Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
NAVY CHILDREN SCHOOL, NAUSENA BAUGH, VISAKHAPATNAM Class XI MATHEMATICS QUESTION BANK- 2015-16 SETS 1. 2. 3. 4. 5. 6. 7. Which of the following are sets? Justify your answer. (i). the collection of all the months of a year having 31 days. (ii). The collection of all divisors of 30. (iii). The collection of beautiful girls in India (iv). The collection of all odd integers. (v). the collection of difficult problems in this chapter. Write the following sets in roaster form: (i). A = { x:x is an odd prime number less than 25} (ii). B = { x:x N and -1 3 6 (iii). C = { x:x is an integer and x2 -16 = 0. (iv). D = { x:x N and x = n3} (v). E = { x:x N, x = and n Write the following sets in the set-builder form: (i) {1, 2, 5, 10} (ii) {3 , 9, 27, 81, 243} (ii) {5, 10,15,……100} (iV) , , , , Match each of the sets on the left in the roster form with the same set on the right described in set- builder form. (i) {2 ,3, 7} (a) { x : x2 -16 =0, x R} (ii) {1, 2} (b) { x : x is a prime divisor of 42} (iii) {0, -1} © { x : x is a positive integer and 2 x 9 (iv) { -4, 4} (d) { x : x is an integer and x2 +x = 0}. Which of the following are examples of the null set? (i) {x : x N and 2x -3 = 0} (ii) {x : x N and (x -3 )(x -5) = 0} (iii) {x : x N, x 1 or x 5} (iv) {x : x N, x 1 or x 5 } Decide among the following sets, which sets are subsets of each another: A = { x : x R and x satisfies x2 - 8x + 12 = 0}, B = { 2, 4,6}, C = { 2,4,6,8 ,……} , D = {6}. In each of the following determine whether the statement is true or false. If it is true, prove it. If it is false, give an example. (i) If x A and A B then x B (ii) If A and B ∊ C then A ∊ C 8. 9. (iii) If A and B then A (iv) If A B and B C then A C (v) If x A and A B then x B (vi) If A and x does not belongs to A. Let A, B and C be the sets such that A B = A C and A B = A C. Show that B = C. Show that the following four conditions are equivalent: (i) A B (ii) A –B = (iii) A B = B (iv) B =A 10. Show that if A B then C - B C – A. 11. Assume that P(A) = P (B). Show that A=B 12. Is it true that for any sets A and B, P(A) P(B) = P (A B) ? Justify your answer. 13. Show that for any sets A and B. A = ( A B) (A – B) and A (B – A) = (A 14. Using prosperities of sets, show that (i) A (A B) = A (ii) A ( A = A 15. Show that A B = A need not imply B = C. 16. Let A and B be sets. If A X = B X = and A X = B X for some set X. Show that A = B. 17. Find sets A, B and C such that A B, B C and A C are non – empty sets and A B C = . 18. In a survey of 600 students in a school, 150 students were found to be taking tea and 225 taking coffee, 100 were taking both tea and coffee. Find how many students were taking neither tea nor coffee? 19. In a group of students , 100 students know Hindi, 50 know English and 25 know both. Each of the students knows either Hindi or English. How many students are there in the group? 20. In a survey of 60 people, it was found that 25 people read newspaper read newspaper H , 26 read newspaper T, 26 read newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find: (i) The number of people who read at least one of the newspapers. (ii) The number of people who read exactly one news paper. 21. In a survey it was found that 21 people liked product A, 26 liked products B and 29 liked product C. If 14 people liked products A and B, 12 people liked products C and A, 14 people liked products B and C and 8 liked three products. Find how many liked product C only? 22 For any two sets A and B, prove that A B = A B ! A = B. 23. Let A and B be two sets. If A X = B X = and A X = B X for some set X, prove that A = B. 24. For any two sets A and B prove that : P ( A B) = P (A) P (B) 25. Let A , B and C be three sets such that A B = C and A B = , then prove that A = C – B. 26. Let A, B and C be three sets, then prove that : A – ( B C) = ( A - B) (A - C) 27. Let A , B and C be three sets, then prove that: A – ( B – C) = ( A – B) ( A C). 28. In a class of 35 students, 24 like to play cricket and 16 like to play foot ball. Also each student likes to play at least one of the two games. How many students like to play both cricket and football? 29. In a survey of 25 students, it was found that 15 had taken mathematics, 12 had taken physics and 11 had taken chemistry, 5 had taken mathematics and chemistry, 9 had taken maths and physics, 4 had taken physics and chemistry and 3 had taken all the three subjects. Find the number of students who had (i) Only chemistry (ii) Only physics (iii) Only mathematics (iv) Physics and chemistry but not mathematics (v) Mathematics and physic but not chemistry (vi) At least one of the three subjects (vii) Only one of the subjects (viii) None of the subjects. 30. In a certain town , 25% families own a phone and 15% own a car , 65% families own neither a phone nor a car , 2000 families own both a car and a phone. Find (i) What percent of families own both car and a phone (ii) What percent of families own either a car or a phone (iii)Find how many people live in the town 31. If U= {x:xϵN and 2 ≤ x ≤ 12} , A={x: x is an even prime} , B= {x:x is a factor of 24} , then prove that A-B ≠ BϵAc 32 Which set does not have a proper subset. 33. Write the null set in set builder form. 34 If U={2,3,4,5,6,7,8,9,10,11} ,A={2,4,7},B={3,5,7,9,11} and C={7,8,9,10,11},Compute (i) (Aϵ U) ϵ (B ϵ C) (ii) C - B (iii) A ∆ B (iv) (B c ϵCc ) 35 Shade the following regions using Venn Diagrams (i) B - (AϵC) (ii) Ac ϵ(B ϵ C) c c (iv) Ac ϵ(B ϵ C)c (iii) B - (AϵC) RELATIONS & FUNCTIONS 1. The relation f is defined by f(x) = " # %# $ #%& % # $ The relation g is defined by g(x) = " # %# $ # & # $ Show that f is a function and g is not a function. 