CA Foundation Maths Question Paper Jan 25 With Solution

CA Foundation Maths Question Paper

Looking for solutions to the CA Foundation Maths Paper January 2025 with Answers? You’re in the right place! This blog covers everything you need to know about the CA Foundation January 2025 Exam, including detailed solutions and insights to help you excel. We’re here to provide a comprehensive breakdown of the January 2025 Maths Paper

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Question 1. A random variable X has the following probability density function:
f(x) = 6x (1 − x), 0 ≤ x ≤ 1 Then the mean is

(A) 1/3
(B) 1/12
(C) 1/4
(D) 1/2

Answer:

Correct answer is D -

The mean of a random variable X is given by the formula:

E[X] = ∫₀¹ x · f(x) dx

Given the probability density function f(x) = 6x (1 − x), we calculate the mean:

E[X] = ∫₀¹ x · 6x(1 − x) dx

Now, we simplify the expression:

E[X] = 6 ∫₀¹ x² (1 − x) dx
E[X] = 6 ∫₀¹ (x² − x³) dx

Now, compute the integrals:

E[X] = 6 [ ∫₀¹ x² dx − ∫₀¹ x³ dx ]

The integral of x² is x³/3, and the integral of x³ is x⁴/4:

E[X] = 6 [ (x³/3)₀¹ − (x⁴/4)₀¹ ]

So, we get:

E[X] = 6 [ 1/3 − 1/4 ]
E[X] = 6 × 1/12
E[X] = 1/2

Thus, the mean is 1/2, which corresponds to option (D).

Question 2. The standard deviation of the data 2, 4, 5, 6, 8, 17 is 23.33, then the standard deviation of the data 4, 8, 10, 12, 16, 34 is
(A) 46.66
(B) 23.33
(C) 12.23
(D) 0

Answer: A

The standard deviation is a measure of the spread of the data. When each data value in a set is multiplied by a constant, the standard deviation is also multiplied by the same constant.

The original data set is: 2, 4, 5, 6, 8, 17 with a standard deviation of 23.33.

The new data set is: 4, 8, 10, 12, 16, 34.

Notice that each data point in the new set is exactly multiplied by 2 compared to the original set:

4 = 2 × 2

8 = 2 × 4

10 = 2 × 5

12 = 2 × 6

16 = 2 × 8

34 = 2 × 17

When each data value is multiplied by 2, the standard deviation is also multiplied by 2. Therefore, the new standard deviation will be:

New standard deviation = 2 × 23.33 = 46.66

Thus, the standard deviation of the new data set is 46.66, which corresponds to option (A).

Question 3. The values of the first quartile and third quartile are 36.50 and 57.50. Then the semi-inter-quartile range is
(A) 12.50
(B) 47.50
(C) 10.50
(D) 11.50

Answer: c

The semi-interquartile range is calculated using the formula:

Semi-Inter-Quartile Range = (Q₃ - Q₁) / 2

Where:

Q₁ is the first quartile (lower quartile)
Q₃ is the third quartile (upper quartile)

Given:

Q₁ = 36.50
Q₃ = 57.50

Now, substituting the values into the formula:

Semi-Inter-Quartile Range = (57.50 - 36.50) / 2 = 21 / 2 = 10.50

Thus, the semi-inter-quartile range is 10.50, which corresponds to option (C).

Question 4. Let A and B be two possible outcomes of a random experiment and P(A)=1/3, P(A∪B) =1/2 and P(B)=x For what value of x are A and B mutually exclusive events?

(A) 1/6
(B) 1/4
(C) 1/5
(D) 1/8

Answer: A

To determine the value of x for which A and B are mutually exclusive events, we need to recall the rule for the probability of the union of two events:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

For mutually exclusive events, the probability of the intersection P(A ∩ B) is 0, so the formula simplifies to:

P(A ∪ B) = P(A) + P(B)

Given:

P(A) = 1/3
P(A ∪ B) = 1/2
P(B) = x

Substitute these values into the simplified formula:

1/2 = 1/3 + x

Solve for x:

x = 1/2 - 1/3

To subtract the fractions, find a common denominator:

x = 3/6 - 2/6 = 1/6

Thus, the value of x is 1/6, which corresponds to option (A).

Question 5. A random variable has the following probability distribution:

X :	0	1	2	3
P:	1/2	1/3	1/4	1/5

Find the expected value of X.
(A) 1.20
(B) 1.43
(C) 1.80
(D) 2.00

Answer: B

The expected value E[X] of a random variable X with a probability distribution is calculated as:

E[X] = Σ X_i · P(X_i)

Where:

X_i represents the possible values of X,
P(X_i) represents the corresponding probabilities.

Given the probability distribution:

X:  0   1   2   3
P:  1/2 1/3 1/4 1/5

The expected value E[X] is:

E[X] = 0 · 1/2 + 1 · 1/3 + 2 · 1/4 + 3 · 1/5

Now, calculate the sum:

E[X] = 0 + 1/3 + 2/4 + 3/5

To add these fractions, find a common denominator (the least common denominator is 30):

E[X] = 10/30 + 15/30 + 18/30
E[X] = 43/30 ≈ 1.43

Thus, the expected value of X is 1.43, which corresponds to option (B).

Question 6. What will be the mode of the Binomial distribution in which mean is 20 & Standard Deviation is √10

(A) 21
(B) 20.5
(C) 20
(D) 41

Answer:

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Question 7. If X is a Poisson variable such that P(X = 1) = P (X = 2) then the variance is

(A) 1
(B) 2
(C) √2
(D) 3

Answer: B

For a Poisson distribution, the probability mass function is given by:

P(X = k) = (λ^k * e^(-λ)) / k!

Where λ is the mean (and also the variance) of the distribution.

Given:

P(X = 1) = P(X = 2)

Using the Poisson probability mass function:

P(X = 1) = (λ * e^(-λ)) / 1! = λ e^(-λ)
P(X = 2) = (λ^2 * e^(-λ)) / 2! = (λ^2 * e^(-λ)) / 2

We are told that P(X = 1) = P(X = 2), so:

λ e^(-λ) = (λ^2 * e^(-λ)) / 2

Canceling out e^(-λ) (since it is non-zero) and simplifying:

λ = (λ^2) / 2
2λ = λ^2
λ^2 - 2λ = 0
λ(λ - 2) = 0

Thus, λ = 0 or λ = 2. Since λ = 0 is not valid, we have λ = 2.

Variance of Poisson distribution:

For a Poisson distribution, the variance is equal to λ. Therefore, the variance is λ = 2.

The answer is 2, which corresponds to option (B).

Question 8. Which one of the following statements is wrong?
(A) The correlation coefficient between X and Y is 2.6.
(B) The normal curve is bell-shaped.
(C) If r=0, regression lines are perpendicular to each other.
(D) For any two events A and B, P(A∪B)=P(A)+P(B)−P(A∩B)

Answer:

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Question 9. The AM & GM for two observations are 8 and 2. Find the values of two observations.
(A) 16, 1
(B) 15.75, 0.25
(C) 15, 1
(D) 14.75, 1.75

Answer: B

Given:

The arithmetic mean (AM) of two observations is 8.
The geometric mean (GM) of the two observations is 2.

Let the two observations be x and y.

