Let the mean and variance of 7 observations 2, 4, 10, x, 12, 14, y, x > y, be 8 and 16 respectively. Two numbers are…

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Q2 jee_main_2026_22_january_evening Mean Deviation about Median

If the mean deviation about the median of the numbers

k, 2k, 3k, , 1000k

500

, then

k²

is equal to:

A. 16
B. 4
C. 1
D. 9

Solution

Related Formula

Mean deviation about median is given by:

M.D. = Σi=1ⁿ |Xᵢ - XM|n

Core Logic

For $1000$ $1000$ numbers in A.P., median $XM = (1001k)/(2)$ $X_M = \frac{1001k}{2}$.

M.D. = 2 ((k)/(2) + (3k)/(2) + (5k)/(2) + + 500 terms)1000

M.D. = (2 \cdot (k)/(2) (500)²)/(1000) = (500k)/(2)

Step 1: Evaluation of k and k^2

Given mean deviation is $500$ $500$:

(500k)/(2) = 500 k = 2

Therefore, $k² = 4$ $k^2 = 4$.

Pattern Recognition

Use symmetry of A.P. around median to quickly evaluate absolute deviation sum.

Chapter Mix

Class 11 Maths: Statistics

Q13 jee_main_2026_23_january_morning Mean and Variance

Let the mean and variance of

8

numbers

-10, -7, -1, x, y, 9, 2, 16

(7)/(2)

and

(293)/(4)

, respectively. Then the mean of

4

numbers

x, y, x + y + 1

|x - y|

is:

A. $11$ $11$
B. $9$ $9$
C. $10$ $10$
D. $12$ $12$

Solution

Related Formula

μ = (Σ xᵢ)/(N)

σ² = (Σ xᵢ²)/(N) - μ²

Core Logic

Calculate $x+y$ $x+y$ using the mean formula:

(-10 - 7 - 1 + x + y + 9 + 2 + 16)/(8) = (7)/(2)

(9 + x + y)/(8) = (7)/(2) \Rightarrow 9 + x + y = 28 \Rightarrow x + y = 19 (1)

Step 1: Calculate sum of squares using Variance

Using the variance formula:

((-10)² + (-7)² + (-1)² + x² + y² + 9² + 2² + 16²)/(8) - ((7)/(2))² = (293)/(4)

(100 + 49 + 1 + x² + y² + 81 + 4 + 256)/(8) - (49)/(4) = (293)/(4)

(491 + x² + y²)/(8) = (293)/(4) + (49)/(4) = (342)/(4) = (684)/(8)

491 + x² + y² = 684 \Rightarrow x² + y² = 193 (2)

Step 2: Solve for x and y

From (1), $y = 19 - x$ $y = 19 - x$. Substitute into (2):

x² + (19 - x)² = 193

x² + 361 - 38x + x² = 193

2x² - 38x + 168 = 0 \Rightarrow x² - 19x + 84 = 0

(x - 12)(x - 7) = 0

Thus, $x = 12$ $x = 12$ and $y = 7$ $y = 7$ (or vice-versa, which doesn't affect absolute differences).

Step 3: Calculate the New Mean

We need the mean of $4$ $4$ numbers: $x, y, x + y + 1, |x - y|$ $x, y, x + y + 1, |x - y|$. Substitute $x = 12, y = 7$ $x = 12, y = 7$: The numbers are $12, 7, (12+7+1), |12-7| = 12, 7, 20, 5$ . New Mean = $(12 + 7 + 20 + 5)/(4) = (44)/(4) = 11$ .

Pattern Recognition

Statistics questions mapping $x+y$ $x+y$ and $x²+y²$ $x^2+y^2$ inherently hide a quadratic symmetric system $(x+y)² - 2xy = x²+y²$ $(x+y)^2 - 2xy = x^2+y^2$. Solving for the roots immediately grants the discrete terms.

Chapter Mix

Class 11 Maths: Statistics

Q18 jee_main_2026_23_january_evening Mean and Variance

If the mean and the variance of the data

Class	4–8	8–12	12–16	16–20
Frequency	3	$λ$ $\lambda$	4	7

are

μ

and 19 respectively, then the value of

λ + μ

A. $18$ $18$
B. $21$ $21$
C. $20$ $20$
D. $19$ $19$

Solution

Related Formula

μ = (Σ fᵢ xᵢ)/(Σ fᵢ)

σ² = (Σ fᵢ xᵢ²)/(Σ fᵢ) - μ²

Core Logic

Midpoints ( $xᵢ$ $x_i$): $6, 10, 14, 18$ $6, 10, 14, 18$. Frequencies ( $fᵢ$ $f_i$): $3, λ, 4, 7$ $3, \lambda, 4, 7$. Sum of frequencies $Σ fᵢ = 3 + λ + 4 + 7 = 14 + λ$ .

