KUET 2023 Question Paper M.A. M.Sc. Statistics

FOR KUET EXAM PREPARATION

KUET 2023
Question Paper · M.A.
M.Sc. Statistics
EXAM YEAR TYPE SUBJECT

KUET 2023 Question Paper M.A. M.Sc. Statistics

Notes · Sample Papers · Previous Year Papers · Mock Tests

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Sr. No. .............................

ENTRANCE TEST-2023
SCHOOL OF PHYSICAL & MATHEMATICAL SCIENCES

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OPTION—1 : STATISTICS

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OPTION—2 : MATHEMATICS FOR STATISTICS
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e6070 Minutes
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Total Questions
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OPTION—1 : STATISTICS
1. The coefficient of variation is 58%. If mean is 10 7. If the coefficient of determination is equal to 1.
then standard deviation is : Then correlation coefficient :
(A) 5.8 (A) Must also be equal to 1
(B) 580 (B) Can either be –1 or +1
(C) 0.17 (C) Can be any value between –1 to 1
(D) None of these
(D) Must be –1
2. If the distribution is negatively skewed, then :
8. If r = 0.8, byx = 1.5, then bxy is approximately :
(A) Mean is more than the mode
(A) 0.32
(B) Mean is less than the mode
(B) 0.42
(C) Median is at right to the mode
(D) Mean is at right to the median (C) 0.75
3. The Median of scores 25, 45, 35, 35, 40, 30 is : (D) 1
(A) 45 9. Suppose P(X) = 0.36 and P(Y) = 0.41. If
(B) 40 P(X|Y) = 0.27, what is P(Y|X) ?
(C) 35
(A) (0.36)(0.41)
(D) 30 (0.27)
4. The arithmetic mean of the first ten whole numbers
is : (0.27)(0.41)
(B)
(A) 5.5 (0.36)
(B) 5 (0.36)(0.27)
(C) 4 (C)
(0.41)
(D) 4.5
5. The correlation coefficient between two variables (0.27)
X and Y is 0.4. The correlation between 2X and (D)
0.36  0.41
(–Y) will be :
10. If P(A) = 0.32 and P(B) = 0.45, what is P(A  B)
(A) 0.4
if A and B are independent ?
(B) –0.8
(A) 0.144
(C) –0.4
(B) 0.626
(D) 0.8
(C) 0.770
6. In regression analysis, the variable that is used to
explain the change in the outcome of an experiment (D) 0.856
is called : 11. The probability of throwing 10 with 2 dice is :
(A) The independent variable (A) 1/6
(B) The predictor variable (B) 1/12
(C) The explanatory variable (C) 2/3
(D) All of the above
(D) 1/4

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12. Which of the following can be the probability of 17. Let X have the Uniform pdf U(0, 100), then
an event ? variance of X is given by :
(A) –1.3 (A) 1/12
(B) 004 (B) 100/12
(C) 3/8 (C) 1000/12
(D) 10/7 (D) 10000/12

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13. Let X have pmf 18. The point of inflexion of normal curve are :

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x (A) 

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f ( x )  , x  1, 2, 3, 4

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10 (B) 3
Then E(X) is equaleto

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(B) 6 a g
(A) 3 (D) None of these
19. Match the following :
(C) 9 Distribution MGF
(D) 12 a. Normal 1. e(et – 1)
2
14. Given E(X + 4) = 10 and E[(X + 4) ] = 116. b. Gamma 2. et + t2  2/2
Then V(X + 4) is :
c. Poisson 3. (1 – t)–
(A) 4
(A) a–2, b–1, c–4

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(B) 16

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(B) a–3, b–1, c–2

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(C) 3

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(D) 9 (C) a–1, b–3, c–2
15. The variance of probability distribution
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X –3 –2 –1 0 1 2 3 20. If X1, X2, X3, ..... Xn are independent and have
normal distributions N( i,  2i ), i = 1, 2, 3 ..... n,
P(X) 1/7 1/7 1/7 1/7 1/7 1/7 1/7
respectively. Then the distribution of
is :
2
(A) 0 n  x i  i 
W   i  1   is :
(B) 8  i 
(C) 4
(A) Gamma distribution
(D) 12

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(B) Chi-square distribution

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16. The Mean of the distribution

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(C) Normal distribution

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x 10  x
10   2   3 
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(D) t-distribution

