Question 1

The mean of a dataset {4, 6, 8, 10, 12} is 8. What is the variance of this dataset?

Accepted Answer

Step 1: Mean = 8 (given). Step 2: Calculate deviations from mean: (4−8)² = 16, (6−8)² = 4, (8−8)² = 0, (10−8)² = 4, (12−8)² = 16. Step 3: Sum of squared deviations = 16 + 4 + 0 + 4 + 16 = 40. Step 4: Variance = 40 ÷ 5 = 8.

Question 2

In how many ways can the letters of the word 'APPLE' be arranged?

Accepted Answer

Step 1: The word 'APPLE' has 5 letters with 'P' repeated twice. Step 2: When objects are repeated, we use: n!/(n₁! × n₂! × ...). Step 3: Number of arrangements = 5!/2! = (5 × 4 × 3 × 2 × 1)/2 = 120/2 = 60.

Question 3

The variance of a dataset is 36. What is the standard deviation of this dataset?

Accepted Answer

Step 1: Recall that Standard Deviation (SD) = √Variance. Step 2: SD = √36 = 6. Therefore, the standard deviation is 6.

Question 4

The standard deviation of five numbers is 2. What is the variance of these five numbers?

Accepted Answer

Step 1: Recall that Variance = (Standard Deviation)². Step 2: Variance = 2² = 4. Therefore, the variance is 4.

Question 5

The median of the dataset {7, 12, 5, 18, 9, 14, 11} is:

Accepted Answer

Step 1: Arrange the data in ascending order: {5, 7, 9, 11, 12, 14, 18}. Step 2: Count the number of values: 7 (odd number). Step 3: The median is the middle value at position (7+1)/2 = 4. Step 4: The 4th value is 11.

Question 6

A team of 4 players is to be selected from 6 batsmen and 5 bowlers such that the team has at least 2 batsmen. How many ways can this be done?

Accepted Answer

Step 1: At least 2 batsmen means: (2 batsmen, 2 bowlers) OR (3 batsmen, 1 bowler) OR (4 batsmen, 0 bowlers). Step 2: Case 1: C(6,2) × C(5,2) = 15 × 10 = 150. Step 3: Case 2: C(6,3) × C(5,1) = 20 × 5 = 100. Step 4: Case 3: C(6,4) × C(5,0) = 15 × 1 = 15. Step 5: Total = 150 + 100 + 15 = 265.

Question 7

How many ways can a committee of 3 people be selected from a group of 8 people?

Accepted Answer

Step 1: We need to select 3 people from 8 where order does not matter. Step 2: This is a combination problem: C(8,3) = 8!/(3! × 5!). Step 3: C(8,3) = (8 × 7 × 6)/(3 × 2 × 1) = 336/6 = 56.

Question 8

In how many ways can 4 identical red balls and 3 identical blue balls be arranged in a line?

Accepted Answer

Step 1: We have 7 balls total: 4 identical red and 3 identical blue. Step 2: When arranging objects where some are identical, use: n!/(n₁! × n₂!). Step 3: Number of arrangements = 7!/(4! × 3!) = 5040/(24 × 6) = 5040/144 = 35.

Question 9

How many 3-digit numbers can be formed using the digits 2, 3, 5, 7, and 9 without repetition?

Accepted Answer

Step 1: We need to form 3-digit numbers using 5 different digits without repetition. Step 2: This is a permutation problem: P(5,3) = 5!/(5-3)! = 5!/2!. Step 3: P(5,3) = 5 × 4 × 3 = 60.

Question 10

If the standard deviation of a dataset is 5, what is the variance of a new dataset formed by multiplying each value by 3?

Accepted Answer

Step 1: Original SD = 5, so original Variance = 25. Step 2: When each value is multiplied by a constant k, the new variance = k² × original variance. Step 3: New Variance = 3² × 25 = 9 × 25 = 225.

Question 11

The variance of a dataset is 49. If each value in the dataset is decreased by 5, what is the variance of the new dataset?

Accepted Answer

Step 1: Original Variance = 49. Step 2: When a constant is added or subtracted from each value, the variance remains unchanged (variance measures spread, not location). Step 3: New Variance = 49.

