Mathematics Booklet- SA1 Chap. No. Name of the chapter Page No. 1 Real Numbers 2-3 2 Polynomials 3-5 3 Pair of linear equations in two variables 5-7 8 Introduction to trigonometry 7-8 14 Statistics 9-11 3 Triangles 11-14 Answers 15-17 Chapter 1 –REAL NUMBERS 1 Mark Questions 1. If two positive integers ‘m’ and ‘n’ can be expressed as m=ab2 and n= a3b; a,b being prime numbers, then LCM ( m , n ) is : 2. If the least factor of ‘a’ is 3, the least prime factor of ‘b’ is 7, then the least prime factor of (a + b) is : 3. After how many places, the decimal form of 125 will terminate? 24.53 4. Write the condition to be satisfied by ‘q’ so that a rational number p/q has terminating decimal expansion. 5. HCF of two numbers is 23 and their LCM is 1449. If one of the numbers is 161, then the other number is : 6. If a = bq + r in Euclid’s division lemma, then r must satisfy : 7. A series of well defined steps which gives a procedure for solving a type of problem is called _______ 8. The unit’s digit of 73 is 3 then what will be the unit’s digit of 711 ? 9. Find the HCF of the smallest composite number and the smallest prime number. 10. If HCF of 65 and 117 is expressible in the form 65m – 117, then the value of ‘m’ is : 2 Mark Questions 11. Find the HCF of 336 and 54 by Euclid’s division algorithm. 12. What is the smallest number by which (√5 - √ 3) be multiplied to make it a rational number? Also find the number so obtained. 13. Find the largest number which divides 70 and 125, leaving remainder 5 and 8, respectively. 14. Explain why 3 x 5 x 7 + 7 + 2 x 7 is a composite number. 15. Given that HCF (135,225) = 45. Find LCM (135,225). 16. Show that every positive odd integer is of the form 4q + 1 or 4q + 3 17. Prove that there is no natural number for which 8n ends with digit zero 18. Show that 5√2 is irrational 19. Find the smallest number which when increased by 17 is exactly divisible by 520 and 468 20. Find the HCF of LCM of 30, 72 and 432 by prime factorization method. 21. The number 525 and 3000 are both divisible only by 3, 5, 15, 25 and 75. What is HCF ( 525, 3000) ? Justify your answer. 22. There is a circular path around a sports field. Ankit takes 18 minutes to drive one round of the field, while Ankita takes 12 minutes for the same. Suppose they both start at the same point, at the same time and go in the same direction. After how many minutes will they meet again at the starting point? 23. Prove 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. 24. Write the condition for terminating of a rational number. And hence, find whether the rational number (13/3125) has a terminating decimal or non-terminating repeating decimal. 25. What is the digit at unit’s place of 9n ? 3 Mark Questions 26. If the HCF of 210 and 55 is expressible in the form 210 X 5 – 55y , then find the value of ‘y’. 27. Show that square of an odd positive integer is of the form 8m + 1, for some whole number m 28. Prove that ( √2 + √ 3 ) is irrational 29. If n is an odd integer, then show that n2 – 1 is divisible by 8 30. Prove that any number of the form 4x + 2 can never be a perfect square. 2|Page 1 31. Prove that (2− √ 5) is an irrational number. 32. Using Euclid’s division, find the largest number that divides 70 and 125, leaving remainder 5 and 8, respectively. 33. Without actually performing the long division, find if 987/10500 will have terminating or non-terminating repeating decimal expansion. Give reasons for your answer. 34. Show that there is no positive integer ‘n’ for which √ − 1 + √ + 1 is rational. 35. State the following: (i) Euclid’s Division Lemma with boundary conditions. (ii) Fundamental Theorem of Arithmetic. 36. Using Euclid’s lemma find the length of the longest tape needed to measure a room of length, breadth, height of 520 cm, 480 cm, 750 cm respectively. 37. Find the HCF and LCM of 72, 120, 360 using prime factorization method. 38. Prove that product of three consecutive positive integers is divisible by 6. −√ 39. If +√ = , then determine whether ‘x’ is rational or irrational. √ 4 Mark questions 40. Prove that one of every three consecutive positive integers is divisible by 3 41. Show that one and only one of n, , + 2, n+ 4 is divisible by 3. 