CBSE 10th Maths Question Paper 2025 (Set 3)

In triangle OAP, OA is the radius and PA is the tangent, so OA is perpendicular to PA (angle OAP = 90 degrees). In triangle OAP, the sum of angles is 180 degrees. Angle APO + angle OAP + angle AOP = 180 degrees. Since angle POB = 115 degrees, angle AOP = 180 - 115 = 65 degrees (linear pair). Thus, angle APO = 180 - 90 - 65 = 25 degrees.

The length of an arc l = (theta/360) * 2 * pi * r. Given length l = 20 cm and radius r = 19.09 (approximately 60/pi). However, assuming standard arc length formula l = r * theta (where theta is in radians), theta = l / r = 20 / (60/pi) = pi/3 radians. pi/3 radians is equal to 60 degrees. Therefore, choice (b) is correct.

Let the three terms in AP be a-d, a, a+d. Their sum is (a-d) + a + (a+d) = 3a. Given the sum is 30, we have 3a = 30, so a = 10. The middle term is 'a', which is 10.

The area of a sector is given by A = (1/2) * l * r, where l is the arc length and r is the radius. Given A = 20 cm^2 and l = 5 cm, we have 20 = (1/2) * 5 * r. This simplifies to 20 = 2.5 * r, so r = 20 / 2.5 = 8 cm.

Substitute x=1, y=2 into the equations: 2(1) - 3(2) + a = 0 => 2 - 6 + a = 0 => a = 4. Substitute into the second: 2(1) + 3(2) - b = 0 => 2 + 6 - b = 0 => b = 8. Comparing a=4 and b=8, we see that 2a = b (2*4 = 8).