MULUNGUSHI UNIVERSITY
SCHOOL OF SCIENCE, ENGINEERING AND TECHNOLOGY
DEPARTMENT OF MATHEMATICS AND SCIENCES
MSM 111 - Mathematical Methods I
Tutorial Sheet 7 - 2020/2021 - Binomial Expansion
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1. Simplify
9
10
9
+
+
3
6
2
giving your answer as a single binomial coefficient of the form
n
.
r
2. Find the expansion of
(i) (3x − 4y)4
3. Expand (a + b)5 . If a =
4.
x−
3
4
1 6
x
and b = 41 , find the value (as a fraction) of the fourth term of the expansion.
(i) Write down the expansion of (2 + x)4 and (2 − x)4 .
(ii) Hence, simplify (2 + x)4 + (2 − x)4 . Use your results to find the exact value of (2 +
√
3)4 + (2 −
√
3)4
5. Determine the value of each of the following correct to five significant figures using the binomial theorem.
(i) (0.98)7
6.
(ii) (1.003)5
(iii) (2.01)9
13
(i) Determine the sixth term in the expansion of 3x + y3
(ii) Determine the middle term of (2x − 5z)8
(ii) Determine the term independent of x in the expansion of x −
1 15
2x2
20
(iv) Determine the term containing x−12 in the expansion of of 2x3 − x1
7
7. Evaluate the coefficients of x4 and x5 in the expansion of x3 − 3 . Hence, evaluate the coefficient of x5 in the
7
expansion of x3 − 3 (x + 6)
8. The coefficient of the second term in the expansion of (1 + 2x)n in ascending powers of x is 40. Find the value of n.
9. The first three terms in the expansion of (1 + bx)n are 1 + 6x + 16x2 . Find the values of n and b.
10. The first three terms in the expansion of (1 + x + bx2 )n are 1 + 7x + 14x2 . Find the values of n and b.
11. In the expansion of (1 + px)n in ascending powers of x, the second term is 18x and the third term is 135x2 . Find
the values of n and p.
12. Find the possible value(s) of x for which the following equation involving binomial coefficients holds:
8
x2
=
8
−4x − 4
13. Expand in ascending powers of x as far as the term in x3 , using the binomial theorem. State in each case the limits
of x for which the expansion is valid.
q
5
√
5
5
1
(ii) (2+x)
(iii) 2x + x2
(iv) 3 + 2x
(v) √5+x
(iii) 1−x
(i) (1+x)
2
3
1+x
1