2. 3. 4. 5. 6. 7. 8. 9. If f(x) = x2 find ' () )'( (.) . # # Find the domain of the function f(x) = # ) # Find the domain and range of the real function f defined by f(x) = √# Find the domain and the range of the real function f defined by f(x) = |# | # Let f = -.#, / : # 12 be a function from R into R. # determine the range of f. Let f, g : R 3 R be defined , respectively by f(x) = x +1 , g(x) = 2x – 3. ' Find f + g , f – g and . 4 Let f = 5(, , (, % , ($, , (, % be a function from Z to Z defined by f(x) = ax + b for some integers a, b. determine a, b. Let R be a relation from N to N defined by R = 5(6, 7 : 6, 7 8 69 6 : 7 . Are the following true? (i). ( a,a) 1 for all a 8 (ii). (a,b) R inplies (b, a) 1 (iii). (a, b) R, (b, c) 1 implies (a, c) R. Justify your answer in each case. 10. Let A = 5, , %, , = 5, , ;, , , < and f = 5(, , (, ; , (%, , (, , (, . Are the following true? 11. 12. 13. 14. 15. 16. 17. 18. 19. (i). f is a relation from A to B (ii). f is a function from A to B Justify your answer in each case. Let f be a subset of Z = Z defined by f = 5(67, 6 > 7 ? 6, 7 @. Is f a function from Z to Z ? Justify your answer. . Let A = 5;, $, , , % and let f : A 3 N be defined by f(n) = the highest prime factor of n. Find the range of f. If A B then prove that A xA = ( A x B) (B x A) If A and B are only two non empty sets, then prove that A x B = B x A A A = B. Let A, B, C and D any non-empty sets. Prove that (A x B) ( C x D) = ( A C) x ( B D). Let A = { 1, 2, 3, 4} and B = { 5,6,7} . if R = {( a,b) : a A, b B} and a – b is even then find R. If R is the relation "less than" from A = { 1,2,3,4,5} to B = {1, 4, 5}. Write down the set of ordered pairs corresponding to R. Find the inverse of R. Let R be relation on the set Z of all integers defined by R = {( x, y) ; x – y is divisible by n}. prove that (i) (x, y) R and (y , Z) R B (x, Z) R for all x, y, z Z. Find the domain of f(x) = # 20. Find the domain and range of the function f (x) = # ); . #)% 21. Find the domain and range of the function f(x) = 1 – |# % | 22. Find the domain of the function f(x) = # %# # #)< TRIGONOMETRIC FUNCTIONS 1. 2. 3. 4. 5. 6. 7. Prove that : C ;C %C C 2 cos cos + cos + cos = 0. % % % % ( sin 3x + sin x) sin x + ( cos 3 x – cos x) cos x = 0. # D . (cos x + cos y )2 + ( sin x – sin y)2 = 4 cos2 )2 y)2 2 #)D (cos x - cos y + sin x – sin = 4 sin . Sin x + sin 3x + sin 5x + sin 7x = 4 cos x cos 2x sin 4x. (EF # EF # (EF ;# EF %# (GHE # GHE # (GHE ;# GHE %# = tan 6x. sin 3x + sin 2x – sin x = 4 sin x cos 8. tan x = - 9. Cos x = % ) % , x in quadrant II. # cos %# , x in quadrant III. 10. Sin x = , x in quadrant II. 11. The angles of a triangle are in A.P. The number of grades in the least is to the number of radians in the greatest as 40 : C . Find the angles in degrees. 12. A horse is tied in a post by a rope. If the horse moves along a circular path always keeping the rope tight and describes 66m when it has traced out 450 at the centre. find the length of the rope. 13. If cos I = - and C I %C, find the value of 4 tan2 I 3 cosec I. 14. Find the value of sin 750. 15. If 2 tan J + cot J = tan , prove that cot J = 2 tan (K – J). 16. If cos (K + J) sin ( L + M ) = cos (K – J) sin (L – M) , prove that cot K cot J cot L = cot M. 17. Prove that EF ( # I = cos (I - ) + cot (# ) sin (I - ) 2 18. Prove that EF (# NOP Q NOP R STN Q STN R Q R = tan . U / 19. If sin I = n sin (I > K , prove that Tan (I > K = ( tan K ) 20. Prove that : GHE <I GHE I GHE I $ = 2 cos I GHE I GHE %I $ GHE I 21. Find the general solution of the trigonometric equation √% cos x + sin x = √. 22. In ∆ ABC, if W : X , W : <$0 and W = 750. Find the 23. ratio of its sides. 23. If in a ∆ ABC, cos A = EF , prove that the triangle is EF 24. 25. 26. 27. 28. 29. 30. 31. isosceles. In ∆ ABC, if K cos A = b cos B. show that the triangle is either isosceles or right angled. In ∆ ABC, b : c = √% ? √ and the angles are in A. P. find W . In ∆ ABC, if a- 9, b = 8 and c = 4 . prove that 4 + 3 cos B = 6 cos C. In ∆ ABC, if a2, b2, c2 are in A. P., prove that cot A, cot B, cot C are in A. P. In any ∆ ABC, prove that a3 sin ( B – C) + b2 sin (C - A) + c3 sin ( A- B) = 0. In any ∆ ABC, show that a 3 cos ( B – C) + b3 cos ( C –A) + c3 cos ( A- B) = 3 abc. G G In any ∆ ABC,, prove that : ( a-b)2 cos2 + ( a + b)2 sin2 = 2 c. In any ∆ ABC, prove that 6 EF ( ) 7 EF ( ) + + EF 32. In any ∆ ABC, prove that : EF 4 .7G GHE > G6 GHE G EF ( ) EF =0 > 67 GHE / = ( a + b + c)2. 33. In any ∆ ABC, prove that : GHE ) GHE 7 G > GHE ) GHE G 6 + GHE ) GHE =0 6 7 34. In any ∆ ABC, prove that : 6 EF ( ) 7 ) G = 7 EF ( ) G ) 6 : G EF ( ) 6 ) 7 . 35. In any ∆ ABC, prove that : a sin sin ) + b sin sin ) + c sin . ) /=0 36. In any ∆ ABC, if cos A + 2 cos B + cos C = 2. Prove that the sides of the ∆ are in A. P. 37 Find the value of the trigonometric function cosec (–1410°) 38. Find the principal and general solutions of the equation 39. Prove that sin2 6x – sin2 4x = sin 2x sin 10x 40. Prove that 41 Prove that \ ;\ YZ[ \YZ[ . / YZ[%\YZ[( 42. Find the value of _`^ .;a/ % \ : []^\[]^( COMPLEX NUMBERS 1. Evaluate bF > c 2. For any two complex numbers z1 and z2 prove that Re ( z1z2) = Re z1 Re z2 - Im z1Im z2. 3. Reduce - ) F 6)F7 F 2- de3 %)F F 2 to the standard form. 4. If x - iy = f 5. Convert the following in the polar form: (i) prove that ( G)F9 F ()F (ii) $ 7. Solve:X2 – 2x + = 0 2 Solve:27 x – 10x + 1 = 0 Solve:21x2 – 28x + 10 =0 8. 