#ssc 10th
- The length of an arc of a circle of radius 21 cm with central angle 60° is \(\left(\pi = \frac{22}{7}\right)\) Answer: D. 22 cm l = 60/360 × 2 × 22/7 × 21 = 22 cm. 1
- To construct a tangent at a point P on a circle with centre O, we draw a line through P that is Answer: A. perpendicular to radius OP A tangent is perpendicular to the radius at the point of contact, so we draw the line through P perpen... 1
- How many tangents can be drawn to a circle from a point outside the circle? Answer: A. 2 Exactly two tangents can be drawn to a circle from an external point. 1
- How many tangents can be drawn to a circle from a point inside the circle? Answer: A. 0 Every line through an interior point cuts the circle at two points, so no tangent can be drawn. 1
- \(\triangle ABC \sim \triangle LMN\) and \(BC : MN = 5 : 3\). Then Answer: D. triangle LMN is smaller than triangle ABC MN = 3/5 BC , so every side of triangle LMN is 3/5 of the corresponding side of triangl... 1
- \(\triangle PQR\) is to be constructed similar to \(\triangle ABC\) such that \(\frac{AB}{PQ} = \frac{3}{4}\). If \(AB = 6\) cm, then \(PQ = \) Answer: D. 8 cm PQ = 4/3 × 6 = 8 cm. 1
- To construct a triangle similar to a given triangle with ratio of corresponding sides 3 : 5 (the new triangle larger), the minimum number of equal parts to be marked on the auxiliary ray is Answer: B. 5 We mark the larger of the two numbers, 5 equal parts, and use the 3rd and 5th points. 1
- To draw tangents to a circle with centre O from an external point P, an auxiliary circle is drawn whose diameter is Answer: D. seg OP We find the midpoint M of OP and draw a circle with centre M and radius MO. This circle has OP as a diameter, and it cuts... 1
- In the construction of tangents from an external point using a circle on diameter OP, the lines drawn are tangents because Answer: B. an angle inscribed in a semicircle is a right angle If A is a point of contact, ∠ OAP is inscribed in a semicircle, so it is 90°.... 1
- A point P is 10 cm from the centre of a circle of radius 6 cm. If tangents are drawn from P, the length of each tangent segment is Answer: B. 8 cm Tangent length = √(10² - 6²) = √(64) = 8 cm. 1
- The construction of a tangent at a point on a circle without using the centre is based on Answer: C. the converse of the tangent secant angle theorem We make the angle between the chord and the line equal to the inscribed angle in... 1
- \(\triangle PQR \sim \triangle XYZ\) and \(PQ : XY = 3 : 4\). Then \(A(\triangle PQR) : A(\triangle XYZ) = \) Answer: C. 9 : 16 The ratio of the areas of similar triangles is the square of the ratio of corresponding sides: 3² : 4² = 9 : 16 . 1
- For two triangles with equal bases, the ratio of their areas is equal to the ratio of their Answer: C. corresponding heights A = 1/2 × base × height . With equal bases, the areas are proportional to the heights. 1
- In \(\triangle ABC\), seg \(DE \parallel\) side \(BC\), with D on AB and E on AC. If \(\frac{AD}{DB} = \frac{2}{3}\) and \(AE = 4\), then \(EC = \) Answer: A. 6 By the basic proportionality theorem, AE/EC = 2/3 ⇒ EC = 3 × 4/2 = 6 . 1
- \(\triangle ABC \sim \triangle PQR\) with \(\angle A = 50°\) and \(\angle B = 60°\). Then \(\angle R = \) Answer: B. 70° Corresponding angles are equal: ∠ R = ∠ C = 180° - 50° - 60° = 70° . 1
- The hypotenuse of a right-angled triangle whose other sides are 9 cm and 40 cm is Answer: B. 41 cm √(81 + 1600) = √(1681) = 41 cm. 1
- In a 30°–60°–90° triangle, the side opposite the 30° angle is 5 cm. The hypotenuse is Answer: B. 10 cm The side opposite 30° is half the hypotenuse, so the hypotenuse is 2 × 5 = 10 cm. 1
- Which of the following is NOT a Pythagorean triplet? Answer: B. (6, 8, 11) 6² + 8² = 100 ≠ 121 . The others check: 64 + 225 = 289 , 121 + 3600 = 3721 , 400 + 441 = 841 . 1
- The measure of a semicircular arc is Answer: B. 180° A semicircle is half of a circle, so its arc measures 360°/2 = 180° . 1
- In cyclic quadrilateral ABCD, \(\angle B = 110°\). Then \(\angle D = \) Answer: D. 70° Opposite angles of a cyclic quadrilateral are supplementary: 180° - 110° = 70° . 1
- \(\triangle PQR\) is constructed similar to \(\triangle ABC\) such that \(BC : QR = 7 : 5\). If \(BC = 7\) cm, then \(QR = \) Answer: A. 5 cm QR = 5/7 × 7 = 5 cm. 1
- A point is 25 cm from the centre of a circle of radius 7 cm. The length of the tangent from the point to the circle is Answer: D. 24 cm √(25² - 7²) = √(576) = 24 cm. 1
- Two circles of radii 7 cm and 3 cm touch each other internally. The distance between their centres is Answer: C. 4 cm For circles touching internally, the distance between the centres is the difference of the radii: 7 - 3 = 4 cm. 1
- While drawing tangents from an external point P to a circle with centre O, the points of contact are obtained where the given circle meets Answer: D. the circle with diameter OP The circle with diameter OP cuts the given circle at A and B. Since ∠ OAP = ∠ OBP = 90° , PA and PB a... 1