Visual SolutionHard

Question

If the mean and variance of eight numbers 3, 7, 9, 12, 14, 20,

x

and

y

are 10 and 25 respectively, then

x \cdot y

is equal to:

Solution Path

From mean:

x+y = 15

. From variance:

x^2+y^2 = 121

. Using

(x+y)^2 = x^2+y^2+2xy

xy = 52

01Question Setup

1/4

Given eight numbers 3, 7, 9, 12, 14, 20,

x

y

with mean 10 and variance 25. Find

x \cdot y

Find

x \cdot y

02Use Mean Condition

2/4

Sum =

8 \times 10 = 80

. Known sum =

3+7+9+12+14+20 = 65

. So

x + y = 15

x + y = 15

03Use Variance ConditionKEY INSIGHT

3/4

Variance =

\frac{\sum x_i^2}{8} - 100 = 25

, so

\sum x_i^2 = 1000

. Known sum of squares = 879. Therefore

x^2 + y^2 = 121

x^2 + y^2 = 121

04Final Answer

4/4

Using

(x+y)^2 = x^2 + y^2 + 2xy

225 = 121 + 2xy

, so

xy = 52

\boxed{xy = 52}

Concepts from this question2 concepts unlocked

★

Variance Shortcut Formula

EASY

Variance = E(X²) - [E(X)]². Compute the mean of squares minus the square of the mean. Avoids computing individual deviations.

\sigma^2 = \frac{\sum f_i x_i^2}{N} - \left(\frac{\sum f_i x_i}{N}\right)^2

Nearly every JEE variance question is faster with the shortcut. The direct formula with (xᵢ - mean)² takes longer and has more room for arithmetic errors.

Variance computationSD from raw dataGrouped data variance

Practice (10 Qs) →

Effect of Linear Transformation on Mean and Variance

STANDARD

If yᵢ = axᵢ + b, then mean(y) = a * mean(x) + b, and Var(y) = a² * Var(x). Adding a constant does not change variance. Multiplying by a constant scales variance by a².

\bar{y} = a\bar{x} + b, \quad \sigma_y^2 = a^2 \sigma_x^2

JEE frequently asks 'if 5 is added to each observation' or 'if each value is doubled'. This single rule answers all such questions instantly.

Change of origin/scaleEffect of adding constantEffect of multiplying constant

Practice (9 Qs) →

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