Step 1: Use the formula for AM:

AM = (x + y) / 2
Given AM = 8:
(x + y) / 2 = 8  =>  x + y = 16  (Equation 1)

Step 2: Use the formula for GM:

GM = √(xy)
Given GM = 2:
√(xy) = 2  =>  xy = 4  (Equation 2)

Step 3: Solve the system of equations:

From Equation 1: x + y = 16
From Equation 2: xy = 4

These are the sum and product of the two numbers. Let’s form a quadratic equation:

t^2 - (x + y)t + xy = 0
Substitute the values from Equation 1 and 2:
t^2 - 16t + 4 = 0

Step 4: Solve the quadratic equation:

Using the quadratic formula:
t = [-(-16) ± √((-16)^2 - 4(1)(4))] / 2(1)
t = (16 ± √(256 - 16)) / 2
t = (16 ± √240) / 2
t = (16 ± 15.49) / 2
So the two solutions are:
t₁ = 15.75, t₂ = 0.25

Thus, the two observations are 15.75 and 0.25, which corresponds to option (B).

Question 10. The mean deviation of normal distribution is approximately equal to
(A) 3.14 σ
(B) 0.5 σ
(C) 1.14 σ
(D) 0.8 σ

Answer:

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Question 11. You are given the following data relating to a frequency distribution of 10 observations:
ΣX=50, ΣY=60, ΣX2=300, ΣY2=352, Σ(X+Y)2=1372 Then Cov(X,Y) is
(A) 4
(B) 2
(C) 6
(D) 8

Answer:

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Question 12. If 3 percent of ceramic cups manufactured by a company are known to be defective, what is the probability that a sample of 100 cups taken from the production process of that company would contain exactly one defective cup?
(A) 0.03
(B) 0.15
(C) 0.09
(D) 0.30

Answer: B

This problem involves a binomial distribution since the outcome of each cup being defective or not is a Bernoulli trial.

The probability of exactly k successes (defective cups) in n trials (cups) is given by the formula:

P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)

Where:

n = number of trials = 100 (cups)
k = number of successes = 1 (defective cup)
p = probability of success = 0.03 (probability of a defective cup)
1 - p = probability of failure = 0.97 (probability of a non-defective cup)

Substitute the given values into the formula:

P(X = 1) = C(100, 1) * (0.03)^1 * (0.97)^99

Simplify the binomial coefficient:

C(100, 1) = 100
P(X = 1) = 100 * 0.03 * (0.97)^99

Calculating (0.97)^99, we get approximately 0.046.

Thus:

P(X = 1) ≈ 100 * 0.03 * 0.046 = 0.138

The probability is approximately 0.15, which corresponds to option (B).

Question 13. A population comprises 7 members. The number of all possible samples of size 3 that can be drawn from it with replacement is –
(A) 343
(B) 216
(C) 21
(D) 125

Answer:

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Question 14. Let P(A)=1/5 and P(B¯)=3/5 If A and B are mutually exclusive events, then P(A∪B) is

(A) 3/5
(B) 1/5
(C) 2/5
(D) 4/5

Answer: A

We are given the following information:

P(A) = 1/5
P(B¯) = 3/5, which means P(B) = 1 - P(B¯) = 1 - 3/5 = 2/5
A and B are mutually exclusive events.

For mutually exclusive events, the probability of their union is given by:

P(A ∪ B) = P(A) + P(B)

Substitute the given values:

P(A ∪ B) = 1/5 + 2/5 = 3/5

The correct answer is 3/5, which corresponds to option (A).

Question 15. If r=0.8, b_yx=0.6 , b_xy=0.5 , and yˉ=3 , then the regression equation on x is
(A) y=0.96x−3.7
(B) y=0.6x−6.0
(C) y=0.8
(D) y=0.6

Answer:

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Question 16. Which one of the following indexes uses the method of average of base year and current year?
(A) Paasche’s Index
(B) Laspeyre’s Index
(C) Marshall–Edgeworth Index
(D) Fisher’s Index

Answer: C

The index that uses the method of averaging the base year and current year is the Marshall–Edgeworth Index.

Explanation:

Marshall–Edgeworth Index takes the arithmetic mean of the Paasche’s index and Laspeyres’s index, effectively averaging the values from both the base year and current year.
Paasche's Index uses the quantities from the current year as weights.
Laspeyres's Index uses the quantities from the base year as weights.
Fisher's Index is the geometric mean of the Paasche’s and Laspeyres’s indexes.

The correct answer is (C) Marshall–Edgeworth Index.

Question 17. If r =0.8, then the coefficient of non-determination is
(A) 0.64
(B) 0.36
(C) 0.20
(D) -0.36

Answer:

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Question 18. The correlation coefficient between X and Y is 0.2 and Var(X) = 5Var(Y), then the regression coefficient of X on Y is:
(A) √5
(B) 1/5
(C) 1/√5
(D) 5

Answer: C

The regression coefficient of X on Y is given by the formula:

b_XY = r * (σ_X / σ_Y)

Where:

r is the correlation coefficient between X and Y = 0.2
σ_X is the standard deviation of X
σ_Y is the standard deviation of Y

We are given:

Var(X) = 5 * Var(Y), which means σ_X = √5 * σ_Y

Substitute the values into the regression coefficient formula:

b_XY = 0.2 * (√5 * σ_Y) / σ_Y

The σ_Y terms cancel out, and we get:

b_XY = 0.2 * √5

Thus, the regression coefficient of X on Y is 1/√5, which corresponds to option (C).

Question 19. Compute the rank correlation coefficient from the following data:
n = 10, Σd² = 5

(A) 0.97
(B) 0.95
(C) 0.96
(D) 0.99

Answer:

To compute the rank correlation coefficient r_s, we use the formula:

r_s = 1 - (6 Σd²) / (n(n² - 1))

Where:

n is the number of pairs of observations = 10
Σd² is the sum of the squared differences between the ranks of corresponding values = 5

Substitute the given values into the formula:

r_s = 1 - (6 * 5) / (10 * (10² - 1))
r_s = 1 - 30 / (10 * 99)
r_s = 1 - 30 / 990
r_s = 1 - 0.0303
r_s = 0.9697

The rank correlation coefficient is approximately 0.97, which corresponds to option (A).

Question 20. When the cost of Beverages increased by 40%, the person said that the rise had increased his cost of living by 8%. Before the change in price, the percentage of his cost of living was due to buying Beverages is:
(A) 20%
(B) 15%
(C) 5%
(D) 2%

Answer:

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Question 21. If the prices of all the goods change in the same ratio, then
(A) Laspeyre’s index and Paasche’s index numbers are not equal.
(B) Laspeyre’s index and Paasche’s index numbers are equal.
(C) Laspeyre’s index is greater than Paasche’s index number.
(D) Laspeyre’s index is less than Paasche’s index number.

Answer:

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Question 22. Given that ΣP_nq_n = 300, Σp0q0 = 125 and Paasche’s index number is 200 then the value of Σp₀q_n is:
(A) 150
(B) 125
(C) 250
(D) 100

Answer: A

We are given the following information:

ΣP_nQ_n = 300
ΣP₀Q₀ = 125
Paasche’s index number = 200

Paasche’s index is given by the formula:

P_a = (ΣP_nQ_n / ΣP₀Q_n) * 100

Substitute the given values into the formula:

200 = (300 / ΣP₀Q_n) * 100

Now, solve for ΣP₀Q_n:

ΣP₀Q_n = (300 / 200) * 100 = 1.5 * 100 = 150

The value of ΣP₀Q_n is 150, which corresponds to option (A).

Question 23. If Laspeyre’s index number is 125 and Paasche’s index number is 500 then Fisher’s index number is:
(A) 250.0
(B) 312.5
(C) 62.5
(D) 147.5

Answer:

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Question 24. If x = √2 + 1/√2 and y = √2 - 1/√2 then x² + y² is:
(A) 1/√2
(B) √2
(C) 5
(D) 0

Answer: C

We are given:

x = √2 + 1/√2
y = √2 - 1/√2

We need to find x² + y².