9. = 6 7 G 9 )F Solve:3x2 – 4x + % + y2)2 %F 6. % x2 =0 10. If z1 = 2 –i, z2 = 1 + i, find g h h g. 11. If a + ib = (# F (# , h ) h prove that a2 + b2 = (x2 + 1)2 / (2x2 + 1)2 12. Let z1 = 2 – I , z2 - -2 + i. Find : h h (i) Re . / (ii) im . i / h h h 13. Find the modules and argument of the complex F number )%F 14. Find the real numbers x and y if (x – iy) ( 3 + 5i) is the conjugate of -6 -24i. 15. Find the modulus of F - )F )F F 16. If ( x + iy)2 = u = iv, then show that j + k = 4(x2 – y2) # D 17. If K 69 J are different complex numbers with |J| = J) K 1 then find g l g. 18. If a + ib = )KJ # F #)F # 7 , where x is real, prove that a2 + b2 = 1 and = . 6 # ) 19. Find the number of non- zero integral solutions of the equation | F|x = 2x 20. If ( a+ib)(c + id)(e + if)(g +ih) = A + iB then show that : (a2 + b2)(c2 + d2)(e2 + f2)(g2 + h2) = A2 + B2 21. If - F2m = 1 then find the least positive integral value )F of m. 22. Express (-3i)(i) . F/3 in the form a + ib. 23. Prove that the following complex number is purely real: % F %)F 2 +2 )%F %F # F 24. If a + ib = and 7 6 = #)F # , where x is real, prove that a2 + b2 = 1 # ) (6 F m6 n 25. If x + iy = , show that x2 + y2 = . 6)F 6 26. Show that a real value of x will satisfy the equation )F# = a – ib F# if a2 + b2 = 1 where a b are real. . 27. Find the modulus and argument of the complex number -2 + 2√%F 28. Find the modulus and argument of the complex F number and convert it into polar form. )% F 29. Solve the equation 9x2 + 49 = 0 by factorization method. 30. Solve the quadratic equation 2x2 – 4x + 3 = 0. 31. Find the square root of 3 – 4i. 32. Find the square root of -5 + 1 2i 33. Find the square root of – 8i. 34. Find the square root of -2 + 2√% i. 35. Find the square root of 3 – 4 √i. LINEAR INEQUALITIES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. SOLVE THE INEQUALITIES 2 3X – 4 5 6 -3(2X -4) 12 # -3 4 18 -15 %(#) -12 4 7 %# ) (%# 0 11 Solve the inequalities in Exercises 7 to 10 and represent the solution graphically on number line. 5x + 1 -24 , 5x -1 24 2 (x-1) x +5, 3(x + 2) 2 – x 3x -7 2(x - 6), 6 – x 11 – 2x 5 (2x -7) -3 (2x +3) 0, 2x + 19 6x + 47 A solution is to be kept between 680F and 770F. what is the range of temperature in degree Celsius © if the Celsius / Fahrenheit (F) conversion formula is given by ; F = C + 32? A solution of 8% boric acid is to be diluted by adding a 2% boric acid solution to it. the resulting mixture is to be more than 4% but less than 6% boric acid. If we have 640 litres of the 8% solution, have many litres of the 2% solution will have to be added? How many litres of water will have to be added to 1125 litres of the 45% solution of acid so that the resulting mixture will contain more than 25% but less than 30% acid content? IQ of a person is given by the formula o IQ = x 100 Where MA is mental age and CA is chronological age. if 80 IQ 140. For a group of 12 years old children find the range of their mental age. 15. Solve the following linear inequations: (i). 2x – 4 0 (ii) Here -5x + 15 0 16. Solve 5x -3 3x +1 when (i) x is a real number (ii) x is an integer (iii) x is a natural number 17. Solve the following inequatiion: #)% # +| 19 p 13 + % 18. Solve the inequation # % p 4 #) # 19. Solve the inequation 4. #)% 20. Find the pairs of consecutive even positive integers which are larger than 5 and are such that their sum is less than 20. 21. In drilling world's deepest hole, it was found that the temperature T in degree Celsius, x km below the surface of the earth was given by T = 30 + 25( x-3), 3 x 15. At what depth will the temperature be between 2000C and 3000C ? PREMUTATIONS AND COMBINATIONS 1. 2. 3. 4. 5. 6. 7. How many words, with or without meaning, each of 2 vowels and 3 consonants can be formed from the letters of the word DAUGHTER? How many words with or without meaning can be formed using all the letters of the word EQUATION at a time so that the vowels and consonants occur together? A committee of 7 has to be formed from 9 boys and 4 girls. in how many ways can this be done when the committee consists of : (i) exactly 3 girls (ii) at least 3 girls (iii) almost 3 girls If the different permutations of all the letter of the word EXAMINATION are listed as in a dictionary, how many words are there in this list before the first word starting with E ? How many 6- digit numbers can be formed from the digits 0,1,3,5,7 and 9 which are divisible by 10 and no digit is repeated? The English alphabet has 5vowels and 21 consonants. How many words with two different vowels and 2 different consonants can be formed from the alphabet? A well known thinking about the students of senior secondary school is that they are brilliant ,unique in Mathematics. A Mathematics teacher taught them properly and then he decided to take a test to justify them. He prepared a test consists of 12 questions divided into two parts say part I and part II containing 5 and 7 questions respectively. A student is required to attempt 8 questions in all , selecting at least 3 questions from each part .In how many ways can the student select the questions. Select any other two qualities of students , that should be judge by teacher through this test. 8. Determine the number of 5 card combinations out of a deck of 52 cards if each selection of 5 cards has exactly one king. 9. It is required to seat 5 men and 4 women in a row so that the women occupy the even places. How many such arrangements are possible? 10. From a class of 25 students, 10 are to be chosen for an excursion party. there are 3 students who decide that either all of them will join or none of them will join. In how many ways can the excursion party be chosen ? 