CHAPTERWISE IIA QUESTIONS final PDF

Question and answers prepared by K.SRINIVAS MATHS LECTURER CHAPTERWISE IMPORTANT QUESTIONS IN IIA CHAPTER 1 COMPLEX NUMBERS 4 MARKS 1 a) Show that the four points in the Argand plane represented by the complex numbers 2+i, 4+3i,2+5i,3i are the vertices of a square. b...

Question and answers prepared by K.SRINIVAS MATHS LECTURER CHAPTERWISE IMPORTANT QUESTIONS IN IIA CHAPTER 1 COMPLEX NUMBERS 4 MARKS 1 a) Show that the four points in the Argand plane represented by the complex numbers 2+i, 4+3i,2+5i,3i are the vertices of a square. b) Show that the four points in the Argand plane represented by the complex numbers -2+7i, -3/2+1/2i,4-3i , 7/2{1+i} are the vertices of a rhombus c) The points P,Q denote complex numbers z1 , z2 in the argand diagram. O is origin. If z1 z2  z1 z2  0 show that POQ=90 d) The complex number z has argument  and satisfying  6 the equation z-3i  3. Then prove that  cot -   i  z 2 a) If z-3+i  4 find the locus of z b) If z+i  z-i  2 find the locus of z 2 2 c) if z+4i  z-4i  10 find the locus of z d) if z-2-3i  5 find the locus of z e) if 2 z-2  z-1 find the locus of z  z-4  f) Determine the locus of z such that Re  0  z-2i  g) If point P denotes the complexnumber z=x+iy in the argand plane z-i and if is purely imaginary then find locus of P z-1 z 1 h) If the part of the complex no. is 1, find the locus of z. z i  z2   i) If the amplitude of   is , find its locus.  z  6i  2 3 a) z=3-5i,show that z 3  10 z 2  58 z  136  0 b) z=2- 7i show that 3z 3  4 z 2  z  88  0  3 (c) If x  iy  then show that x 2  y 2  4 x  3 2  cos   i sin  1 (d) If x  iy  then show that 4x 2  1  0 1  cos   i sin  2i e) If u  iv  where z  x  iy then find u, v z3 2i - 2  11i f) show that and are conjugate to each other (1  2i ) 2 25 3  2isin  g) Find real values of  in order is a (i) real number (ii) purely imaginary 1 - 2isin  2 MARKS 4 (a) z1  (2,1) z 2  (6,3) find z1  z 2 (b) z1  (3,5) z 2  (2,6) find z1  z 2 (c) z1  (6,3) z 2  (2,1) find z1 (d) Find additive inverse of (-6,5)  (10,-4) z2 (e) find Multiplica tive inverse of (7,24) (f) find Multiplica tive inverse of (sin  , cos ) 5 (a) write in the form of A  iB for (2-3i)(3  4i) and also find its Modulus (b) write in the form of A  iB for 3(7  7i)  i(7  7i) and also find its Modulus 4-2i (c) express into A+iB form & find modulus 1-2i (d) write conjugate of (15  3i)-(4-20i) (e ) write conjugate of (2  5i)  (-4  6i) (f) z=5+(3 2)i find zz 6 Transform into modulus amplitude form/Polar form i ) 1-i ii) 1+i 3 iii)- 3  i iv) -1-i 3 v)- 7  i 21 7 Find the square root of a) -5+12i b) -8-6i c) 7+24i z1 8 a) If z1  1, z2  i find Arg b) If z1  1, z2  i Arg(z1 z2 ) z2 c) If z  0 then Arg(z)+Arg z ----   d)if Arg( z1 ) and Arg(z 2 ) are and respectively find Arg(z1 )+Arg(z 2 ) 5 3 1 9 p= find the value of cos +isin in terms of p cos -isin 1  i  n 10 find least positive integer "n" satisfying  1 1  i  11 a) Find the equation of the perpendicular bisector of line segment joining the points 7+7i,7-7i b) Find the equation of straight line joining points -9+6i,11-4i in the Argand plane  CHAPTER 2 DE MOIVER’S THEOREM 7 MARKS n2 n 1 a) If n is a positive integer, show that 1  i   1  i  n n 2 2 cos 4 n 1  i   1  i   2n 1 cos 2n 2n b) If n is a positive integer, show that. 2    n  a) Show that 1  cos   i sin    1  cos   i sin    2 cos   cos n n 1 n . n 2) 2  2  z 2n  1 b) If n is an integer and z=cisθ then show that  i tan n z 2n  1 8    3 1  cos   i sin    1  sin  i cos  c) Show that  i tan 3) Show that  8 8   1 1  cos   i sin  2  1  sin   i cos      8 8  4) If cos + cos + cos = 0 = sin + sin + sin  a) S.T cos3  cos3  cos3  3cos( +  +  ) & Sin3 + sin 3 + sin3  = 3sin( +  +  ) b) S.T cos2 + cos2 + cos2  = 3/2 = sin2 + sin2 + sin2  c) show that cos(   )  cos(    )  cos(   )  0 5 a) If n is positive integer show that 1 1 1 1 q  p  iq  n   p  iq   2( p  q ) cos( arctan ) n 2 2 2n n p b) if  ,  are roots of the equation x 2  2 x  4  0 then for any value n n show that  n   n  2n1 cos( ) 3 6 Find all the solutions of a) x11  x7  x4  1  0 b) x 9  x5  x 4  1  0 c) If z2+z+1=0 where z is complex number. Then find the value 2 2 2 2 2 2  1  2 1   3 1   4 1   5 1   6 1  z    z  2   z  3   z  4   z  5   z  6   z  z   z   z   z   z  d) solve (x-1)n =xn e) Find all roots of (-32)1/5 2 MARKS 12  3 i 7 a) Find the value of    2  2 b) Find the value of (1+i)16   1 1  c) If a  cos   i sin  , b  cos   i sin  ; find  ab   2 ab  1 11    2    2 d) Prove that one value of  cos  i sin    cos  i sin  is zero.  6 6  6 6   1 f) Find all values of  i  6 1 e) Find all values of 1  i 3 3 (cos   i sin  ) 4 g) Find cube root of 8 h) Simplify sin   i cos  8 h) If A,B,C are angles triangle such that x=CisA,y=CisB z=CisC then value of xyz l) if x=cisϴ then value of x6 +1/x6 8 a) If 1, w, w2 are cube roots of unity prove that i) 1  w  w2   1  w2  w   128  1  w  w2   1  w  w2  6 6 7 7 ii)(1-w)(1-w2 )(1-w4 )(1-w8 )  9 iii) (2-w)(2-w2 )(2-w10 )(2-w11 )  49 iv)(a+b)(aw+bw 2 )( aw2  bw)  a 3  b3 1 1 1 v) +  2   1+2 1   vi) S.T x 2  4 x  7  0 If x= w-w 2  2 vii) If  ,  are roots of x 2  x  1  0 then prove that  4   4   1 1  0 CHAPTER 3 QUADRATIC EXPRESSIONS 4 MARKS 1 a) Find the value of “m” if x 2  21  3mx  73  2m  0 have equal roots b) If the equation x 2 -15- m  2x - 8 = 0 has equal roots, find the values of m. c) If the equation (m+1)x 2 +2(m+3)x+(m+8)= 0 has equal roots, find the values of m. x -1 2 a) If ‘x’ real prove that lies beteween 1 and x  5x  9 2 11 x 2  34 x  71 b) Show that none the values of the function over R lies between 5 and 9. x 2  2x  7  1 1 1 c) Prove that   does not lie between 1 and 4, if x is real 3x  1 x  1 (3x  1)( x  1) x p d) If expression is real x  R then find the limits of "p" x  3x  2 2 x2 + x - 1 x+2 e) Determine the range of the following expressions. i) ii) 2 x2 - x + 1 2x + 3x + 6 3 a) If x 2  4ax  3  0 and 2 x 2  3ax  9  0 have a common root, then find the value of “a” and the common root. b) If x 2 - 6x + 5 = 0 and x 2 - 2ax + 35 = 0 have a common root , then find a. and the common root c) Suppose that the quadratic equations ax 2  bx  c  0 and bx 2  cx  a  0 have a common root. Then show that a 3  b 3  c 3  3abc d) If the quadratic equation ax 2 + 2bx + c = 0 and ax 2 + 2cx + b = 0,  b  c  have a common root. Then show that a + 4b + 4c = 0. 4 Let  and  be the roots of the quadratic equations ax 2  bx  c  0, c  0 then 1 1  form the quadratic equation whose roots are and   5. If  ,  are the roots of the equation ax 2  bx  c  0, find the values of 2 1 1 1 1    2  2 i    ii  2  2  iii       iv)    4 7 7 4 v) 2           2 6 a) Solve 71 x  71 x  50 for real x.? x 1 - x 13 x x-3 5 b) Solve   c) Solve: + = 1 x x 6 x-3 x 2 2  1  1  2 1   1 d).Solve: 2  x +  -7  x +  +5 = 0 e). Solve:  x  2   5  x    6  0  x  x  x   x f) Solve: 2x 4 + x 3 - 11x 2 + x + 2 = 0 g) Solve: (x -1) (x – 3) (x – 5) (x – 7) = 9 h) Solve x 1  2x  5  3 7. If x1 ,x 2 are the roots the quadratic equation ax 2 + bx + c = 0 and c  0. Find the value of  ax1 + b  +  ax 2 + b . -2 -2 2 MARKS 8. Form the equation whose roots are m n p-q p+q i) , ii) ,- iii) 7  2 5 iv)-a+ib,-a-ib v)-3  5i n m p+q p-q 9 Find the maximum or minimum value of the quadratic expressions. i) 2 x  7  5 x 2ii)3x 2  2 x  11 iii) x 2 - x + 7 iv) 12x - x 2 - 32 v) 2x +5 - 3x 2 vi) ax 2 + bx + a 10 a) For what