 units.

10. Evaluate:

 

SECTION B

 

11. If, find

OR

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 society?

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

are also coplanar.

16.Find the relationship between “a” and “b” so that the function ‘f’ defined by:

f(x) = is continuous at x= 3.

OR

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?

OR

Consider f: given by f(x) = . Show that f is invertible with the inverse (, where is the set of all non- negative real numbers.

 

18. Evaluate: dx.

OR

Evaluate: dx.

19. Prove the following:

20. The volume of the parallelepiped whose edges are -12

546 cubic units. Find the value of

21. Prove, using properties of determinants:

= 2

22. Prove that: = -

OR

Evaluate: dx

 

SECTION C

23. Find the area of the region bounded by the parabola y = x2 and y =

24.Using matrices, solve the following system of equation:

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?

OR

Show that 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

and perpendicular to the plane , Also find the point of intersection of this line and the plane.

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?

OR

A factory 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?

 

 

 

SOLUTIONS 2012-13

1. The Principal value of = =

2. We have = = = = 1

3. She must fly perpendicular to the given plane.

Shortest distance of A (3,2,8) from the plane is p = = = 11 units

4. The degree of the given differential equation is 1.

5. = 256

=

Now

6. 2

X = 2 and y = -8

7. Given C(x) =

Marginal contentment (MC) = =

MC at x= 4 = = 112

8. The vector are perpendicular to each other if = 0

2 + λ – 6 = 0

λ= 4.

9. The projection of = = =

But the projection of is given as 4 units.

= 4

λ = 5

10. Let = after applying

= 0

11. Given:

Taking logarithm of both sides, we have

y = x

Differetiating both sides w.r.t. x, we get

1

=

OR

 

12. Let N be the number of families having six children. Let probability of survival of a girl child be given

p = 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

P(2) = =

 

Number of families having 2 girls and 4 boys in the backward state = N X P(2) = 729 X = 240

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)

........... (2)

Now differentiating both sides of (2) w.r.t. x, we get

+ 2x + y(2x) +

+


 

14.

It is known that,

From equations (2) and (3), we obtain

15. Since the three vectors are coplanar, therefore = 0 ........(1)

Now,

= , by definition

= , by distributive law

=

=

=

= + by distributive law

=

=

= = 2 = 0 (using 1)

= 0

are coplanar.

 

 

16. L = = 3a + 1

R L = = 3b + 3

Also, f(3) = a(3) + 1 =3a+1

Since f is continuous at x = 3, therefore

= f

3a + 1 = 3b + 3 = 3a+1

3a – 3b – 2 = 0

Hence, the relation between ‘a’ and ‘b’ is 3a-3b-2 = 0

OR

Given .............(1)

Taking log of both sided of (1) , we have

y

y

y

y

y = ..................(2)

Differentiating (2) w.r.t.x

=

=

 

= = =

 

17. The 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 Î N.

\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, and c

\(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.

OR

f: R+ [4, ∞) is given as f(x) = x2 + 4.

One-one:

Let f(x) = f(y).

\ f is a one-one function.

Onto:

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 exists.

Let us define g: [4, ∞) R+ by,

\

Hence, f is invertible and the inverse of f is given by

18. Put , then x = and

dx. = =

= t dt = -t

= -t =

OR

Let I =

Let =

By solving above A = , B = ,C =

Substituting the values of A,B and C, we get

 

=

I = =

= + C

 

19.

20. Let -12 = and =

We know that the volume of a parallelepiped whose three adjacent edges are is

Now, = = 528 - 6λ

So, the volume of the parallelepiped = = cubic units.

But the given volume of the parallelepiped is 546 cubic units.

=

528 - 6λ = 546

λ =-3 .

 

 

 

 

 

·         21)

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 = a

·        when x = a, then a = a

·        

·         Now =

·         =

 

 

 

=

=

=

Integrating by parts by taking as first function and as the second function

= a

= a

= a = a =

OR

 

We have dx

Put x = , then dx =

When x= 0, then = 0 and when x = 1, then =

dx = d =

Let I = d

= d

= d

= d = d

= d - d

= d - I

I + I = =

I =

 

 

 

 

 

 

 

 

 

 

 

23. The area bounded by the parabola, x2 = y,and the line,, can be represented as

The given area is symmetrical about y-axis.

\ Area OACO = Area ODBO

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 =

 

 

= 18

Adj A = =

X =

=

= =

X = 2, y = 1, z = 3

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 = .............................(2)

It is given that y = 0, when x =

.................................(3)

⇒ 0 = 2 + C

⇒ C = -2

Substituting this value of C in (2), we get

y =

⇒ y =

As the required solution.

27.The vector equation of line passing through the Point and is in the direction of is =

= ......(1)

Since the line (1) is perpendicular to the plane,

There fore , it is parallel to the normal = i.e.

Putting = = in (1), we get

= ................. (2)

as required vector equation of a line.

Rewriting (2) as .......... (3)

The equation of the given plane is

................(4)

Since the point of intersection of the line (3) and the plane (4) lies on the plane as well as on the line. Therefore from (3)

lies on the plane (4)

⇒ λ =

Substituting this value of λ in (3) , we get

=

=

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 unbiased coin.

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 coin= 75%

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

P (E1|A).

By using Bayes’ theorem, we obtain

OR

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 (X|E1)

Probability that machine B produced defective items, P (X|E2)

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

 

 

29. Let the cottage industry manufacture x pedestal lamps and y wooden shades. Therefore,

x ≥ 0 and y ≥ 0

The given information can be compiled in a table as follows.

 

Lamps

Shades

Availability

Grinding/Cutting Machine (h)

2

1

12

Sprayer (h)

3

2

20

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,

… (2)

… (3)

x, y ≥ 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

Corner point

Z = 5x + 3y

 

A(6, 0)

30

 

B(4, 4)

32

Maximum

C(0, 10)

30

 

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.

 

 

 

 

 

 

 

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