UPSC CDS 1 2022 Math Paper

Concept:
- Congruence triangle: If two or more triangles are congruent, then all of their corresponding angles and sides are as well congruent.
- 'Side-Angle-Side' triangle congruence theorem: If the two sides of a triangle are equal to the two sides of another triangle, and the angle formed by these sides in the two triangles are equal, then these two triangles are congruent.
- Concentric Circles: Concentric circles are circles with a common center.
- The perpendicular from the center of a circle to a chord bisects the chord.
Calculation:
Statement: 1
Given that, AB and AC are two equal chords of the circle.
In △APB and △APC ,
⇒AB=AC‌‌ [Given ]
⇒∠BAP=∠CAP‌‌[ Given ]
⇒AP=AP‌‌ [Common]
⇒△APB≅△APC‌‌ [by SAS congruence criterion]
⇒ So, by property of congruence triangle we have
⇒BP=CP and ∠APB=∠APC
⇒∠APB+∠APC=180∘ [Linear pair ]
⇒2∠APB=180∘[∠APB=∠APC]
⇒∠APB=90∘
Now, BP=CP and ∠APB=90∘
Therefore, AP is the perpendicular bisector of chord BC .
Hence, AP passes through the center O of the circle.
Statement: 2

Draw a perpendicular from the center of the circle OM to the line AD .
We can see that BC and AD are the chords of the smaller and bigger circle respectively.
We know that a perpendicular drawn from the center of the circle bisects the chord.
⇒BM=MC‌‌⋯(1)
⇒AM=MD‌‌⋯−(2)
Subtracting (2) from (1), we get
AM−BM=DM−CM
⇒AB=CD
⇒AB+BC=CD+bC
⇒AC=BD
∴ Both 1 and 2 are correct.