1 (a)
Find the fourier expansion of f(x) = 4 - x

^{2}in the interval (0,2)
5 M

1 (b)
Find the probability that at most 5 defective fuses will be found in a box of 200 fuses if experience shows that 2% of such fuses are defective.

5 M

1 (c)
Given 6y = 5x + 90, 15x = 8y + 130, (σ

(i) ¯x and ¯y

(ii) r

(iii) (σ

_{x})^{2}= 16. Find :(i) ¯x and ¯y

(ii) r

(iii) (σ

_{y})^{2}
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1 (d)

Solve the two dimensional heat equation\(\dfrac{d^{2}u}{dx^{2}}+\dfrac{d^{2}u}{dy^2}=0\) which satisfies the conditions u(0, y)=u(l, y)=u(x, 0) and u(x, a) = sin \(\dfrac{n \pi x}{l}\)

5 M

2 (a)
Obtain fourier series for f(x) = x - x

^{2}, -π < x < π. Hence deduce that:
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2 (b)
Seven dice are thrown 729 times. How many times do you expect atleast 4 dices to show three or five?

7 M

2 (c)
A continuous random variable X had pdf f(x) = kx

^{2}e^{-x}, x ≥ 0.Find k mean and variance.
6 M

3 (a)
Using normal distribution find the probability that in a group of 100 persons there will be 55 males assuming that the probability of a person being male is 1/2

7 M

3 (b)
Derive wave equation for vibration of string.

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3 (c)
Obtain fourier expansion of f(x)= sin ax in the interval (-l,l) where a is not an integer.

6 M

4 (a)
Calculate the correlation coefficient from the following data.

X : | 23 | 27 | 28 | 29 | 30 | 31 | 33 | 35 | 36 | 39 |

Y : | 18 | 22 | 23 | 24 | 25 | 26 | 28 | 29 | 30 | 32 |

7 M

4 (b)
A die was thrown 132 times and the following frequencies were observed

No. obtained : | 1 | 2 | 3 | 4 | 5 | 6 |

Frequency : | 15 | 20 | 25 | 15 | 29 | 28 |

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4 (c)
Obtain complex form of fourier series for f(x) = cosh 3x + sinh 3x in (-3,3)

6 M

5 (a)
A homogenous rod of conducting material of length l has ends kept at zero temperature and the temperature at centre is T and falls uniformly to zero at the two ends. Find the temperature u(x,t) at any time.

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5 (b)
Obtain half range sine series for f(x) when

f(x) = x, for 0 < x < π/2

f(x) = π-x,for π/2 < x < π

Hence deduce

f(x) = x, for 0 < x < π/2

f(x) = π-x,for π/2 < x < π

Hence deduce

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5 (c)
Two independent samples of sizes 8 and 7 gave the following results

Sample 1 : | 19 | 17 | 15 | 21 | 16 | 18 | 16 | 14 |

Sample 2 : | 15 | 14 | 15 | 19 | 15 | 18 | 16 |

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6 (a)
Find the expansion of f(x) = x(π-x), 0 < x < π as a half range cosine series. Hence show that

(i) (ii)

(i) (ii)

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6 (b)
The diameter of a semicircular plate of radius a is kept at 0°C and the temperature at the semicircular boundary T°C. Find the steady state temperature u(r,θ)

7 M

6 (c)
The average of marks scored by 32 boys is 72 with standard deviation 8 while that of 36 girls is 70 with standard deviation 6. Test at 1% level of significance. Whether the boys perform better than the girls?

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7 (a)
Show that the functions f

_{1}(x)=1; f_{2}(x)=x are orthogonal on (-1,1). Determine the constants a and b such that the function f_{3}(x)=-1+ax+bx^{2}is orthogonal to both f_{1}and f_{2}on that interval.
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7 (b)
Find fourier integral represention of

f(x) = x for 0 < x < a

f(x) = 0 for x > a.

and f(-x) = f(x)

f(x) = x for 0 < x < a

f(x) = 0 for x > a.

and f(-x) = f(x)

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7 (c)
If u=x-y, v=x+y and if x,y are uncorrelated, prove that:

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