values of x the expression x 2  5x  14 is positive b) For what values of x the expression 15+4x-3x 2 is negative 1 1 c) If roots equation x 2  bx  1  0 are  ,  find the equation with roots ,   CHAPTER 4 THEORY OF EQUATIONS 7 MARKS 1 a) Solve the following equations, given that the roots of each are in A.P. (i) 8 x3  36 x 2  18 x  81  0 (ii) x3  3x 2  6 x  8  0 iii) 4 x3  24 x 2  23x  18  0, b) Solve the following equations, given that the roots of each are in G.P. i) 3x 3 - 26x 2 + 52x - 24 = 0 ii) x 3  7 x 2  14 x  8  0, c) Solve the equations, given that the roots of each are in H.P 6 x 3  11x 2  6 x  1  0. d) Solve x 3 - 3x 2 - 16x + 48 = 0, given that the sum of two roots is zero. e) Solve x 3  9 x 2  14 x  24  0, given that two of the roots are in the ration 3:2. f ) Given that one root of 2x 3 +3x 2 - 8x + 3 = 0 is double of another root, find the roots of the equation 2 Solve 18x3  81x 2  121x  60  0, given that a root is equal to half the sum of the remaining roots. 3 a) Given that the sum of two roots of x 4  2 x 3  4 x 2  6 x  21  0 is zero, and find the roots of the equation. b) Solve x 4  5 x 3  5x 2  5x  6  0, given that the product of two of its roots is 3. c) Solve x 4  4 x3  2 x 2  12 x  9  0, given that it has two pairs of equal roots. 4) a) Solve the equation i) 6 x 4  35x 3  62 x 2  35x  6  0 ii) x 4  10 x3  26 x 2  10 x  1  0 iii) x 4  x3  4 x 2  x  1  0  b) Solve: i) x 5 -5x 4 + 9x 3 - 9x 2 + 5x -1= 0 ii) 2 x5  x 4  12 x3  12 x 2  x  2  0 c) solve i) 6 x 6  25 x 5  31x 4  31x 2  25 x  6  0 ii)6 x6  35 x5  56 x 4  56 x 2  35 x  6  0 5 a) Find the condition if roots of the equation x 3  px 2  qx  r  0 in A.P b) Given that roots of x3  3 px 2  3qx  r  0, are in i) A.P show that 2p3  3 pq  r  0 ii) G.P show that p3r  q 3 iii) HP show that 2q 3  r (3 pq  r ) 6 i) Solve the equation x 4 + 2x 3 - 5x 2 + 6x + 2 = 0 given that 1 + i is one of its roots. ii) Solve the equation 3x 3 - 4x 2 + x + 88 = 0 , which has 2  7 as a root. iii). Solve x 4 - 4x 2 + 8x + 35 = 0 , given that 2  i 3 is a root iv) Solve the equation x - 6x + 11x -10x + 2 = 0 , given that 2  4 3 2 3 is a root of the equation. 7 Solve the following equations, given that they have multiple roots. i) x 4 - 6x 3 +13x 2 - 24x + 36 = 0 ii) 3x 4 +16x 3 + 24x 2 -16 = 0 iii) show that x5-5x3+5x2-1= 0 has three equal roots. And find this root. 8 Transform each of the following equations into ones in which the coefficients of the second highest power of x is zero and also find their transformed equations. i) x 3 - 6x 2 +10x -3 = 0 ii) x 4 + 4x 3 + 2x 2 - 4x - 2 = 0 2 MARKS 9. i) If α, β, γ the roots of 4x 2 - 6x 2 + 7x + 3 = 0 , then find the value of αβ + βγ + αγ ii) If 1,1,α are the roots of x 3 - 6x 2 + 9x - 4 = 0 , then find α. iii) Find the value of ‘k” if 1,-1,k are roots of 2 x 3  3x 2  2 x  3  0 iv) If -1, 2 and α are the roots of 2x 3 + x 2 - 7x - 6 = 0 , then find α v) If 1, -2 and 3 are the roots of x 3 - 2x 2 + ax + 6 = 0 , find a. vi If α,β and 1 are the roots of x 3 - 2x 2 - 5x + 6 = 0 , then find α and β. vii) If α,β and γ are the roots of x 3 - 2x 2 + x- 4= 0 , then find (i)  α 2 β 2 (ii)  α 2β+  αβ 2 10. i) If 1, 2, 3 and 4 are the roots of x 4 + ax 3 + bx 2 + cx + d = 0, then find the values of a, b, c and d. ii) Form the polynomial equation of degree a whose roots are 2, 3 and 6. iii) Find the polynomial equation of degree 4 whose roots are the negative of the roots of x 4  6 x3  7 x 2  2 x  1  0. iv) Find the polynomial equation of whose roots are the reciprocals of the roots of the equation x 4  3x 3  6 x 2  2 x  4  0 v) Find the algebraic equation of degree 5 whose roots are the translates of the roots x 5  4 x 3  x 2  11  0 by  3. vi) Find the algebraic equation of degree 4 whose roots are the translates of the roots of 4 x  32 x 3  83x 2  76 x  21  0 by 2. 