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      , x  0,1, 2, ... 10

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 x   5   5

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21. In a test with a standard deviation of 12 and mean

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44 a student scored 41 marks. His Z score is :
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is :

a
(A) 0.50
(B) 5 (B) –0.50
(C) 6 (C) 0.25
(D) 10 (D) –0.25

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22. Arrange the following steps in process of 27. A Medical Assistant sampled the blood pressure
hypothesis testing in proper sequence : of 20 randomly selected patients with high blood
a. Select the level of significance pressure and after receive a dose of a new
b. Setup null and alternative hypothesis medicine, which hypothesis test should she run ?
c. Establish the decision rule (A) t-test for single mean
d. Perform computation (B) F-test for equality of variances
e. Select test statistics (C) Independent t-test for difference of means
f. Draw decision (D) Paired t-test
(A) a, b, c, d, e, f 28. Which of the following are conditions for using
(B) a, b, e, d, c, f the t-distribution for small sample difference
tests ?
(C) b, a, c, d, e, f
I. Samples must be independent.
(D) b, a, e, c, d, f
23. The dividing point between the region where the II. Samples must be drawn from normal
null hypothesis is rejected and the region where populations.
it is not rejected is said to be : III. Samples must be of equal size.
(A) Critical region (A) I only
(B) Significance value (B) I and II
(C) Critical value (C) II only
(D) Acceptance region (D) I and III
24. Which of the following statements best describes 29. The complete list of population, where each
on type I error ? sampling unit is identified by a number is known
(A) Rejecting a null hypothesis when it is true as :
(B) Failing to reject a false null hypothesis (A) Voter list
(C) Accepting a true alternative hypothesis (B) Sampling frame
(D) Rejecting a false alternative hypothesis (C) A list of random numbers
25. For testing of goodness of fit : (D) None of these
(A) The expected frequency should exceed 5 30. The relative efficiency of SRSWOR with SRSWR
(B) The observed frequency should exceed 5 is :
(C) Both the expected and observed frequency
Nn
should exceed 5 (A) N 1
(D) None of the above conditions are necessary
26. What type of data do you need for a Chi square Nn
test ? (B)
N
(A) Ordinal
(B) Interval n
(C)
(C) Ratio N
(D) Categorical N 1
(D)
N

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31. In simple random sampling, the sample mean is : 36. The purpose of stratified sampling is to :
(A) Always zero (A) To ensure that the sample is representative
(B) Smaller than population mean of the population
(C) Equal to population mean (B) To save time and money by not having to
sample the entire population
(D) Random variable
(C) To obtain a biased sample

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32. The finite population correction in usual notation

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is expressed as : (D) To obtain non-random sample

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37. For analysing the completely randomised design

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(A) (N – n)/N

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with t treatments each replicated r times each,

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(B) 1 – (n/N)

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with one missing observation, total degree of

(D) None l a a
(C) Both (A) and (B) freedom are :

33. Which of a
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the following is not true ?
(A) Rt
(B) rt – 1
(A) Var ( y st ) P  Var ( y st ) Ney (C) rt – 2
(D) (r – 1) (t – 1)
(B) Var ( y )SRS  Var ( yst ) P
38. If the total degrees of freedom and between
(C) Var( y)SRS  Var( yst ) Ney b) treatments degrees of freedom in a completely
randomized design are 15 and 4 respectively, the

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(D) All of these degrees of freedom for error will be :

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34. In proportional allocation we have :
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(A) n i  Ni

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N (C) 18
(B) ni = n/k (D) 19
(C) ni = N/k 39. How many factors are involved in a randomized
block design ?
n
(D) n i  N N (A) One
i
(B) Two
35. Which of the following is an example of (C) Three

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systematic sampling ?

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(D) Four

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(A) A researcher selects every 10th person who

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40. In ANOVA, what is the null hypothesis ?

