JEE Main 2020 (7 January Shift 1)

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Animated Solution for Mathematics - Statistics: If the mean and variance of eight numbers $3, 7, 9, 12, 13, 20, x$ and $y$ be 10 and 25 respectively then $x y$ is equal to ________.

Enter Numerical Value:

Visualized Solution

Problem Setup

Given numbers: $3, 7, 9, 12, 13, 20, x, y$
Total count $(n) = 8$
Mean $(\overset{x}{ˉ}) = 10$
Variance $(σ^{2}) = 25$
Objective: Find $x y$

The Mean Formula

Mean formula: $\overset{x}{ˉ} = \frac{\sum x _{i}}{n}$

Substituting Values into Mean

$\frac{3 + 7 + 9 + 12 + 13 + 20 + x + y}{8} = 10$

Simplifying the Sum

Sum of knowns: $3 + 7 + 9 + 12 + 13 + 20 = 64$
Equation becomes: $\frac{64 + x + y}{8} = 10$

Finding $(x + y)$

Multiply by $8$ : $64 + x + y = 80$
Subtract $64$ : $x + y = 80 - 64$
Result: $x + y = 16$

The Variance Formula

Variance formula: $σ^{2} = \frac{\sum x _{i}^{2}}{n} - (\overset{x}{ˉ})^{2}$

Substituting Values into Variance

$25 = \frac{3 ^{2} + 7 ^{2} + 9 ^{2} + 1 2 ^{2} + 1 3 ^{2} + 2 0 ^{2} + x ^{2} + y ^{2}}{8} - 1 0^{2}$

Calculating Sum of Squares

Squares: $9, 49, 81, 144, 169, 400$
Sum: $9 + 49 + 81 + 144 + 169 + 400 = 852$
Equation: $25 = \frac{852 + x ^{2} + y ^{2}}{8} - 100$

Simplifying the Variance Equation

Add $100$ : $125 = \frac{852 + x ^{2} + y ^{2}}{8}$
Multiply by $8$ : $125 \times 8 = 852 + x^{2} + y^{2}$
$1000 = 852 + x^{2} + y^{2}$

Finding $(x^{2} + y^{2})$

Subtract $852$ : $x^{2} + y^{2} = 1000 - 852$
Result: $x^{2} + y^{2} = 148$

The Algebraic Link

Identity: $(x + y)^{2} = x^{2} + y^{2} + 2 x y$

Substituting into the Identity

Substitute $x + y = 16$
Substitute $x^{2} + y^{2} = 148$
Equation: $(16)^{2} = 148 + 2 x y$

Solving for $2 x y$

$(16)^{2} = 256$
$256 = 148 + 2 x y$
$2 x y = 256 - 148$
$2 x y = 108$

Final Calculation for $x y$

$x y = \frac{108}{2}$
$x y = 54$

Summary

Key Takeaway: Mean gives $\sum x_{i}$ , Variance gives $\sum x_{i}^{2}$ .
Algebraic Link: Use $(x + y)^{2} = x^{2} + y^{2} + 2 x y$ to find $x y$ .
Final Result: $x y = 54$

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The Sigma Insight: Variance and Standard Deviation

Analyzing the Setup

You have a crime scene—a set of eight numbers:

3, 7, 9, 12, 13, 20, x, y

—and two clues: the mean and the variance. Your mission is to find the product of the two missing suspects,

x

and

y

The Mean

The Center of Gravity

The mean,

\overset{x}{ˉ} = 10

, is your first clue. It tells you the 'center of gravity' of your data.

By summing all numbers, we set up the following equation:

\frac{3 + 7 + 9 + 12 + 13 + 20 + x + y}{8} = 10

This simplifies to

64 + x + y = 80

, which gives us the first linear constraint:

x + y = 16

The Variance

The Spread

The variance,

σ^{2} = 25

, is the second clue. It tells you how spread out the data is.