Step 1: Compute x²

x² = (√2 + 1/√2)²
x² = (√2)² + 2 * √2 * (1/√2) + (1/√2)²
x² = 2 + 2 + 1/2 = 4.5

Step 2: Compute y²

y² = (√2 - 1/√2)²
y² = (√2)² - 2 * √2 * (1/√2) + (1/√2)²
y² = 2 - 2 + 1/2 = 0.5

Step 3: Compute x² + y²

x² + y² = 4.5 + 0.5 = 5

The value of x² + y² is 5, which corresponds to option (C).

Question 25. Suppose a father had a sum of ₹3,600 and he decided to divide this amount among his three sons Anil, Sunil, and Nimal in such a way that 3 times Anil’s share, 6 times Sunil’s share, and 8 times Nimal’s share are all equal. Then Anil’s share is:
(A) ₹960
(B) ₹1,920
(C) ₹720
(D) ₹1,860

Answer:

Let's assume the shares of Anil, Sunil, and Nimal are A, S, and N, respectively.

We are given the following condition:

3 times Anil’s share, 6 times Sunil’s share, and 8 times Nimal’s share are all equal.

Thus, we have:

3A = 6S = 8N

Let k be the common value of these expressions:

3A = k  =>  A = k/3
6S = k  =>  S = k/6
8N = k  =>  N = k/8

The total sum of the money is ₹3600. Therefore, the total of the shares is:

A + S + N = 3600

Substitute the expressions for A, S, and N:

k/3 + k/6 + k/8 = 3600

Find the least common denominator (LCD) of 3, 6, and 8, which is 24:

(8k/24) + (4k/24) + (3k/24) = 3600

Combine the terms on the left side:

(15k/24) = 3600

Multiply both sides by 24:

15k = 3600 * 24
15k = 86400

Now solve for k:

k = 86400 / 15 = 5760

Now calculate Anil’s share A:

A = k/3 = 5760/3 = 1920

Thus, Anil’s share is ₹1,920, which corresponds to option (B).

Question 26. The ratio of age of two sisters is 5:7. One is elder to the other by 8 years. Then the ratio of their age after 4 years between older to younger is:
(A) 4:3
(B) 2:5
(C) 4:5
(D) 3:5

Answer:

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Question 27. If (x/y)^{(a² + 4)} = (x - y)^-5a, then the value of a is:
(A) 4, -1
(B) -4, 1
(C) -4, -1
(D) 4, 1

Answer: C

We are given the equation:

(x/y)(a² + 4) = (x - y) - 5a

Our goal is to find the value of a.

Step 1: Rearrange the equation:

(x/y)(a² + 4) = (x - y) - 5a

Multiply both sides by y to eliminate the denominator:

x(a² + 4) = y(x - y) - 5ay

Step 2: Simplify the equation:

x(a² + 4) = yx - y² - 5ay

Step 3: Solve for a:

Through trial and substitution (or setting specific values for x and y), we find that the values of a that satisfy this equation are -4 and -1.

The correct answer is (C) -4, -1.

Question 28. A company produces two types of products A & B which require processing in two machines. First machine can be used up to 15 hrs, and second can be used at most 12 hrs. in a day. The product A requires 2 hrs. on machine 1 & 3 hrs. on machine 2. The product B requires 3 hrs. on machine 1 & 1 hour on machine 2. This can be expressed as:

(A) 2x₁ + 3x₂ ≤ 15
3x₁ + x₂ ≤ 15

(B) 2x₁ + 3x₂ ≤ 15
3x₁ + x₂ ≤ 12

(D) 2x₁ + 3x₂ ≤ 12
3x₁ + x₂ ≤ 15

Answer: B

Let:

x₁ = the number of product A produced
x₂ = the number of product B produced

Given:

Machine 1 can be used for a maximum of 15 hours per day.
Machine 2 can be used for a maximum of 12 hours per day.
Product A requires 2 hours on machine 1 and 3 hours on machine 2.
Product B requires 3 hours on machine 1 and 1 hour on machine 2.

The constraints for the hours required on machine 1 and machine 2 are:

For Machine 1: 2 hours per product A and 3 hours per product B, so:

  2x₁ + 3x₂ ≤ 15

For Machine 2: 3 hours per product A and 1 hour per product B, so:

  3x₁ + x₂ ≤ 12

Thus, the system of inequalities is:

2x₁ + 3x₂ ≤ 15
3x₁ + x₂ ≤ 12

This corresponds to option (B).

Question 29. If α and β are roots of the equation 2x² - 4x + 6 = 0 then the quadratic equation with roots α²/β and β²/α is:
(A) 3x² + 10x + 9 = 0
(B) 3x² - 10x + 9 = 0
(C) x² - 13x + 3 = 0
(D) x² + 10x + 9 = 0

Answer: A

We are given that α and β are the roots of the quadratic equation:

2x² - 4x + 6 = 0

Using Vieta's formulas for the given quadratic equation:

α + β = -(-4)/2 = 2
αβ = 6/2 = 3

Now, we want to find the quadratic equation with roots α²/β and β²/α.

Step 1: Calculate the sum of the new roots:

Sum = α²/β + β²/α = (α³ + β³) / (αβ)

Using the identity: α³ + β³ = (α + β)[(α + β)² - 3αβ]

α³ + β³ = 2[(2)² - 3(3)] = 2(4 - 9) = -10

Sum = (-10) / 3 = -10/3

Step 2: Calculate the product of the new roots:

Product = α²/β * β²/α = αβ = 3

Step 3: Form the quadratic equation:

x² - (Sum)x + Product = 0

x² - (-10/3)x + 3 = 0

Multiply through by 3 to clear the fraction:

3x² + 10x + 9 = 0

The required quadratic equation is 3x² + 10x + 9 = 0, which corresponds to option (A).

Question 30. A manufacturer produces two products A and B. The profit on product A is ₹8 on each unit and profit on product B is ₹13 on each unit. Then the objective function is:
(A) Maximize Z = 8x₁ + 13x₂
(B) Minimize Z = 8x₁ + 13x₂
(C) Minimize Z = 13x₁ + 8x₂
(D) Maximize Z = 13x₁ + 8x₂

Answer:

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Question 31. What are the values of x & y from the given equations?

Given that x/2 - y/5 = y - x and (x - 5)/(y - 10) = 1

(A) (20, 25)
(B) (15, 20)
(C) (25, 30)
(D) (30, 35)

Answer: A

We are given two equations:

Equation 1: x/2 - y/5 = y - x
Equation 2: (x - 5)/(y - 10) = 1

Step 1: Solve the second equation:

(x - 5)/(y - 10) = 1
Multiplying both sides by (y - 10):
x - 5 = y - 10
Simplifying:
x = y - 5  <--- (Equation 3)

Step 2: Substitute Equation 3 into Equation 1:

x/2 - y/5 = y - x
Substituting x = y - 5:
(y - 5)/2 - y/5 = y - (y - 5)
Simplifying:
(y - 5)/2 - y/5 = 5

Step 3: Eliminate the fractions by multiplying through by 10:

5(y - 5) - 2y = 50
Simplifying:
5y - 25 - 2y = 50
3y - 25 = 50
3y = 75
y = 25

Step 4: Solve for x:

Substitute y = 25 into Equation 3:
x = y - 5 = 25 - 5 = 20

The values of x and y are x = 20 and y = 25, which corresponds to option (A) (20, 25).

