Class10 Math Assignment B812C6

1. Find the least number divisible by all numbers from 1 to 5. The least number divisible by all numbers from 1 to 5 is the LCM of 1, 2, 3, 4, 5. LCM(1,2,3,4,5) = 60. Answer: 60 (Option B). --- 2. Find the largest number dividing 398, 436, 542 leaving remainders 7, 11, 15 respectively. Subtract remainders: 398 - 7 = 391 436 - 11 = 425 542 - 15 = 527 Find HCF of 391, 425, 527. Prime factors: 391 = 17 × 23 425 = 25 × 17 527 = 17 × 31 HCF = 17. Answer: 17 (Option A). --- 3. Maximum capacity of bag for 490, 588, 882 kg wheat packed exactly. Find HCF of 490, 588, 882. Prime factors: 490 = 2 × 5 × 7² 588 = 2² × 3 × 7² 882 = 2 × 3² × 7² Common factors: 2 × 7² = 2 × 49 = 98. Answer: 98 kg (Option A). --- 4. Greatest 4-digit integer divisible by 97. Find largest 4-digit number divisible by 97. Divide 9999 by 97: $$9999 \div 97 = 103.09...$$ So, largest multiple is 97 × 103 = 9991. Answer: 9991 (Option C). --- 5. HCF of two numbers is 7, product is 168. Find LCM. Formula: $$\text{LCM} = \frac{\text{Product}}{\text{HCF}} = \frac{168}{7} = 24.$$ Answer: 24 (Option A). --- 6. Find HCF of 96 and 404. Prime factors: 96 = 2^5 × 3 404 = 2^2 × 101 Common factors: 2^2 = 4. Answer: 4 (Option C). --- 7. Find HCF of 35 and 22. 35 factors: 5 × 7 22 factors: 2 × 11 No common factors. HCF = 1. Answer: 1 (Option A). --- 8. A quadratic polynomial has: Answer: (C) at most 2 zeroes. --- 9. A consistent pair of linear equations gives: Answer: (A) Unique solution. --- 10. 30th term of A.P. 10, 7, 4, ... Common difference: $$d = 7 - 10 = -3.$$ Formula for nth term: $$a_n = a + (n-1)d = 10 + 29(-3) = 10 - 87 = -77.$$ Answer: -77 (Option A). --- 11. In A.P. 15, 12, 9, 6, ... find common difference. $$d = 12 - 15 = -3.$$ Answer: -3 (Option C). --- 12. In A.P. 54, 58, 62, ... value of a? First term $$a = 54.$$ Answer: 54 (Option D). --- 13. Distance of point (4,4) from origin. Formula: $$\sqrt{(4-0)^2 + (4-0)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}.$$ Answer: 4√2 (Option C). --- 14. For data with $$\bar{x} = 25$$ and $$Z = 25$$, find M. Z-score formula: $$Z = \frac{X - \bar{x}}{\sigma}$$ If Z = 25 and $$\bar{x} = 25$$, then $$X = M = Z + \bar{x} = 25 + 25 = 50$$? But options suggest M = 25 or -25 or 5 or -5. Assuming M = mean = 25. Answer: 25 (Option A). --- 15. If HCF(10,15) = 2a + 1, find a. HCF(10,15) = 5. So, $$5 = 2a + 1 \Rightarrow 2a = 4 \Rightarrow a = 2.$$ Answer: 2. --- 16. Graph of $$p(x) = 3x + 5$$ is a ________. Linear polynomial graph is a line. Answer: Line. --- 17. Zero of linear polynomial $$p(x) = mx + 1$$ where $$m \neq 0$$ is: Set $$p(x) = 0$$: $$mx + 1 = 0 \Rightarrow x = -\frac{1}{m}.$$ Answer: $$-\frac{1}{m}$$. --- 18. If two lines are parallel, linear equation has ________ solution. Parallel lines have no solution. Answer: zero. --- 19. Distance between (12,0) and (0,35). $$\sqrt{(12-0)^2 + (0-35)^2} = \sqrt{144 + 1225} = \sqrt{1369} = 37.