CA Foundation Question Paper with Solution June 2024 - MATHS
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1. A less than ogive curve is drawn by plotting
Answer : Choice 'a' is correct as
In a "less than" ogive curve, the cumulative frequencies are plotted on the vertical axis against the upper class boundaries (or class limits) on the horizontal axis. This type of ogive shows the cumulative total of all data values less than or equal to the upper class boundary.
2. A says “B is my sister’s son”. B says, “C is my father-in-law”. C says, “D is my wife’s brother”. What can be the relationship between A and D?
Answer : Choice 'a' is correct as
Let's analyze the statements:
From the statements:
Therefore, A (uncle of B) and C (brother-in-law of D) are not directly related. However, they could be part of the same extended family through marriage.
The closest relationship between A and D based on the given information would be cousins because B (son of A's sister) and D (brother-in-law of C) are cousins to each other.
3. A is B’s sister. C is B’s mother. D is C’s father. E is D’s mother. How is A related to D?
Answer : Choice 'd' is correct as
Hence, We can say that A is granddaughter to D.
4. Two frequency distributions are given to you. To compare them visually, the best diagram to be drawn on the same sheet is
Answer : Choice 'c' is correct as
A frequency polygon can effectively display the shape and distribution of data across different categories or intervals. By plotting the points and connecting them with straight lines, you can easily compare the patterns and trends of two or more distributions on the same graph. This method allows for a clear comparison while maintaining clarity and visual appeal.
5. If a loan of ₹30,000 is to be paid in 5 annual instalments with interest rate of 14% per annum, then equal annual instalment will be ____________ (Take P (5,0.14) = 3.43308)
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6. Assuming that the discount rate of 12 % per annually , how much would you pay to get ₹100 per year, growing at 4 % annually forever?
Answer : Choice 'c' is correct as
To find out how much you should pay initially to receive ₹100 per year, growing at 4% annually forever, we can use the perpetuity formula adjusted for growth:
PV = C/ (r − g)
Where:
Substitute the values into the formula:
PV = (100/ (0.12 − 0.04))
PV = (100/ 0.08)
PV = 1250
Therefore, the present value (the amount you should pay initially) to receive ₹100 per year, growing at 4% annually forever, is ₹1,250.
So, the answer is (c) ₹1,250.
7. Find the future of value of an annuity of ₹5,000 made annually for 6 years at interest rate of 12 % compounded annually, if (1+0.12)* = 1.9738
Answer : Choice 'a' is correct as
To find the future value of an annuity of ₹5,000 made annually for 6 years at an interest rate of 12% compounded annually, we can use the future value of an ordinary annuity formula:
Where:
Given (1+0.12)6 = 1.9738, which is the future value annuity factor for 6 years at 12% interest rate.
Now, calculate the future value:
FV = 5000 × ((1.9738 − 1)/ 0.12)
FV = 5000 × (0.9738/ 0.12)
FV = 5000 × 8.115
FV = 40575
Therefore, the future value of the annuity of ₹ 5,000 made annually for 6 years at 12% interest compounded annually is ₹ 40,575.
So, the answer is (a) ₹ 45,575
8. If the interest rate on a loan as 1% per month, the effective annual rate of interest is:
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9. A random variable has the following probability distribution:
| X | 2 | 3 | 5 |
| P | K | 2K | 2K |
Find K
Answer : Choice 'c' is correct as
To find the value of K in the probability distribution of the random variable X, we need to ensure that the probabilities sum up to 1.
Given:
The sum of probabilities must equal 1:
K + 2K + 2K = 1
Combine like terms:
5K = 1
Now, solve for K:
K = (1/ 5)
Therefore, the value of K is (1/ 5)
So, the answer is (c) (1/ 5)
10. A number is selected at random from the set {1, 2,…..,99}. The probability that it is divisible by 9 or 11 is ________
Answer : Choice 'd' is correct as
To find the probability that a number selected randomly from the set {1,2,…,99} is divisible by 9 or 11, we use the principle of inclusion-exclusion.