42. Prove that square of any positive integer is either of the form 3m or 3m+1 for some integer m 43. Prove that square of any positive integer is of the form 5q, 5q+1,5q+4 for some integer, q 44. Prove that √2 is irrational. 45. Three sets of English, Hindi and Mathematics books have to be stacked in such a way that all the books are stored topic wise and height of each stack is the same. The number of English books is 96, the number of Hindi books is 240 and the number of Mathematics books is 336. Assuming that books are of same thickness, determine the number f stacks of English, Hindi and Mathematics books respectively. 46. Prove that √7 is irrational. 47. Prove that if x and y are odd positive integers then x2+y2 is even but not divisible by 4. 48. Prove that n3 – n is divisible by 6, for every positive integer ‘n’ 49. Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 of 9m + 8. HOTS 50. Prove that if x and y are both odd positive integers then x2 + y2 is even but not divisible by 4. ( 3 Marks) Chapter 2 - POLYNOMIALS 1 Mark Questions 1. 2. 3. 4. If a and b are the zeroes of 2x2 + 5x – 10, then the value of ab is ______ If one of the zeroes of the quadratic polynomial (k – 1 )x2 + 2kx + 3 is 1, then the value of k is ____ For what values of k is the polynomial f(x) = 2x3 – kx2 + 5x + 9 exactly divisible by (x+2)? The graph of y = p(x) is given below. Find the number of zeroes Y X ’ 3|Page X Y’ 5. p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that p(x) = q(x)g(x) + r(x). Here write the condition that r(x) should satisfy. 6. A polynomial of the form ax5 + bx3 + cx2 + dx + e has at most _________ zeroes 7. The sum and product of zeroes of 2(x2 – 1) + 3x – 9 are : 8. Find a quadratic polynomial, the sum and product of its zeroes are 1 and –6 respectively. 9. Find a quadratic equation, whose roots are(1+√5) and (1-√5) . 10. If (x + 2)(2x - 1)(3x - 2) = 0, find the zeroes of the polynomial. 11. If the sum of the zeroes of the polynomial f(x) = 2x3 - 3kx2 + 4x - 5 is 6, then find the value of k. 2 Mark Questions 12. Find the value of p for which x = b is a zero of the polynomial x2 – (a+b) x + p 13. Find the zeroes of the polynomial a2 – a – 12 and verify relationship between the zeroes and the coefficients. 14. If one zero of the polynomial f(x) = 15 x2 + 14 x – k is reciprocal of the other, then what will be the value of k? 15. If α and β are the roots of the equation ax2 -bx +c = 0, then find the value of 1 + 1 16. If α and β are the roots of the equation 25x2 - 10x + 1= 0, find the value of α2+ β2. 17. Find a quadratic polynomial, the sum of whose zeroes is -1 and the sum of their reciprocals is 1/6 18. If α and β are the zeroes of the polynomial p(x) = x2 – 16x + 63, then find the value of α4β3 +α3β4 19. If the sum of the squares of the zeroes of a quadratic polynomial x2 –18x + p is 180, find the value of p. 20. If one root of the quadratic equation 2x2 + px + 4 = 0 is ‘2’, then find the other root and also find the value of ‘p’. 21. Find the value of the quadratic equation 2x2 - 3x - 2 at x = 1 and x = -2. 3 Mark Questions 22. Find a quadratic polynomial whose zeroes are -2/√3 and (√3)/4. 23. For what value of ‘k’, will the equation 2 − − −1 = +1 have roots reciprocal to each other? 24. If α and β are zeroes of p(x) = x2 + px + q, then find a polynomial having 1/ α and 1/ β as its zeroes: q≠0. 25. On dividing x3 + 3x + 2 by a polynomial g(x), the quotient and the remainder are x – 2 and 16 respectively. Find g(x) 26. What must be subtracted from 4x4 + 2 x3 – 8x2 + 3x – 7 so that it may be exactly divisible by 2x2 + x – 2? 27. If α and β are the roots of the quadratic polynomial 3x2 - 2x – 1 = 0, find the value of + , without finding the values of α and β 28. If α and β are the zeroes of a quadratic polynomial such that α + β = 24 and α - β = 8, then find the quadratic polynomial. 29. If (Z –3) is a factor of Z3 + aZ2 + bZ + 18 and a + b = -7, find a and b. 1 30. If α and β are the zeroes of the quadratic polynomial g(x) = x2 - (a + 12)x + 3(3a + 4), such that (α + β) = 3( α β), then find the value of a. 31. Check whether the polynomial l(x) = x2 – 5x + 1 is a factor of the polynomial m(x) = 4x4 – 13x3 – 31x2 + 35x – 10 by dividing m(x) by l(x). 32. Divide -x3 + 3x2 - 3x + 5 by -x2 + x - 1 and verify the division algorithm. 