4 CHAPTER 5 PERMUTATIONS AND COMBINATIONS 4 MARKS 1 a) Find the number of ways of arranging 6 boys and 6 girls in a row. In how many of these arrangements.(i) All the girls are together (ii) no two girls are together (iii) boys and girls come together alternately. b) Find the no. of ways of seating 5 Indians, 4 Americans and 3 Russians at a round table so that i ) all Indians sit together ii)no two Russians sit together. iii) Persons of the same nationality sit together. c) Find the no. of ways of arranging 6 boys and 6 girls around a circular table so that i ) All the girls come together ii) No two girls come together iii). Boys and girls come alternately. d) Find the no. of ways of arranging the letters of the word BRINGING so that (i) they begin and end with I (ii) the two G’s come together e) Find the no. of ways of arranging the letters of the word PACIFIC. In how many of them (i) do the two C’s come together ?(ii) the two I ’s do not come together ? f) Find the no. of ways of arranging 6 red roses and 3 yellow roses of different sizes into a garland. In how many of them. i. all the yellow roses come together. Ii No two yellow roses come together. 2 a) Find the number of 4 digited numbers that can be formed using the digits 1, 2, 5, 6, 7. How many of them are divisible by i) 2, ii) 3, iii) 4, iv) 5 and v) 25? b) Find the number of 4 digited numbers that can be formed using the digits 2,3,5,6,8 How many of them are divisible by. i) 2, ii) 3, iii) 4, iv) 5 and v) 25? 3 a) Find the sum of all 4 digited numbers that can be formed using the digits 1, 2, 4, 5, 6 without repetition b) Find the sum of all 4 digited numbers that can be formed using the digits 1,3,5,7,9 without repetition c) Find the sum of all 4 digited numbers that can be formed using the digits 0,2,4,7,8 without repetition 4. a) If the letters of the word PRISON are permuted in all possible ways and the words thus formed are arranged in dictionary order, find the ranks of the words (i) PRISON (ii) SIPRON. b) If the letters of the word MASTER are permuted in all possible ways and thus formed are arranged in the dictionary order, then find the ranks of the words (i) REMAST (ii) MASTER. c) If the letters of the word RUBLE are permuted in all possible ways and the words thus formed are arranged in the dictionary order, then find the rank of the word ‘LUBER’. d) If the letters of the word BRING are permuted in all possible ways and thus formed are arranged in the dictionary order, then find 59th word e) If all letters of word “EAMCET” arranged in dictionary order find Rank of “EAMCET”. f) If letters of word “AJANTA” arranged in dictionary order find Ranks of “AJANTA” & “ JANATA” 5 a) Find the no. of ways of forming a committee of 5 persons from a group of 4 Indians and 3 Russians such that there are at least 3 Indians in the committee. b) Find the number of ways of forming a committee of 5 members out of 6 Indians and 5 Americans so that always the Indians will be in majority in the committee.. c) Find number of forming a committee of 5 persons from Group of 5 Indians and 4 Russians such that there are At least 3 Indians. 