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enters a shopping mall to participate in a

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study (A) There is no difference between the means
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of the groups
(B) A researcher selects a random sample of

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the groups
(C) A researcher selects a convenience sample
of participants from a local community center (C) The groups are not normally distributed
(D) None of these (D) The sample size is too small to draw a
conclusion

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41. At every iteration of simplex method, for a 47. The trend values in freehand curve method are
minimization problem, a variable in current basis obtained by :
is replaced with another variable that has : (A) Equation of straight line
(A) A negative value of Zj – Cj (B) Second degree parabola
(B) The value of Zj – Cj = 0 (C) Graph
(C) A positive value of Zj – Cj (D) All of these
(D) None 48. For the given data semi averages for the second
42. A set of values of decision variables which half is given by :
satisfies the linear constraints and non-negativity Year 2010 2011 2012 2013 2014 2015 2016 2017
conditions of a LPP is known as : Sales 20 16 9 11 40 23 21 12
(A) Solution (A) 14
(B) Feasible solution (B) 22
(C) Optimal solution (C) 23
(D) Unbounded solution (D) 24
43. The optimal value of the objective function is 49. The rate computed by adding the age specific
attained at the points : fertility rates of various age groups of child
(A) Given by intersection of lines representing bearing age is known as :
inequations with axes only (A) Crude Birth Rate
(B) Given by intersection of lines representing (B) Net Reproduction Rate
inequations with X-axis only (C) Total Fertility Rate
(C) Given by corner points of the feasible region (D) General Fertility Rate
(D) At the origin 50. In a given year, the Crude Birth Rate of population
44. If the two constraints do not intersect in the positive of size 1,50,000 is 12. The number of births is :
quadrant of the graph, then : (A) 18
(A) The problem is infeasible (B) 180
(B) The solution is unbounded (C) 1800
(C) One of the constraints is redundant (D) 18000
(D) None of these 51. The population will increase, remains stationary
45. In fitting a straight line, the value of slope b or decreasing according to whether the NRR
remains unchanged with the change of : exceeds, equal or is less than :
(A) Scale (A) 0
(B) Origin (B) 0.5
(C) Both (A) and (B) (C) 1
(D) Neither (A) Nor (B) (D) 1.5
46. Secular trend is indicative of long-term variation 52. The data related to births, deaths, marriage and
towards : divorce is called :
(A) Increase only (A) Mortality
(B) Either increase or decrease (B) Morbidity
(C) Decrease only (C) Vital statistics
(D) None of these (D) Survey

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53. If the sample values lie within the control limits 57. What is estimation in Statistics ?
and are in a random way, we say that the process (A) The process of making a prediction or
is : approximation about a population parameter
(A) Under control based on a sample of data
(B) The process of accurately measuring a
(B) Under warning limits population parameter
(C) The process of collecting dat a on a

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(C) Out of control

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population

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(D) Need to stop the process

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(D) The process of analysing data to draw

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conclusions
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54. If a characteristic follows normal distribution, then

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3 sigma limits covers _____ of observations. 58. A confidence interval will be widened if :

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(A) 0.27% (A) The confidence level is decreased and the
sample size is increased
(B) 68.26% (B) The confidence level is increased and the
(C) 95.44% sample size is reduced
(C) The confidence level is increased and the
(D) 99.73% sample size is increased
55. The chart used to monitor attributes is : (D) The confidence level is decreased and the
sample size is decreased
(A) Range chart
59. Sampling distribution is :

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(B) C-chart (A) A distribution of the population parameters

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(C) Mean chart (B) A distribution of the sample statistics

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(C) A distribution of the individual data points

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(D) All of the above

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in sample

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56. Which of the following is a type of control chart (D) A distribution of the differences between the
used in Statistical Quality Control ? sample and population parameters
60. A function for estimating a parameter is called
(A) Histogram
as :
(B) Box and Whisker Plot (A) Estimate
(C) Scatter Plot (B) Estimation
(C) Estimator

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(D) X chart (D) None of these

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a

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OR
OPTION—2 : MATHEMATICS FOR STATISTICS

1. If 5. The angle of intersection of the curves r = a
x a
f (x)  ,x0 and r  is :
|x| 
= 0 x = 0, 
(A)
then lim x  0 f ( x )  4
(A) 0 
(B)
(B) 1 3
(C) –1
2
(D) None of these (C)
3
2. What type of discontinuity does the function
1 

f ( x )  e x have at x = 0 ? (D)
2
(A) Removable discontinuity 6. The angle between the radius vector and the
(B) Discontinuity of the first kind tangent to the curve r = a(1 – cos ) at any point
(C) Discontinuity of the second kind is equal to :
(D) None of these (A) 
3. Which of the following is true for the function
(B) 2

y  log x  1  x 2
 ?