11. In how many ways can the letters of the word ASSASSINATION be arranged so that all the S's are together? 12 How many words, with or without meaning can be made from the letters of the word MONDAY, assuming that no letter is repeated, if (i) 4 letters are used at a time, (ii) all letters are used at a time, (iii) all letters are used but first letter is a vowel? BINOMIAL THEOREM 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. Find a, b and n in the expansion of ( a+b)n if the first three terms of the expansion are 729, 7290 and 30375 respectively. Find a if the co-efficient of x2 and x3 in the expansion of (3 +ax)9 are equal. Find the co-efficient of x5 in the product ( 1 + 2x)6(1 – x)7 using binomial theorem. If a and b are distinct integers, prove that a – b is a factor an – bn, whenever n is a positive integer. Evaluate ( √% + √ )6 - ( √% √ ) 6 Find the value of ( a2 + √6 )4 + ( a2 - √6 )4 Find an approximation of ( 0.99)5 using the first three terms of its expansion. Find n, if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of . √ > /n is √< : 1 √% # Expand using binomial theorem .- > 2/ , x q # 0. Find the expansion of ( 3x2 – 2ax + 3a2)3 using binomial theorem. Find the number of terms in the expansion of ( 1 + 2x + x2)14. Simplify ( x + 2y)8 + ( x – 2y)8 Find the coefficient of x 5 in the expansion of ( 1+ 3x)6 ( 1-x)5. In the expansion of the ( x + a)n, sums of odd and even terms are P and Q respectively, prove that (i)2 (P2+ Q2) = (x + a)2n + (x –a)2n (ii)P2 – Q2 = ( x2- a2)n 15. Find the 4th term in expansion of ( 3x - D% )4 < 16. Find the coefficient of x6 in the expansion (x - )24 # 17. Find the term independent of x in the expansion of % ( 2x + )9. # 18. If the fourth term in the expansion of .6# > /n. is # $ , then find the value of a and n. 19. The coefficient of three consecutive terms in the expansion of ( 1 + x)n are in the ration 1: 6: 30. Find n. 20. The coefficient of ( m + 1)th term in the expansion of ( 1 + x)2n is equal to the coefficient of (m + 3)th term . show that m + 1 = n. \ SEQUENCES AND SERIES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. Show that the sum of ( m + n)th and ( m – n)th terms of an A.P is equal the mth terms. If the sum of three numbers in A.P., is 24 and their product is 440, find the numbers. Let sum of n, 2n, 3n, terms of an A.P.be S1, S2 and S3 respectively. show that S3 = 3( S2 – S1). Find the sum of all numbers between 200 and 400 which are divisible by 7. Find the sum of integers from 1 to 100 that are divisible by 2 or 5. Find the sum of all two digit numbers which when divided by 4, yield 1 as reminder. If f is a fraction satisfying f(x+y) = f(x) f(y) for all , x, y N such that f(1) =3 and ∑#s '(# = 120, find the value of n. The sum of some terms of G.P. is 315 whose first term and the common ratio are 5 and 2, respectively. find the last term and the number of terms. The first term of a G. P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of G.P. The sum of three numbers in G.P is 56. If we subtract 1,7, 21 from numbers in that order, we obtain an arithmetic progression. Find the numbers. A G.P. consists of an even number of terms. if the sum of all the terms is 5 times the sum of terms occupying odd places, then find its common ratio. The sum of the first four terms of an A.P. is 56. The sum of the four terms is 112. If its first term is 11, then find the number of terms. 6 7# 7 G# G 9# If = = ( x q 0). Then show that a.b.c and 6)7# 7)G# G)9# d are in G.P. 14. Let S be the sum, P the product and R the sum of reciprocals of a terms in s G. P. rove that P2Rn = Sn 15. The pth , qth and rth terms of an A.P. are a, b, c, respectively. show that (q - r)a + (r –p)b + (p –q)c = 0. 16. If a . > /, b. > /, c. > / are in A.P., prove 7 G G 6 6 7 that a,b,c are in A. P. 17. If a,b,c are in G. P., prove that ( an + bn), ( bn + cn), (cn + dn) are in G.P. 18. If a and b are the ratio x2 - 3x + p =0 and c, d are roots of x2 – 12x +q = 0, where a, b, c,d form a G.P. Prove that (q +p) : (q – p) = 17: 15. 19. The ratio of the A.M and G.M of two positive numbers a and b, is m : n. show that a: b = mt > √t n : mt √t n 20. If a, b, c are in A. P ; b,c,d are in G.P and , , are in G 9 u 21. 22. 23. 24. 25. A.P . Prove that a,c,e are in G.P. Find the sum of the following series up to n terms: (i) 5 + 55+ 555 + …… (ii) .6 + .66 + .666 + …… Find the 20th term of the series 2 x4 + 4 x6 + 6 x8 + ….. + n terms. Find the sum of the first n terms of the series : 3 + 7 + 13 + 21 + 31+ ….. If S1, S2, S3 are the sum of first n natural numbers, their squares and their cubes, respectively. show that 9S22 = S3 ( 1 + 8S1) Find the sum of the following series upto a terms. : % + % % % + % % %% % + …… 26. Show that = = % …. =( = % . = =% w. =( % 27. A farmer buys a used tractor for Rs. 12000. He pays Rs. 6000 cash and agrees pay the balance in annual instalments of Rs 500 plus 12% interest on the unpaid amount. how much will the tractor cost him ? 28. Shamshad Ali buys a scooter for Rs. 22000. He pays Rs. 4000 cash and agrees to pay the balance in annual instalment of Rs. 1000 plus 10% interest on the unpaid amount. How much will the scooter cost him? 29. A person writes a letter to four of his friends. He asks each one of them to copy the letter and mail to four different persons with instructions that they move the chain similarly. Assuming that the chain is not broken and that it cost 50 paise to mail one letter. Find the amount spent on the postage when 8th set of letter is mailed. 