33. If α and β are the zeroes of 2x2 - 6x + 3, then what is the value of ( + )+ 3 ( 1 + 1 ) + 2 α β 34. Find the zeroes of the quadratic polynomial f(x) = abx2 + (b2 – ac)x - bc and verify the relationship between the zeroes and the coefficients of the polynomial 4|Page 35. Use the division algorithm to find the quotient q(y) and the remainder r(y) when f(y) = 12y3 + 17y2 – 20y – 10 is divided by g(y) = 3y2 + 2y - 5. 36. Find the G.C.D. and L.C.M. of the following polynomials: p(x) = 6(x - 2)(x2 + x - 6) and, q(x) = 3(x2 + 4x 12). 37. Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and (i) deg p(x) = deg q(x) (ii) deg q(x) = deg r(x) (iii) deg r(x ) = 0 4Mark Questions 38. Find the other zeroes of the polynomial x4 – 5x3 + 2x2 + 10x – 8 if it is given that two of its zeroes are – √2 and √2 39. If α and β are the zeroes of the quadratic polynomial 2x2 - 5x +7, find a polynomial whose zeroes are 2 α + 3 β and 3 α + 2 β . 40. Divide ( 3x2 – x3 + 7x + 13) by ( x – 2 – x2 ) and verify the division algorithm. 41. On dividing 3 x3 +x2 + 2x +5 by a polynomial g(x), the quotient and remainder are 3x – 5 and 9x + 10 respectively. Find g(x) 42. If the polynomial x4 + 2 x3 +8x2 + 12x + 18 is divided by another polynomial x2 + 5, the remainder comes out to be px + q. Find the values of p and q. 43. It is given that √2 and -√2 are two zeroes of the polynomial f(y) = 2y4 - 3y3 - 3y2 + 6y - 2, find all the zeroes of f(y). HOTS 44. If α and β are zeroes of p(x) = x2 – 2x + 3, find a polynomial whose zeroes are (α – 1 ) / (α + 1) and (β – 1 ) / (β + 1). 45. If the polynomial f(x) = x4 - 6x3 + 16x2 - 25x + 10 is divided by another polynomial x2 - 2x + k , the remainder comes out to be x + a , find k and a. 46. What must be added to the polynomial 3x4 + 5 x3 -7x2 + 5x + 3 so that the resulting polynomial is exactly divisible by (x2 +3x +1)? The degree of the polynomial to be added must be less than degree of (x2 +3x +1) Chapter 3-PAIR OF LINEAR EQUATONS IN TWO VARIABLES 1 Mark Questions 1. General form of a linear equation in two variable is ________ 2. Geometrical representation of a linear equation in two variables is _________ 3. If a pair of linear equations is consistent, then the lines will be ____________ 4. Find the number solutions for the pair of linear equations x + 2y + 5 = 0 and -4x – 8y + 1 = 0. 5. The pair of equations y = 2 and y = 7 has ________ solution/s 6. Find the value of ‘k’ for which x + 2y = 5, 3x + ky + 15 = 0 are inconsistent. 7. The value of y in the system of equations (x/a) = (y/b), ax + by = a2 + b2 8. The point of intersection of line -3x + 7y = 3 with x-axis is ____________ 9. Find the value of x and y which satisfy the equation x – y = 0; 2x – y = 2 simultaneously. 10. If 4 x-y = 16 and x – 2y = 8 are system of the equations, then the value of x + y is : 11. Find the value of ‘x’ in the following pair of linear equations: 4/x + 5y = 7 and 3/x + 5y = 5. 12. Sum of two numbers in 48 and their difference is 20. Find the numbers 13. If 5x + 7y = 3 and 15x + 21y = k represent coincident lines, then find the value of k. 14. The conditions so that a1x + b1y = c1; a2x + b2y = c2 have unique solution is ________ 15. If a pair of equations are consistent, the lines will be _______ 5|Page 2 Mark Questions 16. Find the values of α and β for which the following system of linear equations has infinite number of solutions. 2x + 3y = 7 ; 2αx + (α + β)y = 28 17. The denominator of a fraction is 7 more than the numerator. If 5 is added to each, the value of the resulting fraction is 1/2. Find the original fraction. 18. Find the values of k for which the system of linear equations has unique solution. 2x + 3y = 7, kx + 9y = 28. 19. Use elimination method to find all possible solutions of the following pair of linear equations: 2x + 3y = 8 and 4x + 6y = 7. 20. Is the pair of equations x + 2y – 3 = 0 and 6y + 3x – 9 = 0 consistent? Justify your answer. 21. Determine the value of u so that the following pair of linear equations have no solution. (3u + 1)x + 3y - 2 = 0 and (u2 + 1)x + (u - 2)y - 5 = 0. 22. Solve the following pair of linear equations by cross multiplication method 3x + y = - 4 and 6x – 2y = - 4 23. Solve the system of linear equations 6 + 2 = 1, 6 - 4 = 1 24. The sum of two numbers is 36 and their difference is 14. Find the numbers. 25. Solve the pair of linear equations by substitution method. x + y = 14, x – y = 4. 