(A) xy2 + y1 = 0 (C)
2
(B) (1 + x2)y2 + y1 = 0
(C) (1 + x2 )y2 + xy1 = 1 
(D)
3
(D) (1 + x2 )y2 + xy1 = 0
7. The radius of curvature of the curve x2 + y2 = 2
 xy 
4. If u  cos1   , then x u  y u  at the point (1, 1) is :
 x y
 x  y
(A) 2
(A) cos u
(B) 2 cot u 1
(B)
2
1
(C) cot u (C) 2
2
1 1
(D)  cot u (D)
2 2

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8. The number of asymptotes to the curve 11. If f(x) = x2 – 3x – 1, x  [1, 3], then the value of
the “c” of the Mean Value Theorem is :
x 2 y2
 1 (A) 2.75
a 2 b2
(B) 2.5
is : (C) 2.3

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(D) 2.1

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(A) 2

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1

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(B) 1 12. The value of lim x   (1  x ) x is :

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(C) 4

(D) None la
(A) 0

g a
a
(B) 1
(C) e
9. Which of the following is/are true for the function
1
  (D)
f(x) = sin x, x  I  0,  ? e
 2
1
1
1 13. The value of  x
dx is :
(A) There is a value of x  I where f ( x )  0 2e  1

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3

c. o(B) log  2e  1 
(A) log 2
(B) The maximum value of f(x) in I is 1

(C) There is a value of x  I where f (x) = 0 em
la s  e 
(D) All of the above
a g (C) 1
(D) 0
10. The coefficient of xn in the Maclaurin’s series of
– log(1 – x) is : 14. The integral of x tan 1 x with respect to x is
equal to :
1
(A) 3
x 1
n (A) x tan
12
x   log (1  x )  c
2 2

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1

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(B) 

c. o 1
3  23 x 1 

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n (B) x tan
1
x   log (1  x )  c

s
2  2 2

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

s g la x 1
la a
2  23 

g
1
x tan x   log (1  x )  c

a
(C) 3  2 2 
1
(D)  n !  23 1 x 1 
(D) 2  x tan x   log (1  x )   c
 2 2 

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15. Which of the following is/are t rue for 18. Which of the following is a necessary and
 sufficient condition for the differential equation
4
M dx + N dy = 0 to be exact ?
I n   tan n xdx ?
0
M N
(A) 
(A) n(I + I ) = 1 x y
n–1 n–2

1 M N
(B) I n  I n  2  (B) 
n 1 y x
(C) Neither (A) Nor (B)
(C) I1  log 2
(D) Both (A) and (B)
(D) All of the above
19. Which of the following is the particular integral

dx of the differential equation (D3 – 3D2 + 4)y = e2x ?
16. What is the value of  2 2 ?
0 (1  x ) (A) x2e2x

 x 2x
(A) (B) e
4 6

 x2 2x
(B) (C) e
2 6

3 x2 2x
(C) (D) e
2 2

3 20. The complementary function of the differential
(D) equation
4
17. Which of the following substitutions reduces the d2y dy
(3x  2) 2 2
 3(3x  2)  36y  3x 2  4x  1
dx dx
dy
differential equation xy (1  xy 2 )  1 to a is :
dx
linear differential equation ? c2
(A) c1 (3x  2)2 
1 (3x  2) 2
(A) z
x c2
(B) c1 (3x  2) 
1 (3x  2)
(B) z
y
(C) c1 (3x  2) 2  c 2 (3x  2)
(C) xy = z
c2
(D) y = x2 (D) c1 (3x  2)  (3x  2)2

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21. The general solution of the differential equation 25. The values of the Bessel’s functions J0(x) and
p2 + 2py cot x = y2 is given by : J1(x) at x = 0 are respectively :
y (A) 0 and 1
(A) 1  cos x  c
(B) 1 and 0
(B) y(1  cos x) = c
(C) 0 and 0

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(C) x(1  cos y) = c

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(D) 1 and 1

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(D) y(1  sin x) = c

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22. The general solution of the differential equation 26. Which of the following is true ?

