Maths MCQs 181 to 210 questions

 

Mathematics MCQs - Part 7

181. The value of 1 - sin²θ is equal to:

sin²θ cos²θ tan²θ sec²θ

Explanation: This is a rearrangement of the fundamental identity sin²θ + cos²θ = 1. Subtracting sin²θ from both sides gives cos²θ = 1 - sin²θ.

182. If a cone is melted and cast into a cylinder of the same radius, and the cylinder's height is 5 cm, the height of the cone was:

15 cm 10 cm 5 cm 20 cm

Explanation: The volumes must be equal. Volume of cone = (1/3)πr²h_cone. Volume of cylinder = πr²h_cyl. So, (1/3)πr²h_cone = πr²(5). This simplifies to h_cone/3 = 5, so h_cone = 15 cm.

183. The point P(-3, -7) lies in which quadrant?

First Second Third Fourth

Explanation: In the Cartesian coordinate system, the third quadrant is where both the x-coordinate and the y-coordinate are negative.

184. A cyclic parallelogram must be a ______.

Rhombus Rectangle Square Trapezium

Explanation: In a cyclic quadrilateral, opposite angles are supplementary (sum to 180°). In a parallelogram, opposite angles are equal. For both conditions to be true, all angles must be 90°, which defines a rectangle.

185. The value of sec 0° is:

1 0 -1 Not defined

Explanation: secθ is the reciprocal of cosθ. Since cos 0° = 1, sec 0° = 1/1 = 1.

186. The ratio of the volume of a cone, a hemisphere, and a cylinder of the same radius and height is:

1:3:2 3:2:1 1:2:3 2:1:3

Explanation: For radius r and height h=r: V_cone = (1/3)πr³, V_hemisphere = (2/3)πr³, V_cylinder = πr³. The ratio is (1/3):(2/3):1, which simplifies to 1:2:3.

187. If two points on a given line subtend equal angles at two distinct points on the same side of the line, then the four points are ______.

Collinear Concyclic Coincident Coplanar

Explanation: This theorem is a condition for four points to be concyclic (lie on the same circle). It is the converse of the theorem that angles inscribed in the same arc are congruent.

188. What is the slope of the line joining points (3, 1) and (5, k) if the line is parallel to the line with slope 2?

3 4 -1 5

Explanation: Parallel lines have equal slopes. So, (k - 1) / (5 - 3) = 2. This gives (k - 1) / 2 = 2, so k - 1 = 4, and k = 5.

189. The value of cosec 45° is:

1/√2 √2 1 2

Explanation: cosecθ is the reciprocal of sinθ. Since sin 45° = 1/√2, cosec 45° = 1 / (1/√2) = √2.

190. The perimeter of a square with a diagonal of 8√2 cm is:

16 cm 16√2 cm 32 cm 64 cm

Explanation: The diagonal of a square with side 's' is s√2. So, s√2 = 8√2, which means the side 's' is 8 cm. The perimeter is 4s = 4 × 8 = 32 cm.

191. If the height of a cone is 24 cm and its base radius is 7 cm, its curved surface area is:

440 cm² 550 cm² 616 cm² 704 cm²

Explanation: First find the slant height (l) using the Pythagorean triplet (7, 24, 25), so l = 25 cm. Curved Surface Area = πrl = (22/7) × 7 × 25 = 550 cm².

192. The point P(0, -3) lies on ______.

the X-axis the Y-axis the origin the first quadrant

Explanation: Since the x-coordinate is 0, the point lies on the Y-axis. Specifically, it is on the negative part of the Y-axis.

193. The value of sec²θ - tan²θ is:

1 0 -1 2

Explanation: This is a rearrangement of the identity 1 + tan²θ = sec²θ. Subtracting tan²θ from both sides gives 1 = sec²θ - tan²θ.

194. If the radius of a circle is 10 cm and the length of a chord is 12 cm, what is the distance of the chord from the center?

6 cm 10 cm 12 cm 8 cm

Explanation: The perpendicular from the center to a chord bisects the chord. This creates a right-angled triangle with hypotenuse 10 cm (radius), one side 6 cm (half the chord), and the other side as the distance (d). By Pythagoras theorem, d² + 6² = 10². So, d² + 36 = 100, d² = 64, and d = 8 cm.

195. The ratio of the perimeter of two similar triangles is equal to the ratio of their corresponding ______.

sides areas square of sides altitudes only

Explanation: For similar triangles, the ratio of their perimeters is the same as the ratio of their corresponding sides, medians, and altitudes.

