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Binomial Theorem · Paper 1 — Free Maths AI SL Practice Questions

1FoundationSAQ-SBinomial expansion and coefficients7 marksPaper 1~11 min

A college basketball team plays

8

games in a season. The probability of winning any particular game is

0.25

, independently of other games.

(a)

State the value of the binomial coefficient

\dbinom{8}{3}

. [1 mark]

(b)

Find the probability that the team wins exactly

3

of the

8

games. Give your answer correct to 3 significant figures. [3 marks]

(c)

Given that the team wins at least

1

game, find the probability that the team wins exactly

3

games. Give your answer correct to 3 significant figures. Binomial probability:

P(X = r) = \dbinom{n}{r} p^{r}(1-p)^{n-r}

[3 marks]

Solutions

2FoundationSAQ-SBinomial expansion and coefficients5 marksPaper 1~8 min

Consider the expansion of

(1+x)^6

(a)

Calculate

\dbinom{6}{2}

. [1 mark]

(b)

Show that the sum of all binomial coefficients in the row

n = 6

of Pascal's triangle equals

64

. [2 marks]

(c)

Find the value of

x

for which the term in

x^2

equals the term in

x^3

in the expansion of

(1+x)^6

, given

x \neq 0

. [2 marks]

Solutions

3MasterySAQ-SBinomial expansion and coefficients5 marksPaper 1~8 min

The coefficient of

x^5

in the expansion of

(1 + ax)^8

1680

, where

a > 0

(a)

Show that

a^5 = 30

. [3 marks]

(b)

Hence find the value of

a

, giving your answer correct to 3 significant figures. [2 marks]

Solutions

4ChallengeSAQ-LBinomial expansion and coefficients8 marksPaper 1~12 min

A triangular number pattern is formed by writing the binomial coefficients in Pascal's triangle. The

2n

th row of Pascal's triangle contains the coefficients

\binom{2n}{0}, \binom{2n}{1}, \ldots, \binom{2n}{2n}

. Consider the expansion of

(1 + x)^{2n}

(a)

Write down the coefficient of

x^n

in the expansion of

(1 + x)^{2n}

in terms of

n

. [1 mark]

(b)

Show that the coefficient of

x^n

(1 + x)^{2n}

equals the middle term of the

2n

th row of Pascal's triangle. [1 mark]

(c)

Given that the coefficient of

x^n

(1 + x)^{2n}

252

, determine the value of

n

and hence find the sum of all coefficients in the expansion. [3 marks]

(d)

A student claims that for all positive integers

n

, the middle term of the

2n

th row of Pascal's triangle is always even. Determine whether this claim is correct, justifying your answer. [3 marks]

Solutions

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