Class JEE Mathematics ALL Q #1173
COMPETENCY BASED
APPLY
4 Marks 2026 JEE Main 2026 (Online) 21st January Morning Shift MCQ SINGLE
If the mean and variance of observations $x, y, 12, 14, 4, 10, 2$ is 8 and 16 respectively, where $x>y$, then the value of $3x-y$ is:
(A) $18$
(B) $20$
(C) $22$
(D) $24$
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AI Tutor Explanation

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Step-by-Step Solution

Step 1: Calculate the sum of observations using the mean.

The mean of the observations is given as 8. The sum of the observations is:

x + y + 12 + 14 + 4 + 10 + 2 = 7 * 8

x + y + 42 = 56

x + y = 14 ---(1)

Step 2: Calculate the sum of squares of observations using the variance.

The variance is given as 16. The formula for variance is:

Variance = (Sum of squares of observations / Number of observations) - (Mean)^2

16 = (x^2 + y^2 + 12^2 + 14^2 + 4^2 + 10^2 + 2^2) / 7 - 8^2

16 = (x^2 + y^2 + 144 + 196 + 16 + 100 + 4) / 7 - 64

80 = (x^2 + y^2 + 460) / 7

560 = x^2 + y^2 + 460

x^2 + y^2 = 100 ---(2)

Step 3: Solve for x and y using equations (1) and (2).

From equation (1), y = 14 - x. Substitute this into equation (2):

x^2 + (14 - x)^2 = 100

x^2 + 196 - 28x + x^2 = 100

2x^2 - 28x + 96 = 0

x^2 - 14x + 48 = 0

(x - 6)(x - 8) = 0

So, x = 6 or x = 8.

If x = 6, then y = 14 - 6 = 8. But we are given that x > y, so this case is not valid.

If x = 8, then y = 14 - 8 = 6. This satisfies x > y.

Therefore, x = 8 and y = 6.

Step 4: Calculate 3x - y.

3x - y = 3(8) - 6 = 24 - 6 = 18

Correct Answer: 18

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AI Suggestion: Option A

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because it requires the student to apply the formulas for mean and variance to solve for unknown variables and then perform a calculation.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to follow a specific procedure to calculate the mean and variance, set up equations, and solve for the unknowns.
Syllabus Audit: In the context of JEE, this is classified as COMPETENCY. It assesses the ability to apply statistical concepts to solve a problem, rather than simply recalling definitions.

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