Question

If the mean of the following grouped data is \(21\): then \(k\) is one of the roots of the equation:

• A. \(2x^2-23x-10=0\)
• B. \(4x^2-35x+24=0\)
• C. \(2x^2-19x-10=0\)
• D. \(2x^2-35x+98=0\)

Answer 1

The Correct Option is A

Concept: For grouped data, the mean is computed using \[ \bar{x}=\frac{\sum f_i x_i}{\sum f_i} \] where \(x_i\) are class midpoints and \(f_i\) are frequencies.
Step 1:Find class midpoints.} \[ 7.5,\;12.5,\;17.5,\;22.5,\;27.5,\;32.5 \]
Step 2:Compute total frequency.} \[ N=2+k+28+54+(k+1)+5 \] \[ N=90+2k \]
Step 3:Compute \( \sum f_i x_i \).} \[ 2(7.5)+k(12.5)+28(17.5)+54(22.5)+(k+1)(27.5)+5(32.5) \] Simplifying, \[ =15+12.5k+490+1215+27.5k+27.5+162.5 \] \[ =1910+40k \]
Step 4:Use the mean formula.} Given mean \(=21\): \[ 21=\frac{1910+40k}{90+2k} \] \[ 21(90+2k)=1910+40k \] \[ 1890+42k=1910+40k \] \[ 2k=20 \] \[ k=10 \]
Step 5:Form the equation whose root is \(k\).} Substituting \(k=10\) into the options, the equation satisfied is \[ 2x^2-23x-10=0 \] Thus the correct option is \[ \boxed{2x^2-23x-10=0} \]