26. Solve the systems of linear equations: 0.4x +0.3 y =1.7 and 0.7x - 0.2y = 0.8 3 Mark Questions 27. Write an equation of a line passing through the point representing solution of the pair of linear equations x + y = 2 and 2x – y = 1. How many such lines can we find? 28. The taxi charges in a city consist of a fixed charge together with the charge for the distance covered. For a distance of 10 km, the charge paid is Rs 75 and for a journey of 15 km, the charge paid is Rs 110. How much does a person have to pay for travelling a distance of 25 km? 29. The path of train A is given by equation x + 2y – 4 = 0 and the path of another train B is given by the equations 2x + 4y – 12 = 0. Represent this situation geometrically. Also check, whether the two paths can cross. 30. Two years ago, Salim was thrice as old as his daughter and six years later, he will be four years older than twice her age. How old are they now? 31. The sum of the digits of a two digit number is 8 and the difference between the number and that formed by reversing the digits is 18. Find the number. 32. Solve the pair of linear equations 3x + 5 y – 13 =0 and 2x - 5 y – 7=0 and hence find the value of ‘m’ for which y = mx–3. 2 3 9 4 9 21 33. Solve the pair of linear equations + = and + = where x ≠ 0, y ≠0. 34. Find the values of m and n for which the system of linear equations has infinite number of solutions. 2x + 3y = 7, 2mx + (m+n) y = 28 1 1 35. Solve the following system of equations : 2 - = -1 and 1 1 - 2 = 8 36. A man starts his job with a certain monthly salary and earns a fixed increment every year. If his salary was Rs 15000 after 4 year of service and Rs 18000 after 10 years of service, what was his starting salary and what is the annual increment? 6|Page 4 Mark Questions 37. A boatman rows his boat 35km upstream and 55km downstream is 12 hours. He can row 30 km upstream and 44km downstream is 10 hours. Find the speed of the stream and that of the boat in still water. Hence find the total time taken by the boatman to row 50km upstream and 77 km downstream. 38. The students of a class are made to stand in rows. If 3 students are extra in a row, there would be 1 row less. If 3 students are less in a row, there would be 2 rows more. Find the number of students in the class. 39. Draw the graphs of the following system of linear equations: 4x +3y – 24 = 0; 2x – y = 2; y + 4 = 0.Obtain the vertices of the triangle so obtained. Also determine its area. 40. (a)Solve: 217x + 131y = 913 and 131x + 217y = 827 (b) For what value of u the system of equations 3x + 5y = 0 and ux + 10 y = 0 has unique solution. 41. Draw the graphs of the following system of linear equations and find unique solutions: 4x – y – 8 = 0 ; 2x – 3y + 6 = 0. Also determine vertices of triangle formed by the lines and the x-axis. 42. A boat covers 32 km upstream and 36 km downstream in 7 hours. Also, it covers 40 km upstream and 48 km downstream in 9 hours. Find the speed of the boat in still water and that of the stream. 43. Solve the following system of linear equations graphically: 2x – y – 4 =0 x + y + 1 =0 Find the points where the line meets the y axis. 44. A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid Rs 27 for a book kept for seven days while Susy paid Rs 21 for the book she kept for five days. Find the fixed charge and the charges for each extra day. 45. By the graphical method, find whether the pair of equations 3x + y + 4 = 0 and 6x – 2y + 4 = 0 are consistent or not. If consistent then find its solution. 46. The sum of a two-digit number and the number obtained by reversing the digits is 66. If the digits of the number differ by 2, find the number. How many such numbers are there? HOTS 47. Two pipes running together can fill a cistern in 10/3 minutes. If one pipe takes 3 minutes more than the other to fill the cistern, find the time in which each would fill the cistern. 48. Solve the following : (a + b)x + ( a – b )y = a2 – b2 ; (a – b )x + ( a + b )y = a2 + b2 Chapter 8- INTRODUCTION TO TRIGONOMETRY 1 Mark Questions 1. If sec θ = 5/4 , then what is the value of tan θ ? 2. Express sinA in terms of cotA. 3. The value of √1− √1+ = _______ Sin2 250 + sin2650 = ____ In a right triangle ABC right angled at B, if sin( A – C) = ½, then angles A, B and C respectively are______ If x = a cos θ and y = b sin θ, then the value of b2x2 +a2y2 is The value of sin2 50 + sin2 100 + …….