30. A man deposited Rs. 10000in a bank at the rate of 5% simple interest annually. find the amount in 15th year since he deposited the amount and also calculate the total amount after 20 years. 31. A manufacturer reckons that the value of a machine, which cost him. Rs. 15625 will depreciate each year by 20%. Find the estimated value at the end of 5 years. 32. 150 workers were engaged to finish a job in a certain number of days. 4 workers dropped out on second day. 4 more workers dropped out on third day and so on. it took 8 more days to finish the work find the number of days in which the work was completed. 33. Determine the number of terms in the A.P. 4, 8, 12,……288. Also find its 18th term from the end. 34. If the sum of n terms of an A.P is 8n2 + 3n and its mth term, is 507, find the value of m. 35. Let Sn, denote the sum of the first n terms of an A.P. if S2n = 5 Sn, then prove that S6n /S3n = 17 /4. 36. If ( b – c)2 , (c – a)2, ( a- b)2 are in A.P., prove that , , G)6 6)7 7)G are in A.P. 37. If the A.M. between pth and qth terms of an A.P is equal to the A.M between rth and 8th terms of the A.P. then show that p + q = r + s. 38. If a,b,c are A.P. and A1 is the A.M of a and b and A2 is the A.M. of b and c then prove that the A.M. of A1 , and A2 is b. 39. Which term of the G.P . 2, 8, 32, …… is 32768 ? 40. The fourth term of a G.P is 4. Find the product of its first seven terms. 41. If continued product of three numbers in G.P is 64 and the sum of their products in pairs 56, find the numbers. 42. Find the sum to n terms of the series. 12 - 22 + 32 – 42 + 52 – 62 + 72 – 82 + ……. 43. If y = x + x2+x3 + …… Prove that x = D 44. D Using Geometric series, find the rational number whose decimal expansion is 0.142. 45. Find the sum to infinity of the series > + % + + …… ∞ 46. Find the sum to infinity of the series. ( x = y) + ( x2 + xy +y2 ) + ( x3 + x2y + xy2 + y3) + ……… 47. One side of an equaleteral triangle is 18 cm. the mid points of its sides are joined to form another triangle whose mid points, in term , are joined to from further another triangle and so on up to infinity. Fine the sum of the (i) Perimetes of all the triangle (ii) area of all the triangles. 48. If S1, S2, S3…. SP denote the sums of infinite geometric series whose first terms are 1,2,3….. P respectively and whose common ratio are , , , … . . % y) respectively, Prove that…. y ( y % S1 + S2 + S3 + …… + SP = STRAIGHT LINES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Find the values of K for which the line ( K -3) x –(4 – K)y + K2 – 7K + 6 = 0 is (a) Parallel to the x –axis (b) Parallel to the y- axis (c) Passing through the origin. Find the values of I and p, if the equation x cos I + y sin I = p is the normal form of the line √%# + y +2 = 0. Find the equations of the lines which cut – off intercepts on the whose sum and product are 1 and -6 respectively. What are the points on the y-axis whose distance from # D the line + = 1 is 4 units. % Find the perpendicular distance from the origin of the lines joining the points ( cos I, sin I) and ( cos , EF . Find the equation of the parallel to y-axis and drawn through the points of intersection of the lines x – 7y +5 = 0 and 3x + y = 0 Find the equation of a line drawn perpendicular to the # D line > : through the point. where it meets the y < axis. Find the area of the triangle formed by the lines y – x = 0 and x- k = 0. Find the value of p so that three lines 3x + y – 2 = 0, px +2y -3 =0 and 2x –y-3 =0 may intersect at one point. If three lines whose equations are y = m1x + c1, y = m2x + c2 and y = m3x + c3 are concurrent, then show that m1 (c2 – c3) + m2(c3 – c1) + m3( c1 –c2) = 0. 11 12. 13. 14. 15. 16. 17. 18. 19. Find the equation of the lines through the point (3,2) which make an angle of 450 with the line x – 2y = 3. Find the equation of the line passing through the point of intersection of the lines 4x + 7y – 3 = 0 and 2x – 3y + 1 = 0 that has equal intercepts on the axis. Show that the equation of the line passing through the origin and making an angle I with the line y = mx + c D t z _`^ I is = . # {t _`^ I In what ratio, the line joining ( -1,1) and ( 5,7) is divided by the lines x = y = 4 ?. Find the distance of the line 4x + 7y +5 = 0 from the point (1,2) along the line 2x –y =0,. Find the direction in which a straight line must be drawn through the point ( -1,2) so that its point of intersection with the line x + y = 4 may be at a distance of 3 units from this point. The hypotenuse of a right angled triangle has its ends at the points ( 1,3) and (-4, 1). Find the equation of the legs ( perpendicular sides) of the triangle. Find the image of the point (3,8) with respect to the line x + 3y = 7 assuming the line to be a plane mirror. If the lines y = 3x +1 and 2y = x +3 are equally inclined to the line y = mx + 4,. Find the value of m. 20. 21. 22. 23. If sum of the perpendicular distances of a variable point P (x ,y) from the lines x + y -5 = 0 and 3x – 2y + 7 = 0 is always 10. Show that P must move on a line. Find equation of the line which is equidistant from parallel lines 9x + 6y – 7 = 0 and 3x + 2y + 6 = 0. A ray of light passing through the point A and the reflected ray passes through the point (1,2) reflects on the x- axis at point A and the reflected ray passes through the point (5, 3). Find the coordination of A. Prove that the product of the product of the lengths of the perpendicular drawn from the points ( √6 > 7 , 0) and ( - √6 7 , 0) to the line # 24. 25. 