Vardaan Learning Institute
Answer Key: Quadratic Equations PYQ
Class: 10 (CBSE)
Subject: Mathematics
Chapter: 04
BOARD EXAM 2025 (Set 30/6/3)
Q12. Which of the following equations does not have a real
root?
Q35 (a). There is a circular park of diameter 65 m where AB is
a diameter. An entry gate is to be constructed at a point P on the boundary such that distance of P
from A is 35 m more than the distance of P from B. Find distance of point P from A and B
respectively.
Let \(BP = x\), \(AP = x + 35\). Angle in semicircle: \(\angle APB = 90�\).
\((x+35)^2 + x^2 = 65^2 \Rightarrow 2x^2 + 70x - 3000 = 0 \Rightarrow x^2 + 35x - 1500 = 0\).
\((x + 60)(x - 25) = 0 \Rightarrow x = 25\). So \(BP = 25\) m, \(AP = 60\) m.
\((x+35)^2 + x^2 = 65^2 \Rightarrow 2x^2 + 70x - 3000 = 0 \Rightarrow x^2 + 35x - 1500 = 0\).
\((x + 60)(x - 25) = 0 \Rightarrow x = 25\). So \(BP = 25\) m, \(AP = 60\) m.
OR
Q35 (b). Find the smallest value of \(p\) for which the
quadratic equation \(x^2 - 2(p+1)x + p^2 = 0\) has real roots. Hence, find the roots.
For real roots: \(D \ge 0 \Rightarrow 4(p+1)^2 - 4p^2 \ge 0 \Rightarrow 8p + 4 \ge 0 \Rightarrow p
\ge -0.5\).
Smallest \(p = -0.5\). Substituting: \(x^2 - x + 0.25 = 0 \Rightarrow x = 0.5, 0.5\).
Smallest \(p = -0.5\). Substituting: \(x^2 - x + 0.25 = 0 \Rightarrow x = 0.5, 0.5\).