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Question Bank Class : XII Subject :Maths

1. Is R defined on the set A={1,2,3,………14,15} defined as R={(x,y):3x-y=0} reflexive?

Ans. No. Let ,

2. Find the angle between the vectors . Ans.

3. Evaluate ?

Ans. 4. Sol

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Ans. (x2 - x + 1) (x + 1) - (x + 1) (x - 1)

= x3 - x2 + x + x2 - x + 1 - (x2 - 1)

= x3 + 1 - x2 + 1

= x3 - x2 + x2

5. Evaluate: tan-l Ans.

=

6. If Find

Ans.

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7. Differentiate

Ans.

8. Find if

Ans. Differentiate both side w.r.t. to x, x3 + x2y + xy2 + y3 = 81

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9. Evaluate:

Ans.

10. Solve the diff. equ. sec2x.tan y dx + sec2 y tan x dy = 0

Ans. Sec2x. tan y dx = -sec2y tan x dy

11. Find the direction ratios and the direction cosines of the vector Ans. D.R of

D.C of

12. In a school there are 1000 students, out of which 430 are girls. It is known that out of

430, 10% of the girls study in class XII. What is the probability that a student chosen

randomly studies in class XII given that the chosen student is a girl?

Ans. Let E, student chosen randomly studies in class XII, F randomly chosen student is girl.

P (EIF) =?

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= = 0.1

13. Find x and y if x + y = and x - y =

Ans. 14. Find the values of K so that the function f is continues at the given value of x.

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Ans.

K=6 15. The length x of a rectangle is decreasing at the rate of 3 cm/ mint and the width y is

increasing at the rate of 2cm/min. when x = 10cm and y = 6cm, find the ratio of change

of (a) the perimeter (b) the area of the rectangle.

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Ans.

(a) Let P be the perimeter

(b)

16. Find the interval in which the function given by f(x) = 4x3 - 6x2 - 72x + 30 is (a)

strictly increasing (b) strictly decreasing.

Ans.

int Sign of f'(x) Result

+ive

Increase

+ive

Decrease

+ive

increase

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Hence function is increasing in and decreasing in (-2, 3)

17. Integrate

.Ans.

18. Two unbiased dice are thrown. Find the probability that neither a doublet nor a

total of 10 will appear?

Ans. Let A denote the event of getting a doublet.

Let B denote the event of getting a total of 10.

For A favorable cases are: {(1,1),(2,2),(3,3),(4,4),(S,S),(6,6)}

Thus, P(A)=6/36

For B favorable cases are: {(4,6),(S,S),(6,4)}

Thus, P(B)=3/36

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19. A fair coin and an unbiased die are tossed. Let A be the event head appear on the

coin and B be the event 3 on the die.

Check weather A and B are independent events or not.

Ans. A: Head appear on the coin

B : 3 appear on the lice

Hence A and B are independent

20. Find the differential equation of the system of circles touching x-axis at the origin

Ans. The circles in the system will have centres on the y-axis. Let (O,a) be the center of a

circle touching the x-axis at the origin.

Thus, radius= . Equation of circles is:

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

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21. Find if .

Ans.

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

22. For any two vectors

Ans.

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23. Find the vector and Cartesian equation of the planes passing through the

intersection of the planes which are at unit distance from the origin.

Ans. Planes passing through the intersection of the given planes are:

Substituting value of , we get the required equations of the plane as:

24. Let A and B be two sets. Show that f: such that f(a, b) = (b, a) is a bijective function.

Ans. Let (al bl) and (a2, b2) A B

(i) f(al bl) = f(a2, b2)

bl = b2 and al = a2 (al bl) = (a2, b2)

Then f(al bl) = f(a2, b2)

(al bl) = (a2, b2) for all

(al bl) = (a2, b2) A B

(ii) f is injective,

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Let (b, a) be an arbitrary

Element of B A. then b B and a A

(a, b) ) (A B)

Thus for all (b, a) B A their exists (a, b) ) (A B)

Hence that

f(a, b) = (b, a)

So f:

Is an on to function.

Hence bijective 25. Find the vector equation of the plane passing through the intersection of planes

And the point (1,1,1)

Ans.

Using the relation

Plane passes through the point (1,1,1)

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26. Integrate

Ans.

27. Solve the following system of equations using matrix method

Ans. Let

the system of equations becomes,

2u + 3v + lOw = 4

4u - 6v + Sw = l

6u + 9v - 2Ow = 2

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, x = 2

, y = 3

, z = 5 28. A dietician wishes to mix two types of foods in such a way that the vitamin contents

of the mixture contain at least 8 units of vitamin A and 10 units of vitamin C. Food 1

contains 2 units per kg of vitamin A and 1 unit per kg of vitamin C. Food 2 contains 1

unit per kg of vitamin A and 2 unit per kg of vitamin C. Food 1 costs Rs.S0 per kg and

Food 2 costs Rs.70 per kg. Using linear programming, find the minimum cost of such a

mixture.

Ans. Suppose the mixture contains x kg of food 1 and y kgs of food2.

Then, Cost Z=50x + 70y

The mathematical formulation of the problem is as follows:

Min Z= 50x+70y

s.t

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We graph the above inequalities. The feasible region as shown in the figure is unbounded.

The corner points areA,B and C. The co-ordinates of the corner points are (0,8), (2,4),(10,0).

Corner Point Z=50x +70y

(0,8) 560

(2,4) 380

(10,0) 500 Thus cost is minimized by mixing 2 units of food 1 and 4 units of food 2 and minimum cost is

380.

29. Draw a rough sketch of the region and find the area enclosed.

Ans. The point of intersection of the curves y2=4x, 4x2+4y2=9: but is not a possible solution ( , not possible)

The shaded area is the required area.