MAH-CET 2026 — Mathematics PYQ

Step 1: Identify properties of the given circle

The equation of the circle is given as:

$$x^2+y^2=25$$

This is a standard circle centered at the origin, $O(0,0)$, with a radius $R=\sqrt{25}=5$.

Step 2: Find the angle subtended by the chord QR at the center

Let us connect the points $Q(3,4)$ and $R(-4,3)$ to the center of the circle $O(0,0)$ to form vectors $\overrightarrow{OQ}$ and $\overrightarrow{OR}$:

• $\overrightarrow{OQ}=3\hat{i}+4\hat{j}$

• $\overrightarrow{OR}=-4\hat{i}+3\hat{j}$

To find the angle $\angle QOR$ subtended at the center, we use the dot product formula:

$$\overrightarrow{OQ}\cdot \overrightarrow{OR}=|\overrightarrow{OQ}|\cdot |\overrightarrow{OR}|\cdot \cos (\angle QOR)$$

Let's compute the dot product:

$$\overrightarrow{OQ}\cdot \overrightarrow{OR}=(3)(-4)+(4)(3)=-12+12=0$$

Since the dot product is $0$, the vectors are perpendicular to each other:

$$\cos (\angle QOR)=0⟹\angle QOR=90^{\circ }=\frac{\pi }{2}$$

Step 3: Apply the inscribed angle theorem

According to the properties of circles, the angle subtended by an arc (or chord) at the center of a circle is twice the angle subtended by the same arc at any point on the remaining part of the circle (circumference).

Therefore:

$$\angle QPR=\frac{1}{2}\cdot \angle QOR$$

$$\angle QPR=\frac{1}{2}\cdot \frac{\pi }{2}=\frac{\pi }{4}$$

Correct Answer

(c) $\frac{\pi }{4}$