6 a) Find the no. of ways of selecting 11 member cricket team from 7 batsmen, 5 bowlers and 3wicket keepers with at least 3 bowlers and two wicket keepers b) Find the number of ways of selecting a cricket team of 11 players from 7 batsmen and 6 bowlers such that there will be at least 5 bowlers in the team. c) If a set A has 12 elements. Find the number of subsets of A having: i) 4 elements, ii) at least 3 elements, iii) at most 3 elements  4 4 7 a) Find the value of 25 C4 +  r=0  29-r  C3. b) Find the value of 34 C5 +  r=0 (38-r) C4 c) if 3  r  n n-3 Cr  3. n-3 Cr-1  3. n-3 Cr-2  n-3 Cr-3  n Cr d) if 2  r  n n Cr-2  2. n Cr-1  n Cr  n+ 2 Cr e) Find the value 10 C5 + 2.10 C4 + 10 C3 8 a) A double Decker minibus has 13 seats in the lower deck and 12 seats in the upper deck. In how many ways can a marriage party of 25 persons be arranged if 4 old people refuse to go to the upper deck and 3 children wish to travel only in the upper deck ? 4n C2n 1.3.5....(4n -1) b) Prove that =. 1.3.5....(2n -1) 2n 2 Cn 2 MARKS ii) If   p5 ;n p6 = 2 : 7 , find n. n 1 9. a) i) If n p 7 = 42. n p 5 , find n. iii) If 18 Pr 1 :17 Pr 1 = 9 : 7 find r iv) n p4 = 5040 find n v) If 56 P r 6 : 54 p r 3  30800 :1, find r ? b) Find the no. of ways of arranging the letters of the word (i) INDEPENDENCE (ii) MATHEMATICS (iii) INTERMEDIATE. c) Find the number of different words that can be formed using 4A’s, 3B’s,2’C and one D.Or 4 A B3C 2 D in expanded form d) Find the number of 7 digited numbers that can be formed using 2,2,2,3,3,4,4 10 a) Find number of 4 digit number that can be formed using the digits 2,3,5,6,8. b) i) Find the number of 4 digited numbers that can be formed using the digits 1, 2, 4, 5, 7, 8, when repetition is allowed.. ii) Find the number of 4 digit numbers that can be formed using the digits 1, 2,3, 4, 5,6 with at least one digit repeated. c) i) Find the number of ways of arranging 7 persons around a circle. ii) Find no of ways arranging 6 roses as garland d) Find all the number of positive divisors of 1080 e) Find number of diagonals of polygon with i) 10 ii) 12 sides f) If polygon has 275 diagonals find number of sides. 13 11 a) Find i) If n C5 = n C 6 then find cn. ii) If n C21 = n C27 , then find 49 Cn.  12 iii) If Cs+1 = 12C2s-5 then find “ s “ iv) If 17 C2t+1 = 17 C3t-5 , find t. 15 v) C2r-1 = 15C2r+4 find r vi) If 10.n C 2 =3. n+1 C3 find n. vii) prove that 10 c3  10 c6  11c4 b) Find the number of ways of selecting 3 vowels and 2 constants from the letters of the word EQUATION.. CHAPTER 6 BINOMIAL THEOREM 7MARKS 1 1.3 1.3.5 1 a) Find the sum of the infinite series 1   .......upto  ? 3 3.6 3.6.9 7 1 1.3 1 1.3.5 1  b) Find the sum of the series 1  2 . 4. 6 ..... ? 5  10 1.2 10 1.2.3 10  3 3.5 3.5.7 c) Find the sum of the series   4 4.8 4.8.12 3.5 3.5.7 3.5.7.9 d) Find the sum of the infinite series  .......upto  ? 5.10 5.10.15 5.10.15.20 1 5 5.7 e) Find the sum of the series   4 4.8 4.8.12 5 5.8 5.8.11 f) Find the sum of the infinite series   ............ 6.12 6.12.18 6.12.18.24 1 1.3 1.3.5 g) If x    .......... then find the value of 3 x 2  6 x. ? 5 5.10 5.10.15 1.3 1.3.5 1.3.5.7 h ) If x=  .......upto  prove that 9x 2  24 x  11 3.6 3.6.9 3.6.9.12 5 5.7 5.7.9 i) If x   2  3 ......... then find the value of x 2  4 x. ? 2!3 3!.3 4!.3 4 4.6 4.6.8 j) if t    .......... then prove that 9t=16 5 5.10 5.10.15 2 a) If 28,56,70 are three successive coefficients in the expansion of 1  x  then find n ? n  a+x  n b) 2nd ,3rd ,4th terms of are 240,720,1080. find a,x,n.? c) If the coefficient of r th ,  r + 1  r + 2 terms in the expansion of 1 + x  are in A.P., th th n then show that n 2 -  4r + 1 n + 4r 2 - 2 = 0 d)If the coefficient of x9,x10,x11 in the expansion of 1 + x  are in A.P., n then show that n 2 -41n +398= 0 11 11  1   1  7 e) If x in the expansion  ax 2 +  and x -7 in  ax- 2   bx   bx  are equation find the relation between a and b? 3 a) If P and Q are the sum ODD terms and the sum of EVEN terms respectively in the expansion of  x + a  , then prove that (i) P 2 - Q2 = x 