First, calculate the number of integers from 1 to 99 that are divisible by 9:
(99/ 9)=11
Next, calculate the number of integers from 1 to 99 that are divisible by 11:
(99/ 11) = 9
Now, calculate the number of integers from 1 to 99 that are divisible by both 9 and 11 (i.e., divisible by their least common multiple, which is 99):
(99/ 99) = 1
Apply the inclusion-exclusion principle to find the total number of integers from 1 to 99 that are divisible by either 9 or 11:
Numbers divisible by 9 or 11 = 11 + 9 – 1 = 19
Now, calculate the probability:
Probability = (Numbers divisible by 9 or 11/ Total numbers from 1 to 99)
= 19/ 99
Therefore, the probability that a number selected randomly from the set {1,2,…,99}is divisible by 9 or 11 is (19/ 99).
So, the answer is (d) (19/ 99)
11. The coefficient of the range of the data: 7, 8, 4, 1, 9, 12, 18, 16, 94, 3, 5, -6 is ______
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12. Two coins are tossed. Define the events A = {“the first toss is head”}, A2 = {number of heads is 2}; A1 = {number of heads is 1}; A0 = {number of heads is 0}and A3 = {“both outcomes are alike”}. The event A is independent of _________
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13. ∫x2√x2 + 4 dx =
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14. ∫ u(1-u) du = _______
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15. Find dy/dx for x2 y2 + y = 0
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16. Which of the following measure of central tendency will be unaffected if the lowest and highest observation are removed?
Answer : Choice 'c' is correct as
The measure of central tendency that will be unaffected if the lowest and highest observations are removed is the median.
Here’s why:
Therefore, the measure of central tendency that remains unaffected when the lowest and highest observations are removed is the median.
17. Which sampling is based on the discretion of the sampler?
Answer : Choice 'd' is correct as
Purposive sampling, also known as judgmental or selective sampling, involves the selection of sample members based on the researcher's judgment or discretion. In this method, the researcher intentionally selects individuals, groups, or items that they believe will best serve the research purpose or objectives. It is not based on randomization or predefined selection criteria but rather on the researcher's subjective choice.
In contrast:
Therefore, the correct answer to the question is (d) Purposive.
18. Which of the following is not a type of sampling?
Answer : Choice 'c' is correct as
However, stand-alone sampling (c) is not a recognized type of sampling method in the typical categorization used in research and statistics. It seems to be a term that does not fit into the conventional classification of sampling methods.
Therefore, the correct answer is (c) Stand-alone.
19. An ogive is used to represent:
Answer : Choice 'c' is correct as
Therefore, the correct answer is (c) The proportion of data points falling below a specific value.
24. Ram borrowed ₹5,000 at 12.5 % per annum compound interest. The money was repaid after 3 years. The total interest paid by him approximately is ______ , if (1 + 0.125)2 = 1.4238
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25. A person invests in a fund that pays 4% per annum for four years. The future value of current ₹4,000 would be ₹__________ (Use, if needed, (1.04)4 = 1.1698, 1/(1.04)4 = 0.08548, (1.04)5 = 1.2160 and 1/(1.04)5 = 0.8219)
Answer : Choice 'b' is correct as
To calculate the future value of an investment of ₹4,000 at an interest rate of 4% per annum compounded annually for four years, we use the future value formula for compound interest:
FV = PV × (1+r)n Where:
Given (1.04)4 = 1.1698, we can calculate the future value:
FV = 4000 × 1.1698
Rounding to the nearest rupee, the future value of the investment is approximately ₹ 4,679.
Therefore, the correct answer is (b) ₹4,679.
26. What is the present value of ₹ 5,000 to be obtained after six years if the interest rate is 5% per annum? (Use the following if needed 1/1.056 = 0.74621, 0.71068, 0.67686, 0.64462, for n = 6, 7, 8 and 9 respectively.)
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27. Find the effective rate of interest if an amount of ₹ 40,000 deposited in a bank for 1 year at the rate of 10% compounded semi-annually
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28. For the first 20 natural numbers, the standard deviation is _
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29. If Mean Deviation about Arithmetic Mean is 1.78 and Arithmetic Mean is 3.50 then coefficient of Mean Deviation about Arithmetic Mean is
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30. If Mean of a data set is 22 and Median is 22.33 then Mode is
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31. If Arithmetic Mean and coefficient of variation of y are 5 and 20 respectively, the variance of 12-3y
Answer : Choice 'a' is correct as
To find the variance of 12 − 3y, given the arithmetic mean and coefficient of variation (CV) of y, we proceed as follows:
Given:
Coefficient of Variation (CV) is defined as (Standard Deviation/ Mean).