+ sin2 850 + sin2 900 is _____ In right triangle PQR right angled at Q, PQ = 8cm, QR = 6cm and PR = 10 cm. Find the value of 25 (sin2θ + 2 cos2 θ - tan θ) 9. The value of tan 10 tan 20 tan 30…… tan 890 is : 10. Find the value of ‘x’ if tan3x = sin 450 . cos 450 + sin 300 2 Mark Questions 11. If sinθ + sin2θ = 1, check the validity of expression: cos2 θ + cos 4 θ = 1. 4. 5. 6. 7. 8. 7|Page 12. If A, B and C are interior angles of ∆ ABC, then show that cot ( + 2 ) = tan 2 13. In right triangle ABC, right angled at B, AB = y cm, BC = 10 cm, AC = x cm and < C = 300. Find the values of x and y 14. ∆ABC is right angled at B and A= C. Is cosA = cosC? Justify your answer 0 0 15. Solve the equation when 0 < θ < 90 ; 3 tan2 θ – 1 = 0 16. If 7sin2A+3cos2 A = 4, show that tan A =1/√3. 17. If sin θ = a/b, then find sec θ + tan θ in terms of a and b. 18. Simplify: sec2x(1-sin2x)+secB(sinB/tanB) 19. Evaluate( cos 600 + sin 300 – cot 300 ) / (tan600 + sec 450 – cosec 450) 20. State whether the following expression is True or False. Justify your answer. sin6 θ + cos 6 θ = 1 – 3 sin2 θ. cos2 θ 21. If A, B are acute angles and sinA= cosB, then find the value of A+B. 22. If 2x=secA and 2/y=tanA, then find the value of 2(x2-1/y2). 23. In ∆ABC, right angled at B , BC = 3 and AC = 6. Determine BCA and BAC. 3 Mark Questions 24. If cosec A = √10, then find other five trigonometric ratios. 25. Prove that: cot2A/(1+cosecA)=(1-sinA)/sinA 26. If 7 cosecA-3cotA = 7, prove that 7cotA - 3cosecA = 3. 27. Prove that (SinA+CosecA)2+(CosA+SecA)2=7+tan2A+cot2A. 28. Evaluate cos (400 + θ) – sin (500 – θ ) + (cos2 400 + cos2 500) / (sin2 400 + sin2 500) 29. sec6x(secxtanx)-sec4x(secxtanx) =sec5xtan3x 30. Prove that (cot A – cos A) / ( cot A + cos A) = (cosec A – 1 ) / (Cosec A + 1) 31. Prove that 1 – 1 = 1 – 1 . sec x – tan x cos x cos x sec x + tan x 32. If cot θ = √7 , show that 2 −sec2 2 +sec2 3 =4 33. If tanA + sinA = m and tanA - sinA = n, prove that (m2 - n2)2 = 16mn. 34. Show that sin1/2xcosx-sin5/2xcosx=cos3xsin1/2x 35. Prove that: secx+tanx=cosx/(1-sinx) 36. Without using trigonometric table, evaluate the following : 530 800 700 2(370 )+ 100 - 3200 + 550 350 50 250 450 650 850 +2 Given 3 cos A – 4 sin A = 0. Evaluate without using table 3−sin . 4Mark Questions 38. If tan A = n . tan B and sin A = m . sin B, then prove that cos2 A = (m2 – 1) / ( n2 – 1) 39. Prove that (tanθ + sinθ) / (tanθ – sinθ) = ( secθ + 1) / ( secθ – 1) = (1+cosθ) / (1 – cosθ) =tan2 θ / (secθ – 1)2 40. If ( a2 – b2 ) sin θ + 2 ab cos θ = a2 + b2, find the value of tan θ 37. tan3 θ cot3 θ 41. Prove that 1+tan2 θ + 1+cot2 θ = sec θ . cosec θ – 2 sin θ . cos θ tan 42. Prove that 1−cot + 1−tan = 1 + cosecAsecA 43. If tanA + sinA = m and tanA – sinA = n, prove that ( m2 – n2 )2 = 16mn 1 1 44. If each of α, β and λ is positive acute angle such that sin (α+ β –λ) = 2 , cos (β +λ- α) = 2 and tan (λ +α+β)=1. Find the value of α, β and λ. 45. If x = r sinAcosC, y = r sinA sinC and z = r cos A, prove that r2 = x2 + y2 + z2. 8|Page Chapter 14- STATISTICS 1 Mark Questions 1. Construction of a cumulative frequency table is useful in determining the __________ 2. ̅ is the mean of x1, x2, x1, ……….. , xn , then the mean of 1, 2, ‘’’’’, where k ≠0 is ______ 3. The mean of n observations is ̅ . If the first term is increased by 1, second term by 2 and so on, then the new mean is _________ 4. The relationship between mean, median and mode for a moderately skewed distribution is ______ 5. Median and mode of distribution are 21.2 and 21.4 respectively. Then, its mean is : _____ 6. The median of a frequency distribution is found graphically with the help of _______ 7. Find the mode for the following series : 7.5, 7.3, 7.2, 7.2, 7.4, 7.7, 7.7, 7.5, 7.3, 7.2, 7.6, 7.2 8. If ̅ is the mean of n observations x1, x2, x1, ……….. , xn then ∑ (xi - ̅ ) ,where i= 1 to n , is equal to _____ 9. For what value of k, the mode of the following data is 7? 3,5,6,7,4,6,9,7,6,2k – 1 ,10, 7, 12 2 Mark Questions 10. The mean of five numbers is 18. If one number is excluded, their mean is 16, then what will be the excluded number? 