1 (CIE 2012, s, paper 11, question 9)
Find the values of the positive constants p and q such that, in binomial expansion of (p+qx)10, the coefficient of x5 is 252 and the coefficient of x3 is 6 times the coefficient of x2.
[8]
2 (CIE 2012, s, paper 22, question 6)
(a) Find the coefficient of x3 in the expansion of
(i) (1−2x)7, [2]
(ii) (3+4x)(1−2x)7. [3]
(b) Find the term independent of x in the expansion of (x+3x2)6. [3]
3 (CIE 2012, w, paper 11, question 6)
(i) Find the first 3 terms, in descending powers of x, in the expansion of (x+2x2)6. [3]
(ii) Hence find the term independent of x in the expansion of (2−4x3)(x+2x2)6. [2]
4 (CIE 2012, w, paper 13, question 6)
In the expansion of (p+x)6, where p is a positive integer, the coefficient of x2 is equal to 1.5 times the coefficient of x3.
(i) Find the value of p. [4]
(ii) Use your value of p to find the term independent of x in the expansion of (p+x)6(1−1x)2. [3]
5 (CIE 2012, w, paper 22, question 4)
(i) Find the coefficient of x5 in the expansion of (2−x)8. [2]
(ii) Find the coefficient of x5 in the expansion of (l+2x)(2−x)8. [3]
6 (CIE 2013, s, paper 12, question 9)
(i) Given that n is a positive integer, find the first 3 terms in the expansion of (1+12x)n. in ascending powers of x. [2]
(ii) Given that the coefficient of x2 in the expansion of (1−x)(1+12x)n is 254, find the value of n. [5]
7 (CIE 2013, s, paper 21, question 7)
(i) Find the first four terms in the expansion of (2+x)6 in ascending powers of x. [3]
(ii) Hence find the coefficient of x3 in the expansion of (1+3x)(1−x)(2+x)6. [4]
8 (CIE 2013, w, paper 21, question 6)
(a) (i) Find the coefficient of x3 in the expansion of (1−2x)6. [2]
(ii) Find the coefficient of x3 in the expansion of (1+x2)(1−2x)6. [3]
(b) Expand (2√x+1√x)4 in a series of powers of x with integer coefficients. [3]
9 (CIE 2013, w, paper 23, question 6)
The expression 2x3+ax2+bx+21 has a factor x+3 and leaves a remainder of 65 when divided by x−2.
(i) Find the value of a and of b. [5]
(ii) Hence find the value of the remainder when the expression is divided by 2x+1. [2]
10 (CIE 2014, s, paper 13, question 5)
(i) The first three terms in the expansion of (2−5x)6, in ascending powers of x, are p+qx+rx2. Find the value of each of the integers p,q and r. [3]
(ii) In the expansion of (2−5x)6(a+bx)3, the constant term is equal to 512 and the coefficient of x is zero. Find the value of each of the constants a and b. [4]
11 (CIE 2014, s, paper 21, question 6)
(a) Find the coefficient of x5 in the expansion of (3−2x)8. [2]
(b) (i) Write down the first three terms in the expansion of (1+2x)6 in ascending powers of x. [2]
(ii) In the expansion of (1+ax)(1+2x)6, the coefficient of x2 is 1.5 times the coefficient of x. Find the value of the constant a. [4]
12 (CIE 2014, s, paper 22, question 5)
(i) Find and simplify the first three terms of the expansion, in ascending powers of x, of (1−4x)5. [2]
(ii) The first three terms in the expansion of (1−4x)5(1+ax+bx2) are 1−23x+222x2. Find the value of each of the constants a and b. [4]
13 (CIE 2014, w, paper 11, question 6)
(i) Given that the coefficient of x2 in the expansion of (2+px)6 is 60, find the value of the positive constant p. [3]
(ii) Using your value of p, find the coefficient of x2 in the expansion of (3−x)(2+px)6. [3]
14 (CIE 2014, w, paper 13, question 9)
(a) Given that the first 3 terms in the expansion of (5−qx)p are 625−1500x+rx2, find the value of each of the integers p,q and r. [5]
(b) Find the value of the term that is independent of x in the expansion of (2x+14x3)12. [3]
15 (CIE 2015, s, paper 11, question 3)
(i) Find the first 4 terms in the expansion of (2+x2)6 in ascending powers of x. [3]
(ii) Find the term independent of x in the expansion of (2+x2)6(1−3x2)2. [3]
16 (CIE 2015, s, paper 22, question 7)
In the expansion of (1+2x)n , the coefficient of x4 is ten times the coefficient of x2. Find the value of the positive integer, n. [6]
17 (CIE 2015, w, paper 13, question 8)
(a) Given that the first 4 terms in the expansion of (2+kx)8 are 256+256x+px2+qx3, find the value of k, of p and of q. [3]
(b) Find the term that is independent of x in the expansion of (x−2x2)9. [3]
18 (CIE 2015, w, paper 21, question 2)
(i) Find, in the simplest form, the first 3 terms of the expansion of (2−3x)6, in ascending powers of x. [3]
(ii) Find the coefficient of x2 in the expansion of (1+2x)(2−3x)6. [2]
19 (CIE 2016, march, paper 22, question 5)
(i) Find, in ascending powers of x, the first 3 terms of the expansion of (3+kx)7, where k is a constant. Give each term in its simplest form. [3]
(ii) Given that, in the expansion of (3+kx)7, the coefficient of x2 is twice the coefficient of x, find the value of k.
20 (CIE 2016, s, paper 12 , question 2)
(i) The first 3 terms in the expansion of (2−14x)5 are a+bx+cx2. Find the value of each of the [3] integers a,b and c.
