Calculus and Analytical Geometry Paper 2067 (BSc CSIT)

Tribhuvan University
Institute of Science and Technology
2067

Bachelor Level/ First Year/ First Semeter/ Science                                            Full Marks: 80
Computer Science and Information Technology (MTH 104)                       Pass Marks: 32
(Calculus and Analytical Geometry)                                                                      Time: 3 hours.
Candidates are required to give their answers in their own words as far as practicable.
The figures in the margin indicate full marks.
Attempt all questions.

Group A(10x2=20)

1. Define a relation and a function from a set into another set. Give suitable example.


2. Show that the series  converses by using integral test.


3. Investigate the convergence of the series .


4. Find the foci, vertices, center of the ellipse .


5. Find the equation for the plane through (-3, 0, 7) perpendicular to .


6. Define cylindrical coordinates (r, v, z). Find an equation for the circular cylinder  in cylindrical coordinates.


7. Calculate   for  f(x, y) = 1 – 6x²y,    R : 0 ≤ x ≤ 2,  -1 ≤ y ≤ 1.


8. Define Jacobian determinant for   x = g(u, v, w),   y = h(u, v, w),   z = k(u, v, w).


9. What do you mean by local extreme points of   f(x, y)? Illustrate the concept by graphs.


10. Define partial differential equations of the first index with suitable examples.

Group B(5x4=20)

11. State the mean value theorem for a differentiable function and verify it for the function    on the interval  [-1, 1].


12. Find the Taylor series and Taylor polynomials generated by the function f(x) = cos x  at  x = 0.


13. Find the length of cardioid  r = 1 – cosθ.


14. Define the partial derivative of f(x, y) at a point (x0, y0) with respect to all variables. Find the derivative of   f(x, y) = xe^y + cos(x, y) at the point (2, 0) in the direction of  A = 3i – 4j.


15. Find a general solution of the differential equation .

Group C(5x8=40)

16. Find the area of the region in the first quadrant that is bounded above by    and below by the  x - axis and the line y = x – 2.
    
    OR
    Investigate the convergence of the integrals
    1. 
    2. 
17. Calculate the curvature and torsion for the helix r(t) = (a cos t)i + (a sin t)j + btk, a, b ≥ 0, a²+ b²≠ 0.


18. Find the volume of the region D enclosed by the surfaces  z = x² + 3y² and z = 8 – x² – y².


19. Find the absolute maximum  and minimum values of f(x, y) = 2 + 2x + 2y – x² – y² on the triangular plate in the first quadrant bounded by lines  x = 0,  y = 0  and  x + y = 9.
    
    OR
    Find the points on the curve  xy² = 54 nearest to the origin. How are the Lagrange  multipliers defined?


20. Derive D' Alembert’s solution satisfying the initials conditions of the one-dimensional wave equation.

Calculus and Analytical Geometry Question Paper 2066 (BSc CSIT)

Tribhuvan University
Institute of Science and Technology
2066

Bachelor Level/ First Year/ First Semester/ Science                                   Full Marks: 80
Computer Science and Information Technology (MTH 104)                Pass Marks: 32
(Calculus and Analytical Geometry)                                                                Time: 3 hours.

Candidates are required to give their answers in their own words as for as practicable.
The figures in the margin indicate full marks.
Attempt all the questions.

Group A (10x2=20)

1. Find the length of the curve  from x = 0  to x = 4.


2. Find the critical points of the function  .


3. Does the following series converge?
   


4. Find the polar equation of the circle (x + 2)² + y² = 4.


5. Find the area of the parallelogram where vertices are A(0, 0), B(7, 3), C(9, 8) and D(2, 5).


6. Evaluate the integral .


7. Evaluate the limit


8. Find  if ω = x² + y - z + sin t  and  x + y = t.


9. Solve the partial differential equation p + q = x.


10. Find the general integral of the linear partial differential equation z(xp - yq) = z² - x².

Group B (5x4=20)

11. State and prove Rolle’s theorem.


12. Find the length of the cardioid r = 1 + cos θ.


13. Define unit tangent vector of a differentiable curve. Find the unit tangent vector of the curve r(t) = (cos t + t sin t) i + (sin t - t cos t) j, t > 0.


