IB DP Math MAA SL : IB Style Mock Exams – Set 10 Paper 1

Question

Consider the following sequence of figures.

Figure 1 contains 6 line segments.

(a) Given that Figure n contains 101 line segments, show that n = 20.

(b) Find the total number of line segments in the first 20 figures.

Answer/Explanation

Ans:

(a) We have an arithmetic sequence with u1 = 6, u2 = 11 and u3 = 16.

The common difference is

D = u2 – u1

    = 11 – 6

    = 5

Hence, if we substitute un = 101, u1 = 6 and d = 5 in un = u1 + (n -1)d and solve the resulting equation for n, we get

101 = 6 + (n – 1) (5)

101  = 5n + 1

100 = 5n

N = 20

(b) Using the sum of n terms formula \(S_{n}=\frac{n}{2}(u_{1}+u_{n})\) with n = 20, we obtain

\(S_{20}=\frac{20}{2}(u_{1}+u_{20})\)

\(S_=\frac{20}{2}(6+101)\)

= 10(107)

=1070

Question

A box contains 8 green and 4 blue marbles. Two marbles are selected at random (without replacement). Find the probability that selected marbles are:

(a) of the same colour;

(b) of different colours.

Answer/Explanation

Ans:

(a) We have

P(of the same colour) = P(both green) + P(both blue)

\( =\frac{8}{12}\times \frac{7}{11}+\frac{4}{12}\times \frac{3}{11}\)

\( =\frac{17}{33}\)

(b) Using the complementary events formula, we get

P(of different colours) = 1 – P(of the same colour)

\( =1-\frac{17}{33}\)                                 [by part (a)]

\( =\frac{16}{33}\)

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