STUPIDSID
Answer the following questions in brief
1(a)
An unbiased coin is tossed 3 times. What is the probability of obtaining two heads?
1 M
1(b)
A 4 sided fair die is thrown twice. What is the probability that the sum of the two outcomes is equal to 6?
1 M
1(c)
Write at least two differences between Secant method and False Position method.
1 M
1(d)
Which method is also known as the method of Tangents? Write down the iterative formula for it.
1 M
1(e)
Define curve fitting.
1 M
1(f)
Which method is used to find the dominant Eigen Value.
1 M
1(g)
If the value of the coefficient of correlation is negative than what does it signify about the relationship of two variables?
1 M
1(h)
Define Mode and also give the relationship between Mean, Median and Mode.
1 M
1(i)
Determine the point of intersection of the regression line of y on x and regression line of y on x.
1 M
1(j)
Suppose you're taking another multiple choice test in Mathematics. The test consists of 40 questions, each having 5 options. If you guess at all 40 questions, what are the mean and
standard deviation of the number of correct answers?
1 M
1(k)
State at least two differences between Newton's Divided difference and Newton's Forward Interpolation method.
1 M
1(l)
In the Gauss elimination method for solving the system of linear equations, name the matrix which is obtained after triangularization.
1 M
1(m)
Determine f ( x , y ) for solving the following differential equation by Euler's method \( 3\dfrac{dy}{dx}+5y^2=\sin x,y(0)=5. \)
1 M
1(n)
Using Picard's method determine the first approximation y1 of the initial value problem \( \dfrac{dy}{dx}=x^2+y^2, y(0)=0. \)
1 M
2(a)
Use trapezoidal rule to evaluate \( \int ^1_0 x^3 dx \) considering five sub intervals.
3 M
2(b)
In a certain assembly plant, three machines, B1, B2, and B3, make 30%, 45%, and 25%, respectively, of the products. It is known from past experience that 2%, 3%,and 2% of the products made by each machine, respectively, are defective. Now,
suppose that a finished product is randomly selected. What is the probability that it is defective?
4 M
Solve any one question from Q.2(c) & Q.2(d)
2(c)
Solve the following system of equations by Gauss - Seidel method:
10x - y - z = 13; x + 10y + z = 36; x + y ' 10z = -35.
10x - y - z = 13; x + 10y + z = 36; x + y ' 10z = -35.
7 M
2(d)
Solve the following system of equations by Gauss elimination method:
x1 + 2x2 + 3x3 = 10; 6x1 + 5x2 + 2x3 = 30; x1 + 3x2 + x3 = 10.
x1 + 2x2 + 3x3 = 10; 6x1 + 5x2 + 2x3 = 30; x1 + 3x2 + x3 = 10.
7 M
Solve any three question from Q.3(a), Q.3(b), Q.3(c) & Q.3(d), Q.3(e), Q.3(f)
3(a)
Derive Newton-Raphson's formula for finding the cube root of a positive number N . Hence find \( \sqrt[3]{12}. \)
3 M
3(b)
From the Taylor's series for y(x), find y(0,1) correct to four decimal places if y(0, 1) correct to four decimal places if y(x) satisfies \( \dfrac{dy}{dx}=x-y^2 \) and y(0) = 1. Also find y(0, 2).
4 M
3(c)
Using Milne's Method, solve \( \dfrac{dy}{dx}=1+y^2 \) with y(0) = 0, y(0, 2) = 0.2027, y(0, 4) = 0.4228, y(0, 6) = 0.6841, obtail y(0, 8) and y(1).
7 M
3(d)
Use secant method to find root of the equation cos x ' x ex = 0 upto four decimal places.
3 M
3(e)
Find by Taylor's series method the value of y at x = 0 . 1 and x = 0 . 2 to four places of decimal from \( \dfrac{dy}{dx}=x^2-1;y(0)=1. \)
4 M
3(f)
Using Runge ' Kutta method to fourth order, solve \( \dfrac{dy}{dx}=\dfrac{y^2-x^2}{y^2+x^2} \) with y(0) = 1 at x = 0.2 with a step ' size of 0.1.
7 M
Solve any three question from Q.4(a), Q.4(b), Q.4(c) & Q.4(d), Q.4(e), Q.4(f)
4(a)
Prove the relation: a) (1+Δ)(1-∇)=1 b) ∇ - Δ = -∇Δ
3 M
4(b)
The speed, v m/s, of a car, t seconds after it starts, is shown in table:
Using Simpson's 1/3rd rule, find the distance traveled by the car in 2 minutes.
| t | 0 | 12 | 24 | 36 | 48 | 60 | 72 | 84 | 96 | 108 | 120 |
| v | 0 | 3.6 | 10.08 | 18.90 | 21.60 | 18.54 | 10.26 | 5.40 | 4.50 | 5.40 | 9.00 |
Using Simpson's 1/3rd rule, find the distance traveled by the car in 2 minutes.