14. What do you mean by critical point of a function f(x, y) in a region? Find local extreme values of the function f(x, y) = xy - x² - y² -2x - 2y + 4.


15. Find a particular integral of the equation

Group C (5x8=40)

16. Graph the function .


17. What do you mean by Taylor’s polynomial of order n? Obtain Taylor’s polynomial and Taylor’s series generated by the function f(x) = cos x at  x = 0.


18. Find the volume of the region enclosed by the surface z = x² + 3y² and  z = 8 - x² - y².


19. Obtain the absolute maximum and minimum values of the function f(x, y) = 2 + 2x + 2y - x² - y² on the triangular plate in the first quadrant bounded by lines x = 0, y = 0, y = 9 - x.
    OR
    Evaluate the integral  .


20. Show that the solution of the wave equation   is and deduce the result if the velocity is zero.
    OR
    Find a particular integral of the equation    where A, l, m are constants.

Calculus and Analytical Geometry Question Paper 2065 (BSc CSIT)

Tribhuvan University
Institute of Science and Technology
2065
Bachelor Level/ First Year/ First Semester/ Science                                 Full Marks: 80
Computer Science and Information Technology (MTH 104)             Pass Marks: 32
(Calculus and Analytical Geometry)                                                            Time: 3 hours.
Candidates are required to give their answers in their own words as for as practicable.
The figures in the margin indicate full marks.
Attempt all the questions:

Group A [10x2=20]

1. Verify Rolle’s theorem for the function  on the interval [-3, 3].


2. Obtain the area between two curves y = sec²x and y = sin x from x = 0 to .


3. Test the convergence of p – series  for p > 1.


4. Find the eccentricity of the hyperbola 9x² - 16y² = 144.


5. Find a vector perpendicular to the plane of P(1, -1, 0), C(2, 1, -1) and R(-1, 1, 2).


6. Find the area enclosed by the curve r² = 4 cos2θ.


7. Obtain the values of  and   at the point (4, -5) if  f(x, y) = x² + 3xy + y - 1.


8. Using partial derivatives, find   if   x² + cosy - y² = 0.


9. Find the partial differential equation of the function (x - a)² + (y - b)² + z² = c².


10. Solve the partial differential equation x²p + q = z².

Group B [5x4=20]

11. State and prove the mean value theorem for a differential function.


12. Find the length of the Astroid   x = cos³t, y = sin³t  for 0 ≤ t ≥ 2π.


13. Define a curvature of a curve. Prove that the curvature of a circle of radius a is 1/a.


14. What is meant by direction derivative in the plain? Obtain the derivative of the function f(x, y) = x² + xy at P(1, 2) in the direction of the unit vector .


15. Find the center of mass of a solid of constant density δ, bounded below by the disk: x² + y² = 4 in the plane z = 0 and above by the paraboloid z = 4 - x² - y².

Group C[5x8=40]

16. Graph the function f(x) = - x3 + 12x + 5 for  -3 ≤ x ≤ 3.


17. Define Taylor’s polynomial of order  n. Obtain Taylor’s polynomial and Taylor’s series generated by the function f(x) = e^x at x = 0.


18. Obtain the centroid and the region in the first quadrant that is bounded above by the line y = x and below by the parabola y = x².


19. Find the maximum and the minimum values of f(x, y) = 2xy – 2y² – 5x² + 4x – 4. Also find the saddle point if it exists.
    
    OR
    Evaluate the integral   .


20. What do you mean by d’ Alembert’s solution of the one-dimensional wave equation? Derive it.
    
    OR
    Find the particular integral of the equation (D2 – D1)z = 2y – x2 where  

Calculus and Analytical Geometry Model Question Paper (BSc CSIT)

Tribhuvan University
Institute of Science and Technology
Bachelor of Science in Computer Science and Information Technology
Model Question Paper
Bachelor Level/ First Year/ First Semester/ Science                                         Full Marks: 80
Computer Science and Information Technology (MTH 104)                      Pass Marks: 32
(Calculus and Analytical Geometry)                                                                     Time: 3 hours.
Candidates are required to give their answers in their own words as for as practicable.
Attempt all questions.