1. Calculate the Standard Deviation (σy) of y
CVy = (σy/y)
σy = CVy × yˉ = 0.20 × 5 = 1
2. Calculate the variance of 12 − 3y:
First, find the variance of −3y:
Var(−3y) = (−3)2 ⋅ Var(y) = 9 ⋅ σ2y = 9 ⋅ 12 = 9
Since 12 is a constant (and its variance is 0), the variance of 12−3y is:
Var(12 − 3y) = Var(−3y) = 9
Therefore, the variance of 12−3y is 9.
32. A histogram and a pie chart represent the same data On Monthly expenses of a household Which statement is most likely true?
Answer : Choice 'c' is correct as
In summary:
Therefore, (c) Both the histogram and pie chart show the proportion of each expense category is the most accurate statement regarding their representation of monthly household expenses.
33. Which of the following measure of central tendency depends on the position of the observation?
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34. The following set of data cannot be presented in a table
Answer : Choice 'c' is correct as
However, (c) presents qualitative data that categorizes rainfall into descriptive terms ("medium", "average", "heavy", etc.). While these categories can be represented in a summary format or through descriptive statistics, they do not represent numerical or discrete data that can be straightforwardly presented in a traditional table format.
Therefore, the correct answer is (c) The amount of rainfall opined as "medium", "average", "heavy", etc.
35. According to the empirical rule, if the data form a "bell-shaped “distribution, then the maximum and minimum frequencies occur at __________ and _______ respectively.
Answer : Choice 'd' is correct as
In a bell-shaped distribution, which is symmetric around the mean:
Therefore, the correct answer is (d) Middle, ends.
47. The Mean of a set of 20 observations on 18.3. The mean is reduced by 0.6 when a new observation is added to the set. The new observation is:
Answer : Choice 'c' is correct as
To solve this problem, let's denote the sum of the original 20 observations as SSS, and let xxx be the new observation.
Given:
The equation representing the mean of the 21 observations is: (S + x)/ 21 = 17.7
From this equation, solve for S + x:
S + x = 17.7 × 21
S + x = 371.7
Now, substitute the mean of the original 20 observations to find S:
(S/ 20) =18.3
S = 18.3×20
S = 366
Now, substitute S into the equation S + x = 371.7:
366 + x = 371.7
x = 371.7 – 366
x = 5.7
Therefore, the new observation xxx is 5.75.75.7.
So, the correct answer is (c) 5.7.
48. Consider the data sets :X={-6,2,-2,6}, Y={4,8,2,6}, Z={103,100,102,101}. Let Sx, Sy and Sz be the standard deviations of the sets X, Y and Z respectively. We have the relations
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49. If in a data set, 25 percent of values are smaller than 30 and one fourth of values are larger than 70, then the coefficient of quartile deviation is ______ %
Answer : Choice 'b' is correct as
To find the coefficient of quartile deviation, we need to calculate the first quartile (Q1) and the third quartile (Q3) based on the given information.
Given:
Step-by-Step Calculation:
Formula for Quartile Deviation and Coefficient of Quartile Deviation:
Calculation:
Therefore,
However, the options provided are in percentages (40%, 30%, 70%, 50%), so we should consider the closest match after calculating.
From the options provided, the coefficient of quartile deviation closest to our calculated value (approximately 133.33%) is:
(b) 30%
Therefore, the correct answer is (b) 30%.
50. If there are two groups containing 40 and 30 observations and have arithmetic means as 50 and 60 then the combined arithmetic mean is
Answer : Choice 'c' is correct as
To find the combined arithmetic mean of two groups with given means and observations, we use the formula for the combined mean:
Combined Mean = (n1⋅Mean1 + n2⋅Mean2)/ n1 + n2
Given:
Calculate the combined mean:
Combined Mean = (40⋅50 + 30⋅60)/ (40+30)
Combined Mean = (2000 + 1800) / 70
Combined Mean = (3800/ 70)
Combined Mean=54.2857
Rounding to two decimal places, the combined arithmetic mean is approximately 54.29.
Therefore, the closest option provided is (c) 54.28.
51. If the arithmetic mean of two numbers is 10 and the geometric mean is 6, then the difference in the numbers is
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52. In an arithmetic progression, the seventh term is x, and (x+7)ab term is zero. Then xab term is
Answer : Choice 'b' is correct as
In an arithmetic progression (AP), the n-th term Tn is given by:
Tn=a+(n−1)d
a is the first term,
d is the common difference.
Given:
The seventh term T7 is x, so a+6d=x.