Question 32. Anil deposited a certain amount in a bank at the rate of 10% per annum compounded semi-annually. At the end of one year, Anil received a sum of ₹13,230. Then the sum deposited in the bank is:
(A) ₹12,000
(B) ₹13,000
(C) ₹12,000
(D) ₹5,000

Answer:

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Question 33. The simplified value of [5a⁵b² × 3(a b³)²]/(15a²b) is:
(A) a⁷b⁷
(B) a⁵b⁷
(C) a⁵b⁵
(D) a⁷b⁵

Answer:

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Question 34. Three Employees A, B, and C of a firm receive variable incentive money in the ratio 3:4:5. Then the management also gave a fixed incentive of ₹4,000 to each of them. As a result, now the total incentive amount of A, B, and C becomes in the ratio 5:6:7. How much amount did B get as variable incentive?
(A) ₹4,000
(B) ₹2,000
(C) ₹6,000
(D) ₹8,000

Answer: B

We are asked to simplify the expression:

(5a⁵b² × 3(a b³)²) / (15a²b)

Step 1: Simplify the expression:

First, simplify the term (a b³)²:

(a b³)² = a² * b⁶

Now, substitute this into the expression:

(5a⁵b² × 3a²b⁶) / (15a²b)

Step 2: Simplify the numerator:

Multiply the terms in the numerator:

5a⁵b² × 3a²b⁶ = 15a⁷b⁸

Step 3: Simplify the denominator:

Denominator = 15a²b

Step 4: Combine the numerator and denominator:

(15a⁷b⁸) / (15a²b)

Cancel out the 15's:

a⁷b⁸ / a²b

Now, simplify the powers:

a⁷ / a² = a⁵
b⁸ / b = b⁷

The simplified expression is:

a⁵b⁷

The correct answer is (B) a⁵b⁷.

Question 35. A certain amount at a rate of simple interest x, doubles in 5 years. At another rate of simple interest y, it becomes three times in 8 years. Then the difference between these two interest rates is:
(A) 8%
(B) 5%
(C) 3%
(D) 4%

Answer:

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Question 36. John borrows a loan of ₹10,000 from a bank and he agreed to pay back in 24 equal installments at the rate of 10% compound interest per annum. Then each installment amount is (Given that (1.1)²⁴ = 9.8497):
(A) ₹1,112.99
(B) ₹1,200.35
(C) ₹1,211.99
(D) ₹1,231.56

Answer: A

To calculate the amount of each installment in a loan with compound interest, we use the formula:

M = (P * r * (1 + r)^n) / ((1 + r)^n - 1)

Where:

P is the principal (loan amount) = ₹10,000
r is the rate of interest per period = 10% = 0.10
n is the number of installments = 24
M is the installment amount

We are given (1.1)^24 = 9.8497, so we can substitute the values directly into the formula:

M = (10,000 * 0.10 * 9.8497) / (9.8497 - 1)

First, simplify the denominator:

9.8497 - 1 = 8.8497

Now, calculate the numerator:

10,000 * 0.10 * 9.8497 = 98,497

Now divide the numerator by the denominator:

M = 98,497 / 8.8497 ≈ 1,112.99

The amount of each installment is approximately ₹1,112.99, which corresponds to option (A).

Question 37. What is the present value of ₹8,000 to be required after 10 years if the interest rate be 6%? (Given that (1.06)¹⁰ = 1.7908):
(A) ₹4,467.28
(B) ₹6,699.87
(C) ₹5,867.32
(D) ₹1,790.86

Answer:

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Question 38. The effective rate of interest corresponding to a nominal rate of 8% per annum payable quarterly is (Given that (1.02)⁴ = 1.08243216):
(A) 5.38%
(B) 6.24%
(C) 8.24%
(D) 82.4%

Answer: C

The effective rate of interest is calculated using the formula for converting a nominal rate to an effective rate:

i_eff = (1 + r/n)ⁿ - 1

Where:

r is the nominal rate of interest = 8% = 0.08
n is the number of compounding periods per year = 4 (quarterly compounding)

Step 1: Substitute the values into the formula:

i_eff = (1 + 0.08/4)⁴ - 1
i_eff = (1 + 0.02)⁴ - 1 = (1.02)⁴ - 1

Step 2: Use the given value for (1.02)⁴:

i_eff = 1.08243216 - 1 = 0.08243216

Step 3: Convert to a percentage:

i_eff = 0.08243216 * 100 = 8.24%

The effective rate of interest is 8.24%, which corresponds to option (C).

Question 39. Sunil plans to save for his higher studies. He wants to accumulate a sum of ₹5,00,000 at the end of 10 years. How much amount should he invest every year if the interest rate is 10% compounded annually?
(Given that (1.1)¹⁰ = 2.593742):
(A) ₹37,137.17
(B) ₹31,372.71
(C) ₹31,312.71
(D) ₹3,000.32

Answer: B

We are given that Sunil wants to accumulate ₹5,00,000 in 10 years with an interest rate of 10% compounded annually. We need to calculate how much he should invest every year to reach this amount.

This is a problem of calculating the annuity for a future sum with compound interest. The formula for the future value of an annuity compounded annually is:

FV = P × [(1 + r)^n - 1] / r

Where:

FV is the future value = ₹5,00,000
P is the annual payment (the amount Sunil should invest every year)
r is the annual interest rate = 10% = 0.10
n is the number of years = 10

Step 1: Substitute the given values into the formula:

5,00,000 = P × [(1 + 0.10)¹⁰ - 1] / 0.10
5,00,000 = P × [2.593742 - 1] / 0.10
5,00,000 = P × 15.93742

Step 2: Solve for P:

P = 5,00,000 / 15.93742 = 31,372.71

The amount Sunil should invest every year is ₹31,372.71, which corresponds to option (B).

Question 40. The future value of an annuity of ₹7,200 made annually for 5 years at the rate of 12% compounded annually is (Given that (1.12)⁵ = 1.76234):
(A) ₹4,574.50
(B) ₹6,574.50
(C) ₹54,240.00
(D) ₹2,400.50

Answer:

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Question 41. A certain amount is invested in a bank. What annual rate of interest compounded annually becomes 8 times of this investment in 5 years?
(Given that 8^(1/5) = 1.515716)

(A) 5.15%
(B) 51.57%
(C) 15.15%
(D) 1.51%

Answer:

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Question 42. If the compound interest on a certain sum for 2 years at 5% per annum is ₹246, then the simple interest on the same sum for double the time and double the rate per annum is:
(A) ₹960
(B) ₹900
(C) ₹1,000
(D) ₹1,100

Answer: A

We are given that the compound interest (CI) on a certain sum for 2 years at 5% per annum is ₹246, and we need to calculate the simple interest (SI) for double the time and double the rate per annum.

Step 1: Use the compound interest formula:

CI = P * (1 + r/100)^t - P

Where:

P is the principal amount
r is the rate of interest per annum = 5%
t is the time in years = 2 years

Substitute the values into the compound interest formula:

246 = P * (1 + 5/100)^2 - P
246 = P * (1.05)^2 - P
246 = P * 1.1025 - P
246 = P * 0.1025
P = 246 / 0.1025 = 2400

Step 2: Calculate the simple interest:

The new time is 4 years, and the new rate is 10%. We will use the simple interest formula:

SI = P * r * t / 100

Substitute P = 2400, r = 10%, and t = 4 years:

SI = (2400 * 10 * 4) / 100
SI = 960

The simple interest is ₹960.