2 - a 2   n n ii) (x+a)2n  ( x  a)2 n  4 PQ b) If the coefficient of 4 consecutive terms in the expansion of 1 + x  are a1 ,a 2 ,a 3 ,a 4 n a1 a3 2a 2 respectively, then show that + = a1 + a 2 a3 + a 4 a 2 + a3   = I + f , then show that (i) I is an odd n 4) If I, n are positive integers, 0 < f < 1 and 7 + 4 3 integer and ii) (I + f) (1 – f) = 1 5 a) Using binomial theorem how that 50n  49n  1 is divisible by 49 for all positive integers n b) Prove that  625   48n  1 is divisible by 576 for all natural numbers n c) Prove that  5   52n  1 is divisible by 676 for all natural numbers of n. 4n 6 a) Prove that (i), c0.cr+ c1.cr+1+c2.cr+2+………….cn-r.cn=2nc(n+r) And deduce that (ii) C0  C1  C2 ......  Cn  2 2 2 2 2n 2n cn (iii)c0.c1+c1.c2+c2.c3+………..+cn-1.cn= cn 1 b) If n is positive integer and x is any non zero real number then prove that www.studentsmilestone.com x x2 x n (1+x) n+1 -1 C0 +C1 +C2 +...........+Cn = 2 3 n+1  n+1 x hence deduce that C1 C2 Cn (2) n+1 -1 3C1 9C2 3n C n (4) n+1 -1 a)C0 + + +...........+ = b)C0 + + +...........+ = 2 3 n+1  n+1 2 3 n+1 3  n+1 c) prove that  n C0  C  C2  C3.........  (1) Cn  (1) 2. nC n if n is even 2 2 2 2 n 2 1  2 = 0 if n odd 7 compute the numerically greatest term(s) in the expansion of i)  5 x  6 y  14 when x  2 and y  1 ii) 2 x  3 y  12 5 x  1, y . 3 5 2 iii)  3 + 7x  when x = 3/5, n = 10 n 1 1 4 8.a) Find approximate value of i) 3 ii) (627) up to 5 decimel 999 b) If x is so small that x 2 and higer powers of x may be neglected then find approximate value of 1 3 1 (4+3x) 2 (1-2x) (32  5 x) 2 5 i) ii) (3-2x) 2 (3-x)3 2 MARKS 9  2x 3y   2x+3y+z  7 th 9 a) i) Find number terms of ii) Find 6 term of  +   3 2  10 11  3x   3b  b) Find the middle term (s) in the expansion of i)  - 2y  ii)  4a+   7   2  13 8  2 3  11  -4 3 c) Find the coefficient of i) x in  2x + 3  ii) x in  2x -   x   x 10 10.  x 5  7 1/2  25  x 3  d) Find the term independent of x in i)  -  ii)  - 3x  iii)     7 x  2  x 1/3   3 2x 2      e) Find the values of x for which the binomial expansions of the following are valid.     3 5 i)  2  3x  ii)  4  5x  7/3 iii) 9  4x 2 iv) 4  x 2 3/4 CHAPTER 7 PARTIAL FRACTIONS 4 MARKS 2x + 3 3x  7 1. Resolve the following into partial fractions. a) b) 2  x + 1 x - 3 x  3x  2 x+4 d) 9 x 2  13x  15 5x 2 +3x+2 c) e) f)  x -4   x + 3 2 ( x  1)( x  2)2 2 x  3x  32 x 3 +x 2 x 2  4 x  15 3x 3  8 x 2  10 g) h) x  33 x  14  x2 1 x2  3 2 x 2  3x  4   2 Resolve into partial fractions a) b) c) x 2  4 x  2  x  2  ( x 2  1)  x  1 ( x 2  2) 5x2  2 3x  1 3x 3  2 x 2  1 d) 3 e) f) x x 1  x  x 2   x  2  x 4  x 2 1 x3 2 x  1x  2x  3 3 a) Resolve into partial fractions. ? x3 b) Resolve the following into partial fractions.  x - a  x - b  x - c  x4 x  1x  2 c) Resolve into partial fractions. ? x4 xn 3x 2  2 x   4 a) Find the coefficients of and in the expansion of ? x 2  2 x  3 x b) Find the coefficients of x4 and xn in the expansion of x  1 x  2 2 CHAPTER 8 MEASURES OF DISPERSION 7 MARKS 1 a) Find mean deviation about median for the following distribution Class 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 interval frequency 5 8 7 12 28 20 10 10 b) Find mean deviation about mean for the following distribution(using step deviation method) Height 95-105 105-115 115-125 125-135 135-145 145-155 (in Cms) Number of 9 13 26 30 12 10 boys c) Find mean deviation about mean for the following distribution(using step deviation method) Marks 0-10 10-20 20-30 30-40 40-50 50-60 60-70 No.of 6 5 8 15 7 6 3 students d) find mean deviation from median for following data xi 6 9 3 12 15 13 21 22 fi 4 5 3 2 5 4 4 3 2 a) Find the variance and standard deviation of following distribution xi 6 10 