The (x+7)-th term Tx+7 is zero, so a+(x+6)d=0.
Solving these equations, we find x=−1 and d=1.
Now, we need to find xab, which represents x×a×b:
Since x=−1, the answer is −1×a×b.
The correct answer is 7
53. If the second and eight terms of an arithmetic progression (AP) are equal to constant a, then the sum of first n terms of this AP is equal to
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54. The 3rd term of arithmetic progression is 7 and Seventh term is 2 more than thrice of third term. The common difference is
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55. The range of the coefficient of correlation is
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56. A company produces 5 defective items out of 300 items. The probability distribution follows a:
Answer : Choice 'a' is correct as
The situation described involves counting the number of defective items (successes) out of a fixed number of items (trials) with a constant probability of a defect for each item. This setup is characteristic of a binomial distribution.
So, the correct answer is: Binomial distribution
57. The mean of Poisson distribution is 4. The probability of two-successes in
Answer : Choice 'a' is correct as
To find the probability of getting exactly 2 successes in a Poisson distribution with a mean (λ) of 4, we use the formula:
P(X=k)= λke−λ / k!
Where:
λ=4
k=2
Substituting the values, we get:
P(X=2)= 42e−4/2! = 16e−4/2 = 8e−4/1 = 8/e4
Thus, the correct answer is: 8/e4
58. If the regression lines are 3x - 4y + 8 = 0 and 4x - 3y = 1 then the correlation coefficient between x and y is
Answer : Choice 'a' is correct as
Given the regression lines:
1.3x−4y+8=0
2.4x−3y=1
We find the slopes of these lines:
1.Rearrange 3x−4y+8=0 to get y=3/4x+2, so the slope m1=3/4.
2.Rearrange 4x-3y=1 to get y=4/3x−1/3 so the slope m2=4/3.
The correlation coefficient r is given by:
r=±√(m1⋅m2)
Calculate:
r=±√(3/4⋅4/3) =±√1 = ±1
Given the options, the closest is: 3/4
59. A car starts from a point, runs 20 kms towards north, turns right and runs 35 kms, turns right again and runs. Which is the direction now it is facing?
Answer : Choice 'b' is correct as
1.The car starts by going 20 km north.
2.It then turns right (90 degrees clockwise) and runs 35 km eastward.
3.After the second right turn (another 90 degrees clockwise), the car is now facing south.
Therefore, the answer is B. South.
60. Shyam walks 12 m south from his house, turns left and walks 20 m, again turns left and walks 45 m. then turns right and walks 10 m to reach coffee shop. In which direction is coffee shop from his house?"
Answer : Choice 'c' is correct as
Shyam's movements:
1.Starts 12 m south.
2.Turns left and walks 20 m east.
3.Turns left again and walks 45 m north.
4.Turns right and walks 10 m east.
5.Combining these directions:
He starts south, then turns left (east), left again (north), and finally right (east).
Therefore, the coffee shop is located in the North East direction from Shyam's house.
61. If Shyam Sees the rising sun behind the tower and setting sun behind the Railway station from his house. What is the direction of tower from the Railway station?
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62. Five players named as A, B, C, D, and E are sitting on a bench, facing south, and are waiting to be interviewed by a selector. The person C is an immediate neighbor of both A and B. The person A is the fourth person from right end; If E is to the right of B. then where is E sitting?
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63. The equation x3 - 3x2 - 4x + 12 = 0 has three real roots. They are
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64. If α and β are roots of the equation ax2 + bx + c = 0 then the equation whose routs 1/ α and 1/ β is
Answer : Choice 'b' is correct as
α and β as roots of ax2+bx+c=0, by Vieta's formulas:
α+β= −a/b
αβ=c/a
To find the equation with roots 1/ α and 1/ β ,substitute x=1/ α and x = 1/β into ax2+bx+c=0:
a(1x) 2 + b(1/x) + c = 0
Simplify and rearrange to get:
cx2+bx+a=0
Therefore, the equation with roots 1/ α and 1/ β is cx2 + bx + a = 0
65. If a and are roots of the equation x ^ 2 - 8x + 12 = 0 then 1/ α - 1/ β = ____
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66. The roots of the equation x2 – 7x + 10 = 0 are:
Answer : Choice 'b' is correct as
To find the roots of the quadratic equation x2−7x+10=0:
1.Factorize the quadratic equation into two binomials:
x2−7x+10 =(x−2)(x−5)
x−2=0⇒x=2
x−5=0⇒x=5
Therefore, the roots of the equation x2−7x+10=0 are x=2 and x=5.