196. The point P divides the segment AB with A(-1, 7) and B(4, -3) in the ratio 2:3. What are the coordinates of P?

(1, 1) (1, 3) (2, 5) (3, 1)

Explanation: Using the section formula: x = (2*4 + 3*(-1))/(2+3) = (8-3)/5 = 1. y = (2*(-3) + 3*7)/(2+3) = (-6+21)/5 = 15/5 = 3. The coordinates are (1, 3).

197. The value of cosec²θ - cot²θ is:

1 0 -1 2

Explanation: This is a rearrangement of the identity 1 + cot²θ = cosec²θ. Subtracting cot²θ from both sides gives 1 = cosec²θ - cot²θ.

198. The capacity of a bucket in the shape of a frustum of a cone is its ______.

Curved surface area Total surface area Volume Slant height

Explanation: The capacity of any container refers to the amount it can hold, which is measured by its volume.

199. Two poles of height 18m and 7m are on the ground. A wire of length 22m is fastened to their tops. What is the distance between the poles?

11 m 11√3 m 22 m 29 m

Explanation: This forms a right-angled triangle where the hypotenuse is the wire (22m), one side is the difference in pole heights (18-7=11m), and the other side is the distance between the poles (d). So, d² + 11² = 22². d² + 121 = 484. d² = 363. d = √363 = √(121×3) = 11√3 m.

200. The angle subtended by a diameter at any point on the circle is:

45° 90° 180° 60°

Explanation: An angle inscribed in a semicircle is always a right angle (90°). The diameter subtends a semicircle arc of 180°, and the inscribed angle is half of that.

201. If the slope of line AB is 1/2, what is the slope of a line parallel to AB?

1/2 -1/2 2 -2

Explanation: Parallel lines have the same slope. Therefore, the slope of the parallel line is also 1/2.

202. The value of sin 45° is:

1/2 √3/2 1/√2 1

Explanation: In a 45°-45°-90° triangle, if the equal sides are 1, the hypotenuse is √2. Since sinθ = opposite/hypotenuse, sin 45° = 1/√2.

203. The volume of a cube is 1 m³. Its side length is:

1 cm 10 cm 100 cm 1000 cm

Explanation: The volume of a cube is l³. So, l³ = 1 m³. Taking the cube root, l = 1 meter. Since 1 meter = 100 cm, the side length is 100 cm.

204. An equilateral triangle is inscribed in a circle. What is the measure of the arc subtended by each side?

60° 90° 120° 180°

Explanation: An equilateral triangle has three equal sides, which means it will cut the circle into three equal arcs. Since the total measure of a circle is 360°, each arc measures 360°/3 = 120°.

205. The point P divides segment AB with A(3, 8) and B(-9, 3) in the ratio 3:1. What are the coordinates of P?

(-6, 4.25) (-3, 5.5) (0, 6.75) (-1.5, 4.5)

Explanation: Using the section formula: x = (3*(-9) + 1*3)/(3+1) = (-27+3)/4 = -24/4 = -6. y = (3*3 + 1*8)/(3+1) = (9+8)/4 = 17/4 = 4.25. The coordinates are (-6, 4.25).

206. If the radius of a sphere is doubled, its volume increases by a factor of:

2 4 8 16

Explanation: Volume V = (4/3)πr³. If the new radius is 2r, the new volume V' = (4/3)π(2r)³ = (4/3)π(8r³) = 8 × [(4/3)πr³] = 8V. The volume increases by a factor of 8.

207. The value of sec²45° is:

1 2 1/2 √2

Explanation: cos 45° = 1/√2. Since secθ = 1/cosθ, sec 45° = √2. Therefore, sec²45° = (√2)² = 2.

208. A line segment joining the center of a circle to the midpoint of a chord is ______ to the chord.

Parallel Perpendicular Equal Longer than

Explanation: This is a property of circles. The line segment from the center to the midpoint of a chord is always perpendicular to the chord.

209. What is the slope of a line with an inclination of 135°?

1 0 -1 Not defined

Explanation: The slope m = tan(θ). tan(135°) = tan(180° - 45°) = -tan(45°) = -1.

210. The total surface area of a cube is 96 cm². What is the length of its side?

6 cm 4 cm 16 cm 8 cm

Explanation: Total Surface Area = 6l². So, 96 = 6l². This gives l² = 96/6 = 16. Therefore, the side length l = √16 = 4 cm.

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