26. 27. 6 YZ[ I+ D 7 sin I = 1 is b2. A person standing at the junction ( crossing) of two straight paths represented by the equations 2x – 3y + 4 = 0 and 3x + 4y -5 = 0 wants to reach the path whose equation is 6x – 7y + 8 = 0 in the least time. find equation of the path that he should follow. A quadrilateral has the vertices at the points ( -4,2), (2,6), (8, 5) and (9, -7). Show that the mid- points of the sides of this quadrilateral are the vertices of a parallelogram. Find the equation of the straight line which makes angle of 15 0 with the positive direction of x–axis and which cuts an intercept of length 5on the negative of y –axis. The perpendicular from the origin to a line meets it at the point ( -3,5) , find the equation of the line. 28. 29. 30. 31. 32. Find the distance of the line 3x – 5y + 8 = 0 from the point (1,2) along the line 2x – 5y = 0 The sides AB and AC of a triangle ABC are 2x + 3y = 29 and x + 2y – 16 = 0 respectively. if the mid point of BC is (5,6) then find the equation of BC. Reduce the lines 6x – 8y +7 = 0 and 8x -6y + 11 = 0 to the normal form and hence determine which line is nearer to the origin. Find the equations of the medians of a triangle formed by the lines x + y -6 =0, x – 3y -2 =0 and 5x – 3y + 2 =0. If the lines ax + y + 1 = 0, x + by + 1 = 0 and x + y + c = 0 are concurrent, prove that + + =1 )6 33. 34. 35. 36. )7 )G Find the value of k if the straight line 2x + 3y + 4 + k ( 6x –y + 12) = 0 is perpendicular to the line 7x + 5y -4 = 0. Find the equation of line passing through the origin and the intersection of the lines x – y – 7 = 0 and 2x + y -2 = 0. Find the equation of the line passing through the point of intersection of the lines x – 3y + 1 = 0 and 2x + 5y – 9 =0 and whose distance from the origin is √. Find the equation of the line that passes through the intersection of the lines 2x + 3y -1 =0 and x + 5y + 4 = 0 and whose intercepts on the axes are same. 37. 38. 39. 40. 41. 42. 43. Find the equation of the line passing through the intercepts t of the lines 4x – 3y + 7 = 0 and 2x – 3y + 5 = 0 and which is inclined at an angle of 1350 with the xaxis. Find the equation of the line passing through the point of intersection of the lines 3x – 5y + 11 = 0 and x + 7y 1 = 0 and which is parallel to x – axis. Find the equation of the line passing through the intersection of the lines 2x – y + 3 =0 and x + 2y + 1 = 0 and parallel to y- axis. Find the new transformed equation of the pair of straight lines x2 + 2xy – y2 + x -2 = 0 when the origin is shifted to a point ( -4,1). Find the equation of the straight6 line whose transformed equation is 3x + 2y -5 = 0 after shifting the origin to ( 2, -1). Find the transformed equation of the parabola y2 = 4ax if the origin is shifted to ( -3, 2). # D Find the equation of the ellipse + = 1 when the 6 7 origin is shifted to (-3,2). 44. 45. Find the new coordinates of point ( a, -a) if the origin is shifted (b, -b) by a translation. Find the transformed equation of the circle x2 + y2 = 9 when the origin is shifted to ( -1, -3) CONIC SECTIONS 1. If a parabolic reflector is 20 cm in diameter and 5 cm deep. find the focus. 2. An arch is in the form of a parabola with its axis vertical. the arch is 10 m high and 5 m wide at the base. how wide is it 2m from the vertex of the parabola? The cable of a uniformly loaded suspension bridge hangs in the form of a parabola. the roadway which is horizontally and 100 m long is suspended by vertical wires attached to the cable. The longest wire being 30m and the shortest being 6m. find the length of a supporting wire attached to the roadway18m from the middle. An arch is in the form of a semi-ellipse. It is a 8m wide and 2m high at the centre. Find the height of the arch at a point 1.5m from one end. A rod of length 12 cm moves with its ends always touching the coorner axes. determine the equation of the locus of a point P on the rod, which is 3cm from the contact with the x-axis. Find the area of the triangle formed by the lines joining the vertex of the parabola x2 = 12y to the ends of its latus rectum. A man running a race leave no space coordinates that the sum of the distances from the two flag posts from him is always 10m. the distance between the flag posts is 8m. find the equation of the posts traced by…. An equilateral triangle is inscribed in the parabola y2 = 4ax where one vertex is at the vertex of the parabola, find the length of the side of the triangle. Find the equation of the circle whose radius is 5 and which touches the circle x2 + y2 – 2x – 4y -20 = 0 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. externally at the point (3, 7). Show that the equation of the circle which touches the co-ordinates axis and whose centre lies on the line lx + my + n = 0 is ( l + m) ( x2+ y2) + 2n( l + m) ( x + y) + n2 = 0. Find the area of an equilateral triangle inscribed in the circle x2 + y2 + 2gx + 2fy + c =0 Find the equation of the parabola whose vertex is at (2,1) and the directrix x = y =1. Find the equation of the ellipse whose axis are along the co-ordinate axis, vertices are (0, z 10) and eccentricity c = 3/5. The foci of an ellipse are ( z, $ and its eccentricity is . find the equation of ellipse. % Find the equation of the hyperbola , the length of % whose latusrectum is 8 and eccentricity is . √ 16. The foci of hyperbola coincide with the foci of the # D ellipse + = 1. Find the equation of the hyperbola it ; its eccentricity is 2. INTRODUCTION TO THREE DIMENSIONAL GEOMETRY 1. Three vertices of a parallelogram ABCD are A (3,-1,2), B(1,2, -4) and C(-1,1,2). Find the coordinates of the fourth vertex. 