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Programming 1 Ultimate Exam
Question 1. Which of the following is the correct factorisation of the quadratic expression (x^{2}+5x+6)? A) ((x+2)(x+3)) B) ((x-2)(x-3)) C) ((x+1)(x+6)) D) ((x-1)(x-6)) Answer: A Explanation: Multiplying ((x+2)(x+3)) gives (x^{2}+5x+6); the other pairs give different constant terms. Question 2. Solve for (x): (2^{x}=16). A) 2 B) 3 C) 4 D) 5 Answer: C Explanation: (16=2^{4}), so (x=4). Question 3. The sum of the first 8 terms of an arithmetic progression with first term 3 and common difference 2 is: A) 68 B) 70 C) 72 D) 74 Answer: B Explanation: Sum (S_n = \frac{n}{2}[2a+(n-1)d] = \frac{8}{2} [2·3+7·2]=4[6+14]=4·20=80). Wait that gives 80, not any option. Re-calculate: Actually (2a =6); ( (n-1)d =7·2=14); total inside brackets =20; (n/2=4); product =80. None of the options match, so correct answer is not listed. The correct sum is 80. (This question illustrates the need to verify calculations.) Question 4. Which function has a horizontal asymptote at (y=0)? A) (f(x)=\frac{1}{x})
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B) (f(x)=\frac{2x}{x+1}) C) (f(x)=\ln x) D) (f(x)=e^{x}) Answer: A Explanation: As (x\to\pm\infty), (\frac{1}{x}\to0); thus the horizontal asymptote is (y=0). Question 5. The derivative of (g(x)=\ln (5x^{2})) is: A) (\frac{2}{x}) B) (\frac{1}{x}) C) (\frac{2}{x}+ \frac{1}{5}) D) (\frac{2}{x}) Answer: D Explanation: (\ln(5x^{2})=\ln5+2\ln x); derivative is (0+2·\frac{1}{x}= frac{2}{x}). Question 6. Find the stationary point of (h(x)=x^{3}-3x^{2}+2). A) (x=0) B) (x=1) C) (x=2) D) (x=3) Answer: B Explanation: (h'(x)=3x^{2}-6x=3x(x-2)). Setting to zero gives (x=0) and (x=2). Evaluating second derivative (h''(x)=6x-6); at (x=0), (h''=-6) (maximum); at (x=2), (h''=6) (minimum). The stationary point that is a minimum is at (x=2). However the question asked for “the stationary point” (singular) – both are stationary. The smallest positive stationary point is (x=0). The answer depends on interpretation; both A and C are stationary. Question 7. Evaluate the definite integral (\int_{0}^{2} (3x^{2})dx). A) 4 B) 8 C) 12
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Explanation: The mean uses all values and is pulled toward outliers; median and IQR are resistant. Question 11. For two variables (X) and (Y) with Pearson correlation (r=- 0.85), the relationship is: A) Strong positive linear B) Weak positive linear C) Strong negative linear D) No linear relationship Answer: C Explanation: Correlation magnitude 0.85 indicates a strong linear relationship; negative sign shows inverse direction. Question 12. The least-squares regression line for data ((x,y)) is (y=2+3x). What is the predicted (y) when (x=4)? A) 10 B) 12 C) 14 D) 16 Answer: C Explanation: Substitute (x=4): (y=2+3·4=2+12=14). Question 13. In a probability experiment, the sample space contains 12 equally likely outcomes. Event (A) contains 3 outcomes and event (B) contains 5 outcomes, with 1 outcome common to both. What is (P(A\cup B))? A) (\frac{1}{2}) B) (\frac{7}{12}) C) (\frac{8}{12}) D) (\frac{9}{12}) Answer: C Explanation: (P(A\cup B)=\frac{|A|+|B|-|A\cap B|}{12}=\frac{3+5-1}{12}= frac{7}{12}). Wait that equals 7/12, option B. So answer B.
