161. A girl walks 200 meters to the east and then 150 meters to the north. What is the girl’s distance from the starting point?
Answer: 250 m
162. If 2 sin²β – cos²β = 2, what is the value of β?
Answer: 90°
163. The lines x = a and y = b are
Answer: Parallel lines
164. How many rotations are required for a circular wheel with a radius of 0.7 meters to cover a distance of 176 meters?
Answer: 22
165. An alphabet from the English alphabet is randomly selected. What is the probability that it is a letter from the word ‘MATHEMATICS’?
Answer: 9/26
166. What is the value of p for which the system of equations px – y = 2; 6x – 2y = 3 has only one solution (i.e., a unique solution)?
Answer: p ≠ 3
167. OBAC is a rectangle with three vertices A(0, 3), O(0, 0), and B(5, 0). What is the length of its diagonal?
Answer: √34 units
168. What is the sum of the first 20 odd natural numbers?
Answer: 400
169. If sin(x – 20°) = cos(3x – 10°), where 3x – 10° is an acute angle, what is the value of x?
Answer: 35°
170. What is the area of the triangle with vertices (a, b + c), (b, c + a), and (c, a + b)?
Answer: 0
171. If tan 30° = sin 45° cos 45° + sin 30°, then θ =
Answer: 15°
172. If the sum of the first p terms of an A.P. is ap² + bp, then its common difference is
Answer: 2a
173. The curved surface area of a cylinder is 264 m² and its volume is 924 m³. The ratio of its diameter to its height is
Answer: 1:2
174. The height of an equilateral triangle with side ‘a’ is equal to
Answer: √3a/2
175. The ratio of the LCM and HCF of the smallest composite and smallest prime numbers is
Answer: 1:2
176. In the given figure, MN || AB, BC = 7.5 cm, AM = 4 cm, and MC = 2 cm. Find the length of BN.
Answer: 2.5 cm
177. From a lighthouse, the angle of depression of two ships in opposite directions is 30° and 45°. If the height of the lighthouse is h meters, find the distance between the ships.
Answer: (√3+1)h
178. If the mode of a distribution is 12 more than its mean, by how much is the mode greater than the median?
Answer: 4
179. If A(4, 2), B(6, 5), and C(1, 4) are the vertices of ΔABC and AD is the median, what are the coordinates of D?
Answer: (5/2, 3)
180. If the cubic polynomial ax³ + bx² + cx + d has two zeros, each of which is a root, what is the third root?
Answer: -b/a