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 p  (A) J2(x) = J1(x) – J0(x)

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p  tan  x   is given by :
 1  p2  (B) J2(x) = J1(x) + J0(x)

1 p 1 x
(A) x  tan p  2
,yc (C) J 2 ( x )  J1 ( x )  J 0 ( x )
1 p 1  p2 2

1 p 2
(B) x  c  2
, y  tan1 p  (D) J 2 ( x )  J1 ( x )  J 0 ( x )
1 p 1  p2 x

m
(C) Neither (A) Nor (B) 27. Which of the following is not true for the Legendre
(D) Both (A) and (B)
c. o polynomial Pn(x) ?

em (A) P (1) = 1
las
23. The solution of the differential equation
n

ag
yp + px – p2x – y = p
(B) Pn(–x) = (–1)nPn(x)
is given by :
(C) P2n + 1(0) = 0
(A) y = x + c
( 1)n (2n ) !
x2 (D) P2 n (0) 
(B) y  c ( n ! )2
2
1
c
(C) y  cx  28. For m = n,  Pm (x ) Pn (x ) dx 
m
c 1

c. o
0

m
c. o m
(D) y = c x

e
(A) 0

em a s
24. The substitution X = x2, Y = y2 reduces the

s l
(B) 1

la (A) Linear form ag
equation (px – y) (py + x) = 2p to :

ag
2
(C) 2n  1
(B) Bernoulli’s form
(C) Clairut’s form 2
(D) None of these (D) n  1

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29. The order and the degree of the partial differential 33. Which of the following sets is not countable ?
3
 2u  u  (A) The set of natural numbers
equation    are respectively :
xy  z  (B) The set of rational numbers
(A) 2 and 2 (C) The set of complex numbers with rational
(B) 2 and 1 real and imaginary parts

(C) 1 and 2 (D) The set of points in the closed interval [1.2]

(D) 2 and 3 34. Which of the following is a bounded set ?

30. The solution of the partial differential equation (A) The set of natural numbers

 2z (B) The set of rational numbers
2
 z  0 , given that when x = 0, z = ey and
x  1 1 1 
(C) The set  1, , , , ......
z  2 3 4 
 1 , is :
y
(D) The set of positive real numbers
x
(A) z = e cos y + sin y 35. Which of the following is true ?
y
(B) z = e cos y + sin x (A) Every infinite set has a limit point
y
(C) z = e cos x + sin x (B) Every bounded set has a limit point
x
(D) z = e sin y + cos x
(C) Every finite set has a limit point
31. The general solution of the partial differential
(D) Every infinite bounded set has a limit point
equation x2(y – z)p + y2(z – x)q = z2(x – y) is :
36. If for any real numbers x, y and a, |x – a| < |y – a| < ,
(A) x2 + y2 + z2 = f(x + y + z)
then |x – y| <
(B) xyz = f(x2 + y2 + z2)
(A) 
 2 2 2 1 1 1 
(C) f  x  y  z ,     0 (B)
 x y z 2

1 1 1  (C) 2
(D) f    , xyz   0
x y z  (D) a + 
32. The complete solution of partial differential 37. For what value of an the sequence {an} oscillates
equation p = eq is : infinitely ?
(A) z = ax + y log a + c (A) an = (–1)n
(B) z = ax + log y + c (B) an = (–1)n n
(C) z = ax + by + c (C) an = n2
(D) z = ay + x log a + c (D) an = –2n

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m
m c. o
c. o

e m
em l a s
s ag
38. The limit of the sequence {an}, where 41. Which of the following is not true ?

1 1 1 (A) A positive term series either converges or
an  1   2  ......  n ,
3 3 3 diverges to +

is : 
(B) The series  a n is convergent if and only

m
(A) 1 n 1

m .co
if limn  = 0

co m
1
. s e
(B)

m la
3
e


s g
(C) The series  a n is convergent if and only

l a a sums is
ag
n 1
2
(C) if the sequence of its partial
3
convergent
3
(D) (D) The sum of two convergent series is
2
convergent
39. For what of an is the sequence {an} a monotonic
decreasing sequence that is not convergent ? 42. For what value of x is the series

m
c. odivergent ?
1 1 + x + x2 + x3 + .....
(A) a n 

em
n

(B) an = –n
l a s
g
(A) x < –1

(C) a n 
n 1
n
a (B) x = –1

(C) x  1
n
(D) a n  2
n 1 (D) –1 < x < 1

40. If b and c are positive real numbers, then the 
n

m
43. For what value of x is the series  xn

c. o
cn 2
  n 1

m
b n 1

c. o
sequence 1    converges to :

e m
 n  convergent ?