4 M
4(c)
The compressive strength of samples of cement can be modeled by a normal distribution with a mean 6000 kg cm2 and a standard deviation of 100 kg cm2.
(i) What is the probability that a sample's strength is less than 6250 kg cm2 ?
(ii) What is the probability if sample strength is between 5800 and 5900 kg cm2 ?
(iii) What strength is exceeded by 95% of the samples?
[P(z = 2.5) = 0.9938, P(z = 1) = 0.8413, P(z = 2) = 0.9772, P(z = 1.65) = 0.95]
(i) What is the probability that a sample's strength is less than 6250 kg cm2 ?
(ii) What is the probability if sample strength is between 5800 and 5900 kg cm2 ?
(iii) What strength is exceeded by 95% of the samples?
[P(z = 2.5) = 0.9938, P(z = 1) = 0.8413, P(z = 2) = 0.9772, P(z = 1.65) = 0.95]
7 M
4(d)
Prove that: \( a.\ \mu \delta =\dfrac{1}{2}(\Delta + \nabla)\ \ \ \ \ b. \ \ \Delta =E\nabla=\nabla E=\delta E^{1/2} \)
3 M
4(e)
Find \( \int ^6_0\dfrac{e^x}{1+x}dx \) approximately using Simpson's 3/8th rule with h = 1.
4 M
4(f)
In a photographic process, the developing time of prints may be looked upon as a random variable having the normal distribution with a mean of 16.28 seconds and a
standard deviation of 0.12 second. Find the probability that it will take
(i) anywhere from 16.00 to 16.50 seconds to develop one of the prints;
(ii) at least 16.20 seconds to develop one of the prints;
(iii) at most 16.35 seconds to develop one of the prints.
[P(z = 1.83) = 0.9664, P(z = 0.66) = 0.7454, P(z = 0.58) = 0.7190]
(i) anywhere from 16.00 to 16.50 seconds to develop one of the prints;
(ii) at least 16.20 seconds to develop one of the prints;
(iii) at most 16.35 seconds to develop one of the prints.
[P(z = 1.83) = 0.9664, P(z = 0.66) = 0.7454, P(z = 0.58) = 0.7190]
7 M
Solve any three question from Q.5(a), Q.5(b), Q.5(c) & Q.5(d), Q.5(e), Q.5(f)
5(a)
Find the real root of the equation x log10x = 1.2 by Regula - Falsi method correct to four decimal places.
3 M
5(b)
By the method of least square fit a curve of the form y = axb to the following data:
| x: | 2 | 3 | 4 | 5 |
| y: | 27.8 | 62.1 | 110 | 161 |
4 M
5(c)
Find the value of tan 33° by Lagrange's interpolation formula if
tan 30°= 0 . 5774 , tan 32° = 0 . 6249 , tan 35° = 0 . 7002 , tan 38° = 0 . 7813 .
7 M
5(d)
Potholes on a highway can be a serious problem. The past experience suggests that there are, on the average, 2 potholes per mile after a certain amount of usage. It is assumed that the Poisson process applies to the random variable 'number of
potholes.' What is the probability that no more than 4 potholes will occur in a given section of 5 miles?
3 M
5(e)
The pH of a solution is measured eight times by one operator using the same instrument. She obtains the following data: 7.15, 7.20, 7.18, 7.19, 7.21, 7.20, 7.16,
and 7.18. Calculate the sample mean, the sample variance and sample standard deviation.
4 M
5(f)
A study of the amount of rainfall and the quantity of air pollution removed produced the following data:
(i) Find the equation of the regression line to predict the particulate removed from the amount of daily rainfall.
(ii) Find the amount of particulate removed when daily rainfall is x = 4.8 units.
| Daily rainfall x(0.01 cm) | 4.3 | 4.5 | 5.9 | 5.6 | 6.1 | 5.2 | 3.8 | 2.1 | 7.5 |
| Particulate removed, y(μg / m3) | 126 | 121 | 116 | 118 | 114 | 118 | 132 | 141 | 108 |
(i) Find the equation of the regression line to predict the particulate removed from the amount of daily rainfall.
(ii) Find the amount of particulate removed when daily rainfall is x = 4.8 units.
7 M
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