#ssc 10th
- To divide a line segment internally in the ratio 3 : 4 by construction, the number of equal parts marked on the auxiliary ray is Answer: D. 7 We mark 3 + 4 = 7 equal parts and join the 7th point to the other end of the segment. 1
- The distance between the points (1, 1) and (4, 5) is Answer: D. 5 √((4-1)² + (5-1)²) = √(9 + 16) = 5 . 1
- The slope of a line parallel to the Y-axis is Answer: A. not defined For a line parallel to the Y-axis, x₂ - x₁ = 0 , so the slope y₂ - y₁/x₂ - x₁ is not defined. 1
- The midpoint of the segment joining \((-4, 6)\) and \((2, -2)\) is Answer: A. (-1, 2) (-4+2/2, 6-2/2) = (-1, 2) . 1
- The centroid of the triangle with vertices (0, 0), (6, 0) and (0, 9) is Answer: A. (2, 3) (0+6+0/3, 0+0+9/3) = (2, 3) . 1
- \(\cos^2\theta\,(1 + \tan^2\theta) = \) Answer: A. 1 cos²θ × sec²θ = 1 . 1
- If \(\tan\theta = \frac{3}{4}\) and \(\theta\) is acute, then \(\sin\theta = \) Answer: C. 3/5 Take opposite side 3 and adjacent side 4; the hypotenuse is 5. sinθ = 3/5 . 1
- From the top of a tower 50 m high, the angle of depression of a car on the ground is 45°. The distance of the car from the foot of the tower is Answer: A. 50 m tan 45° = 50/d = 1 ⇒ d = 50 m. 1
- A point is 13 cm from the centre of a circle of radius 5 cm. The length of a tangent drawn from this point to the circle is Answer: A. 12 cm The radius is perpendicular to the tangent, so tangent length = √(13² - 5²) = √(144) = 12 cm. 1
- Two circles of radii 5.5 cm and 4.2 cm touch each other externally. The distance between their centres is Answer: C. 9.7 cm For circles touching externally, the distance between the centres is the sum of the radii: 5.5 + 4.2 = 9.7 cm. 1
- The measure of an angle inscribed in a semicircle is Answer: B. 90° An angle inscribed in a semicircle is a right angle, because its intercepted arc is 180°. 1
- In cyclic quadrilateral ABCD, \(\angle A = 75°\). Then \(\angle C = \) Answer: C. 105° Opposite angles of a cyclic quadrilateral are supplementary: 180° - 75° = 105° . 1
- The distance of the point (3, 4) from the origin is Answer: A. 5 √(3² + 4²) = √(25) = 5 . 1
- The midpoint of the segment joining (2, 6) and (4, 8) is Answer: D. (3, 7) (2+4/2, 6+8/2) = (3, 7) . 1
- The slope of the X-axis is Answer: B. 0 The X-axis makes an angle of 0° with the positive X-axis and tan 0° = 0 . 1
- Line PA touches a circle at A, and a secant through P meets the circle at B and C (with B between P and C). If \(PB = 4\) and \(PC = 9\), then \(PA = \) Answer: B. 6 By the tangent secant segments theorem, PA² = PB × PC = 4 × 9 = 36 , so PA = 6 . 1
- Chords AB and CD of a circle intersect at an interior point E. If \(AE = 6\), \(EB = 4\) and \(CE = 3\), then \(ED = \) Answer: A. 8 By the theorem of chords intersecting inside a circle, AE × EB = CE × ED ⇒ 24 = 3 × ED ⇒ ED = 8 . 1
- The angle between a tangent to a circle and a chord through the point of contact is 55°. The measure of the arc intercepted by this angle is Answer: B. 110° By the tangent secant angle theorem, the angle is half the intercepted arc, so the arc is 2 × 55° = 110° . 1
- The distance between the points \((-3, 2)\) and \((5, -4)\) is Answer: B. 10 √((5+3)² + (-4-2)²) = √(64 + 36) = √(100) = 10 . 1
- The slope of the line passing through (2, 3) and (4, 7) is Answer: B. 2 m = 7 - 3/4 - 2 = 4/2 = 2 . 1
- The centroid of the triangle with vertices (1, 2), (3, 4) and (5, 6) is Answer: D. (3, 4) (1+3+5/3, 2+4+6/3) = (3, 4) . 1
- The point dividing the segment joining A(2, 3) and B(8, 9) internally in the ratio 1 : 2 (from A) is Answer: A. (4, 5) x = 1(8) + 2(2)/3 = 4 , y = 1(9) + 2(3)/3 = 5 . 1
- The line through the points (1, k) and (3, 5) has slope 2. Then \(k = \) Answer: D. 1 5 - k/3 - 1 = 2 ⇒ 5 - k = 4 ⇒ k = 1 . 1
- The points (1, 1), (2, 3) and (3, k) are collinear. Then \(k = \) Answer: D. 5 Slope of the first two points = 3-1/2-1 = 2 . For collinearity, k-3/3-2 = 2 ⇒ k = 5 . 1