em (A) e l a s
s
la (B) e ag
(A) x < 1

ag b
(B) x > 1
c
(C) e (C) x = 1
(D) ebc (D) None of these

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44. Which of the following is true for the series 47. Which of the following is not true for the sequence
2
 
( 1) n  1 {fn}, where fn(x) = nxe–nx , n = 1, 2, 3, .... and
 a n and  b n , where a n  n and f(x) = limn  fn(x) ?
n 1 n 1
1
(  1) n  1 n
bn 
2n  1
? (A)  f ( x ) dx  0
0

1
  1
(A) Both  a n and  b n are convergent (B) lim n    f n ( x ) dx 
n 1 n 1 0 2

 
(C) f ( x )  0, x
(B) Neither  a n Nor  b n is convergent (D) {fn} does not converge uniformly to f
n 1 n 1
48. The radius of convergence of the power series
 
(C)  a n is convergent but  b n is divergent 1 + 2x + 3x2 + 4x3 + ..... is :
n 1 n 1 (A) 1
  (B) 
(D)  a n is divergent but  b n is convergent (C) 0
n 1 n 1

(D) 2
45. Which of the following series is/are uniformly
convergent for all real values of x and 0 < r < 1 ? 49. Which of the following is a group ?
(A) The set of natural numbers under addition


(A)  r n cos nx (B) The set of real numbers under addition
n 1
(C) The set of real numbers under multiplication

(D) The set of complex numbers under
(B)  r n sin nx multiplication
n 1

50. Which of the following conditions makes any

n 2
(C)  r cos n x multiplicative group G into an Abelian group ?
n 1
(A) a , b  G, (a · b)2  a 2 · b2
(D) All of the above
(B) Each element of G is idempotent
46. Which of the following is true for the sequence (C) Each element of G is its own inverse
{ f n}, where fn(x) = tan–1(nx), x  0, n = 1, 2, 3, .... ?
(D) All of the above
(A) {fn} is not uniformly convergent in any
51. What is the order of the symmetric group S4 ?
interval [a, b], a > 0
(A) 4
(B) {fn} is uniformly convergent in [0, b]
(B) 6
(C) {fn} is not point-wise convergent in [0, b] (C) 24
(D) None of these (D) 120

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m
m c. o
c. o

e m
em l a s
s ag
52. The number of elements in the quaternion group Z
56. The order of the quotient group , where Z is
is equal to : N
(A) 4 the additive group of integers and N  3 , the
(B) 6 subgroup of Z consisting of all multiples of 3, is :
(C) 8 (A) 1

m
.co
(B) 2

m
(D) 24

o
(C) 3

.c e m
(A) A non-emptymsubset S of a group G 57. Which of the following rings is not free from s
53. Which of the following is not true ? (D) Infinity

s e g l a
a
zero

l a
bg
is a subgroup of G if and only if divisors ?
a , a 1
 S, ab  S (A) Ring of integers
(B) Ring of all square matrices of order 2
(B) For any subgroup S of a group G, O(S)|O(G)
(C) Ring of integers modulo a prime number
(C) For any element a in a group G, aO(G) = e
(D) Ring of complex numbers
(D) For any two subgroups A and B of G, A  B 58. If f and g are any two non-zero polynomials over
is also a subgroup of G a ring R without proper zero divisors, then

m
c. o(B) is less than deg f + deg g
54. Which of the following is a generator of the cyclic degree(f.g) :

em
th
group formed by the n roots of unity ? (A) is less than max(deg f, deg g)

l a s
2 i

g
(A) e n (C) is equal to deg f + deg g

(B) e
i
n
a (D) is equal to deg f. deg g
59. Which of the following is not true ?
(C) e2n (A) Every finite integral domain is a field
(D) None of these (B) Every field is an integral domain
(C) Every ideal of a ring R is a subring of R
55. Which of the following is not true for the
(D) The ring of integers is an ideal of the ring of

m
multiplicative group G = {1, –1, i, –i}, where

c. o
rational numbers

m
i  1 ?

c. o m
60. Which of the following is true for any element a

s e
em (B) G is a cyclic group with two generators
in a ring R and the subset S = {x  R; xa = 0} ?
(A) S is a subringaof R
(A) G is an abelian group

s g l of R
gla (C) O(i) = 4 a
a
(B) S is a right ideal
(C) S is an ideal of R
(D) G has no subgroup of order 2 (D) None of these

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ROUGH WORK

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