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Class 10 › Mathematics › Quadratic Equations
CBSE2026Class Class 10 · Mathematics5 Marks · Long✅ Verified
(a) ABCD is a rectangle of dimensions 80 cm × 60 cm. Another rectangle PQRS is drawn inside ABCD leaving space of equal width x cm along the edges of ABCD. If area PQRS is half of the area ABCD, then find the value of x.
OR
(b) A train covers a distance of 90 km at a uniform speed. Had the speed been 15 km/h more, it would have taken 30 minutes less for the same journey. Find the original speed of the train.
✅ Answer & Solution
**(a)** Area of ABCD $=80\times60=4800\ cm^2$. Dimensions of PQRS $=(80-2x)\times(60-2x)$, with area $=2400$.
$$(80-2x)(60-2x)=2400 \Rightarrow 4800-280x+4x^2=2400$$ $$4x^2-280x+2400=0 \Rightarrow x^2-70x+600=0$$ Discriminant $=4900-2400=2500=50^2$. $$x=\frac{70\pm50}{2}=60\text{ or }10$$ Since $x=60$ is not feasible (dimensions would become negative), $$x=10\ cm$$
**(b)** Let original speed $=s$ km/h. $$\frac{90}{s}-\frac{90}{s+15}=\frac12$$ $$90\times\frac{15}{s(s+15)}=\frac12 \Rightarrow 2700=s^2+15s$$ $$s^2+15s-2700=0$$ Discriminant $=225+10800=11025=105^2$. $$s=\frac{-15+105}{2}=45\ km/h$$
