All questions
- Out of the dates given below which date constitutes a Pythagorean triplet? (A) 15/08/17 (B) 16/08/16 (C) 3/5/17 (D) 4/9/15 Answer: (A) 15/08/17 Check (15, 8, 17): 8² + 15² = 64 + 225 = 289 = 17² . It is a Pythagorean triplet. (16, 8, 16): 64 + 256 = 320 ≠ 256 (3,... 1
- If a, b, c are sides of a triangle and a² + b² = c², name the type of triangle. (A) Obtuse angled triangle (B) Acute angled triangle (C) Right angled triangle (D) Equilateral triangle Answer: (C) Right angled triangle By the converse of Pythagoras theorem, if the square of one side equals the sum of the squares of the othe... 1
- Find perimeter of a square if its diagonal is 10√2 cm. (A) 10 cm (B) 40√2 cm (C) 20 cm (D) 40 cm Answer: (D) 40 cm For a square of side s, diagonal = s√(2) . s√(2) = 10√(2) ⇒ s = 10\ cm Perimeter = 4 × 10 = 40\ cm 1
- Altitude on the hypotenuse of a right angled triangle divides it in two parts of lengths 4 cm and 9 cm. Find the length of the altitude. (A) 9 cm (B) 4 cm (C) 6 cm (D) 2√6 cm Answer: (C) 6 cm By the theorem of geometric mean, the altitude on the hypotenuse is the geometric mean of the two parts: h² = 4 × 9 = 36 h... 1
- Height and base of a right angled triangle are 24 cm and 18 cm find the length of its hypotenus (A) 24 cm (B) 30 cm (C) 15 cm (D) 18 cm Answer: (B) 30 cm h² = 24² + 18² = 576 + 324 = 900 h = 30\ cm 1
- In ΔABC, AB = 6√3 cm, AC = 12 cm, BC = 6 cm. Find measure of ∠A. (A) 30° (B) 60° (C) 90° (D) 45° Answer: (A) 30^° AB² + BC² = 108 + 36 = 144 = AC² So by the converse of Pythagoras theorem, ∠ B = 90^° and AC is the hypotenuse. BC = 6 is h... 1
- Find the length of the hypotenuse of a right-angled triangle if remaining sides are 9 cm and 12 cm. Answer: Hypotenuse = 15 cm By the Pythagoras theorem, hypotenuse² = 9² + 12² = 81 + 144 = 225. ∴ hypotenuse = √(225) = 15 cm. 1
- A side of an isosceles right angled triangle is x. Find its hypotenuse. Answer: The hypotenuse is x√(2) . In an isosceles right-angled triangle, the two sides making the right angle are equal, each x. By Pythagor... 1
- In ΔPQR; PQ = √8, QR = √5, PR = √3. Is ΔPQR a right angled triangle? If yes, which angle is of 90°? Answer: Yes, triangle PQR is right angled and ∠ PRQ = 90^° . PQ² = 8, QR² = 5, PR² = 3 QR² + PR² = 5 + 3 = 8 = PQ² By the converse of Pythag... 1
- In ΔRST, ∠S = 90°, ∠T = 30°, RT = 12 cm then find RS and ST. Answer: RS = 6 cm and ST = 6√(3) cm triangle RST is a 30^° – 60^° – 90^° triangle with hypotenuse RT = 12 cm ( ∠ R = 60^° ). Side opposite 3... 1
- Find the diagonal of a rectangle whose length is 16 cm and area is 192 sq.cm. Answer: The diagonal is 20 cm. breadth = 192/16 = 12\ cm By Pythagoras theorem, d² = 16² + 12² = 256 + 144 = 400 d = 20\ cm 1
- Find the length of the side and perimeter of an equilateral triangle whose height is √3 cm. Answer: Side = 2 cm, perimeter = 6 cm For an equilateral triangle of side a, the altitude bisects the base and makes a 30^° – 60^° – 90^° tr... 1
- In ΔABC seg AP is a median. If BC = 18, AB² + AC² = 260 Find AP. Answer: AP = 7 P is the midpoint of BC, so BP = 9. By Apollonius theorem, AB² + AC² = 2AP² + 2BP² 260 = 2AP² + 2 × 81 2AP² = 98, AP² = 49 AP... 1
- ΔABC is an equilateral triangle. Point P is on base BC such that PC = (1/3) BC, if AB = 6 cm find AP. Answer: AP = 2√(7) cm Draw AD ⊥ BC. In an equilateral triangle, D is the midpoint of BC, so DC = 3 cm. AD = √(3)/2 × 6 = 3√(3)\ cm PC = 1/3... 1
- From the information given in the figure 2.31, prove that PM = PN = √3 × a Answer: triangle PQR is equilateral with side a, so PS = √(3)/2a and S is the midpoint of QR; then MS = NS = 3a/2 and Pythagoras theorem giv... 1
- Prove that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides. Answer: Apply Apollonius theorem to the two triangles formed by one diagonal, using the fact that the diagonals bisect each other. Let □ABCD... 1
- Pranali and Prasad started walking to the East and to the North respectively, from the same point and at the same speed. After 2 hours distance between them was 15√2 km. Find their speed per hour. Answer: Their speed is 7.5 km per hour. Let the speed be x km/hr. In 2 hours each walks 2x km. East and North are perpendicular, so the dist... 1
- In ΔABC, ∠BAC = 90°, seg BL and seg CM are medians of ΔABC. Then prove that: 4(BL² + CM²) = 5 BC² Answer: Pythagoras theorem in triangle BAL , triangle CAM and triangle BAC gives the result. L is the midpoint of AC, so AL = AC/2 ; M is th... 1
- Find the correct answer: Distance covered per litre (km) 12-14, 14-16, 16-18, 18-20; No. of cars 11, 12, 20, 7. The median of the distances covered per litre is in the group ... Answer: (C) 16–18 N = 50, so N/2 = 25 . Cumulative frequencies: 11, 23, 43, 50. The first cumulative frequency greater than 25 is 43, which... 1
- Find the correct answer: No. of trees planted by each student 1-3, 4-6, 7-9, 10-12; No. of students 7, 8, 6, 4. For a frequency polygon, the coordinates of the point for the class 4-6 are ... (A) (4, 8) (B) (3, 5) (C) (5, 8) (D) (8, 4) Answer: (C) (5, 8) A point of a frequency polygon is (class mark, frequency). Class mark of 4–6 = 4 + 6/2 = 5 and its frequency is 8, so the... 1
- The following table shows the income of farmers in a grape season. Find the mean of their income. Answer: Mean income = 52.5 thousand rupees, i.e. ₹ 52,500 ClassClass mark x_i Frequency f_i f_ix_i 20–302510250 30–403511385 40–504515675 50... 1
- The loans sanctioned by a bank for construction of farm ponds are shown in the following table. Find the mean of the loans. Answer: Mean loan = 65.4 thousand rupees, i.e. ₹ 65,400 ClassClass mark x_i Frequency f_i f_ix_i 40–504513585 50–6055201100 60–7065241560 70... 1
- The weekly wages of 120 workers in a factory are shown in the following frequency distribution table. Find the mean of the weekly wages. Answer: Mean weekly wage = ₹ 4250 ClassClass mark x_i Frequency f_i f_ix_i 0–200010001515000 2000–4000300035105000 4000–6000500050250000 600... 1
- The following frequency distribution table shows the amount of aid given to 50 flood affected families. Find the mean of the amount of aid. Answer: Mean aid = 72.4 thousand rupees, i.e. ₹ 72,400 ClassClass mark x_i Frequency f_i f_ix_i 50–60557385 60–706513845 70–8075201500 80–90... 1