Mathematics MCQs - Part 8
211. The value of 1 - cos²Î¸ is equal to:
Explanation: This is a rearrangement of the fundamental identity sin²Î¸ + cos²Î¸ = 1. Subtracting cos²Î¸ from both sides gives sin²Î¸ = 1 - cos²Î¸.
212. If the radius of a circle is 10 cm, the area of a sector with a 90° central angle is:
Explanation: Area of sector = (θ/360) × Ï€r² = (90/360) × Ï€(10)² = (1/4) × 100Ï€ = 25Ï€ cm².
213. The point P(2, -5) lies in which quadrant?
Explanation: The fourth quadrant is where the x-coordinate is positive and the y-coordinate is negative.
214. If two triangles are congruent, their areas are ______.
Explanation: Congruent figures have the same size and shape, which means all corresponding sides and angles are equal. Therefore, their areas must also be equal.
215. The value of tan 60° is:
Explanation: In a 30°-60°-90° triangle, the side opposite 60° is √3 and the side adjacent is 1 (in ratio). tanθ = opposite/adjacent, so tan 60° = √3/1 = √3.
216. The volume of a pyramid is (1/3) × Base Area × ______.
Explanation: The general formula for the volume of a pyramid (including a cone) is V = (1/3) × Base Area × Height.
217. A line segment joining two points on a circle is called a ______.
Explanation: A chord is a line segment whose endpoints both lie on the circle. A secant is a line that intersects the circle at two points.
218. The distance between points L(5, -8) and M(-7, -3) is:
Explanation: Using the distance formula: d = √[(-7-5)² + (-3-(-8))²] = √[(-12)² + 5²] = √[144 + 25] = √169 = 13.
219. The value of cosec 60° is:
Explanation: cosecθ is the reciprocal of sinθ. Since sin 60° = √3/2, cosec 60° = 1 / (√3/2) = 2/√3.
220. If the perimeter of a square is 40 cm, its area is:
Explanation: Perimeter = 4s = 40 cm, so the side 's' is 10 cm. Area = s² = 10² = 100 cm².
221. If the height of a lighthouse is 90 m and an observer sees a ship at an angle of depression of 60°, how far is the ship from the lighthouse?
Explanation: The angle of elevation from the ship to the lighthouse is also 60°. Let d be the distance. tan(60°) = height/distance = 90/d. So, √3 = 90/d. d = 90/√3 = (90√3)/3 = 30√3 m.
222. The point P(1, -3) lies in which quadrant?
Explanation: The fourth quadrant is where the x-coordinate is positive and the y-coordinate is negative.
223. The value of sec 30° is:
Explanation: secθ is the reciprocal of cosθ. Since cos 30° = √3/2, sec 30° = 1 / (√3/2) = 2/√3.
224. Two circles are congruent if they have the same ______.
Explanation: Congruent circles are identical in size. The size of a circle is determined by its radius (or diameter).
225. The sum of the lengths of any two sides of a triangle is always ______ the third side.
Explanation: This is the Triangle Inequality Theorem, a fundamental property of all triangles.
226. If the radius of a sphere is doubled, its surface area increases by a factor of:
Explanation: Surface Area A = 4Ï€r². If the new radius is 2r, the new area A' = 4Ï€(2r)² = 4Ï€(4r²) = 4 × (4Ï€r²) = 4A. The area increases by a factor of 4.
227. The line segment AB is divided into four equal parts. If A is (-14, -10) and B is (6, -2), what are the coordinates of the first point of division from A?
Explanation: The first point divides the segment in the ratio 1:3. x = (1*6 + 3*(-14))/(1+3) = (6-42)/4 = -36/4 = -9. y = (1*(-2) + 3*(-10))/(1+3) = (-2-30)/4 = -32/4 = -8. The coordinates are (-9, -8).
228. The value of sin²30° + cos²30° is:
Explanation: The identity sin²Î¸ + cos²Î¸ = 1 is true for all values of θ, including 30°.
229. If the diagonals of a rhombus are 12 cm and 16 cm, what is the length of its side?
Explanation: The diagonals of a rhombus are perpendicular bisectors of each other. They form four right-angled triangles with the sides of the rhombus as hypotenuses. The legs of these triangles are half the lengths of the diagonals (6 cm and 8 cm). By Pythagoras theorem, side² = 6² + 8² = 36 + 64 = 100. So, the side is 10 cm.
230. A circle with center O has a chord AB. If ∠AOB = 60° and the radius is 6 cm, what is the length of chord AB?
Explanation: Triangle AOB is an isosceles triangle with OA = OB = 6 cm. Since the angle between them is 60°, the other two angles are also 60°, making it an equilateral triangle. Therefore, the length of chord AB is also 6 cm.
231. The slope of the line x = 5 is:
Explanation: The line x = 5 is a vertical line. Vertical lines have an undefined slope because the change in x (the "run") is zero.
232. The value of cot 30° is:
Explanation: cotθ is the reciprocal of tanθ. Since tan 30° = 1/√3, cot 30° = 1 / (1/√3) = √3.
233. A solid metal sphere of radius 9 cm is melted to make a wire of diameter 4 mm. What is the length of the wire?
Explanation: Volume of sphere = (4/3)Ï€(9)³ = 972Ï€ cm³. Radius of wire = 2 mm = 0.2 cm. Volume of wire (cylinder) = Ï€(0.2)²L = 0.04Ï€L. Equating volumes: 972Ï€ = 0.04Ï€L. L = 972 / 0.04 = 24300 cm = 243 m.
234. If two circles are concentric, they have the same ______.
Explanation: Concentric circles are circles that share the same center but have different radii.
235. The sum of the angles in a quadrilateral is:
Explanation: Any quadrilateral can be divided into two triangles by a diagonal. Since the sum of angles in a triangle is 180°, the sum for a quadrilateral is 2 × 180° = 360°.
236. If the radius of a cylinder is doubled and its height is halved, its volume will:
Explanation: Original Volume V = Ï€r²h. New radius = 2r, new height = h/2. New Volume V' = Ï€(2r)²(h/2) = Ï€(4r²)(h/2) = 2Ï€r²h = 2V. The volume will double.
237. The point P divides segment AB with A(2, 7) and B(-4, -8) into three equal parts. Which of the following could be the coordinates of P?
Explanation: The two points of trisection are at ratios 1:2 and 2:1. For 1:2 ratio: x=(1*(-4)+2*2)/3=0, y=(1*(-8)+2*7)/3=6/3=2. So (0, 2) is one point. For 2:1 ratio: x=(2*(-4)+1*2)/3=-6/3=-2, y=(2*(-8)+1*7)/3=-9/3=-3. So (-2, -3) is the other point.
238. The value of sec²Î¸ - 1 is:
Explanation: This is from the identity 1 + tan²Î¸ = sec²Î¸. Subtracting 1 from both sides gives tan²Î¸ = sec²Î¸ - 1.
239. The diagonal of a rectangle is 13 cm and its breadth is 5 cm. What is its length?
Explanation: The diagonal and sides form a right-angled triangle. Using Pythagoras theorem, l² + 5² = 13². So, l² + 25 = 169. l² = 144, and l = 12 cm. This is a 5-12-13 Pythagorean triplet.
240. The length of an arc of a circle with radius 18 cm and central angle 80° is:
Explanation: Length of arc = (θ/360) × 2Ï€r = (80/360) × 2Ï€(18) = (2/9) × 36Ï€ = 8Ï€ cm. Using Ï€ ≈ 3.14, the length is 25.12 cm.
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