$$ Answer: 37 units. --- 20. Mean of first five even numbers. First five even numbers: 2, 4, 6, 8, 10. Mean: $$\frac{2 + 4 + 6 + 8 + 10}{5} = \frac{30}{5} = 6.$$ Answer: 6. --- 21. Class mark of class 30–40. Class mark = $$\frac{30 + 40}{2} = 35.$$ Answer: 35. --- 22. Probability of a certain event. Certain event probability = 1. Answer: 1. --- 23. Probability of event E + Probability of not E = Sum = 1. Answer: 1. --- 24. Product of zeroes of $$P(x) = 6x^2 - x + k$$ is -3. Product of zeroes = $$\frac{c}{a} = \frac{k}{6} = -3 \Rightarrow k = -18.$$ Options given: 2, -2, 6. None matches -18, so answer is none of these. --- 25. If $$\alpha$$ and $$\beta$$ are zeroes of $$p(x) = ax^2 + bx + c$$, then $$\alpha + \beta = -\frac{b}{a}$$. Answer: $$-\frac{b}{a}$$. --- 26. If $$(x + 1)$$ is factor of $$x^2 - 3ax + 3a - 7$$, find $$a$$. Substitute $$x = -1$$: $$(-1)^2 - 3a(-1) + 3a - 7 = 0 \Rightarrow 1 + 3a + 3a - 7 = 0 \Rightarrow 6a - 6 = 0 \Rightarrow 6a = 6 \Rightarrow a = 1.$$ Answer: 1. --- 27. If $$\alpha$$ and $$\beta$$ are zeroes of $$x^2 - 16$$, find $$\alpha \beta (\alpha + \beta)$$. Zeroes of $$x^2 - 16 = 0$$ are $$\pm 4$$. Sum: $$\alpha + \beta = 4 + (-4) = 0$$. Product: $$\alpha \beta = -16$$. So, $$\alpha \beta (\alpha + \beta) = -16 \times 0 = 0.$$ Answer: 0. --- 28. If zeroes of $$ax^2 + bx + b$$ are reciprocals, then $$b = a$$. Because product of zeroes = $$\frac{c}{a} = \frac{b}{a}$$ and if zeroes are reciprocals, product = 1. So, $$\frac{b}{a} = 1 \Rightarrow b = a.$$ Answer: b = a (not in options, so none). --- 29. Pair of equations $$3x + 2ky = 2$$ and $$2x + 5y + 1 = 0$$ has no solution, find $$k$$. Condition for no solution: $$\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$$ Here: $$a_1 = 3, b_1 = 2k, c_1 = 2$$ $$a_2 = 2, b_2 = 5, c_2 = -1$$ Set: $$\frac{3}{2} = \frac{2k}{5} \Rightarrow 3 imes 5 = 2 imes 2k \Rightarrow 15 = 4k \Rightarrow k = \frac{15}{4}.$$ Answer: 15/4. --- 30. If $$3x + 2y = 5$$ and $$2x + 3y = 10$$, find $$x + y$$. Multiply first by 3: $$9x + 6y = 15$$ Multiply second by 2: $$4x + 6y = 20$$ Subtract: $$5x = -5 \Rightarrow x = -1.$$ Substitute in first: $$3(-1) + 2y = 5 \Rightarrow -3 + 2y = 5 \Rightarrow 2y = 8 \Rightarrow y = 4.$$ Sum: $$x + y = -1 + 4 = 3.$$ Answer: 3. --- 31. Probability of getting even number on one die throw. Even numbers: 2, 4, 6. Probability = $$\frac{3}{6} = \frac{1}{2}.$$ Answer: 1/2. --- 32. Probability of an event cannot be: Negative or greater than 1. Answer: -1. --- 33. If $$P(A) = \frac{3}{4}$$, then $$P(\bar{A}) = 1 - P(A) = 1 - \frac{3}{4} = \frac{1}{4}.$$ Answer: 1/4. --- 34. If $$P(A) = \frac{x}{3}$$ and $$P(\bar{A}) = \frac{2}{5}$$, find $$x$$. Sum of probabilities = 1: $$\frac{x}{3} + \frac{2}{5} = 1 \Rightarrow \frac{5x + 6}{15} = 1 \Rightarrow 5x + 6 = 15 \Rightarrow 5x = 9 \Rightarrow x = \frac{9}{5} = 1.8.$$ Answer: 1.8. --- 35. 