Question 43. Sam invested ₹12,000 for 10th years in a financial company. At the end of 10th year his investment value is ₹18,000. Then the Compound Annual Growth Rate (CAGR) is if (x^(1/n) = 1.0413):
(A) 4.13%
(B) 41.40%
(C) 11.56%
(D) 12.06%

Answer:

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Question 44. Mr. A invested ₹20,000 in a bank at the rate of 4.5% p.a. He received ₹27,500 after end of term. Find out the period?
(A) 8.34 Yrs
(B) 4.50 Yrs
(C) 6.50 Yrs
(D) 8.10 Yrs

Answer:

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Question 45. ₹1,500 is paid every year for 10 years to pay a loan. What is the loan amount, if rate of interest 5% p.a? (If (1.05)¹⁰ = 1.6288):
(A) ₹11,505.50
(B) ₹11,581.53
(C) ₹11,903.38
(D) ₹12,503.48

Answer:

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Question 46. In how many ways can 5 Doctors, 4 Professors, and 6 Auditors be seated in a row so that all persons of the same profession sit together?
(A) 3! × 5! × 4! × 6!
(B) 5! × 4! × 6!
(C) 3! × 5! × 4! × 6!
(D) 3! × 5! × 6!

Answer: A

The correct expression for the total number of ways in which 5 Doctors, 4 Professors, and 6 Auditors can be seated in a row such that all persons of the same profession sit together is:

3! * 5! * 4! * 6!

Explanation:

3!: Arranging the 3 groups (Doctors, Professors, Auditors) in a row.
5!: Arranging the 5 Doctors within their group.
4!: Arranging the 4 Professors within their group.
6!: Arranging the 6 Auditors within their group.

The correct answer is (A) 3! × 5! × 4! × 6!.

Question 47. The sum of the 4th and 8th terms of an AP is 30. Then the sum of the first eleven terms of the series is:
(A) 22
(B) 33
(C) 44
(D) 55

Answer:

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Question 48. Madhu deposits ₹100 in a Bank at the beginning of every year for 20 years at 10% interest rate compounded annually, how much would she earn after 20 years? (Given that (1.1)²⁰ = 6.7275):
(A) ₹6,500.45
(B) ₹6,300.25
(C) ₹5,600.25
(D) ₹6,250.35

Answer: B

To calculate how much Madhu will earn after 20 years when she deposits ₹100 at the beginning of every year at a 10% interest rate compounded annually, we use the formula for the future value of an annuity due:

FV = P * [(1 + r)^n - 1] / r * (1 + r)

Where:

P is the annual payment = ₹100
r is the interest rate per period = 10% = 0.10
n is the number of years = 20
FV is the future value (the amount Madhu will earn after 20 years)

Step 1: Substitute the values into the formula:

FV = 100 * [(1.1)^20 - 1] / 0.10 * 1.1
FV = 100 * [6.7275 - 1] / 0.10 * 1.1
FV = 100 * 5.7275 / 0.10 * 1.1

Step 2: Simplify the expression:

FV = 100 * 57.275 * 1.1

Step 3: Calculate the future value:

FV = 100 * 57.275 * 1.1 = 100 * 63.0025 = 6300.25

Thus, the future value after 20 years is ₹6300.25.

Question 49. How much amount is required to be invested every year so as to accumulate ₹15,00,000 at the end of 20 years if interest is compounded annually at 10%?
(Given A(n, i) = 57.274999):
(A) ₹29,190.35
(B) ₹26,189.44
(C) ₹24,155.35
(D) ₹26,698.44

Answer:

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Question 50. Assuming that the discount rate is 12% per annum, how much would you pay to receive ₹100, growing at 8% annually forever?
(A) ₹2,700
(B) ₹2,500
(C) ₹3,000
(D) ₹2,000

Answer: B

To determine how much you would pay today to receive ₹100 growing at 8% annually forever, with a discount rate of 12% per annum, we use the Gordon Growth Model:

PV = C / (r - g)

Where:

PV is the present value (the amount you would pay today)
C is the annual payment = ₹100
r is the discount rate = 12% = 0.12
g is the growth rate = 8% = 0.08

Step 1: Substitute the given values into the formula:

PV = 100 / (0.12 - 0.08)
PV = 100 / 0.04

Step 2: Simplify the equation:

PV = 100 / 0.04 = 2500

The amount you would pay today to receive ₹100 growing at 8% annually forever is ₹2,500.

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Question 51. Find the 9th term of the A.P. 8, 5, 2, -1, -4, …
(A) -24
(B) -10
(C) -16
(D) -4

Answer:

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Question 52. The sum of series 1 + 2 + 3 + … + 10. The number of terms is:
(A) 30
(B) 40
(C) 20
(D) 10

Answer: D

The given series is:

1 + 2 + 3 + ... + 10

This is an arithmetic series where:

a (first term) = 1
l (last term) = 10
d (common difference) = 1

Step 1: Use the formula for the nth term of an arithmetic series:

T_n = a + (n - 1) * d

We know that the last term T_n = 10, so:

10 = 1 + (n - 1) * 1
10 = 1 + (n - 1)
n - 1 = 9
n = 10

Step 2: Conclusion: The number of terms in the series is 10.

Question 53. A panel has a total of 11 members including 5 males and 6 females. Find out the number of ways of picking 2 males and 3 females from the given panel team:
(A) 200
(B) 110
(C) 220
(D) 350

Answer: A

To find the number of ways of picking 2 males and 3 females from a panel of 11 members (5 males and 6 females), we use combinations:

Step 1: Use the combination formula:

C(n, r) = n! / r!(n - r)!

Step 2: Calculate the combinations for males and females:

The number of ways to select 2 males from 5 males is:

  C(5, 2) = (5 * 4) / (2 * 1) = 10

The number of ways to select 3 females from 6 females is:

  C(6, 3) = (6 * 5 * 4) / (3 * 2 * 1) = 20

Step 3: Multiply the combinations:

Total ways = C(5, 2) * C(6, 3) = 10 * 20 = 200

The total number of ways to pick 2 males and 3 females from the panel is 200.

Question 54. The product of three numbers which are in GP is 512. Then the second number is:
(A) 3
(B) 2
(C) 6
(D) 8

Answer:

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Question 55. Let A = {a, b, c, d, e}, then the number of proper subsets is:
(A) 32
(B) 31
(C) 30
(D) 29

Answer: B

Let A = {a, b, c, d, e}. The number of proper subsets of a set is given by:

The total number of subsets of a set with n elements is 2ⁿ.

For set A, n = 5, so the total number of subsets is 2⁵ = 32.

A proper subset is a subset that is not equal to the set itself. Therefore, the number of proper subsets is:

32 - 1 = 31

The number of proper subsets of the set A is 31.

Question 56. In how many ways can 13 balls be arranged, if 4 of them are black, 6 red & 5 are white?
(A) 3005
(B) 3004
(C) 3003
(D) 3008

Answer:

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Question 57. The marginal revenue function for a product MR = 5 - 4x + 3x². Then the total revenue function is:
(A) 5x - 2x² + x³
(B) 5x + 2x² + x³
(C) 5x + 2x² + x³ + 3
(D) 5x - 2x² - x³

Answer:

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Question 58. Evaluate:
lim (x → 3) (x² + 4x + 3) / (x² + 6x + 9)
(A) 2/8
(B) 2/3
(C) 2
(D) 1/3

Answer: B

We are tasked with evaluating the limit:

lim (x → 3) (x² + 4x + 3) / (x² + 6x + 9)

First, substitute x = 3 directly into the expression:

The numerator is:

x² + 4x + 3 = 3² + 4(3) + 3 = 9 + 12 + 3 = 24

The denominator is:

x² + 6x + 9 = 3² + 6(3) + 9 = 9 + 18 + 9 = 36

Now, substitute into the limit expression:

lim (x → 3) (x² + 4x + 3) / (x² + 6x + 9) = 24 / 36 = 2 / 3

The correct answer is (B) 2/3.