Using the computational formula

σ^{2} = \frac{\sum x _{i}^{2}}{n} - (\overset{x}{ˉ})^{2}

, we calculate the sum of squares. Substituting the known values:

25 = \frac{3 ^{2} + 7 ^{2} + 9 ^{2} + 1 2 ^{2} + 1 3 ^{2} + 2 0 ^{2} + x ^{2} + y ^{2}}{8} - 1 0^{2}

Calculating the sum of the known squares, we get

9 + 49 + 81 + 144 + 169 + 400 = 852

. Substituting this back into the equation:

25 = \frac{852 + x ^{2} + y ^{2}}{8} - 100

Adding

100

to both sides gives

125 = \frac{852 + x ^{2} + y ^{2}}{8}

. Multiplying by

8

yields

1000 = 852 + x^{2} + y^{2}

, which simplifies to:

x^{2} + y^{2} = 148

The Algebraic Bridge

Now you have two powerful pieces of information:

x + y = 16

and

x^{2} + y^{2} = 148

. You need to find the product

x y

The identity

(x + y)^{2} = x^{2} + y^{2} + 2 x y

acts as your bridge. Substituting the known values:

(16)^{2} = 148 + 2 x y

This simplifies to

256 = 148 + 2 x y

. Solving for

2 x y

2 x y = 256 - 148 = 108

Finally, dividing by

2

, we find the product of the suspects:

$x y = 54$

Common Traps & FAQs

Why do we use the formula

σ^{2} = \frac{\sum x _{i}^{2}}{n} - (\overset{x}{ˉ})^{2}

instead of the definition

\frac{\sum ( x _{i} - x ˉ ) ^{2}}{n}

Great question! While both are mathematically equivalent, the computational formula

\frac{\sum x _{i}^{2}}{n} - (\overset{x}{ˉ})^{2}

is significantly more efficient when you have unknown variables. It allows you to aggregate the knowns and unknowns separately, saving you from expanding binomials like

(x - 10)^{2}

and

(y - 10)^{2}

, which would only complicate your algebra.

Do I need to find the individual values of

x

and

y

to solve this?

Not at all! This is a common trap. Students often waste precious exam time trying to solve for

x

and

y

individually using a quadratic equation. Remember, the question only asks for the product

x y

. By using the identity

(x + y)^{2} = x^{2} + y^{2} + 2 x y

, you can find the product directly. Work smarter, not harder!

What if the variance was zero? Would that change the approach?

If the variance were zero, it would imply that all numbers in the set are identical (i.e.,

x = y = \overset{x}{ˉ} = 10

). While the approach remains valid, the problem would become trivial. In JEE problems, variance is usually non-zero to ensure you engage with the algebraic relationships between the variables.

Animated Solution for Mathematics - Statistics: If the mean and variance of eight numbers $3, 7, 9, 12, 13, 20, x$ and $y$ be 10 and 25 respectively then $x y$ is equal to ________.

Enter Numerical Value:

Visualized Solution

Problem Setup

The Mean Formula

Substituting Values into Mean

Simplifying the Sum

Finding $(x + y)$

The Variance Formula

Substituting Values into Variance

Calculating Sum of Squares

Simplifying the Variance Equation

Finding $(x^{2} + y^{2})$

The Algebraic Link

Substituting into the Identity

Solving for $2 x y$

Final Calculation for $x y$

Summary

The Sigma Insight: Variance and Standard Deviation

Analyzing the Setup

The Mean

The Variance

The Algebraic Bridge

Similar Questions

If the mean and variance of eight numbers 3, 7, 9, 12, 13, 20, $x$ and $y$ be 10 and 25 respectively, then $x \cdot y$ is equal to....

Let the mean and variance of 8 numbers $x, y, 10, 12, 6, 12, 4, 8$ be 9 and 9.25 respectively. If $x > y$ , then $3 x - 2 y$ is equal to ______.

