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(Pages : 3)
Reg. No. : ..................................... Name : ........................................
Combined First and Second Semester (S1S2) B.Arch. Degree Examination, June 2009 (2008 Scheme) MATHEMATICS Time : 3 Hours
Max. Marks : 100 PART – A
Answer all questions. Each question carries 5 marks. 3x
1. Find the 8th derivative of e Sin 3x .
(
)
2. Find the radius of curvature of 4ay2 = (2a – x)3 at a , a 2 . 3. Find the equation of the ellipse whose focus is (3, 1), directrix is x – y + 6 = 0 and eccentricity is 1 2 . 4. Find the condition that the line lx + my + n = 0 may be a normal to the hyperbola x 2 y2 − = 1. a2 b2 5. Compute mean of the following distribution : Marks
:
0−10 10−20 20−30 30−40 40−50
No. of Students :
5
10
20
5
10
6. A sample of 12 fathers and their eldest sons gave the following data about their heights in inches. Father
:
65
63
67
64
68
62
70 66
68 67
69
71
Son
:
68
66
68
65
69
66
68 65
71 67
68
70
Calculate the co-efficient of rank correlation. P.T.O.
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7. An urn contains 4 white and 3 red balls. Three balls are drawn with replacement, from this urn. Find the mean of the number of red balls drawn. 8. Fit a Poisson distribution to the following : x
:
0
1
2
3
4
f
:
192
100
24
3
1
PART – B Answer one question from each Module. Each question carries 15 marks. Module – I −1 a Sin x
9. a) If y = e
prove that (1 – x2) yn+2 – (2n + 1) ) xy n+1 – (n2+a2) y n = 0.
b) Find the evolute of the curve given by x = a Cos3 θ and y = a Sin3 θ for x
2
3
+y
2
3
=a
2
3.
10. a) Find the length of the arc of the parabola y2 = 4x from the vertex upto the point (4, 4). b) Find the volume generated by revolving the loop of the curve y2 (a + x) = x2 (a – x) about the x-axis. Module – II 11. a) Derive the equation of the parabola in the form x2 = 4ay. b) If the normal at the point ‘t1’ to the rectangular hyperbola xy = c2 meets it again at the point ‘t2’, prove that t 2 = −
1 . t13
12. a) Find the co-ordinates of the centre, foci and the equations to the directrices of the ellipse 9x2 + 25y2 – 18x – 100y – 116 = 0. x 2 y2 b) Show that the angle between the asymptotes of the hyperbola 2 − 2 = 1 is a b Sec–1(e).
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Module – III 13. a) Find the mode for the following : Mid value
:
15
20
25
30
35
40
45
50
55
Frequency
:
2
22
19
14
3
4
6
1
1
b) Calculate the coefficient of correlation from the following data : x : 23 27
28
28 29
30
31
33
35 36
y : 18 20
22
27 21
29
27
29
28 29
14. a) The scores of two golfers for 10 rounds each are : A : 58 59
60
54
65
66
52
75
69 52
B : 84 56
92
65
86
78
44
54
78 68
Which may be regarded as the more consistent player ? b) By the method of least squares, fit a second degree curve y = a + bx2 to the following data : x: 1
2
3
4
5
y: 2
6
7
8
10 Module – IV
15. a) A continuous random variable X has a probability density function f (x) = 3x2, 0 ≤ x ≤ 1. Find a and b such that 1) P (X ≤ a) = P (X > a) 2) P(X > b) = 0.05 b) Find the probability that atmost 5 defective bolts will be found in a box of 200 bolts if it is known that 2% of such bolts are expected to be defective. 16. a) Find the mean and variance of a binomial distribution. b) In a certain examination, the percentage of candidates passing and getting distinctions were 45 and 9 respectively. Evaluate the average marks obtained by the candidate, the minimum pass and distinction marks being 40 and 75 respectively. (Assume the distribution to be normal) ––––––––––––––––