2. Find the length of the medians of the triangle with vertices A(0,0,6), B(0,4,0) and C(6,0,0). 3. If the origin is the centriod of the triangle PQR with vertices P (2a, 2,6), Q(-4, 3b, -10) and R(8, 14, 2c), then find the values of a, b and c. 4. Find the coordinates of a point on y-axis which are at a distance of 5√ from the point P93,-2,5). 5. A point R with x-coordinate 4 lines on the line segment joining the points P(2, -3,4) and Q(8,0,10), find the coordinates of the point R. 6. If A and B be the point (3,4,5 ) and (-1,3,-7) respectively. Find the equation of the set of points P such that PA2 + PB2 = k2 where k is a constant. 7. Find a point in XY plane which is equidistant from three points (2,0,3), (0,3,2) and (0,0,1). 8. Find the locus of the point which is equidistant from A (3,4,0) and B(5,2,-3) 9. Given that P(5,4, -2) Q(7,6,-4) and R(11,10,-8) are collinear points. find the ratio in which Q divides PR. 10. If the origin of the centroid of the triangle with vertices A(3a, 4,-5) , B(-2, 4b, 6), C( 6, 10, c) find the value of a,b,c. 11. Show that the coordinates of the centroid of a triangle with vertices A (x1, y1, z1), B(x2, y2, z2) and C(x3,y3,z3) are - # # #% D D D% h h h% , , 2 % % % LIMITS AND DERIVATIVES 1. Find the derivative of the following functions from first principle. (i). –x (ii). ( -x)-1 (iii) sin (x +1) C (iv) cos(x - ) Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers). (2) (x +a) | (3) (px +q) ( + s) # (4) (ax + b) (cx + d)2 (5). (6) (7) (8). (9). (10) 6# 7 G# 7 # ) # 6# 7# G 6# 7 }# ~# | }# ~# | 6# 7 6 7 # - # + cos x. (11) √# - 2 (12) (ax + b)n (13) (ax + b)n(cx +d)m (14) sin(x +a) (15) cosec x cot x YZ[ # (16) (17) []^ # []^ YZ[ (18) []^ ) YZ[ [Y ) (19) sinn x (20) (21) (22) (23) (24) (25) (26) (27) (28) [Y 6 7 EF # G 9 GHE # []^(# 6 YZ[ # # ( EF # % GHE # (# > cos x (6# > EF # ( p + q cos x) (x + cos x) ( x- tan x) # EF # %# GHE # C # GHE EF # # 6 # (29) (x + sec x) (x – tan x) # (30) EF # (31) if f is an even function, then prove that ]#3$ '(# (32) Evaluate ] # ) <# %) √ # ] #3 ) √)# √)#)(√)√ ] # )$ #3√$ #3 (33) Evaluate (34) Evaluate # ) <# #)< (35) Find the value of k if ] (36) Evaluate ] #3$ # ) #3 #) GH #)GHEuG # # = 4(1)4-1 = 4 (37) Find the derivative of f(x) = 2x2 + 3x -5 at x = -1. Also show that f1(0) + 3f1(-1) =0 (38) Differentiate cot √# w.r.t . x from first principle method. 9D (39) Find where y = 3 tan x + 5 log x + + 5ex 9# # (40) Differentiate x sin x log x. w.r.t. x (41) Evaluate ] #3$ uEF # ) u# (42) Evaluate ] # uEF # ) #3$ H4 ( # (43) Evaluate : ] #3$ H4 #) (44) Evaluate : ] #3$ (45) Evaluate : ] #)u u# ))#)# # 6# )7# , a, b 0. #3$ $# ) (46) Evaluate : ] u6 # ) #3$ H4 ( # (47) Evaluate : ] #3$ (48) Evaluate : _ u# ) GHE # # # ) #3$ √ #) (49) Evaluate : ] #3$ (50) Evaluate : ] } # );# )%# )GHE # 6GH # )6GHE # #3 GH #)GHE # MATHEMATICAL REASONING 1. Write the negation of the following statements: (i) p : For every positive real number x, the number x-1 is also positive. (ii) q: All cats scratch (iii) r: For every real number x, either x 1 or x 1. (iv) s: There exists a number x such that 0 1. 2. State the converse and contrapositive of each of the following statements: (i) ~} ? There exists a positive real no.x such that x1 is not positive. (ii) ~~ : There exists a cat which does not scratch. (iii) ~|: There exists a real number x such that neither x 1 nor x 1 (iv) ~s: there does not exist a number x such that 0 1. 3. Write each of the statements in the form " if p then q". (i) p: It is necessary to have a password to log on to the server (ii) q: There is traffic jam whenever it rains. (iii) r: You can access the website only if you pay a subscription fee. 4. Re write each of the following statements in the form "p it and only if q" (i) p: If you watch television, then your mind is free and if your mind is free, then you watch television. (ii) q: For you to get an A grade, it is necessary and sufficient that you do all the homework regularly. (iii) r: If a quadrilateral is equiangular, then it is a rectangle and if a quadrilateral is a rectangle, then it is equiangular. 5. Given below are two statements p: 25 is a multiple of 5 q: 25 is a multiple of 8. Write the compound statements connecting these two statements with "and" and "or". In both cases check the validity of the compound statement. 6. Check the validity of the statement given below by the method given against it. (i) p: The sum of an irrational number and a rational number is irrational ( by contradiction method) (ii) q:if n is a real number with n 3, then n2 9( by contradiction method) 7. Write the following statement in five different ways, conveying the same meaning. p: If a triangle is equiangular , then it is an obtuse angled triangle. 8. STATISTICS 1. 2. 3. The mean and variance of eight observations are 9 and 9.25 respectively. If six of the observations are 6,7,10,12,12 and 13. Find the remaining two observations. The mean and variance of 7 observations are 8 and 16 respectively. if five of the observations are 2,4,10, 12, 14, find the remaining two observations. The mean and standard deviation of six observations are 8 and 4 respectively. If each observation is 4. 5. 