14 18 24 28 30 fi 2 4 7 12 8 4 3 b) Find Mean and variance of using step variance of following data Age(in Yrs) 20-30 30-40 40-50 50-60 60-70 70-80 80-90 No. of members 3 61 132 153 140 51 2 3 a) An analysis of monthly wages paid to the workers of two firms A and B belonging to the same gives the following data Firm A Firm B No of workers 500 600 Average daily Wages(Rs) 186 175 Variance of distribution of 81 100 wages I) which Firm A or B has greater variability In individual wages ii) which firm has larger wage bill b) The scores of two cricketers A and B in 10 innings given below. find who is better run getter and who is consistent player Scores 40 25 19 80 38 8 67 121 66 76 of A xi Scores 28 70 31 0 14 111 66 31 25 4 of B yi 2 MARKS 4 a) Find mean deviation about mean for following data i) 38,70,48,40,42,55,63,46,54,44 ii) 3,6,10,4,9,10 b) Find mean deviation about Median for following data i) 13,17,16,11,13,10,16,11,18,12,17 ii) 4,6,9,3,10,13,2 5 a) find mean deviation about mean for following distribution xi 10 30 50 70 90 fi 4 24 28 16 8 b) find mean deviation about median for following distribution xi 5 7 9 10 12 15 fi 7 6 2 2 2 6 6 Find variance for discrete data below a) 6,7,10,12,13,4,8,12 b) 350,361,370,373,376,379,385,387,394,395 7. The coefficient of variation of two distributions are 60 and 70 and their standard deviations are 21 and 16 respectively. Find their arithmetic mean. 8.The variance of 20 observations is 5. If each each of observation is multiplied by 2, find the variance resulting observations. CHAPTER 9 PROBABILITY 7MARKS 1 A, B, C are 3 newspapers from a city. 20% of the population read A, 16% read B, 14% read C, 8% both A and B, 5% both A and C, 4% both B and C and 2% all the three. Find the percentage of the population who read at least one newspaper and also find probability 2 If A,B,C are three independent events of an experiment such that P  A  Bc  Cc   , P  A c  B  CC   , P  A c  Bc  Cc   , find P(A), P(B) &P(C). 1 1 1 4 8 4 3 a) Suppose A and B are independent events with P(A) = 0.6, P(B) = 0.7. Compute B i) P  A  B , ii) P  A  B , iii) P   iv) P  A  B  C C   A b) The probabilities Three events A,B,C such that P(A)=0.3 P(B)=0.4 P(C)=0.8 , P( A  B)  0.08 P(A  C)=0.28 P(A  B  C )  0.09 and P(A  B  C )  0.75 show that P(B  C) lies in the interval [0.23,0.48] 4 A bag B1 contains 4 white and 2 black balls, bag B 2 contains 3 white and 4 black balls. A bag is drawn at random and a ball is chosen at random from it. Then what is the probability that the ball is white. b) There are 4 white and 3 black balls in one bag , 3 white and 4 black balls in second bag. Die is rolled and the first bag selected if the die shows up 1 or 3 and second bag selected for the rest.Find the probability of drawing a black ball from the bag selected 5 a) Suppose that an urn B1 contains 2 white and 3 black balls and another urn B2 contains 3 white and 4 black balls. Out of these two, one urn is selected at random and a ball is drawn from it. If the ball drawn is black find the probability that urn B1 was chosen. b) Three boxes numbered I,II,III contains the Balls as follows WHITE BLACK RED BOX1 1 2 3 BOX2 2 1 1 BOX3 4 5 3 One box is selected and a ball is selected from it If Ball is Red then find the Probability that it is from Box II c) Three boxes numbered B1, B2, B3,contains the Balls as follows WHITE BLACK RED B1 2 1 2 B2 3 2 4 B3 4 3 2 A die is thrown B1 is chosen if either 1 or 2 is turns up B2 is chosen if 3 or 4 chosen B3 is chosen 5 or 6 turns up. Having chosen a box in this way , a ball is chosen at random from this box. If the ball drawn is found to be red find the probability that it