11. If median = 15 and mean = 16, the mode is ____ 12. Find the mean by direct method for the following data: Classes Frequency 10-20 4 20-30 8 30-40 10 40-50 12 50-60 10 60-70 4 70-80 2 13. The arithmetic mean of the numbers 7, 11, x, 3, y and 2 is 11. Find the arithmetic mean of x and y. 14. Find the arithmetic mean of the 10 prime numbers. 15. The average mark scored by girls is 68 and that of the boys is 62. The average mark of the whole class is 64. Find the ratio of the girls & boys in the class. 16. The mean, of x – 5y, x – 3y, x – y , x + y , x + 3y & x + 5y is 12. Find the value of x. 17. Find the median class of the following data: Marks Obtained Frequency 0-10 10-20 20-30 30-40 40-50 50-60 8 10 12 22 30 18 3 Mark Questions 18. Find the unknown entries a, b, c, d, e, f in the following distribution of heights of students in a class Height ( in cm) 135-140 140-145 145-150 150-155 155-160 160-165 9|Page Frequency Cumulative frequency 4 b 18 d e 5 a 11 c 40 46 f 19. Consider the following : Class interval 0-5 5-10 10-15 15-20 Frequency 10 15 12 20 Find the sum of lower limits of the median class and the modal class. 20. The total number of marks scored by class in test is given below. Find the mean. Below 20 4 Below 40 12 Below 60 30 Below 80 44 Below 100 50 20-25 9 21. Find the mode for the following frequency distribution of marks obtained by 80 students : Marks 0-10 10-20 20-30 30-40 No. of students 6 10 12 32 22. The mode of the following frequency distribution is 32. Find the missing frequency in it : Class 0-10 10-20 20-30 30-40 40-50 50-60 interval Frequency 8 10 x 16 12 6 23. Find the median from the following distribution. Class interval 4-6 6-8 8-10 10-12 Frequency 5 4 10 7 24. If the mean of the following data is 20, find the value of p. x: f: 15 2 17 3 19 4 21 5p 40-50 20 60-70 7 12-14 4 23 6 4Mark Questions 25. The mean of the following frequency distribution is 62.8 and the sum of ll frequency is 50. Compute the missing frequencies 1 and 2 . Class 0-20 20-40 40-60 60-80 80-100 100-120 Total Frequency 5 10 7 8 50 1 2 26. The table below gives the distribution of villages under different heights from sea level in a certain region. Compute the mean height of the region : Height ( in 200 meters) No. of 142 villages 600 1000 1400 1800 2200 265 560 271 89 16 27. Draw less than and more than ogives for the following distribution on the same graph and hence find the median : Marks 30-39 40-49 50-59 60-69 70-79 80-89 90-99 No. of 14 6 10 20 30 8 12 students 10 | P a g e 28. Find the mean and mode of the following data : Classes 0-10 10-20 20-30 30-40 40-50 50-60 60-70 Frequency 3 8 10 15 7 4 3 29. During the medical check-up of 35 students of a class, their weights were recorded as follows. Draw less than type ogive for the given data. Hence, obtain the median weight from the graph and verify the result by using the formula. Less than Less than Less than Less than Less than Less than Less than Less than Wt. in Kg 38 40 42 44 46 48 50 52 No. of 0 3 5 9 14 28 32 35 students 30. The median of the following data is 28.5. Find the values of x and y, if the total frequency is 60. Class interval No. of students 31. 0-10 10-20 20-30 30-40 40-50 50-60 5 x 20 15 y 5 The frequency distribution of scores obtained by 230 candidates in an engineering entrance test is as follows : Scores 400 – 450 – 500 – 550 – 600 – 650 – 700 – 750 450 500 550 600 650 700 750 800 No. of 20 35 40 32 24 27 18 34 candidates Draw cumulative frequency curves by less than and more than method on the same axes. Also, draw the two types of cumulative frequency polygons. Chapter – 3 TRIANGLES 1 Mark Questions 1. In the adjoining figure, LM ║ AB. If AL = x – 3, AC = 2x, BM = x – 2 and BC = 2x + 3, the value of x is : C L M B A 2. Let ∆ABC ~ ∆DEF and their areas be respectively 64 cm2 and 121 cm2. If EF = 15.4 cm, then the value of BC is _____ 3. In ∆LMN, <L = 500 and < N = 600. If ∆LMN ~ ∆PQR, then < Q is equal to : 4. If ∆ABC & ∆DEF are similar such that 2 AB = DE, and BC = 8 cm, then EF = ___ cm. 5. Area of an equilateral triangle with sides 4 cm = ___ __ 2 Marks Questions 6. In the adjoining figure, = and <TQS = <PRS, show that ∆ PQS ~ ∆ TQR P T Q 11 | P a g e R S 7. In the given figure, AD I. BC. Prove that AB 2 + CD 2 = BD2 + AC2 C D B A 8. ABC is an isosceles triangle, right-angled at B. Equilateral