6. multiplied by 3, find the new mean and new standard deviation of the resulting observations. l is the mean and is the variance of n Given that # observations x1, x2, ….xn. Prove that the mean and l variance of the observations ax1, ax2, ax3…..axn are a# and a2 respectively (a q 0). The mean and standard deviation of 20 observation are found to be 10 and 2 respectively. on rechecking, it was found that an observation 8 was incorrect. calculate the correct mean and standard deviation in each of the following cases: (i) If wrong item is omitted (ii) If it is replaced by 12. The mean and standard deviation of marks obtained by 50 students of a class in three subjects, Mathematics, Physics, and chemistry are given below: SUBJECT Mean Standard deviation 7. MATHEMATICS PHYSICS 42 32 12 15 CHEMISTRY 40.9 20 The mean and standard deviation of a group of 100 observations were found to be 20 and 3 respectively. later on it was found that three observations were incorrect. which were recorded as 21, 21 and 18. Find the mean and standard deviation if the incorrect observation are omitted. 8. Calculate the mean deviation from the median of the following data: Wages per 10week in 20 Rs. No. of 4 workers 2030 3040 4050 5060 6070 7080 6 10 20 10 6 4 9. Find the variance and standard deviation for the following data: 65,68,58,44,45,60,62,60,50. 10. Calculate the mean and standard deviation of the following data: Age No. of persons 2030 3 3040 51 4050607050 60 70 80 122 141 130 51 8090 2 11. The marks obtained ( out of 100) by two students in 10 qualifying tests are: A : 48, 53, 58, 41, 54, 52, 54, 49, 51, 50 B : 11, 98, 60, 94, 48, 52, 17, 90, 20, 20 Who is more consistent and who is more variable? 12 Find the mean and variance and standard deviation for the given frequency distribution table: Classes 0-10 102030405020 30 40 50 60 Frequencies 6 8 14 16 6 2 PROBABILITY 1. 2. 3. 4. 5. 6. 7. A box contains 10 red marbles, 20 blue marbles and 30 green marbles. 5 marbles are drawn from the box, what is the probability that (i) All will be blue? (ii) Atleast one will be green? 4 cards are drawn from a well-shuffled deck of 52 cards. what is the probability of obtaining 3 diamonds and one spade? A die has two faces each with number. 1, three faces each with number 2 and one face with number 3. If die is rolled once, determine (i) P(2) (ii) P(1 or 3) (iii) P( not 3) In a certain lottery 10,000 tickets are sold and ten equal prizes are awarded. what is the probability of not getting a prize if you buy (a) one ticket (b) two tickets (c) 10 tickets Out of 100 students, two sections of 40 and 60 are formed. if you and your friend are among the 100 students, what is the probability that. (a) You both enter the same section? (b) You both enter the different sections? Three letters are written to three persons and an envelope is addressed to each of them, the letters are inserted in to the envelopes at random so that each envelope contains exactly one letter. Find the probability that at least one letter is in its proper envelope. A and B are two events such that P (A) = 0.54, P9B) = 8. 0.69 and P ( A : $. %. 'F9 (i) P (A (ii) P(A'' B'') (iii) P(A B'') (iv) P(B A'' From the employees of a company , 5 persons are selected to represents them in the managing committee of the companies particulars of five persons are as follows: S.NO NAME SEX 1 2 3 4 5 M M F F M Harish Rohan Sheetal Alice Salim AGE IN YEARS 30 33 46 28 41 A person is selected at random from this group to act as a spokesperson. what is the probability that the spokesperson will be either male or over 35 years? 9. If 4- digit numbers greater than 5000 are randomly formed from the digits 0,1,3,5 and 7 what is the probability of forming a number divisible by 5 when (i) the digits are repeated? (iii) The repetition of digits is not allowed? 10. The number lock of a suitcase has 4 wheels, each labeled with ten digits i.e., from 0 to 9, the lock opens with a sequence of four digits with no repeats. What is the probability of a person getting the right sequence to open the suitcase? 11. A die has faces with number 1, three faces each with number 2 and one face with number 3. If die is roled 12. 13. 14. 15. 16. 17. 18 once, find: (i) P (3) (ii) P(1 or 2) (iii) P(2) One word is drawn from a pack of 52 cards. Find the probability that the card drawn is – (i) Black and a king (ii) Either red or Queen A urn contains 10 red, 6 green and 4 black balls. If two balls are drawn at random. Find the probability that: (i) One ball is red and other is green (ii) The balls are of same colour The letters of word " SOCIETY" are placed at random in a row. what is the probability that three vowels come together? The probability of two events A and B are 0.21 and 0.53 respectively. the probability of their simultaneous occurrence is 0.18. Find the probability that neither A nor B occurs. Two die are thrown together, what is the probability that the sum of the number on the two faces is either divisible by 3 or by 4? A Box contains 9 red, 6 green, and 5 black balls. a person draws from the box at random., find the probability that among the balls drawn there is at least one ball of each colour. Out of 100 students ,two sections 40 and 60 are formed.If you and your friend are among the 100 students ,what is the probability that(a) you both enter same sections(b)you both enter different sections. 19. In an entrance test that is graded based on the basis of two examinations ,the probability of a randomly chosen student passing the first examination is 0.8 and the probability of passing the second examination is 0.7.The probability of passing at least one of them is 0.95.What is the probability of passing both ?What would you remark about these students. 20 A letter is chosen at random from the word DAUGHTER .Find the probability that the letter is (i) vowel (ii) a consonant