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1.By using vector method find the locus of a point which is equidistant from the points A (3, 4, –5) and B (–2, 1, 4)

2. Prove that by vector method

sin (A + B) = sin A . cos B + cos A . sin B

3. Find the vector equation of a sphere described on the join of the points A (2, –3, 4) and B (–5, 6, –7) as the opposite ends of a diameter Deduce the equation in cartesian form. Also find the center and radius of the sphere.

4. There are two points A (1, 3, 4) and B (1, –2, –1). A point p moves Such that 3PA = 2PB find the locus of P and prove that it is a sphere.

5. Prove by vector method

sin (α – β) = sinα. cosβ – cosα .sinβ

6. AB is the diameter of the sphere x^{2} + y^{2} + z^{2} –3x –2y + 2z – 15 = 0. the coordinate of A are (–1, 4, –3). find the coordinate of point B

7. Find the equation of the sphere passing through the points (3, 0, 0), (0, –1, 0), (0, 0, –2) and having the centre on the plane 3x + 2y + 4z = 1.

8. One particle is moving in straight line the distance s travelled by its given relation s = 4t ^{3} + 2t ^{2}. Find the velocity and acceleration of the particle after t = 4 sec.

9. A dice is thrown twice in throw getting even number is taken success, find the probability distribution of the success.

10. A Coin is tossed twice. find the probability distribution of the number of head.

11. Two cubical dice are thrown simultaneously. find the probability of getting an odd number on the first dice or the sum of 9′.

12. Find the probability of drawing two spades from A well shuffled pack of 52 cards if cards are drawn

(a) with replacement

(b) without replacement.

13. Two cards are drown from a well shuffled pack of 52 cards find the probability that both cards are Red or Ace

14. Solve the differential equation (1 + y^{2} ) dx = (tan^{–1}y – x)dy

15. Find the equation of the plane passing through the point (–1, –1, 2) and perpendicular to the planes 3x + 2y – 3z = 1 and 5x – 4y + z = 5.

16. Find the probability distribution of the number of sixes in three throws of a dice.

17. Solve the following differential equation

(1 + x^{2}) dy = (1 + y^{2}) dx

18. Find the equation of the plane bisecting the acute angle between the planes 3x – 4y + 12z = 26 and 2x – y + 2z + 3 = 0

19. Find the equation of planes possing through the intersection of the planes x + 3y + 6 = 0 and 3x – y – 4z = 0 whose distance from origin is 1.

20. Solve the following differential equation

(e^{x} + e^{–x}) dy = (e^{x} – e^{–x}) dx

21. A plane intersects the co-ordinate axes at point A, B and C respectively. If the centroid of the ∆ABC is (–2, 4, 6) then find the equation of the plane.

22. Find the correlation coefficient between x and y on the basis of two regression lines x + 3y = 11 and 2x + y = 7 calculate the value of x then y = 4

23. Find the angle between the two lines whose direction cosines are given by the *equations l + m + n = 0 and 2l + 2m – mn *= 0

24. Find the value of y from following data, when x = 70 and coefficient of correlation is 0.8.

25. Prove that arithmetic mean of the regression coefficient is greater than the coefficient of correlation.

26. Find the correlation coefficient of the following data:

x 1 2 3 4 5 6 7 8 9

y 9 8 10 12 11 13 14 16 15

27. Following data are related to the expenditure on advertisement and sell at farm:

coefficient of correlation r = 0.9 if the proposed advertising expenditure is ` 10 Caror then find out the expected sell.

28. Verify the Rolle’s theorem for the function f (x) = x^{3} – 6x^{2} + 11x – 6 on [1, 3]

29. The radius of a circle is increasing at the rate of 2 cm/sec. At what rate is the area increasing when the radius is 8 cm.

30. Estimate the value of y from the following data when x = 12

Series x y

Mean 7-6 14-8

S.D. 3-6 2-5

Coefficient of correlation ρ = 0.99

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Please give me maths notes and Imp question for 2020 MP board 12th exam