If the mean and the variance of 6, 4, a, 8, b, 12, 10, 13 are 9 and 9.25 respectively, then $a + b + ab$ is equal to :

The mean and variance of the data $4, 5, 6, 6, 7, 8, x, y$ where $x < y$ are $6$ and $\frac{9}{4}$ respectively. Then $x^{4} + y^{2}$ is equal to

If the mean and variance of the frequency distribution $x_{i} f_{i} 2444 6 α 815108 12 β 144165$ are 9 and 15.08, respectively, then the value of $α^{2} + β^{2} - α β$ is _____.

The mean and variance of 7 observations are 8 and 16 respectively. If one observation 14 is omitted and $a$ and $b$ are respectively mean and variance of remaining 6 observation, then $a + 3 b - 5$ is equal to ______.

If the mean of the data : 7, 8, 9, 7, 8, 7, $λ$ , 8 is 8, then the variance of this data is :-

The mean and variance of seven observations are 8 and 16, respectively. If 5 of the observations are 2, 4, 10, 12, 14, then the product of the remaining two observations is :

Let the six numbers $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}$ be in A.P. and $a_{1} + a_{3} = 10$ . If the mean of these six numbers is $\frac{19}{2}$ and their variance is $σ^{2}$ , then $8 σ^{2}$ is equal to

Animated Solution for Mathematics - Statistics: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the mean and variance of eight numbers 3,7,9,12,13,20,x and y be 10 and 25 respectively then xy is equal to ________.

Enter Numerical Value:

Visualized Solution

Problem Setup

The Mean Formula

Substituting Values into Mean

Simplifying the Sum

Finding .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } (x+y)

The Variance Formula

Substituting Values into Variance

Calculating Sum of Squares

Simplifying the Variance Equation

Finding .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } (x2+y2)

The Algebraic Link

Substituting into the Identity

Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 2xy

Final Calculation for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } xy

Summary

The Sigma Insight: Variance and Standard Deviation

Analyzing the Setup

The Mean

The Variance

The Algebraic Bridge

Similar Questions

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the mean and variance of eight numbers 3, 7, 9, 12, 13, 20, x and y be 10 and 25 respectively, then x⋅y is equal to....

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let the mean and variance of 8 numbers x,y,10,12,6,12,4,8 be 9 and 9.25 respectively. If x>y, then 3x−2y is equal to ______.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the mean and the variance of 6, 4, a, 8, b, 12, 10, 13 are 9 and 9.25 respectively, then a+b+ab is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The mean and variance of the data 4,5,6,6,7,8,x,y where x<y are 6 and 49​ respectively. Then x4+y2 is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the mean of the data : 7, 8, 9, 7, 8, 7, λ, 8 is 8, then the variance of this data is :-

The mean and variance of seven observations are 8 and 16, respectively. If 5 of the observations are 2, 4, 10, 12, 14, then the product of the remaining two observations is :

Animated Solution for Mathematics - Statistics: If the mean and variance of eight numbers $3, 7, 9, 12, 13, 20, x$ and $y$ be 10 and 25 respectively then $x y$ is equal to ________.

Finding $(x + y)$

Finding $(x^{2} + y^{2})$

Solving for $2 x y$

Final Calculation for $x y$

If the mean and variance of eight numbers 3, 7, 9, 12, 13, 20, $x$ and $y$ be 10 and 25 respectively, then $x \cdot y$ is equal to....

Let the mean and variance of 8 numbers $x, y, 10, 12, 6, 12, 4, 8$ be 9 and 9.25 respectively. If $x > y$ , then $3 x - 2 y$ is equal to ______.

If the mean and the variance of 6, 4, a, 8, b, 12, 10, 13 are 9 and 9.25 respectively, then $a + b + ab$ is equal to :

The mean and variance of the data $4, 5, 6, 6, 7, 8, x, y$ where $x < y$ are $6$ and $\frac{9}{4}$ respectively. Then $x^{4} + y^{2}$ is equal to

If the mean of the data : 7, 8, 9, 7, 8, 7, $λ$ , 8 is 8, then the variance of this data is :-