is drawn from box B2 6 The Probability of Australia wins a match against India in cricket is 1/3. If India and Australia play 3 matches find Probability that i) Australia will loose all games ii) Australia wins Atleast one match 4 MARKS 7 In a committee of 25 members, each member is proficient either in mathematics or in statistics or in both. If 19 of these are proficient in Mathematics, 16 in statistics, find the probability that a person selected from the committee is proficient in both 8 a) Find the probability that a non-leap year contains i), 53 Sundays ii) 52 Sundays only. b) what is probability that a leap year will contains 53 Sundays. 9 If E1, E2 are any two events of a random experiment then P(E1  E2)=P(E1)+P(E2)-P(E1  E2) { addition theorem} 10 a) In a box containing 15 bulbs, 5 are defective. If 5 bulbs are selected at random from the box, find the probability of the event, that i) none of them is defective. ii) Only one of them is defective. iii) At least one of them is defective b) If two numbers are selected randomly from 20 consecutive natural numbers, find the probability that the sum of the two numbers is (i) an even number, (ii) an odd number. c) A, B, C are three horses in a race. The probability of A to win the race is twice that of B and probability of B is twice that of C. What are the probabilities of A,B and C to win the race? d) A bag contains 12 two rupee coins, 7 one rupee coins and 4 half a rupee coins. If three coins are selected at random, then find the probability that (i) the sum of three coins is maximum, (ii) the sum of three is minimum (iii) each coin is different value. e) “A” speaks truth in 75% of the cases and “B” speaks truth 80% of cases. What is the probability of their statement about a incident is do not match f) If one ticket is drawn from ticket numbered 1 to 30 then find probability that the number on ticket is i) a multiple of 5 or 7 ii) a multiple of 3 or 5 11 a) A problem is three students A,B,C. The chances of solving are 1/3,1/4,1/5 respectively. Find the probability of that problem solved b) If A,B are independent events with P(A B) =0.65 P(AB)=0.15 Then find P(AC)+P(BC) c) A and B are events with P(A)=0.5 P(B)=0.4 and P(AB)=0.3 Find the probability that i) A does not occur ii) neither A nor B occur d) Two persons A,B are roll a dice on the condition that the person who gets 3 will win game. IF A starts game then find probability of A and B respectively win the game. e) Let A and B be independent events with P(A)=0.2 P(B)=0.5 find i) P(A/B) ii) P(B/A) iii) P(AB) iv) P(A B) f) If A and B are independent events of random experiment show that A C and BC are also independent 12 a) A bag contains 10 identical balls of which 4 are blue and 6 are red. Three balls are taken out of random from the bag one after the other. find the probability that all three balls drawn are red. b) i) If 6 cards are drawn at random from a pack of cards, then probability to get 3 red and 3 black cards ii) If one card is drawn from pack of cards then find probability of getting king or queen iii) what is the probability of drawing an ace or a spade from a well shuffled pack of 52 cards ? c) i) Two fair dice are rolled. What is the probability that the sum on the faces of two dice is 10 ii) Two fair dice are rolled. What is the probability that the sum on the faces of two dice is atleast 8 CHAPTER 10 RANDOM VARIABLE AND PROBABILITY DISTRIBUTIONS. 7 MARKS 1. a) The probability distribution of a random variable X is given below: Find the value of K, mean and variance of X. b) A random variable X has the following probability distribution. Find (i) k (ii) the mean and (iii) P(0