Group A [10x2=20]

1. Verify Rolle’s theorem for the function   on [-1, 1] and hence find the corresponding point.


2. Find the length of the curve   from  x = 2 to x = 3.


3. Test the p-series    for p a real constant.


4. Find the polar equation of the circle x² + (y - 3)² = 9.


5. Find a spherical coordinate equation for x² + y² +z² = 4.


6. Use double integral to find the area of the region bounded by y = x and y = x² in the forst quadrant.


7. Verify the Euler’s theorem for mixed partial derivatives:  w = x sin y + y sin x + xy .


8. Use the chain rule to find the derivative of  w = xy  with respect to t along the path x = cos t, y = sin t.


9. Form a partial differential equation by eliminating the constants a  and b  from the surface (x - a)² + (y - b)² + z² = c².


10. Solve the partial differential equation  p + q = x , where the symbols have their usual meanings.

Group B [5x4=20]

11. State and prove the mean value theorem  for definite integral. Apply the theorem to calculate the average value of f(x) = 4 - x² on [0, 3].


12. Find the area of the region that lies inside the circle r = 1 and outside the cardioid  r = 1 - cos θ.


13. Find the curvature and principal unit normal for the helix r(t) = (a cos t) i + (a sin t) j + (bt)k  with a, b ≥ 0 and  a² + b² ≠ 0, where the symbols have their usual meanings.


14. What do you mean by directional derivative in the plane? Find the derivative of  f(x, y) = xe^y + cos(xy) at the point (2, 0) in the direction of te vector .


15. Find a particular integral of the equation .

Group C [5x8=40]

16. Graph the function .


17. Find the Taylor’s series and the Taylor’s polynomial generated by f(x) = e^ax and g(x) = x cos x at x = 0.


18. Evaluate the double integral   by applying te transformation  and integrating over an appropriate region in the uv-plane.
    OR
    Find the volume of the region D enclosed by z = x² + 3y² and z = 8 - x² - y².


19. Find the local minima, local maxima and saddle points of the function f(x, y) = 2xy - 5x² - 2y² + 4x +4y - 4.
    OR
    Find the maximum and minimum of the function f(x, y) = 3x - y + 6 subject to the constraint x² + y² = 4 and explain its geometry.


20. Show that the solution of the wave equation   is And deduce the result if the initial velocity is zero.

Probability and Statistics Model Question Paper (BSc CSIT)

Tribhuvan University
Institute of Science and Technology
Model Question Paper

Bachelor Level/ First Year/ First Semester/ Science                             Full Marks: 60
Computer Science and Information Technology (Stat. 103)           Pass Marks: 24
(Probability and Statistics)                                                                        Time: 3 hours.
Candidates are required to give their answers in their own words as for as practicable.
All notations have the usual meaning.

Group A
Attempt any Two:                                                                                               (2x10=20)

1. Scores of 20 students on a test are summarized below. Compute the followings:  mean, median, mode, Q₁, Q₃, and standard deviation. Comment on the shape of the distribution.
   

   X	95	81	59	68	100	92	75	67	85	79
   Y	71	88	100	91	87	65	93	72	83	91



2. 1. Give the classical definition of probability and comment on the limitations of this definition. Describe how probability is used in testing of hypothesis?
   2. The incidence of a disease in a locality is such that the individuals have a 20% chance of suffering from it. What is the probability out of six individuals chosen at random four or more will suffer from the disease?
3. 1. Write the likelihood function of n independent random sample drawn from a Bernolli distribution f(x, θ). Explain this likelihood function in terms of probability.
   2. Obtain the maximum likelihood estimate of the parameter, proportion of success, in binomial distribution.

Group B
Answer any eight questions:                                                                             (8x5=40)

4. A consulting firm rents cars from three agencies, 20% from agency A, 20% from agency from B, and 60% from agency C. If 10% of the cars from A,  12% of the cars from B, and 4% of the cars from C had bad tires, what is the probability that a car with bad tires rented by the firm came from agency C?