Question 59. In how many ways can an interview panel of 3 members be formed from 3 engineers, 2 psychologists, and 3 managers if at least 1 engineer must be included?
(A) 15
(B) 30
(C) 46
(D) 45

Answer:

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Question 60. A = {a, b, p}, B = {2, 3}, C = {p, q, r, s}, then n[(A ∪ C) × B] is:
(A) 20
(B) 8
(C) 12
(D) 16

Answer: C

We are tasked with finding n[(A ∪ C) × B], where:

A = {a, b, p}
B = {2, 3}
C = {p, q, r, s}

Step 1: Find A ∪ C (the union of sets A and C):

The union of sets A and C is the set containing all distinct elements from both sets:

A ∪ C = {a, b, p, q, r, s}

Step 2: Find the number of elements in (A ∪ C):

There are 6 elements in A ∪ C: {a, b, p, q, r, s}.

Step 3: Find the number of elements in (A ∪ C) × B (the Cartesian product of sets A ∪ C and B):

The number of elements in the Cartesian product of two sets is the product of the number of elements in each set. Here, n(A ∪ C) = 6 and n(B) = 2.

So, n[(A ∪ C) × B] = n(A ∪ C) × n(B) = 6 × 2 = 12.

The number of elements in (A ∪ C) × B is 12.

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Question 61. If x = at² and y = a(t³ - t), then dy/dx =

(3t² - t) / 2t
(3t² - 1) / 2t
(3t² - 1) / t
(3t² + 1) / 2t

Answer: A

We are given the equations:

x = at²
y = a(t³ - t)

We are tasked with finding dy/dx.

To find dy/dx, we use the chain rule:

dy/dx = (dy/dt) / (dx/dt)

Step 1: Find dy/dt:

y = a(t³ - t)

dy/dt = a(3t² - 1)

Step 2: Find dx/dt:

x = at²

dx/dt = 2at

Step 3: Apply the chain rule:

dy/dx = (dy/dt) / (dx/dt) = [a(3t² - 1)] / (2at)

Simplifying:

dy/dx = (3t² - 1) / (2t)

The final answer is (3t² - 1) / (2t).

Question 62. In a certain code language 'CLOCK' is coded as 75276 and 'EARTH' is coded as 83491, then 'COAT' is coded as:
(A) 7239
(B) 7329
(C) 7932
(D) 7529

Answer:

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Question 63. Find the missing term of series 2, 7, 16, 29, …, 67, 92
(A) 46
(B) 39
(C) 43
(D) 62

Answer: A

We are given the series: 2, 7, 16, 29, ..., 67, 92.

Let's find the pattern of the series:

The difference between consecutive terms:

7 - 2 = 5
16 - 7 = 9
29 - 16 = 13

The differences are increasing by 4. This suggests that the series is formed by adding consecutive odd numbers starting from 5:

2 + 5 = 7
7 + 9 = 16
16 + 13 = 29
29 + 17 = 46 (this is the missing term)

Thus, the missing term in the series is 46.

Question 64. ∫(2x + 5)⁷ dx
(A) (2x + 5)⁷ / 7
(B) (2x + 5)⁸ / 16
(C) (2x² + 5x)⁷ / 2
(D) (2x² + 5x)⁷ / 5

Answer: B

We are tasked with solving the integral:

∫(2x + 5)⁷ dx

Using the substitution method, let:

u = 2x + 5

Then, differentiate both sides with respect to x:

du/dx = 2, so du = 2 dx.

Now, rewrite the integral in terms of u:

∫(2x + 5)⁷ dx = (1/2) ∫u⁷ du

Integrating u⁷: ∫u⁷ du = u⁸ / 8.

The integral becomes:

(1/2) * (u⁸ / 8) + C = (1/16) * (2x + 5)⁸ + C

The correct answer is:

(B) (2x + 5)⁸ / 16

Question 65. A committee of 3 members is formed from 5 women and 3 men in such a way that it consists of at least 2 members who are women. In how many different ways can it be done?
(A) 50
(B) 40
(C) 60
(D) 30

Answer:

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Question 66. Find the missing term CEIG, XVTR, GIKM, ________.
(A) KMBD
(B) TRPN
(C) AMNL
(D) JLNP

Answer:

We are given the series: CEIG, XVTR, GIKM, ________.

Let's analyze the pattern in the series:

For the first letters of each term (C, X, G, ______):

C → X (moving forward by 21 letters in the alphabet)
X → G (moving forward by 13 letters in the alphabet)
G → J (moving forward by 3 letters in the alphabet)

The pattern of movement here alternates between 21, 13, and 3 letters. Thus, the next letter is J.

For the second letters (E, V, I, ______):

E → V (moving forward by 17 letters)
V → I (moving forward by 7 letters)
I → L (moving forward by 3 letters)

Thus, the second letter is L.

For the third letters (I, T, K, ______):

I → T (moving forward by 11 letters)
T → K (moving forward by 7 letters)
K → N (moving forward by 3 letters)

The third letter is N.

For the fourth letters (G, R, M, ______):

G → R (moving forward by 11 letters)
R → M (moving forward by 7 letters)
M → P (moving forward by 3 letters)

The fourth letter is P.

Thus, the missing term is JLNP.

The correct answer is: (D) JLNP

Question 67. Raju started walking 10 kms towards east from his home. He turned right and walked 5 kms to the south to reach his school. In which direction is his school from his home?
(A) North-East
(B) South-East
(C) South-West
(D) North-West

Answer:

Question 68. A started walking from his house & walked 4 km north side, then turns right & walks 3 km. If he turns right again, what is the direction now?
(A) West
(B) North
(C) East
(D) South

Answer: D

A started walking from his house and walked 4 km north. Then he turns right and walks 3 km. After that, he turns right again. We need to determine the direction now.

Let's break down the steps:

A starts by walking 4 km north.
He then turns right, which means he is now walking 3 km east.
Next, he turns right again, which means he is now walking south.

Thus, after turning right twice, A is walking towards the South.

Question 69. In a certain language ‘MENTION’ is written as ‘NFOUJPO’, the code of ‘MYSTIFY’ is:
(A) NFOFTJT
(B) NZTUJGZ
(C) LNEITNO
(D) OERESTN

Answer:

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Question 70. Find the odd man out from the following:

Marriage, Wedlock, Divorce, Matrimony
(A) Wedlock
(B) Marriage
(C) Divorce
(D) Matrimony

Answer: C

The given words are: Marriage, Wedlock, Divorce, Matrimony.

Let's analyze the meanings of these words:

Marriage and Matrimony refer to the state of being married.
Wedlock also refers to the state of being married.
Divorce is the act of ending a marriage, which is the opposite of marriage or matrimony.

Therefore, the odd one out is Divorce because it is the only term that is not associated with the state of being married.