3.14... is irrational? False, 3.14... (pi) is irrational. Answer: True. --- 36. Quadratic polynomial has at least two zeroes? False, it can have 0, 1 or 2 real zeroes. Answer: False. --- 37. Number of zeroes of $$p(x) = x^3 - x$$ is 3. True, cubic polynomial has 3 zeroes (real or complex). Answer: True. --- 38. Product of zeroes of $$x^3 - 3x^2 - x + 3$$ is -3. For cubic $$ax^3 + bx^2 + cx + d$$, product of zeroes = $$-\frac{d}{a} = -\frac{3}{1} = -3.$$ Answer: True. --- 39. Graph of $$3y - 8 = 7$$ is line parallel to x-axis. Simplify: $$3y = 15 \Rightarrow y = 5.$$ Horizontal line parallel to x-axis. Answer: True. --- 40. Sum of probabilities of all elementary events is 1. True. --- 41. Probability of Sun rising in West is -1. False, probability cannot be negative. --- 42. From rectangle ABCD, values of x and y are 22 and 8. Assuming true as per figure. Answer: True. --- 43. Algebraic representation of total cost of 5 pencils and 7 pens is $$5x = 50 - 7y$$. False, should be $$5x + 7y = 50$$. Answer: False. --- 44. Midpoint of (8, -4) and (-4, 8) is (6, 6). Midpoint: $$\left(\frac{8 + (-4)}{2}, \frac{-4 + 8}{2}\right) = (2, 2).$$ Answer: False. --- 45. Distance of (3,4) from origin is 5. $$\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5.$$ Answer: True. --- 46. Which term of A.P. 13, 20, 27, ... is 384? Common difference: $$d = 7.$$ Formula: $$a_n = a + (n-1)d = 13 + (n-1)7 = 384.$$ Solve: $$7(n-1) = 371 \Rightarrow n-1 = 53 \Rightarrow n = 54.$$ Answer: 54th term. --- 47. Sum of first 15 multiples of 3. Multiples: 3, 6, 9, ..., 45. Sum: $$S_n = \frac{n}{2}[2a + (n-1)d] = \frac{15}{2}[2 \times 3 + 14 \times 3] = \frac{15}{2}[6 + 42] = \frac{15}{2} \times 48 = 15 \times 24 = 360.$$ Answer: 360. --- 48. Centre of circle with diameter endpoints (10, -8) and (-6, 2). Centre is midpoint: $$\left(\frac{10 + (-6)}{2}, \frac{-8 + 2}{2}\right) = (2, -3).$$ Answer: (2, -3). --- 49. Mean of 5, 12, 8, 15, 16, 21 and x is 13. Sum: $$5 + 12 + 8 + 15 + 16 + 21 + x = 13 \times 7 = 91.$$ Calculate sum without x: $$5 + 12 + 8 + 15 + 16 + 21 = 77.$$ So, $$77 + x = 91 \Rightarrow x = 14.$$ Answer: 14. --- 50. Mean of 45, 49, 53, 32, 63 and x is 47. Sum: $$45 + 49 + 53 + 32 + 63 + x = 47 \times 6 = 282.$$ Sum without x: $$45 + 49 + 53 + 32 + 63 = 242.$$ So, $$242 + x = 282 \Rightarrow x = 40.$$ Answer: 40. --- 51. Probability of scoring 80 marks in 80-mark test. Assuming scoring exactly 80 marks is certain event. Probability = 1. Answer: 1. --- 52. Two numbers sum 14, difference 4. Bigger number? Let numbers be $$x$$ and $$y$$, $$x > y$$. $$x + y = 14$$ $$x - y = 4$$ Add: $$2x = 18 \Rightarrow x = 9.$$ Answer: 9. --- 53. For pair of linear equations $$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$$, number of solutions? Infinite solutions. Answer: Infinite. --- 54. Triangle with vertices (5,5), (5,0), (0,5) is? Right-angled isosceles triangle. Answer: Right-angled isosceles triangle. --- 55. Distance of point from Y-axis is called? Abscissa or x-coordinate. Answer: Abscissa. --- q_count: 55