Explanation
1. Identify the Terms of the Arithmetic Progression
Let the first term of the arithmetic progression be a and the common difference be d=23.
The n-th term of an AP is given by Tn=a+(n−1)d.
2. Find the First Term (a)
We are given that the sum of the first 11 terms (S11) is 88. The formula for the sum of n terms is Sn=2n[2a+(n−1)d].
S11=211[2a+10(23)]=88
211[2a+15]=88
Divide both sides by 11:
21[2a+15]=8
2a+15=16
2a=1⟹a=0.5
Now, calculate T10 and T11:
T10=0.5+13.5=14
T11=0.5+15=15.5
3. Determine p and q
The roots of the quadratic equation 3x2−px+q=0 are 14 and 15.5.
Comparing this to the standard form Ax2+Bx+C=0, where the sum of roots is −AB and the product of roots is AC:
Sum of roots:
14+15.5=3p
29.5=3p
p=29.5×3=88.5
Product of roots:
14×15.5=3q
217=3q
q=217×3=651
4. Calculate q−2p
Substitute the values of p and q:
q−2p=651−2(88.5)
q−2p=651−177
q−2p=474
Correct Option: 2 (474)
