# ASSIGNMENT NO -1 (SIMILAR TRIANGLES)

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1 ASSIGNMENT NO -1 (SIMILAR TRIANGLES) 1. In an equilateral Δ ABC, the side BC is trisected at D. Prove that 9AD2 = 7AB2 2. P and Q are points on sides AB and AC respectively, of ΔABC. If AP = 3 cm,pb = 6 cm, AQ = 5 cm and QC = 10 cm, show that BC = 3 PQ. 3. The image of a tree on the film of a camera is of length 35 mm, the distance from the lens to the film is 42 mm and the distance from the lens to the tree is 6 m. How tall is the portion of the tree being photographed? 4.. Prove that in any triangle the sum of the squares of any two sides is equal to twice the square of half of the third side together with twice the square of the median, which bisects the third side. 5. If a straight line is drawn parallel to one side of a triangle intersecting the othertwo sides, then it divides the two sides in the same ratio. 6. If ABC is an obtuse angled triangle, obtuse angled at B and if AD ^ CB Prove that AC 2 =AB 2 + BC 2 +2 BC x BD 7. If a straight line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side. 8. ABCD is a quadrilateral with AB =AD. If AE and AF are internal bisectors of ΔABC, D and E are points on AB and AC respectively such that AD/ DB = AEC/EC and ΔABC is isosceles. 9. In a ΔABC, points D, E and F are taken on the sides AB, BC and CA respectively such that DE IIAC and FE II AB Prove that three times the sum of the squares of the sides of a triangle is equal to four times the sum of the squares of the medians of the triangle. 11. If a perpendicular is drawn from the vertex of a right angled triangle to its hypotenuse, then the triangles on each side of the perpendicular are similar to the whole triangle. 12. A man sees the top of a tower in a mirror which is at a distance of 87.6 m from the tower. The mirror is on the ground, facing upward. The man is 0.4 m away from the mirror, and the distance of his eye level from the ground is 1.5 m. How tall is the tower? (The foot of man, the mirror and the foot of the tower lie along a straight line). 13. In a right Δ ABC, right angled at C, P and Q are points of the sides CA and CB respectively, which divide these sides in the ratio 2: 1. Prove that (I) 9AQ 2 = 9AC 2 +4BC 2 (II) 9 BP 2 = 9 BC 2 + 4AC 2 (III) 9 (AQ 2 +BP 2 ) = 13AB ABC is a triangle. PQ is the line segment intersecting AB in P and AC in Q such that PQ parallel to BC and divides Δ ABC into two parts equal in area. Find BP: AB. 15. P and Q are the mid points on the sides CA and CB respectively of triangle ABC right angled at C. Prove that4(aq 2 +BP 2 ) = 5 AB 2

2 ASSIGNMENT NO -2 (POLYNOMIALS)al MCQ Assignments in Mathematics Class X (Term I) 1. If α,β are zeroes of the polynomial f(x) = x 2 + px + q, then polynomial having 1/α and 1/β as its zeroes is (a) x 2 + qx + p (b) x 2 px + q (c) qx 2 + px + 1 (d) px 2 + qx If α and β are zeroes of x 2 4x + 1, then 1/α + 1/β αβ is (a) 3 (b) 5 (c) 5 (d) 3 3. The quadratic polynomial having zeroes as 1 and 2 is : (a) x 2 x + 2 (b) x 2 x 2 (c) x 2 + x 2 (d) x 2 + x If α, β are zeroes of x 2 6x + k, what is the value of k if 3α+2β=20? (a) 16 (b) 8 (c) 2 (d) 8 5. If one zero of 2x 2 3x + k is reciprocal to the other, then the value of k is (a) 2 (b) 23 (c) 32 (d) 3 6. The quadratic polynomial whose sum of zeroes is 3 and product of zeroes is 2 is (a) x 2 + 3x 2 (b) x 2 2x + 3 (c) x 2 3x + 2 (d) x 2 3x 2 7. If (x + 1) is a factor of x 2 3ax + 3a 7, then the value of a is : (a)1 (b) 1 (c) 0 (d) 2 8. The number of polynomials having zeroes 2 and 5 is : (a)1 (b) 2(c)3 (d) more than 3 9. The quadratic polynomial p(y) with 15 and 7 as sum and one of the zeroes respectively is : (a) y 2 15y 56 (b) y 2 15y + 56 (c) y y + 56 (d) y y The value of p for which the polynomial x 3 + 4x 2 px + 8 is exactly divisible by (x 2) is : (a) 0 (b) 3 (c) 5 (d) If 1 is a zero of the polynomial p(x) = ax 2 3(a 1)x 1, then the value of a is : (a) 1 (b) 1 (c) 2 (d) If 4 is a zero of the polynomial x 2 x (2 + 2k), then the value of k is : (a) 3 (b) 9 (c) 6 (d) 9 13.The degree of the polynomial (x + 1)(x 2 x x 4 + 1) is : (a) 2 (b) 3 (c) 4 (d) If (x + 1) is a factor of x 2 3ax + 3a 7, then the value of a is : (a) 1 (b) 1 (c) 0 (d) If sum of the squares of zeroes of the quadratic polynomial f(x) = x 2 8x + k is 40, the value of k is (a) 10 (b) 12 (c) 14 (d) 16