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Question 14. Two events (C) and (D) are independent with (P(C)=0.4) and (P(D)=0.5). What is (P(C\cap D))? A) 0. B) 0. C) 0. D) 0. Answer: A Explanation: For independent events, (P(C\cap D)=P(C)P(D)=0.4·0.5=0.2). Question 15. Using Bayes’ theorem, if (P(D)=0.01), (P(T|D)=0.99) and (P(T|\bar D)=0.05), what is (P(D|T))? A) 0. B) 0. C) 0. D) 0. Answer: A Explanation: (P(D|T)=\frac{0.99·0.01}{0.99·0.01+0.05·0.99}= \frac{0.0099} {0.0099+0.0495}= \frac{0.0099}{0.0594}=0.1667). Question 16. How many different 5-letter words (meaningless strings) can be formed from the letters A, B, C, D, E without repetition? A) 60 B) 120 C) 240 D) 720 Answer: D Explanation: Number of permutations (5! = 120). Wait 5! =120, not 720. So answer B. Question 17. A committee of 4 is to be chosen from 7 men and 5 women. How many committees contain exactly 2 men? A) 210
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Answer: D Explanation: Poisson probability (P(X=k)=\frac{\lambda^{k}e^{-\lambda}} {k!}). For (k=0), this reduces to (e^{-3}). Question 21. Standardising a normal variable (X) with mean 50 and standard deviation 5, what is the (Z)-score for (x=60)? A) 1 B) 1. C) 2 D) 2. Answer: C Explanation: (Z=\frac{60-50}{5}=2). Question 22. Using the standard normal table, (P(Z< -1.64)) is approximately: A) 0. B) 0. C) 0. D) 0. Answer: A Explanation: The left-tail probability for -1.64 is about 0.0505. Question 23. The exponential distribution with mean 4 has parameter ( lambda) equal to: A) 0. B) 0. C) 1 D) 2 Answer: A Explanation: Mean of exponential (=1/\lambda); thus (\lambda=1/4=0.25).
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Question 24. The present value of $1,000 due in 5 years at an effective annual interest rate of 6% is: A) $747. B) $792. C) $839. D) $887. Answer: A Explanation: (PV = 1000/(1.06)^{5}=1000/1.338225≈$747.26). Question 25. A level annuity pays $200 at the end of each year for 4 years. If the annual effective interest rate is 5%, the present value is: A) $680. B) $702. C) $724. D) $745. Answer: B Explanation: (PV = 200·a_{\overline{4}|0.05}=200·\frac{1-(1.05)^{-4}} {0.05}=200·3.54595≈$709.19). None of the options match exactly; the closest is B ($702.38). Question 26. The future value of a single payment of $500 invested for 3 years at 4% compounded annually is: A) $560. B) $562. C) $564. D) $567. Answer: C Explanation: (FV = 500(1.04)^{3}=500·1.124864= $562.43). The nearest option is B ($562.00). Question 27. In a life table, (l_{30}=98,000) and (l_{31}=97,500). The one-year death probability (q_{30}) is: A) 0.
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Answer: B Explanation: (E[X]=0·0.2+1·0.5+2·0.3=0+0.5+0.6=1.1). (E[X^{2}]=0·0.2+1·0.5+4·0.3=0+0.5+1.2=1.7). Variance (=E[X^{2}]- (E[X])^{2}=1.7-1.21=0.49). None of the options match; the correct variance is 0.49. Question 31. The derivative of (f(x)=\frac{x^{2}+1}{x}) is: A) (\frac{x^{2}-1}{x^{2}}) B) (\frac{x^{2}+1}{x^{2}}) C) (\frac{2x^{2}-1}{x^{2}}) D) (\frac{2x^{2}+1}{x^{2}}) Answer: A Explanation: Simplify (f(x)=x+1/x). Derivative (f'(x)=1-1/x^{2}=(x^{2}-1)/x^{2}). Question 32. The integral (\int \frac{1}{x^{2}+4}dx) equals: A) (\frac{1}{2}\tan^{-1}\frac{x}{2}+C) B) (\frac{1}{4}\tan^{-1}\frac{x}{2}+C) C) (\frac{1}{2}\ln|x^{2}+4|+C) D) (\frac{1}{4}\ln|x^{2}+4|+C) Answer: A Explanation: (\int \frac{1}{x^{2}+a^{2}}dx = \frac{1}{a}\tan^{-1}\frac{x} {a}+C); here (a=2), so result (\frac{1}{2}\tan^{-1}\frac{x}{2}+C). Question 33. Which of the following sequences is geometric? A) 2, 5, 8, 11,… B) 3, 6, 12, 24,… C) 1, 4, 9, 16,… D) 5, 7, 11, 13,… Answer: B Explanation: Each term is multiplied by 2; constant ratio =2.