Calculate Mean ( $μ$ $\mu$):

μ = (6(3) + 10(λ) + 14(4) + 18(7))/(14 + λ)

μ = (18 + 10λ + 56 + 126)/(14 + λ) = (10λ + 200)/(λ + 14) = 10 + (60)/(λ + 14)

Since data is standard, $μ$ $\mu$ and $λ$ $\lambda$ typically hold integer values, hinting $λ + 14$ $\lambda + 14$ must be a factor of 60. Therefore $λ$ $\lambda$ could be $1, 6, 16$ $1, 6, 16$.

Step 1: Applying Variance

Variance $σ² = 19$ $\sigma^2 = 19$.

σ² = (Σ fᵢ xᵢ²)/(14 + λ) - μ²

Σ fᵢ xᵢ² = 3(36) + λ(100) + 4(196) + 7(324) = 108 + 100λ + 784 + 2268 = 100λ + 3160

Substitute into variance:

19 = (100λ + 3160)/(λ + 14) - ((10λ + 200)/(λ + 14))²

Step 2: Testing Candidates

Let's test the potential integer $λ = 6$ $\lambda = 6$ (a factor of 60 is $14+6=20$ $14+6=20$): If $λ = 6$ $\lambda = 6$:

μ = 10 + (60)/(20) = 10 + 3 = 13

Let's verify variance for $λ = 6$ $\lambda = 6$:

σ² = (100(6) + 3160)/(20) - 13² = (3760)/(20) - 169 = 188 - 169 = 19

This perfectly matches the given variance.

Thus, $λ = 6$ $\lambda = 6$ and $μ = 13$ $\mu = 13$.

λ + μ = 6 + 13 = 19

Pattern Recognition

Instead of solving a horrible cubic/quadratic equation for the variance parameter, use the integer division trait of the mean to test valid candidates.

Chapter Mix

Class 11 Maths: Statistics

Q18 jee_main_2026_24_january_morning Variance and Mean Replacement

The mean and variance of a data of

10

observations are

10

and

2

, respectively. If an observation

α

in this data is replaced by

β

, then the mean and variance become

10.1

and

1.99

, respectively. Then

α + β

equals.

A. $10$ $10$
B. $15$ $15$
C. $5$ $5$
D. $20$ $20$

Solution

Related Formula

Mean x = (Σ xᵢ)/(n)

Variance σ² = (Σ xᵢ²)/(n) - x²

Core Logic

Let the 9 unchanged numbers be $x₁, x₂, , x₉$ $x_1, x_2, \dots, x_9$. Case 1 (with $α$ $\alpha$):

Σi=1⁹ xᵢ + α = 10 \times 10 = 100 \Rightarrow Σi=1⁹ xᵢ = 100 - α

Σi=1⁹ xᵢ² + α²10 - (10)² = 2 \Rightarrow Σi=1⁹ xᵢ² + α² = 1020 \Rightarrow Σi=1⁹ xᵢ² = 1020 - α²

Step 1: Applying Replacement Conditions

Case 2 (with $β$ $\beta$): Mean = 10.1

Σi=1⁹ xᵢ + β10 = 10.1 \Rightarrow 100 - α + β = 101 \Rightarrow β - α = 1

Variance = 1.99

Σi=1⁹ xᵢ² + β²10 - (10.1)² = 1.99

Σi=1⁹ xᵢ² + β² = 10(1.99 + 102.01) = 10(104) = 1040

Step 2: Solving for Alpha and Beta

Substitute $Σi=1⁹ xᵢ² = 1020 - α²$ $\sum_{i=1}^9 x_i^2 = 1020 - \alpha^2$:

1020 - α² + β² = 1040 \Rightarrow β² - α² = 20

(β - α)(β + α) = 20

Since $β - α = 1$ $\beta - \alpha = 1$:

(1)(α + β) = 20 \Rightarrow α + β = 20

(Solving gives $α = 19/2, β = 21/2$ $\alpha = 19/2, \beta = 21/2$).

Pattern Recognition

When a single term is replaced, immediately establish the difference of sums and difference of sum-of-squares. The identity $β² - α²$ $\beta^2 - \alpha^2$ gracefully factors to use the established $β - α$ $\beta - \alpha$ value.

Chapter Mix

Class 11 Maths: Statistics

Solution & Explanation

Related Formula

Core Logic

Step 1: Solve for x and y

Step 2: Construct the set and evaluate Probability

Step 3: Final Calculation via Complement

Pattern Recognition

Chapter Mix

Reference Study Guides

More Statistics and Probability Previous-Year Questions

Solution

Related Formula

Core Logic

Step 1: Evaluation of k and k^2

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Calculate sum of squares using Variance

Step 2: Solve for x and y

Step 3: Calculate the New Mean

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Applying Variance

Step 2: Testing Candidates

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Applying Replacement Conditions

Step 2: Solving for Alpha and Beta

Pattern Recognition

Chapter Mix

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