3 ASSIGNMENT NO -3 (REAL NUMBERS) 1. Express 140 as a product of its prime factors 2. Find the LCM and HCF of 12, 15 and 21 by the prime factorization method. 3. Find the LCM and HCF of 6 and 20 by the prime factorization method. 4. State whether13/3125 will have a terminating decimal expansion or a non-terminating repeating 5. State whether 17/8 will have a terminating decimal expansion or a non-terminating repeating 6. Find the LCM and HCF of 26 and 91 and verify that LCM HCF = product of the two numbers. 7. Use Euclid s division algorithm to find the HCF of 135 and Use Euclid s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m 9. Prove that 3 is irrational. 10. Show that 5 3 is irrational 11. Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer. 12. An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march? 13. Express 156 as a product of its prime factors. 14. Find the LCM and HCF of 17, 23 and 29 by the prime factorization method. 15. Find the HCF and LCM of 12, 36 and 160, using the prime factorization method. 16. State whether 6/15 will have a terminating decimal expansion or a non-terminating repeating 17. State whether35/50 will have a terminating decimal expansion or a non-terminating repeating Find the LCM and HCF of 192 and 8 and verify that LCM HCF = product of the two numbers. 20. Use Euclid s algorithm to find the HCF of 4052 and Show that any positive odd integer is of the form of 4q + 1 or 4q + 3, where q is some integer. 22. Use Euclid s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m. 23. Prove that is irrational Prove that 1/ 2 is irrational. (3 marks)

4 25. In a school there are tow sections- section A and Section B of class X. There are 32 students in section A and 36 students in section B. Determine the minimum number of books required for their class library so that they can be distributed equally among students of section A or section B. 26. Express 3825 as a product of its prime factors. 27. Find the LCM and HCF of 8, 9 and 25 by the prime factorization method. 28. Find the HCF and LCM of 6, 72 and 120, using the prime factorization method. 29. State whether 29/343 will have a terminating decimal expansion or a non-terminating repeating 30. State whether 23/ 23 52will have a terminating decimal expansion or a non-terminating repeating decimal 31. Find the LCM and HCF of 336 and 54 and verify that LCM HCF = product of the two numbers 32. Use Euclid s division algorithm to find the HCF of 867 and Show that every positive even integer is of the form 2q, and that every positive odd integer is of the form 2q + 1, where q is some integer. 34. Use Euclid s division lemma to show that the cube of any positive integer is of the form 9m,9lm + 1 or 9m Prove that 7 5 is irrational. 36. Prove that 5 is irrational. 37. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point? 38. Express 5005 as a product of its prime factors. 39. Find the LCM and HCF of 24, 36 and 72 by the prime factorization method. 40. Find the LCM and HCF of 96 and 404 by the prime factorization method 41. State whether 64/455 will have a terminating decimal expansion or a non-terminating repeating decimal 42. State whether15/ 1600 will have a terminating decimal expansion or a non-terminating repeating 43. Find the LCM and HCF of 510 and 92 and verify that LCM HCF = product of the two numbers. 44. Use Euclid s division algorithm to find the HCF of 196 and (3 marks) 45. Use Euclid s division lemma to show that the cube of any positive integer is of the form 9m,9m + 1 or 9m + 8

5 46. Show that every positive odd integer is of the form 2q, and that every positive odd integer is of the form 2q + 1, where q is some integer Show that 3 2 is irrational. 48. Prove that is irrational. 49. A sweet seller has 420 kaju barfis and 130 badam barfis. She wants to stack them in such a way that each stack has the same number, and they take up the least area of the tray. What is the maximum number of barfis that can be placed in each stack for this purpose? 50. Use Euclid s division algorithm to find the HCF of : (i) 135 and 225 (ii) 196 and (iii) 867 and Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer. 52. An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march? Sol. Hints: Find the HCF of 616 and Find the largest number which divides 615 and 963 leaving remainder 6 in each case. 54. show that is an irrational number. 55. Find the HCF of 52 and 117 and express it in form 52 x + 117y. 56. find the (HCF LCM) for the numbers 100 and the HCF of 45 and 105 is 15. Write their LCM. 58. Write a rational number between 2 aaaaaa Divide 4x 3 + 2x 2 + 5x 6 by 2x 2 + 3x find the zeros of 4x 2 7 and verify the relationship between the zeros and its coefficients. 61. find a quadratic polynomial whose zeros are 5 + 2aaaaaa if one zero of the polynomial 5x x P is reciprocal of the other then find p. 63. if the product of two zeros of polynomial 2x 3 + 3x 2 5x 6 is 3 then find its third zero. 64. Find the zero of 4x x Obtain all other zeros of the polynomial x 4 3x 3 x 2 + x + 9x 6, if two of its zeros are 3aaaaaa 3.

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