Binominal Exercise | Answer Key

University:
Texas A&M University
Course:
MATH 102 | Algebra
Academic year:

2022
Views:

95

Pages:

1
Author:

opisivanetmi

BINOMIAL EXERCISE Using binomial theorem, expand each of the following : Q.1 (3x + 2y)4 Q.2 (2x  3y)4 11 Q.3  1 x  y  Q.5 Evaluate : ( Q.6 Evaluate : (2 + 3 ) + (2 – 3 ) Q.7 Evaluate : (1001) 5 Q.4 ( x  y )10 6 6 2 + 1) + ( 2 + 1) 7 Q.19 If the coefficients of 2nd, 3rd and 4th terms in the expansion of (1 + x)2n are in AP, show that 2n2  9n + 7 = 0. Q.20 In the expansion of (1 + x) n, the three successive coefficients are 462, 330 and 165 respectively. Find the values of n and r. Q.21 The first three terms in the binomial expansion of (x + y)n are 1, 56 and 1372 respectively. Find the values of x and y. The coefficients of three consecutive terms in the expansion of (1 + x)n are in the ratio 1 : 7 : 42. Find n. 7 Q.22 8  4x 5    . Q.8 (i) Find the 7th term in the expansion of   5 2x  12  a 2b  (ii) Find the 10th term in the expansion of   2  b a  Q.23 In the binomial expansion of (a + b)n, the coefficients of 4th and 13th terms are equal to each other. Find n. ( x  y )17 Q.24 In the expansion of (1 + x)m + n, where m and n are positive integers , prove that the coefficients of xm and xn are equal. (iii) Find the 16th term in the expansion of Q.9 Write the general term in the expansion of (x2y)6. Q.10 Q.11 Find the 5th term from the end in the expansion 12 1  of  x   . x  Find the coefficients of ANSWER KEY 1. 81 x + 216 x3y + 216 x2y2 + 96xy3 + 16y4 2. 16x4  96x3y + 216x2y3 – 216xy2 + 81y4 3.  11 11x10 55 x 2 165 x8 330 x7  x  y  y 2  y3  y 4  4 11  2 1 (i) x2 in the expansion of  x   ; x  6 (ii) x2 in the expansion of Q.12 1   3x   ; x   Find the middle term in the expansion of; 14  x2  (i) (3 + x) (ii) 1   2  Find the two middle terms in the expansion of 15 2  2 2 5 3x  (i) (x + a ) (ii)   x2   Show that the term containing x3 does not 6 Q.13 Q.14 8 Q.15 Q.16 1   exist in the expansion of  3 x   . 2x   Find the term independent of x in the expansion of; 10 10 1 1    (i)  x   (ii)  x  2  x 3x    Show that the ratio of the coefficient of x10 in (1  x2)10 and the term independent of x in 10 Q.17 2   x   is (1 : 32). x  Simplify the following expressions: 6 Q.18 6 1  1   x    x   x  x  Find the coefficient of x5 in the expansion of (1 + x)3 (1  x)6. 4. 3 2 462 x 6 462 x5 330 x 4 165 x 55 x 11x 1     10  11    y8 y9 y y  y5 y6 y7 5. x5 +10x9/2y1/2 +45x4y +120x7/2y3/2 + 210x3y2 + 252x5/2y5/2 + 210x2y3 + 120x3/2y7/2 + 45xy4 + 10x1/2y9/2 + y5 198 6. 9884 7. 1005010010005001 8. (i) 9. (–1)r 6Cr x12–2r yr 10. 495x4 11. (i) 462 (ii) 1215 12. 4375 (366080)b5 (ii) (iii) –136 xy15/2 x4 a14 (i) 540 x3 13. (i) 10x6a4 (ii) (ii) 429 14 x 16 6435  38  27 6435  37  28 and x6 x9   15. (i) 252 (ii) 5 17. 12  x 20. n = 11 and r = 7 22. 55 23. 15 4  1 10    x4 3  21. x = 1, y = 7 18. 6