triangles ACD and ABE are constructed on sides AC and AB. Find the ratio between the areas of ∆ ABE and ∆ ACD. A D E C B 9. In the adjoining figure, ∆ABC is right- angled at C and DE I AB, AD = 3 cm, DC = 2 cm, BC = 12 cm. Prove that ∆ABC ~ ∆ADE and hence, find the lengths of AE and DE. A E D B C 1 10. In ∆PQR, DE ║ QR intersecting PQ and QR in points D and E, respectively. If DE = 3 QR, then show 1 that PD = 2 DQ 11. In the given figure, E is a point on the side CB produced of an isosceles triangle ABC with AB=AC. If AD BC and EF AC, prove that ABD ECF 12. The perimeters of two similar triangles are 36 cm and 48 cm respectively. If one side of the first triangles is 9 cm, what is the corresponding side of the other triangle? 13. In ∆ABC, D and E are points on the sides AB and AC respectively such that DE ║ BC. If AD = 4x – 3, AE = 8x – 7, BD = 3x – 1 and CE = 5x – 3, then find x 14. If ∆ABC ~ ∆DEF, AB = 4 cm, DE = 6 cm, EF = 9 cm and FD = 12 cm. Find the perimeter of ∆ABC. 15. ABC is a right triangle right-angled at B. Let D and E be any points on AB and BC respectively. Prove that AE2 + CD2 = AC2 + DE2. 16. Any point X inside the DEF is joined to its vertices. From a point P in DX, PQ is drawn parallel to DE meeting XE at Q and QR is drawn parallel to EF meeting XF in R. Prove that PR || DF. 12 | P a g e 3 Marks Questions 17. In the adjoining figure ∆ABC and ∆AMP are 2 right angled triangles at B and M respectively. Prove that (1) ∆ABC ~ ∆AMP (2) = C M A P B 18. In the adjoining figure, D and E are points on the sides CA and CB respectively of a ∆ABC, right angled at C. Prove that AE2 + BD2 = AB2 + DE2 A D B E A C 19. 2 isosceles triangles have equal vertical angles and their areas are in the ration 16:25. Find the ratio of their corresponding heights 20. ABC is an isosceles triangle is which AB=AC=10cm and BC=12. PQRS is a rectangle inside the isosceles triangle. Given PQ=SR=y cm, PS=QR=2x. Prove that x = 6 -(3/4)y. 21. In the given figure, <AEF=<AFE and E is the mid-point of CA. Prove that BD/CD=BF/CE. 22. ABC is a right triangle right-angled at C. Let BC = a, CA = b, AB = c and let P be the length of perpendicular form C on AB prove that (i) cp = ab 1 1 1 (ii)2 = 2 + 2 13 | P a g e 4 Marks Questions 23. State and prove Basic proportionality Theorem. In a ∆ABC, D and E are points on sides AB and AC respectively, such that BD = CE, <B = <C, then show that DE ║ BC using converse of the above theorem 24. In a right triangle ABC, right angled at C, P and Q are points of the sides CA and CB respectively that divide these sides in the ratio 2: 1. Prove that (i) 9AQ2= 9AC2 +4BC2 (ii) 9BP2= 9BC2 + 4AC2 (iii) 9 (AQ2+BP2) = 13AB2 25. State and prove converse of Pythagoras theorem. Using the theorem, prove that ∆ABC is right angled in the given figure A 6 cm 24 cm D B 8 cm 26 cm B 26. Prove that “the ratio of the areas of 2 similar triangles is equal to the squares of the ratio of their corresponding sides”. Using this theorem, solve the following : If ∆ABC ~ ∆DEF, area (∆ABC) = 64 cm2 , area (∆DEF) = 121 cm2, and EF = 15.4 cm, find BC 27. O is any point in the interior of the ∆ABC. OD I BC, OE I AC and OF I AB. Show that AF2 + BD2 + CE2 = AE2 + CD2+BF2 1 1 1 28. In the adjoining figure, PA, QB AND RC are each perpendicular to AC. Prove that + = P R Q x z y A 14 | P a g e B C ANSWERS Chapter 1- Real numbers 1 Mark questions a3b2 2 four places prime factorization of ‘q’ must be of the form 2m X 5n, where m and n are non-negative integers 5) 207 6) 0≤ r < b 7) an algorithm 8) 3 9) 2 10) 2 1) 2) 3) 4) 11) 6 12) (√5 + √ 3) ; 2 13) 13 14) given no. is a multiple of 7 15) 675 16) Prove 17) 5 does not occur in the prime factorization of 8 3 Marks Questions 26) 19 27) . 28) . 15 | P a g e 12) p = ab 13) zeroes are -3 and 4 14) k = - 15 15) –b/c 16) 2/25 17) k(x2 + x – 6) 18) 4000752 19) P = 72 20) 1, p = -6 21) P(1) = -3, p(-2) = 12 4 Marks Questions 23) 40) take consecutive integers as n, n+1,n+2 41) . 42) . 43) . 44) . 45) English = 2, Hindi = 5, Mathematics = 7 46) . 47) . 48) . 49) . 2 Marks question 18) 19) 4663 ( Hint : LCM (520, 468 ) – 17 ) 20) 2160 21) 75 22) 36 minutes 23) . 24) Dr. is of the form 2m X 5n; terminating decimal 25) Even power – 1 , odd power - 9 29) Any positive odd integer is of the form n = 4q + 1 or 4q + 3 30) . 