I PUC MATHEMATICS CHAPTER - 08 Binomial Theorem. x 1. Expand x + using binomial theorem and hence find the coefficient of

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1 Two Maks Questios I PU MATHEMATIS HAPTER - 08 Biomial Theoem. Epad + usig biomial theoem ad hece fid the coefficiet of y y. Epad usig biomial theoem. Hece fid the costat tem of the epasio.. Simplify + ) + ). Simplify + ) + ). Usig biomial theoem, evaluate 9). Usig biomial theoem evaluate 0.99) coect to fou decimal places.. Usig biomial theoem evaluate 0.) 8. Fid the th tem i the epasio of 9. Fid the th tem i the epasio of y 0. Fid the coefficiet of i the epasio of b. Fid the coefficiet of i the epasio of + b a a. Fid the atio of the coefficiet of 8 to the coefficiet of i the epasio of ) 0.. If the fouth tem i the epasio of a + is, evaluate a. Fid the middle tem i the epasio of +. y. Fid the middle tem i the epasio of b y b. Fid the middle tem i the epasio of a a. Fid the tem idepedet of i the epasio of 8. Fid the costat tem i the epasio of

2 9. Show that thee is o tem idepedet of i the epasio of 0. Pove that thee is o tem ivolvig i the epasio of. Pove that the coefficiets of m ad ae equal i the epasio of +) m+ whee m ad ae positive iteges. mak questios:. The d, th ad th tems i the epasio of +a) ae espectively 8, 80 ad 0. Fid the values of, a ad.. Usig biomial theoem pove that 8 9 is divisible by 8 whee is a positive itege.. The coefficiets of thee cosecutive tems i the epasio of +) ae i the atio :: fid.. Fid the middle tems i the epasio of +.. Fid the coefficiet of i )... Fid the coefficiet of 0 i the epasio of ++ ) ).. Show that the coefficiets of i the epasio of + ) +) is Give that the coefficiets of m +) th ad m+) th tems i the epasio of +) ae equal, fid m 9. Fid the tem idepedet of i the epasio of + ) 9 0

3 Solutio fo two maks questios. Solutios Biomial Theoem. + = y y y y = y y 8y y y + y + y = y y y y y The coefficiet of y is.. = + ) + ) = ) ) + + = The costat tem of the epasio = T =. ) + ). oside + ) = ) + ) + ) + ) + + ) = + 9 ) + 9) + 0 ) + ) + + ) = ) +. oside ) = ) + ) + ) + = [ ] = [08] = + ) = ) + ) + = + ) + ) ) = ) = ) - ) = [0 + ] = { ] = 8 ) + ) + ) + ) +

4 . 9) = 00 ) = = 00) 00) ) + 00) ) 00) ) + 00)) -) = ) ) )9) ) ) + 00) 8) = ) = 0.0) = 0.0) + 0.0) 0.0) + 0.0) 0.0) + 0.0) = ) )+ eglectig highe powes of 0.0) = ) = ) = 0 + 0) 0.) + 0) 0.) + 0)0.) ) = )0.) + 00) 0.0) +0)0.008) ) = 08.. The geeal tem i the epasio of is T + = Put =, T + = T = 0 =. [ = = T = 9 = ]. T + = 0 0 ) y Put =, T + = 0 ) y T = y. T + = 0 = T + = )

5 0 If = 0 = = Substitutig = i ) [ = = ] T = 0 = = 9 coefficiet of is 8. Geeal tem = T + = 8 b) a T + = 8 8 b) 8 a By data, the powe of b must be. [ 8 8 = = ] fo = T + = 8 b) a b = ) ) ) a b T = 88 a b coefficiet of is 88 a. T + = 0 ) 0- - ) = 0 - ) lealy coefficiet of 8 is 0 ad the coefficiet of is the equied atio = = = the atio is :. T + = a) - Puttig =

6 T + = a T = 0 a Give T = [ = 0] 0 a = a = 8 a = 8. T + = 0 ) 0 Sice = 0 thee ae tems i the epasio. T is the middle tem. Puttig = 0, we get T = 0 0 ) = T = is the middle tem. 9. T + = y b b y = Thee ae tems i the epasio T is the middle tem. Puttig = we get T + = y b b y T = y b b y T = = 9 is the middle tem 0. T + = 0 a) 0- a = 0 the middle tem is T = T + puttig =, T + = 0 a) 0- a T = 0 a a 0 ) ) a = -

7 = - a 0 is the middle tem 0. T + = 0 = 0 0 ) 0 T + = 0 ) 0 0 Equatig the powe of to zeo. 0 = 0 = Fom equatio ) T + = 0 ) 0 = T = is the costat tem =0. T + = = ) 0 T + = -) - ) 0+ Fo costat tem, 0 + = 0 Solvig we get = = = = T + = -) = 8 T = 0 is the costat tem.. T + = 0 ) 0 0 ) = ) = ) 90 Fo the costat tem, = 0 90 = 0 = 90 = which is a factio.

8 Sice the value of caot be a factio, thee does t eist costat tem i the epasio.. T + = ) - = 0- ) T + = -) 0- Equatig the powe of to, 0 = = R = which is a factio. thee does t eist tem cotaiig i the epasio.. T + = m+) ) m+- T + = m+) oefficiet of m is m+) m oefficiet of is m+ But m+) m = m+) m+-m m+) m = m+) Hece poved. Solutios fo maks questios. T + = X - a Give T = 8, T = 80, T = 0 Now T = 8 - a = 8 ) ) a = 8 -) - a = 8 ) T = 80 - a = 80 ) ) - a = 80 -) -) - a = 80 ) T = 0 - a = 0 ) ) ) - a = 0 -) -) -) - a = 0 ) ) ) ) a 0 = 80 )a = 8 ) 8

9 ) ) ) a 80 = 8 ) a = 0 ) ) ) = -) = ) = Puttig = i ) we get a = Substitutig = ad a = i ), we get - ) = 8 8 = X = = Sice a = a = =, a =, =. oside + 8) = ) = ) = ) =8 [- + 8) ] =8 K Whee K = ) is a itege is divisible by 8. i.e. 8 9 is divisible by 8. T + = Let coefficiets of T -, T, T + be i the atio : : coeff. oft coeff. oft = ad = coeff. oft coeff. oft = ad + = ad = + ad = = k k +) = 8 = + ad = Solvig these equatios we get =, =. = 9 Thee ae 0 tems i the epasio. T ad T ae the middle tems. k = k + 9

10 T + = 9 ) Puttig = we get, 9 T + = 9 T = 9. 8 ) T = 00 = Puttig = i ) we get, 9 T + = 9 T = 9. ) 0 T = )) 80 T = 80 ae the middle tems. 00 ad. [ ] = [ + + +)] = + ) + ) = [ ] [ + + ).+ ) ] coefficiet of = = ) ) = + + ) ) ) = ) ) = ) + + ) The coefficiet of 0 = 0 - = - = 00 = -. + ) + ) = [ + + ) +.+ ) ] [ ] oefficiet of = + = 0 + = = 0 8. T + = 0

11 Puttig = m, T m+ = m m Fo = m + T m+ = m+ m+ Give coefficiet of T m+ = coefficiet of T m+ m = m+ m = m + o m + m + = m = o m = 9. ostat tem = [- coefficiet of - i + [ coefficiet of - i is T + = - Geeal tem of T + = -) - If = - = 8 If = - = 9 coefficiet of - is 8 Ad coefficiet of - is - 9 fom ) ostat tem = [- 8 ] + [ - 9 )] = - ) + - ) = = -8 ) *******************************************