06 - Binomial Theorem
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St Joseph’s Institution Secondary Four Mathematics REVISION TOPIC − BINOMIAL THEOREM Name:_____________________________________ (
) Class: ___________
8
Q1)
x Expand 2 − in ascending powers of x up to and including the term in 4 8 3 x . Hence, evaluate (1.995 ) correct to 4 decimal places. 10
Q2)
a The term independent of x in the binomial expansion of x 3 + 2 is 210. x Find the value of a. 9
Q3)
1 Find the fifth and sixth terms of the binomial expansion of 1 − x 2 . 3 9
Hence, find the coefficient of x
Q4)
10
(
)
1 in 1 − x 2 3 x 2 + 1 . 3
Given that p, in terms of a, is the coefficient of
1 in the expansion of x2
4
1 a − and q, in terms of a, is the coefficient of x in the expansion of x 6
x 2 + and if q = 4 p , find the value of a. a 6
Q5)
1 Find the first three terms in the expansion of 1 − x in ascending 3 powers of x, simplifying the coefficient. Given further that the first three 6 1 terms of the expansion of ( a + bx ) 1 − x are 2 − x + cx 2 , state the value 3 of a and hence find the value of b and of c.
© Jason Ingham 2009
1
n
Q6)
b Write down the forth term in the expansion of ax − . x (i) If this term is independent of x, find the value of n. (ii)
Q7)
With this value of n and given further that the fourth term is − 4320 and b − a = 1 , where a and b are both positive, find the value of a and of b.
n The first four terms in the expansion of (1 + px ) , where n > 0 are
1 + qx + 66 p 2 x 2 + 5940 x 3 . Calculate the value of n, of p and of q.
Q8)
Find, in ascending powers of x, the first three terms in the expansion of (1 − px ) 5 . Given that the first two non-zero terms in the expansion of
(1 + qx )(1 − px ) 5
are r and − 135x 2 , state the value of r and find the possible values of p and q. 10
Q9)
Q10)
(a)
x7 1 Find the coefficient of 3 in the expansion x − . y 4 y
(
)
8
(b)
Given that the first four terms in the expansion of 1 − x + ax 2 1 − 8 x + px 2 − qx 3 , express p in terms of q.
(a)
In the expansion of ( 2 + 5 x ) , the coefficient of x 3 and x 4 are in the ratio 8 : 15. Find the value of n.
(b)
is
n
1 Find the term independent of x in the expansion of 9 x − 3 x .
2
18
) Class: ___________
8
Q1)
x Expand 2 − in ascending powers of x up to and including the term in 4 8 3 x . Hence, evaluate (1.995 ) correct to 4 decimal places. 10
Q2)
a The term independent of x in the binomial expansion of x 3 + 2 is 210. x Find the value of a. 9
Q3)
1 Find the fifth and sixth terms of the binomial expansion of 1 − x 2 . 3 9
Hence, find the coefficient of x
Q4)
10
(
)
1 in 1 − x 2 3 x 2 + 1 . 3
Given that p, in terms of a, is the coefficient of
1 in the expansion of x2
4
1 a − and q, in terms of a, is the coefficient of x in the expansion of x 6
x 2 + and if q = 4 p , find the value of a. a 6
Q5)
1 Find the first three terms in the expansion of 1 − x in ascending 3 powers of x, simplifying the coefficient. Given further that the first three 6 1 terms of the expansion of ( a + bx ) 1 − x are 2 − x + cx 2 , state the value 3 of a and hence find the value of b and of c.
© Jason Ingham 2009
1
n
Q6)
b Write down the forth term in the expansion of ax − . x (i) If this term is independent of x, find the value of n. (ii)
Q7)
With this value of n and given further that the fourth term is − 4320 and b − a = 1 , where a and b are both positive, find the value of a and of b.
n The first four terms in the expansion of (1 + px ) , where n > 0 are
1 + qx + 66 p 2 x 2 + 5940 x 3 . Calculate the value of n, of p and of q.
Q8)
Find, in ascending powers of x, the first three terms in the expansion of (1 − px ) 5 . Given that the first two non-zero terms in the expansion of
(1 + qx )(1 − px ) 5
are r and − 135x 2 , state the value of r and find the possible values of p and q. 10
Q9)
Q10)
(a)
x7 1 Find the coefficient of 3 in the expansion x − . y 4 y
(
)
8
(b)
Given that the first four terms in the expansion of 1 − x + ax 2 1 − 8 x + px 2 − qx 3 , express p in terms of q.
(a)
In the expansion of ( 2 + 5 x ) , the coefficient of x 3 and x 4 are in the ratio 8 : 15. Find the value of n.
(b)
is
n
1 Find the term independent of x in the expansion of 9 x − 3 x .
2
18
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