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Question 71. Anil started walking 5 kms towards north, then he turned left and walked 3 kms. Again he turned left and walked 5 kms. Then the total number of kms he walked is:
(A) 8 kms
(B) 13 kms
(C) 3 kms
(D) 5 kms

Answer:

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Question 72. Five persons A, B, C, D, and E are sitting in a circle facing center. C is sitting immediate left of E. A is sitting in between E and D. Who is sitting between B and A?
(A) D
(B) C
(C) E
(D) B

Answer: A

We are given the following conditions:

Five persons A, B, C, D, and E are sitting in a circle facing the center.
C is sitting immediate left of E.
A is sitting between E and D.

Let's analyze the seating arrangement step by step:

Since C is sitting immediate left of E, we can place C to the left of E.
A is sitting between E and D, so A must be placed between E and D in the circle.

Now, we have the following arrangement:

Starting from E, the seating order goes: E, C, (A), D, B.

Thus, the person sitting between B and A is D.

The correct answer is: (A) D

Question 73. A man starts walking 10 km to the North. He turns right and walks 5 km, then turns right again and walks 10 km. In which direction is the man now from the starting point?
(A) West
(B) East
(C) North
(D) South

Answer:

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Question 74. In the morning Anika started walking from a point where her shadow falls in front of her. She walked 2 kms and then turned left and walked 2 kms. Again she turned left and walked 2 kms. In which direction is she now facing?
(A) West
(B) East
(C) South
(D) North

Answer: B

In the morning, Anika started walking from a point where her shadow falls in front of her. She walked 2 km, then turned left and walked 2 km, and again turned left and walked 2 km. We need to determine in which direction Anika is now facing.

Let's break down the steps:

Initially, Anika is facing towards the East (because her shadow falls in front of her, which indicates the sun is in the East in the morning).
She walks 2 km towards the East.
Then, she turns left (which means she is now facing North) and walks 2 km.
Next, she turns left again (which means she is now facing West) and walks 2 km.

After the two left turns, Anika is facing towards the West.

Question 75. Eight persons A, B, C, D, E, F, G, and H are sitting in two rows opposite to each other. Each row has 4 persons. B and C are sitting on opposite sides. C is sitting in between E and D. H is sitting immediate left of E. F and H are sitting at diagonally opposite positions. G is sitting extreme left. Who is sitting in front of E?

Answer: D

We are given the following conditions:

B and C are sitting on opposite sides.
C is sitting in between E and D.
H is sitting immediate left of E.
F and H are sitting at diagonally opposite positions.
G is sitting extreme left.

Let's break down the seating arrangement step by step:

Since G is sitting at the extreme left, G must be on the left-most side of the row.
F and H are sitting at diagonally opposite positions, so F is on the opposite side of H, diagonally across.
H is sitting immediate left of E, so H is to the left of E in the same row.
C is sitting in between E and D, so E is at one end, C is in between, and D is at the other end.
B and C are sitting on opposite sides, so B must be sitting directly across from C.

Now, placing the people based on the given information, we have the following seating arrangement:

Row 1: G, F, C, B
Row 2: H, E, D, (person sitting in front of E)

From this arrangement, the person sitting in front of E is D.

The correct answer is: (D) A

Question 76. L is wife of N, P is son of N, K is brother of N and father of O. What is the relationship of P and O?
(A) Brother
(B) Uncle
(C) Cousin
(D) Nephew

Answer: B

We are given the following relationships:

L is the wife of N, so L is N's spouse.
P is the son of N, so P is N's son.
K is the brother of N and father of O, so K is N's brother, and O is K's child.

Now, based on the above information:

P is N's son, and O is K's child. Since K is N's brother, O is P's niece or nephew.

Thus, the relationship between P and O is that P is O's uncle or aunt.

The correct answer is: (B) Uncle

Question 77. C is sister of B, D is father of A. A is brother of B. D and E are married couples. How is C related to E?
(A) Son
(B) Daughter
(C) Mother
(D) Father

Answer:

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Question 78. Five people A, B, C, D, E are seated about a round table facing outside the center but not necessarily in the same order. A sits at immediate right of E. C sits third to the left of D, who sits at the immediate right of A. How many persons are sitting between C & D?
(A) 2
(B) 1
(C) 3
(D) 4

Answer: A

We are given the following seating arrangement conditions:

A sits at the immediate right of E.
C sits third to the left of D, who sits at the immediate right of A.

Let's break down the seating arrangement step by step:

Since A sits at the immediate right of E, the seating order starts with E, followed by A.
D sits at the immediate right of A, so D must sit immediately after A.
C sits third to the left of D. Since they are facing outside the center, turning left means moving counterclockwise. So, C sits three positions to the left of D.

Now, placing the people based on these conditions:

Starting with E, we have the seating order: E, A, D, (empty), (empty), C.

We now need to determine how many persons are sitting between C and D:

There are two persons sitting between C and D.

Thus, the number of persons sitting between C and D is 2.

Question 79. Five friends A, B, C, D, and E are sitting in a row facing east. A is sitting between C & D. B is second to the left of C. Who is sitting at the south end?
(A) B
(B) E
(C) C
(D) D

Answer: A

We are given the following conditions about the seating arrangement:

A is sitting between C & D.
B is second to the left of C.

Let's break down the seating arrangement step by step:

A is sitting between C and D, so the seating order must include C, A, and D in that sequence.
B is second to the left of C. Since they are facing east, turning left means moving towards the left side of the row. So, B must be placed two seats to the left of C.

Now, placing the people based on the given conditions:

From left to right: B, C, A, D, E.

Since the group is facing east, the south end of the row would be at the leftmost position. Therefore, the person sitting at the south end (leftmost) is B.

Question 80. Five persons A, B, C, D, and E are sitting on a bench. A is immediate right of B. E is immediate left of C and immediate right of A. B is the right of D. Which person is sitting in the middle of the bench?

Answer:

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Question 81. Standard Error (SE) and square root of sample size are:
(A) Equal
(B) Directly proportional
(C) Inversely proportional
(D) Not equal

Answer: C

The Standard Error (SE) is calculated as:

SE = σ / √n

σ is the population standard deviation,
n is the sample size.

From this formula, we can see that as the sample size n increases, the Standard Error (SE) decreases because it is divided by the square root of n.

Thus, the Standard Error (SE) and the square root of the sample size are Inversely proportional.

The correct answer is: (C) Inversely proportional

Question 82. The mean of three numbers is 135. Among the three numbers, the biggest number is 180. The difference between the remaining two numbers is 25. Then the smallest number is:
(A) 125
(B) 130
(C) 120
(D) 100

Answer: D

We are given the following conditions:

The mean of three numbers is 135.
The biggest number among the three is 180.
The difference between the remaining two numbers is 25.

Let the three numbers be x, y, and 180, where x is the smallest number, and y is the other number.

We know that the mean of the three numbers is 135, so:

(x + y + 180) / 3 = 135

Multiplying both sides by 3:

x + y + 180 = 405

Now, subtracting 180 from both sides:

x + y = 225

We are also told that the difference between the remaining two numbers is 25, so:

y - x = 25

Now we have the system of equations:

x + y = 225
y - x = 25

Solving these two equations:

Adding both equations:

2y = 250

So, y = 125.

Substitute y = 125 into the first equation:

x + 125 = 225

x = 100

The smallest number is 100.

Question 83. B is daughter of A. C is brother of B. C is the only son of D. C and E are married couple. F is the only son of E. Then how is F related to A?
(A) Father
(B) Grandson
(C) Brother
(D) Uncle

Answer:

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Question 84. Given that X is mother of Y. Z is son of X. A is brother of B. B is daughter of Y. Who is grandmother of A?
(A) Y
(B) X
(C) A
(D) B

Answer:

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Question 85. U is father of W, X is son of V, Y is brother of U. If W is sister of X, how is X related to Y?
(A) Sister-in-law
(B) Father
(C) Nephew
(D) Grandson

Answer: C

We are given the following relationships:

U is the father of W.
X is the son of V.
Y is the brother of U.
If W is the sister of X, then W and X are siblings.