31) . 32) HCF of 65 and 117 = 13 33) Terminating decimal expansion 34) . 35) . 36) HCF = 10 cm 37) HCF = 24; LCM = 360 38) . 39) . HOTS 50) Odd +ve integers x = 2m+1 and y = 2n + 1 Chapter 2 - POLYNOMIALS 1 Mark questions 1) 2) 3) 4) 5) 6) 7) –5 – 2/3 – 17/4 4 r(x) = 0 or deg r(x) deg q(x) 5 zeroes – 3/2 , - 11/2 8) k[x - x - 6] 9) x2 -2x – 4 = 0 2 10) x = -2 or, x = 1/2 or, x = 2/3 11) k = 4 2 Marks Questions 3 Marks Questions 22) p(x) = 4√3 x2 + 5x - 2√3. +1 −1 24) qx2 + px + 1 25) x2 + 2x + 7 26) 5x – 11 27) -10/3 28) x2 - 24x + 128 29) a = -4, b = -3 30) a = 4 31) Remainder = 28x – 10 ≠0, not a factor 32) quotient = x - 2, remainder = 3 33) 13 34) –b/a and c/b 35) q(y) = 4y + 3 and r(y) = 5 – 6y 36) GCD = 3 (x-2), L C M = 6 ( x+3)(x+6)(x-2)2 37) (i) p(x) = 6x2 + 3x + 2, g(x) = 2x2 + x, r(x) = 2 (ii) p(x) = 8x3 +6x2 - x + 7, g(x) = 2x2+ 1, q(x)= 4x + 3, r(x) = -5x + 4 (iii)p(x) = 9x2+6x + 5, g(x) = 3x+ 2, q(x) = 3x, r(x) = 5 4 Marks Questions 38) 1,4 39) k(x2 - 25 2 x + 41) 40) . 41) x2 + 2x + 1 42) px + q = 2x + 3, p = 2, q = 3 43) √2 , -√2, 1 and 1/2 HOTS 44) 3x2 – 2x + 1 45) K = 5, a = -5 46) -3x – 1 , degree of added polynomial = 1 Chapter 3-Pair of linear equations in two variables 1 Mark Questions 1) ax + by + c = 0, where a, b, c are real numbers, and a and b are not both zero. 2) Straight line 3) Intersecting or coincident 4) No solution. 5) No solution 6) k = 6 7) y = b 8) (-1,0) 9) X = 2, y = 2 10) - 10 11) x=1/2 12) x = 34 and y = 14 13) k = 9 14) 15) ≠ Intersecting or coincident 2 Marks Questions α = 4, β = 8 17) the original fraction is 2/9. 18) k≠6 19) the pair of equations has no solution 20) Coincident lines, hence consistent 21) u = -1 22) x = -1, y = - 1 23) x = 6, y = 0 24) x = 25, y = 11 25) x = 4, y = 2 26) x = 2, y = 3 16) 3 Marks Questions 16 | P a g e 27) x = y , infinitely many 28) Rs. 180 29) The path being parallel cannot cross. 30) Salim’s age = 38 yrs, daughter’s age = 14yrs. 31) No is 53 32) X = 4, y = 1/5 , m = 4/5 33) X = 1, y = 3 34) m = 4, n= 8 35) x = 1/6 , y = ¼ 36) starting salary =Rs 13000 annual increment = Rs 500. 4 Marks Questions 37) Speed of the boat in still water = 8 km/hr. Speed of the stream = 3 km/hr. 38) No. of students = 36 39) A(3,4), B (_1, -4), C (9, -4); Area = 40 sq. units 40) (a) x = 3, y = 2 (b) u ≠6 41) x= 3, y = 4; (3,4), (2,0), (3,0) 42) the speed of boat in still water = 10 km/hr and speed of stream = 2km/hr 43) (0, -4) for the first equation and (0, -1) for the second equation. 44) Rs. 15 and Rs. 3 45) x = -1, y = -1 46) the two numbers are 42, 24. HOTS 47) 5 minutes, 8 minutes 48) x = (a2 + b2 )/ 2a , y = (a2 + b2 )/ 2a Chapter 8- INTRODUCTION TO TRIGONOMETRY 1 Mark Questions 1) ¾ 1 2) Sin A = √1+ 2 3) 4) 5) 6) 7) 8) 9) 10) cosec θ – cot θ 1 600 , 900 and 300 a2b2 9.5 2/3 1 150 2 Marks Questions 11) . 12) . 13) x = 14) 15) 16) 17) 18) 19) 20) 20√3 3 ,y= 10√3 3 Yes θ = 300 . √(b+a) / √(b – a ) 2 (√3 – 3 ) / 3 False . (sin 800 – sin100= positive , as θ increases, value of sin θ increases ) 21) A + B = 900 22) ½ 23) 600 and 300 3 Marks Questions 24) Sin A = 1/√10, cos A = 3/√10, tan A = 3, cot A = 3, sec A = √10/3 25) . 26) . 27) . 28) 1 29) . 30) . 31) . 32) . 33) . 34) . 35) . 36) 1 37) Take 4 = 4 = k, Ans : 11/9 4 Marks Questions 38) . 39) 18) a = 4, b = 7, c = 29, d = 11, e = 6, f = 51 19) 10 + 15 = 25 20) Mean = 60 21) 36.25 22) x =15 23) 9.2 24) P = 1 2 −2 2 4 Marks Questions 41) 42) 43) 44) . . . 45) . 25) 26) 27) 28) 29) 30) 40) α = 37.50, β=450 and λ= 52.50 1 = 8 and 2 = 12 Mean height = 984.51 69.5 Mean = 32.8, mode = 33.85 Median = 46.5 x = 8, y = 7 Chapter – 3 TRIANGLES Chapter 14- STATISTICS 1 Mark Questions 1) Median ̅ 2) 3) ̅ + 4) 5) 6) 7) 8) 9) +1 2 Mode = 3 median – 2 mean 21.1 Ogives Mode = 7.2 0 K=4 1 Mark Questions 1) 2) 3) 4) 5) 2 Marks Questions 2 Marks Questions 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 10) 11) 12) 13) 14) 15) 16) 26 13 Mean = 42.2 21.5 12.9 1 :2 x = 12 x=9 11.2 cm 700 EF = 4 cm Area = 4√3. . , ½ DE = 36/13cm and AE = 15 / 13 cm . . 12 cm x=1 Perimeter = 18 cm . . 17) 30-40 is the median class. 3 Marks Questions 3 Marks Questions 17) . 18) . 17 | P a g e 19) 20) 21) 22) . . . . 4 Marks Questions 23) 24) 25) 26) 27) 28) . . AB = 10 cm BC =11.2 cm . .

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