5. The following data shows the number of hours jet aircraft engines have been used and the number of hours required for repair.
   

   Aircraft used in 100 hours (X)	1	2	3	4	5
   Repair time in hours (Y)	10	40	30	80	90

   
   Fit a least square line to estimate mean repair time at X = 4.5.


6. Suppose . Find the probability that X lies (a) between 43 and 54, (b) between 40 and 45 (c) between 60 and 75. If population size is 1000, find the number of units between 60 and 75.


7. A book of 200 pages contains 400 misprints. Selecting pages at random, find the probability (i) 3 misprints in 10 pages, (ii) no misprints in 10 pages, and (iii) more than 2 misprints on a single page.


8. A random sample of 16 observations from normal population has mean of 41.5 and sum of square deviation from sample mean is 135. Find 95% confidence interval for population mean.


9. What are the criteria for good estimators? Show that the sample mean is and unbiased and consistent estimator of a population mean.


10. State and prove the additive law of probability. Hence, derive the additive law when two events are (a) mutually exclusive, and (b) independent.


11. Two discrete random variables X and Y have the following joint distribution:
    
    
    Examine whether X and Y are independent.


12. Give the canonical definitions of t-distribution and its density function. Discuss some of its properties.


13. What do you mean by power of the test? The mean height of 100 individuals is also 66 inches; can it reasonably be regarded as a sample from population with mean height 65 inches and standard deviation 2.6 inches?

Probability and Statistics Paper 2067 (BSc CSIT)

Tribhuvan University
Institute of Science and Technology
2067

Bachelor Level/ First Year/ First Semester/ Science                                            Full Marks: 60
Computer Science and Information Technology (Stat. 103)                          Pass marks: 24
(Probability and Statistics)                                                                                        Time: 3 hours.
Candidates are required to give their answers in their own words as for as practicable.
All notations have the usual meanings.

Group A
Attempt any Two:                                                                                                   (2x10=20)

1. State Baye’s Theorem.In a certain assembly plant, three machines B₁, B₂, and B₃ make 30%, 45% and 25% respectively, of the product. It is known from past experience that 2%, 3% and 2% of the products made by each machine, respectively, are defective. If a product were chosen randomly and found to be defective, what is the probability that it was made by machine B₃?


2. 1. Explain point estimation and interval estimation. What are the criteria for good estimators?
   2. If , and assuming that the population is normally distributed, estimate the standard error of the sample mean and estimate 99% confidence interval for the population mean π.
3. 1. Define Karl Pearson’s correlation coefficient and state its properties.
   2. The following table shows the production of coal and the number of wage earners in the coal industry over a ten year period during which the capital equipment has remained constant.
      

      Output in tons (Y)	21	21	20	18	17	17	14	13
      No. of Workers (X)	70	68	65	50	47	47	44	43

      Determine the fitted regression line and predict Y for X = 55.

Group B
Answer any eight questions:                                                                                  (8x5=40)

4. The following data represent the total fat for burgers from a sample of fast-food chains.
   19           31           34           35           39           39           43
   Compute mean, median and mode then describe the shape of the distribution.


5. What is axiomatic definition of probabilities and what are its properties?


6. If two random variables X₁ and X₂ have the joint probability density
   
   Find the conditional density of X₁ given X₂= x₂.


7. Prove that Var(X+Y) = Var(X) + Var(Y) + 2 Cov(X,Y).


8. Find the first and second moments of binomial distribution and also compute variance for the binomial distribution.


9. Service calls come to a maintenance center according to a Poisson process and on the average 2.7 calls come per minute. Find the probability that no more than 4 calls come in any period.


10. 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.


11. Obtain the maximum likelihood estimate for mean (μ) and variance (σ²) of the normal distribution.


12. Define canonical definition of t-distribution. Discuss some of its properties.


13. It is claimed that an automobile is driven on the average more than 20,000 kilometers per year. To test this claim, a random sample of 100 automobiles owners are asked to keep a record of the kilometers they travel. Would you agree with this claim if the random sample showed as average of 23,500 kilometers with a standard deviation of 3900 kilometers?