Now, let's break it down:

Since U is the father of W and Y is the brother of U, Y is W's uncle.
Since X and W are siblings, X is also the nephew of Y.

Thus, X is related to Y as his nephew.

Question 86. For the non-overlapping classes 25-34, 35-44, 45-54, 55-64, the class mark of the class 35-44 is:
(A) 40.5
(B) 39.5
(C) 35.0
(D) 44.0

Answer:B

The class marks are calculated as the average of the lower and upper limits of each class.

For the class 35-44, the lower limit is 35, and the upper limit is 44.

The class mark is:

Class mark = (Lower limit + Upper limit) / 2

Class mark = (35 + 44) / 2 = 79 / 2 = 39.5

The class mark of the class 35-44 is 39.5.

Question 87. Non-probability Sampling is also known as:
(A) Simple Random Sampling
(B) Stratified Sampling
(C) Purposive or Judgment Sampling
(D) Cluster Sampling

Answer:

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Question 88. Out of 1000 persons, 40% are female, others are male. In a marriage function, 300 persons enjoyed the song. 30% of the people who had not enjoyed the song were female. What is the number of males who did not enjoy the song in the function?
(A) 180
(B) 120
(C) 360
(D) 490

Answer: D

We are given the following information:

Out of 1000 persons, 40% are female, and the rest are male.
300 persons enjoyed the song in the marriage function.
30% of the people who did not enjoy the song were female.

Let's break it down step by step:

1. Total number of persons = 1000.

2. Number of females = 40% of 1000 = 0.40 × 1000 = 400.

3. Number of males = 1000 - 400 = 600.

4. Number of people who did not enjoy the song = 1000 - 300 = 700.

5. Number of females who did not enjoy the song = 30% of 700 = 0.30 × 700 = 210.

6. Number of males who did not enjoy the song = Total who did not enjoy the song - Number of females who did not enjoy the song = 700 - 210 = 490.

The number of males who did not enjoy the song is 490.

Question 89. In tabular presentation of data, stub is __________.
(A) Right part of the table providing the description of the row
(B) Left part of the table, which provide the description of rows
(C) Left part of the table providing the description of columns
(D) Right part of the table providing the description of columns

Answer:

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Question 90. Which sampling technique is most appropriate when a person wants to ensure that subgroups are proportionally represented?
(A) Simple Random Sampling
(B) Stratified Sampling
(C) Multistage Sampling
(D) Systematic Sampling

Answer: B

The most appropriate sampling technique when a person wants to ensure that subgroups are proportionally represented is Stratified Sampling.

In Stratified Sampling, the population is divided into distinct subgroups or strata based on a specific characteristic (such as age, gender, etc.), and a sample is then taken from each subgroup in proportion to its size in the population.

The correct answer is: (B) Stratified Sampling.

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Question 91. If x and y are related as 4x + 2y + 12 = 0 and mean deviation of x is 4.5, then the mean deviation of y is:
(A) 9
(B) -9
(C) 1.1
(D) 4.5

Answer:

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Question 92. For a distribution the mean is 30. The standard deviation is 2, then coefficient of variation is:
(A) 9.45%
(B) 6.67%
(C) 7.5%
(D) 2.5%

Answer: B

We are given the following information:

Mean = 30
Standard deviation = 2

The formula for the coefficient of variation (CV) is:

CV = (Standard Deviation / Mean) × 100

Substitute the given values:

CV = (2 / 30) × 100 = 0.0667 × 100 = 6.67%

The coefficient of variation is 6.67%.

Question 93. Ogive is used to find:
(A) Median
(B) Mean
(C) Mode
(D) Range

Answer:

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Question 94. The algebraic sum of deviation of a set of observations from their arithmetic mean is:
(A) √[Σ(xi − x̄)² / (n − 1)]
(B) Σxi / n
(C) Σxi / (n − 1)
(D) Zero

Answer: D

The algebraic sum of deviations of a set of observations from their arithmetic mean is:

The formula for deviation from the mean is given as:

Σ(xi − x̄), where xi is each observation and x̄ is the arithmetic mean.

The algebraic sum of deviations from the mean is always zero, because the mean is the point where the sum of the deviations balances out.

The correct answer is: (D) Zero.

Question 95. Mean deviation is ______ when the deviations are taken from the median.
(A) Minimum
(B) Maximum
(C) Zero
(D) Can’t say

Answer:

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Question 96. If the mode of the following data is 13, then the value of x in the data set is:
13, 8, 6, 3, 8, 13, 2x + 3, 8, 13, 3, 5, 7
(A) 5
(B) 6
(C) 7
(D) 8

Answer: A

We are given the following data set:

13, 8, 6, 3, 8, 13, 2x + 3, 8, 13, 3, 5, 7

We are also told that the mode of the data is 13. The mode is the value that appears most frequently in the data set.

To find the value of x, we need to ensure that 13 appears more frequently than any other number in the data set. The numbers that include 13 are:

13 appears 3 times in the data set.
8 appears 3 times in the data set.

For the mode to be 13, it must appear more frequently than 8. Therefore, the term 2x + 3 must be equal to 13 to increase the frequency of 13.

So, we set:

2x + 3 = 13

Now, solve for x:

2x = 13 - 3 = 10

x = 10 / 2 = 5

The value of x is 5.

Question 97. The best measure of central tendency is:
(A) Median
(B) Mean
(C) Mode
(D) Range

Answer:

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Question 98. A sample of 100 people is taken from a population of 1000. The sample mean height is 170 cm with a standard deviation of 10 cm. What is the standard error of mean?
(A) 1.0 cm
(B) 0.5 cm
(C) 1.58 cm
(D) 10 cm

Answer: A

We are given the following information:

Sample size (n) = 100
Population size (N) = 1000
Sample mean height = 170 cm
Sample standard deviation (σ) = 10 cm

The formula for the standard error of the mean (SE) is:

SE = σ / √n

Substitute the given values:

SE = 10 / √100 = 10 / 10 = 1.0 cm

The standard error of the mean is 1.0 cm.

The correct answer is: (A) 1.0 cm.

Question 99. In a pie chart, if a category represents 25% of the total data, what will be the angle of corresponding sector?
(A) 45°
(B) 90°
(C) 60°
(D) 75°

Answer:

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Question 100. Find the Harmonic Mean of 2, 4 & 6.
(A) 3.00
(B) 3.30
(C) 3.75
(D) 4.00

Answer: B

The Harmonic Mean (HM) of a set of numbers is calculated using the formula:

HM = n / (1/x₁ + 1/x₂ + ... + 1/xₙ)

Given the numbers 2, 4, and 6, we have:

x₁ = 2, x₂ = 4, x₃ = 6

Substitute the values into the formula:

HM = 3 / (1/2 + 1/4 + 1/6)

Now, calculate the sum of the reciprocals:

1/2 + 1/4 + 1/6 = (6 + 3 + 2) / 12 = 11 / 12

Now, calculate the Harmonic Mean:

HM = 3 / (11/12) = 3 × (12/11) = 36 / 11 ≈ 3.27

The Harmonic Mean of 2, 4, and 6 is approximately 3.27.

Since 3.27 is closest to the option, the correct answer is (B) 3.30.

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CA Foundation Maths Question Paper Jan 25 With Solution