Question 12

A shopkeeper recorded daily sales (in ₹100s) for 6 days: 12, 15, 18, 12, 20, 15. What is the mean daily sales?

Accepted Answer

Step 1: Sum all values: 12 + 15 + 18 + 12 + 20 + 15 = 92. Step 2: Count the number of days: 6. Step 3: Mean = 92 / 6 = 15.33 (in ₹100s). Step 4: In actual rupees: 15.33 × 100 = ₹1533.33 ≈ ₹1533.

Question 13

Two datasets have variances 16 and 9 respectively. What are their standard deviations?

Accepted Answer

Step 1: For first dataset, SD₁ = √16 = 4. Step 2: For second dataset, SD₂ = √9 = 3. Step 3: The standard deviations are 4 and 3 respectively.

Question 14

The mode of the dataset {15, 22, 15, 18, 22, 15, 25, 22} is:

Accepted Answer

Step 1: Count the frequency of each value. 15 appears 3 times, 22 appears 3 times, 18 appears 1 time, 25 appears 1 time. Step 2: Both 15 and 22 have the highest frequency of 3. Step 3: When two values share the highest frequency, the dataset is bimodal. Step 4: The modes are 15 and 22.

Question 15

The mean of 8 numbers is 32. If one number is removed, the mean of the remaining 7 numbers becomes 30. What is the number that was removed?

Accepted Answer

Step 1: Sum of 8 numbers = Mean × Count = 32 × 8 = 256. Step 2: Sum of 7 numbers = 30 × 7 = 210. Step 3: The removed number = 256 - 210 = 46.

Question 16

In how many ways can 5 different books be arranged on a shelf?

Accepted Answer

Step 1: We need to arrange 5 different books in a line. Step 2: This is a permutation problem where all 5 objects are used. Step 3: Number of arrangements = 5! = 5 × 4 × 3 × 2 × 1 = 120.

Question 17

The variance of a set of 6 numbers is 25. The sum of the numbers is 60. If one number is removed, the sum becomes 54. What is the variance of the remaining 5 numbers? (Assume the removed number was 6)

Accepted Answer

Step 1: Original: n=6, Sum=60, Variance=25. Step 2: Sum of squares: Var = (ΣX²/n) - (Mean)². Mean = 60/6 = 10. So 25 = (ΣX²/6) - 100 ⟹ ΣX² = 750. Step 3: Removed number = 6. New sum = 54, n = 5. Step 4: New mean = 54/5 = 10.8. Step 5: New ΣX² = 750 - 36 = 714. Step 6: New Variance = (714/5) - (10.8)² = 142.8 - 116.64 = 26.16.

Question 18

Two datasets A and B have the same mean of 20. Dataset A has 8 observations with variance 16, and Dataset B has 12 observations with variance 9. What is the combined variance of both datasets?

Accepted Answer

Step 1: For combined variance when means are equal: Combined Variance = (n₁·Var₁ + n₂·Var₂) / (n₁ + n₂). Step 2: Combined Variance = (8×16 + 12×9) / (8+12) = (128 + 108) / 20 = 236 / 20 = 11.8.

Question 19

The standard deviation of five numbers is 6. If the sum of the numbers is 50, what is the sum of their squares?

Accepted Answer

Step 1: Use the formula: Variance = (Sum of Squares / n) - (Mean)². Step 2: Standard deviation = 6, so Variance = 36. Step 3: Mean = 50/5 = 10. Step 4: 36 = (Sum of Squares / 5) - 100. Step 5: Sum of Squares / 5 = 136. Step 6: Sum of Squares = 680.

Question 20

The mean of a set of 10 numbers is 35. If the mean of the first 6 numbers is 32, what is the mean of the remaining 4 numbers?

Accepted Answer

Step 1: Sum of all 10 numbers = 10 × 35 = 350. Step 2: Sum of first 6 numbers = 6 × 32 = 192. Step 3: Sum of remaining 4 numbers = 350 − 192 = 158. Step 4: Mean of remaining 4 numbers = 158 ÷ 4 = 39.5.

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