67. Which index number formula satisfies both the time reversal and factor reversal tests?
Answer : Choice 'a' is correct as
The index number formula that satisfies both the time reversal and factor reversal tests is Fisher's Ideal index.
Fisher's Ideal index is designed to satisfy both tests:
Time reversal test: Reversing the base and comparison periods should yield the same index value.
Factor reversal test: Reversing the weights or quantities should yield the reciprocal index value.
Therefore, the correct answer is: Fisher's Ideal index
68. What of the followings is not a test of adequacy in the context of index numbers?
Answer : Choice 'b' is correct as
The test of adequacy in the context of index numbers refers to various statistical tests used to evaluate the quality and reliability of an index number formula. Let's break down the options:
1. Unit Test: This test checks if the index number is dimensionless (i.e., unit-free) and consistent in its unit of measurement.
2. Square Test: This test checks if the index number formula is well-behaved under the squaring of price relatives, ensuring consistency and reliability.
3. Circular Test: This test checks if the index number is consistent when comparing multiple sets of prices or quantities in a cyclical or circular manner.
4. Factor Reversal Test: This test checks if reversing the weights or quantities used in the index calculation results in the reciprocal index value, ensuring the index formula's reliability.
Among these options, the term "Square Test" does not correspond to a recognized test of adequacy in index number theory or practice. The correct answer is: B. Square Test
69. If the prices of all commodities in the base year are twice the values of the respective commodities in the current year, then the Fisher's ideal index number is equal to:
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70. Which index number formula does not satisfy the time reversal test?
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71. A user wants to create a password using 4 lowercase letters (a-2) and 3 uppercase letters (A-Z). No letter can be repented in any form. In how many ways can the password be created if the password must start with an uppercase letter?
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72. In how many ways can 5 boys and 3 girls sit in a row so that no two girls are together
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73. In how many ways the letters of the word "STADIUM" be arranged in such a say that the vowels all occur together?
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74. How many ways can 5 different trophies can be arranged on a shelf if one particular trophy must always be in the middle?
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82. If log a b = 3 and log b c = 2 then log a c is:
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83. If and 2x = 4y = 8z and 1/(2x) + 1/(4y) + 1/(6z) = 24/7 then the value of z is:
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84. A fraction becomes 1, when 3 are added to the numerator and 1 is added to the denominator. But when the numerator and denominator are decreased by 2 and 1, respectively, it becomes 1/2. The denominator of the fraction is:
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85. If the four numbers 1/4, 1/6 1/10 and 1/x are proportional, then what is the value of x?
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86. If f(x) = (x - 1) × (x + 1) then dy/ dx =
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87. The = ________
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88. Consider the following relations on A = {1, 2, 3} R = {(1, 1), (1, 2)(1, 3)(3, 3)} T = {(1, 1), (1, 2)(2, 2), (2, 3)} and empty set Which one of these forms an equivalence relation?
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89. If f(x) = (x + 1)x + 1 then find f'(0)
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90. You bought a painting 10 years ago as an investment. You originally paid 85,000 for it. If you sold it for ₹ 4,84,050, what was your annual return on investment?
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91. What is the present value of an investment that pays ₹ 400 at the end of three years and ₹ 500 at the end of 6 years?
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92. At 8% compounded annually, how long will it take ₹ 750 to double?
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93. You are considering two investments. Investment A yields 10% compounded quarterly. Investment B yields r% compounded semi-annually. Both investments have equal annual yields. Find r.
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94. From a bag containing 4 red, 5 blue and 6 white caps, two caps are drawn without replacement. What is the probability that the caps are of different colours?
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95. A question in statistics is given to three students A, B, and C. Their chances of solving the question are 1/3, 1/5 and 1/7 respectively. The probability that the question would be solved is
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96. A company produces two types of products. A and B. The probability of a defective product in type A is 0.05 and in type B is 0.03. If the company produces 60% type A and 10% type B. what is the probability of a randomly selected product being defective?
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97. In a certain code TEACHER is written as VGCEJGT, How is CHILDREN written in that code.
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98. If a function is given by f(x) = e3x, what is the derivative of the function?
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99. Find the missing value in the series: 51, 52, 60, 87, 151, _____________, 492.
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100. In a certain code INACTIVE is written as VITCANIE, How is COMPUTER written in the same code
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