Probability and Statistics Paper 2066 (BSc CSIT)

Tribhuvan University
Institute of Science of Technology
2066

Bachelor Level/ First Year/ First Semester/ Science                                     Full Marks: 60
Computer Science and Information Technology (Stat. 103)                   Pass Marks: 24
Probability and Statistics)                                                                                  Time: 3 hours.
Candidates are required to give their answers in their own words as for as practicable.
All notations have the usual meanings.

Group A

Attempt any Two:                                                                                             (2x10=20)

1. Define the following three measures of locations – mean, median and mode – and clearly state their properties. Write down a situation where mode is preferred to mean. Score obtained by 14 students in a test are given below. Compute mean, median and mode.
   

   42

   39

   45

   55

   38

   35

   60

   55

   55

   65

   40

   43

   35

   37



2. Explain the terms – sample space and events of a random experiment. State the classical and the statistical definition of probability. Which of the two definitions is the most useful in statistics and why? A survey of 300 families was conducted to study income level versus brand preference. The data are summarized below.
   

   	Brand
   Income level	Brand 1	Brand 2	Brand 3	Total
   High	55	45	20	120
   Medium	45	25	25	95
   Low	25	35	25	85
   Total	125	105	70	300

   
   If a family is selected at random, then compute the probability that (a) the family belongs to high income group, (b) the family prefers Brand 3, and (c) the family belongs to the low income group and prefers Brand 3.


3. Make a clear distinction between correlation coefficient and slope regression coefficient. A school teacher believes that there is a linear relationship between the verbal test score (Y) for eighth graders and the number of library books checked out (X). Following are the data collected on 10 students.
   

   X	12	15	3	7	10	5	22	9	13	7
   Y	77	85	48	59	75	41	94	65	79	70

   
   The above data reveal the following statistics:
   
   
   1. Compute the correlation coefficient r between X and Y. Interpret the meaning of r².
   2. Fit a simple linear regression model of Y on X using the least square method. Interpret the estimated slope regression coefficient.

Group B

Attempt any eight questions:                                                                                (8x5=40)

4. State with suitable examples the role played by the computer technology in applied statistics and also the role of statistics in Information Technology.


5. Define discrete and continuous random variables with suitable examples. A continuous random variable X has the following density function.
   
   Find the value of k show that the total probability would be 1. Also find E(X).


6. Assume that the two continuous random variables X and Y have the following density function
   
   Find (a) marginal density function of X and (b) conditional probability P(2<y<3|x=1).


7. In a binomial distribution with parameters n and p, prove that mean and variance in binomial distribution are correspondingly np and npq, where q = 1 - p.


8. The systolic blood pressure of 18 years old women (X) is normally distributed with a mean of 120 mm Hg and a standard deviation of 12 mm Hg randomly selected 18 years old women. Compute the following probabilities:
   
   1. P(X>150)
   2. P(X<115)
   3. P(110<X<130)


9. If X₁, X₂,.........,X_n are n independent random variables each is distributed as normal with mean μ and variance σ², then derive the distribution of .


10. Write the density function of negative exponential distribution, and derive its mean and variance.


11. Obtain the maximum likelihood function of n independent random sample drawn from a normal population with unknown mean μ and unknown variance σ², and, using the principle of maximum likelihood method of estimation derive the estimators of μ and σ².


12. A survey of 100 percents of first and second grade children revealed that the number of hours per week their children watch television (X) had an average of 25.8 hours and standard deviation of 4.0 hours. The problem is to determine whether there is statistical evidence to conclude that μ (population mean of X) exceeds 25 hours. Set up appropriate null and alternative hypothesis and carry out appropriate test at 5% level of significance.


13. A standardized psychology exam has a mean of 70. A research psychologist wised to see whether a particular drug had an effect on performance on the exam. He administered exam to 18 volunteers who had taken the drug, and obtained the following scores: 68, 71, 71, 65, 64, 70, 70, 64, 71, 73, 62, 78, 70, 69, 76, 67, 69, 72, which yielded  and . The problem is to determine whether there is statistical evidence suggesting that taking drug reduces one/s score on the exam. Set up appropriate null and alternative hypothesis and carry out the test at 5% level.