(ii) Hence find the term independent of x in the expansion of (2−14x)5(3+4x). [2]
21 (CIE 2016, s, paper 21, question 8)
(a) (i) Use the Binomial Theorem to expand (a+b)4, giving each term in its simplest form. [2]
(ii) I Ience find the term independent of x in the expansion of (2x+15x)4. [2]
(b) The coefficient of x3 in the expansion of (1+x2)n equals 5h12. Find the value of the positive integer n.
22 (CIE 2016, w, paper 11, question 4)
(i) Find the first 3 terms in the expansion of (2x2−13x)5, in descending powers of x. [3]
(ii) Hence find the coefficient of x7 in the expansion of (3+1x3)(2x2−13x)5.
23 (CIE 2016,w, paper 13 , question 4)
(i) Find, in ascending powers of x, the first 3 terms in the expansion of (2−x4)6. [3]
(ii) Hence find the term independent of x in the expansion of (4+2x+3x2)(2−x4). [3]
24 (CIE 2017, march, paper 12, question 3) The first three terms in the expansion of (a+x4)5 are 32+bx+cx2. Find the value of each of the constants a,b and c.
25 (CIE 2017, s, paper 12, question 4) The first 3 terms in the expansion of (3−x6)n are 81+ax+bx2. Find the value of each of the constants n,a and b.
26 (CIE 2017, s, paper 21, question 5)
(i) Given that a is a constant, expand (2+ax)4, in ascending powers of x, simplifying each term of your expansion. [2]
Given also that the coefficient of x2 is equal to the coefficient of x3,
(ii) show that a=3, [1]
(iii) use your expansion to show that the value of 1.974 is 15.1 to 1 decimal place.
27 (CIE 2017, s, paper 23, question 6) The first three terms of the binomial expansion of (2−ax)n are 64−16hx+100hx2. Find the value of each of the integers n,a and b.
28 (CIE 2017, w, paper 12, question 3)
(i) Find, in ascending powers of x, the first 3 terms in the expansion of (2−x24)5 [3]
(ii) Hence find the term independent of x in the expansion of (2−x24)5(1x−3x2)2. [3]
29 (CIE 2017, w, paper 13, question 7)
(i) Find, in ascending powers of x, the first 3 terms in the expansion of (2−x24)6. Give each term in its simplest form. [3]
(ii) IIcnce find the cocfficient of x2 in the cxpansion of (2−x24)6(1x+x)2. [4]
30 (CIE 2017, w, paper 21, question 9)
(i) Expand (1+x)4, simplifying all cocfficicnts.
(ii) Expand (6−x)4, simplifying all cocfficients. [2]
(iii) Hence express (6−x)4−(1+x)4=175 in the form ax3+bx2+cx+d=0, where a,b,c and d are integers. [2]
(iv) Show that x=2 is a solution of the equation in part (iii) and show that this equation has no other real roots. [5]
31 (CIE 2018, march, paper 12, question 5) The first 3 terms in the expansion of (2+ax)n are equal to 1024−1280x+bx2, where n,a and b are constants.
(i) Find the value of each of n,a and b. [5]
(ii) Hence find the term independent of x in the expansion of (2+ax)n(x−1x)2. [3]
32 (CIE 2018, s, paper 11, question 9)
(i) Find the first 3 terns in the expansion of (2x−116x)8 in descending powers of x. [3]
(ii) Hence find the coefficient of x4 in the expansion of (2x−116x)8(1x2+1)2. [3]
33 (CIE 2018, s, paper 12, question 5)
(i) The first three terms in the expansion of (3−19x)9 can he written as a+bx+cx2, Find the value of each of the constants a,b and c. [3]
(ii) Use your values of a,b and c to find the term independent of x in the expansion of
(3−19x)9(2+9x)2. [3]
Answers
1. p=2/3;q=3/2
2. (a) −280;−504 (b) 135
3. (i) x6+12x3+60+⋯
(ii) 72
4. (i) p=2,(ii) −80
5. (i) −448 (ii) 1792
6. (i) 1+n(x/2)+n(n−1)2(x/2)2
(ii) n=10
7. (i) 64+192x+240x2+160x3
(ii) 64
8. (a)(i)−160 (ii) −130
(b) 16x2+32x+24+8x+1x2
9. (i) a=5;b=4(ii)20
10. (i) 64−960x+6000x2
(ii) a=2;b=10
11. (a) −48384
(b)(i) 1+12x+60x2
(ii) −4
12. (i) 1−20x+160x2
(ii) a=−3;b=2
13. (i) p=1/2 (ii) 84
14. (a) p=4;q=3;r=1350 (b) 1760
15. (i) 64+192x2+240x4+160x6, (ii) 1072
16. n=8
17. (a) k=1/4;p=112;q=28
(b) −672
18. (i) 64−576x+2160x2
(ii) 1008
19. (i) 2187+5103kx+5103k2x2
(ii) k=2
20. (i) 32−20/x+5/x2
(ii) 16
21. (ai) a4+4a3b+6a2b2+4ab3+b4
(ii) 24/25 (b) n=6
22. (i) 32x10−80x7/3+80x4/9
(ii) −48
23. (i) 64−48x+15x2
(ii) 205
24. a=2,b=20,c=5
25. a=−18,b=3/2
26. (i) 16+32ax+24a2x2+8a3x3
a4x4
27. n=6,a=5,b=60
28. (i) 32−20x2+5x4
(ii) 25
29. (i) 64−48x2+15x4
(ii) −17
30. (i) 1+4x+6x2+4x3+x4
(ii) 1296−864x+216x2−24x3+x4
(iii) 28x3−210x2+868x−1120=0
31. (i) n=10,a=−1/4,b=720
(ii) −1328
32. 256x8−64x6+7x4
135
33. (i) a=242,b=−45,c=10/3
(ii) −378
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