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1.Two sides of a triangle are given. Find the angle between them such that the area shall be maximum.

2. Calculate the correlation coefficient from the following data:

x: 3 4 6 8 9

y: 90 100 130 160 170

3. Prove that the value of correlation coefficient lies between – 1 to + 1

4. solve the differential equation:

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

5. Two dice are thrown simultaneously. Find the probability of getting an even number on the first die or a sum of 8

6. Find the equation of the sphere with center (3, 6, – 4) and touching the plane 2x – 2y – z – 10 = 0

7. The following data are related to the expenditure on advertisement and sale of a firm:

| Sales in Crores | Advertisement in crores |

Mean | 40 | 6 |

Standard Deviation | 10 | 1.5 |

Coefficient of Correlation ϒ = 0.9. if target of sale of a firm is Rs. 60 crore, then how much money will be spent on advertisement?

8. Find the differential Coefficient of x log_{a} x with respect to ‘x’

9. The radius of a balloon is increasing at the rate of 10 cm per sec. At what rate is the surface area of the balloon increasing when its radius is 15 cm?

10. Prove by vector method that mid-point of the hypotenuse of a right angled triangle is equidistant from its vertices.

11. Find two positive numbers, whose product is 64 and the sum is minimum.

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

13. . Find the area of ΔABC, whose vertices are A(1, -1, 2) B(2,1,-1) and C(3,-1,2)

14. If the regression line of y on x is ax +by + c = 0 and that x on y is a_{1}x +b_{1}y +c_{1} = o, then prove that: ab_{1} ≤ a_{1}b.

15. In a class, 30% students have offered \maths, 20% chemistry and 10% have offered both the subjects. One student is to be selected at random. Find the probability of his being a student of Maths or chemistry.

16. A bag contain 50 bolts and 150 nuts. Half of the bolts and half of the nuts are rusted. If one item is taken out at random, what is the probability that it is rusted nut or is a bolt.

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

18. Find the area bounded by the Curve x^{2} = 4y and the line x = 4y – 2

19. Find the acute angle between two lines, whose direction ratios are 2, 3, 6 and 1,2,2 respectively.

20. A ball is thrown vertically upwards. Its equation of motion is S = 490t – 4.9t^{2}. Find the maximum height reached by it.

21. Prove that the Arithmetic means of Regression coefficients is greater than correlation coefficient

22. Two cards are drawn at random from a well-shuffled pack of 52 cards. Find the probability that either both are red or both are Ace.

23. A die is tossed four times. Getting a number greater than 4 is considered a success. Find the probability distribution of the number of successes.

24. Find the angle between two lines, whose direction cosines are given by the equations:

*l + m + n = 0 and 2lm – mn = 0*

25. A directed line makes angle 45° and 60° with x axis and y axis respectively. What angle does it make with z axis?

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

Series | X | Y |

Mean | 7.6 | 14.8 |

Standard deviation | 3.6 | 2.5 |

Coefficient of correlation r = .99.

27. Prove that by vector method:

Sin (α – β) = sin α cos β – cos α sin β.

28. A bag contains 13 ball numbered 1 to 13. An even number is considered success. Two balls are drawn one by one with replacement, from the bag. Find the probability of getting:

i) Two successes

ii) At least one success

iii) No success.

29. A coin is tossed twice. Find the probability distribution of the number of getting heads.

30. A line makes angles α, β, ϒ, δ with four diagonals of a cube, prove that:

Cos^{2} α + cos^{2} β cos^{2} ϒ + cos2 δ = 4/3.

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