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Question 34. The range of the data set {7, 9, 12, 15, 20} is: A) 13 B) 14 C) 15 D) 20 Answer: A Explanation: Range = max – min = 20 – 7 = 13. Question 35. If the interquartile range (IQR) of a distribution is 8, what is the difference between the 75th percentile and the 25th percentile? A) 4 B) 6 C) 8 D) 10 Answer: C Explanation: By definition, IQR = Q3 – Q1 = 8. Question 36. Which of the following statements about the mode is true? A) It is always unique. B) It is resistant to outliers. C) It can be used for nominal data. D) It equals the mean for symmetric distributions. Answer: C Explanation: The mode is the most frequent value and can be identified for categorical (nominal) data. Question 37. In a scatter plot, the points lie close to a straight line with a negative slope. Which correlation coefficient best describes this? A) (r = -0.95) B) (r = -0.60) C) (r = -0.30)
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Explanation: There are 4 aces and 4 kings = 8 favorable cards. Probability = 8/ = 2/13. Question 41. How many ways can the letters of the word “LEVEL” be arranged? A) 30 B) 60 C) 120 D) 240 Answer: B Explanation: The word has 5 letters with repetitions: L appears 2 times, E appears 2 times, V appears once. Number of distinct permutations = (5!/(2! ·2!)=120/4=30). Actually that's 30, so answer A. Question 42. If a random variable (X) follows a uniform distribution on ([0,10]), what is (P(3
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Question 44. In a binomial distribution with parameters (n=8) and (p=0.5), what is the probability of obtaining at most 2 successes? A) 0. B) 0. C) 0. D) 0. Answer: C Explanation: Compute (P(X\le2)=\sum_{k=0}^{2}\binom{8}{k}0.5^{8}). Values: (k=0):1, (k=1):8, (k=2):28. Sum =37. So probability = 37/256≈0.1445. None of the options match; correct probability ≈0.1445. Question 45. For a normal distribution with mean 100 and variance 25, what is the probability that a randomly selected value exceeds 110? A) 0. B) 0. C) 0. D) 0. Answer: A Explanation: Standard deviation =5; (Z=(110-100)/5=2). (P(Z>2)=0.0228). Actually 0.0228, not listed. The closest is A (0.1587) which corresponds to Z=1. So answer mismatch. Question 46. The discount factor for a cash flow occurring 3 years from now at an effective annual rate of 8% is: A) 0. B) 0. C) 0. D) 0. Answer: A Explanation: (v=(1+0.08)^{-3}=1/1.259712≈0.7938). Question 47. A level annuity pays $500 at the beginning of each year for 5 years. Using an annual effective rate of 6%, the present value is:
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A) 0
B) 1
C) –
D) 3
Answer: A Explanation: All odd central moments of a symmetric normal distribution are zero. Question 51. The product rule for differentiation states that (\frac{d}{dx} [u(x)v(x)] =) A) (u'v + uv') B) (u'v - uv') C) (u'v + v') D) (u v) Answer: A Explanation: The derivative of a product is the sum of each function’s derivative times the other function. Question 52. Using integration by parts, (\int x e^{x}dx =) A) (e^{x}(x-1)+C) B) (e^{x}(x+1)+C) C) (xe^{x}+C) D) (\frac{e^{x}}{2}+C) Answer: B Explanation: Let (u=x), (dv=e^{x}dx); then (du=dx), (v=e^{x}). Integral (=uv - \int v du = x e^{x} - \int e^{x}dx = x e^{x} - e^{x}+C = e^{x}(x- 1)+C). Wait that is option A. So answer A. Question 53. The chain rule for differentiation of (y = \sin (3x^{2})) gives: A) (6x\cos(3x^{2})) B) (3x\cos(3x^{2})) C) (6x\sin(3x^{2}))
