If the average score is 3, find the: A) Value of x B) The total number of students C) Standard deviation

Here's how to solve the problem:

Step 1: Set up expressions for the sum of frequencies and the sum of (mark × frequency).
The given marks ($m$) and frequencies ($f$) are:

Marks ($m$)	Frequencies ($f$)
1	$x^2+2$
2	1
3	$x^2+1$
4	3
5	$x+3$

The total number of students is the sum of frequencies ($\sum f$):

$$\sum f = (x^2+2) + 1 + (x^2+1) + 3 + (x+3)$$ $$\sum f = 2x^2 + x + 10$$

The sum of (mark × frequency) is ($\sum (m \cdot f)$):

$$\sum (m \cdot f) = 1(x^2+2) + 2(1) + 3(x^2+1) + 4(3) + 5(x+3)$$ $$\sum (m \cdot f) = x^2+2 + 2 + 3x^2+3 + 12 + 5x+15$$ $$\sum (m \cdot f) = 4x^2 + 5x + 34$$

Step 2: Use the given average score to find the value(s) of x.
The average score ($\bar{m}$) is given as 3. The formula for the average is:

$$\bar{m} = \frac{\sum (m \cdot f)}{\sum f}$$

Substitute the expressions and the given average:

$$3 = \frac{4x^2 + 5x + 34}{2x^2 + x + 10}$$

Multiply both sides by the denominator:

$$3(2x^2 + x + 10) = 4x^2 + 5x + 34$$ $$6x^2 + 3x + 30 = 4x^2 + 5x + 34$$

Rearrange into a quadratic equation:

$$6x^2 - 4x^2 + 3x - 5x + 30 - 34 = 0$$ $$2x^2 - 2x - 4 = 0$$

Divide by 2:

$$x^2 - x - 2 = 0$$

Factor the quadratic equation:

$$(x-2)(x+1) = 0$$

This gives two possible values for $x$: $x=2$ or $x=-1$.
We must check if these values result in non-negative frequencies.
For $x=2$: Frequencies are $2^2+2=6$, 1, $2^2+1=5$, 3, $2+3=5$. All are positive.
For $x=-1$: Frequencies are $(-1)^2+2=3$, 1, $(-1)^2+1=2$, 3, $-1+3=2$. All are positive.
Both values are mathematically valid. However, in the context of such problems, $x$ is typically a positive value. We will proceed with $x=2$.