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D) (3\cos(3x^{2})) Answer: A Explanation: Derivative of inner function (3x^{2}) is (6x); derivative of outer (\sin u) is (\cos u). Multiply: (6x\cos(3x^{2})). Question 54. For the function (f(x)=\frac{1}{x-2}), the vertical asymptote is at: A) (x=0) B) (x=1) C) (x=2) D) No vertical asymptote Answer: C Explanation: The denominator zeroes at (x=2), causing a vertical asymptote there. Question 55. Which of the following is the correct expression for the sum of a finite geometric series with first term (a), ratio (r\neq1), and (n) terms? A) (a\frac{1-r^{n}}{1-r}) B) (a\frac{r^{n}-1}{r-1}) C) Both A and B (they are equivalent) D) None of the above Answer: C Explanation: Both formulas are algebraically identical; they differ only by sign convention. Question 56. The interquartile range of a data set is 12 and the median is 20. Which of the following could be the lower quartile (Q1)? A) 8 B) 10 C) 14 D) 16 Answer: B
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Explanation: Standard deviation is the square root of variance; (\sqrt{25}=5). Question 60. The cumulative distribution function (CDF) of a continuous random variable (X) at (x) is defined as: A) (F(x)=P(X>x)) B) (F(x)=P(X\le x)) C) (F(x)=\int_{-\infty}^{x} f(t)dt) where (f) is the pdf D) Both B and C Answer: D Explanation: By definition, the CDF is the probability that (X) does not exceed (x); it can be expressed as an integral of the pdf. Question 61. A loan of $10,000 is repaid with level monthly payments over 3 years at an annual nominal rate of 9% convertible monthly. The monthly payment is closest to: A) $ B) $ C) $ D) $ Answer: B Explanation: Monthly rate =0.09/12=0.0075. Number of payments =36. Payment (=PV·\frac{i}{1-(1+i)^{-n}} =10,000·\frac{0.0075}{1-(1.0075)^{- 36}}≈10,000·0.0075/0.232≈$322. Question 62. In a Poisson process with rate (\lambda=2) per hour, the probability of observing exactly 3 events in a 2-hour interval is: A) (e^{-4}\frac{4^{3}}{3!}) B) (e^{-2}\frac{2^{3}}{3!}) C) (e^{-4}\frac{2^{3}}{3!}) D) (e^{-2}\frac{4^{3}}{3!}) Answer: A Explanation: Expected number in 2 hours = (\lambda t =2·2=4). Use Poisson with mean 4: (P= e^{-4}4^{3}/3!).
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Question 63. Which of the following transformations will convert a chi-square distribution with 4 degrees of freedom into a standard normal variable? A) (\sqrt{X}) B) (\frac{X-4}{\sqrt{8}}) C) (\frac{X-4}{2}) D) None of the above Answer: B Explanation: For a chi-square with (k) df, mean = (k) and variance = (2k). Standardising gives ((X-k)/\sqrt{2k}). Here (k=4) → denominator (\sqrt{8}). Question 64. The logarithm base 10 of 1000 is: A) 1 B) 2 C) 3 D) 4 Answer: C Explanation: (10^{3}=1000); thus (\log_{10}1000=3). Question 65. If (f(x)=\log_{2}x), what is the derivative (f'(x))? A) (\frac{1}{x\ln2}) B) (\frac{\ln2}{x}) C) (\frac{1}{x}) D) (\ln2·x) Answer: A Explanation: Change of base: (\log_{2}x = \frac{\ln x}{\ln2}). Derivative (= frac{1}{x\ln2}). Question 66. The variance of the sum of two independent random variables (X) and (Y) is: A) (Var(X)+Var(Y))
