OCR A Level Math Exam
Questions and Mark Scheme
,OCR A Level Math Exam
Questions and Mark Scheme
1. i. The first three terms of an arithmetic progression are 2x, x + 4 and 2x − 7 respectively. Find
the value of x.
[3]
ii. The first three terms of another sequence are also 2x, x + 4 and 2x − 7 respectively.
a. Verify that when x = 8 the terms form a geometric progression and find the sum to
infinity in this case.
[4]
b. Find the other possible value of x that also gives a geometric progression.
[4]
2. Sarah is carrying out a series of experiments which involve using increasing amounts of a
chemical. In the first experiment she uses 6 g of the chemical and in the second experiment she
uses 7.8 g of the chemical.
i. Given that the amounts of the chemical used form an arithmetic progression, find the total
amount of chemical used in the first 30 experiments.
[3]
ii. Instead it is given that the amounts of the chemical used form a geometric progression.
Sarah has a total of 1800 g of the chemical available. Show that N, the greatest number of
experiments possible, satisfies the inequality
1.3N ⩽ 91,
and use logarithms to calculate the value of N.
[6]
,3. a. The first term of a geometric progression is 50 and the common ratio is 0.8. Use logarithms
to find the smallest value of k such that the value of the kth term is less than 0.15.
[4]
b. In a different geometric progression, the second term is −3 and the sum to infinity is 4.
Show that there is only one possible value of the common ratio and hence find the first
term.
[8]
4. A geometric progression has first term 3 and second term − 6.
i. State the value of the common ratio.
[1]
ii. Find the value of the eleventh term.
[2]
iii. Find the sum of the first twenty terms.
[2]
5. An arithmetic progression u1, u2, u3, … is defined by u1 = 5 and un+1 = un + 1.5 for n ⩾ 1.
i. Given that uk = 140, find the value of k.
[3]
A geometric progression w1, w2, w3, … is defined by wn = 120 ×(0.9) n−1 for n ⩾ 1.
ii. Find the sum of the first 16 terms of this geometric progression, giving your answer
correct to 3 significant figures.
[2]
iii. Use an algebraic method to find the smallest value of N such that
[6]
, 6. Business A made a £5000 profit during its first year.
In each subsequent year, the profit increased by £1500 so that the profit was £6500 during
the second year, £8000 during the third year and so on.
Business B made a £5000 profit during its first year.
In each subsequent year, the profit was 90% of the previous year’s profit.
(a) Find an expression for the total profit made by business A during the first n years.
[2]
Give your answer in its simplest form.
(b) Find an expression for the total profit made by business B during the first n years.
[3]
Give your answer in its simplest form.
(c) Find how many years it will take for the total profit of business A to reach £385
000. [3]
(d) Comment on the profits made by each business in the long term. [2]
7. In this question you must show detailed reasoning.
It is given that the geometric series
is convergent.
(a) Find the set of possible values of x, giving your answer in set notation. [5]
(b) Given that the sum to infinity of the series is , find the value of x. [3]
8(i). The seventh term of a geometric progression is equal to twice the fifth term. The sum of
Questions and Mark Scheme
,OCR A Level Math Exam
Questions and Mark Scheme
1. i. The first three terms of an arithmetic progression are 2x, x + 4 and 2x − 7 respectively. Find
the value of x.
[3]
ii. The first three terms of another sequence are also 2x, x + 4 and 2x − 7 respectively.
a. Verify that when x = 8 the terms form a geometric progression and find the sum to
infinity in this case.
[4]
b. Find the other possible value of x that also gives a geometric progression.
[4]
2. Sarah is carrying out a series of experiments which involve using increasing amounts of a
chemical. In the first experiment she uses 6 g of the chemical and in the second experiment she
uses 7.8 g of the chemical.
i. Given that the amounts of the chemical used form an arithmetic progression, find the total
amount of chemical used in the first 30 experiments.
[3]
ii. Instead it is given that the amounts of the chemical used form a geometric progression.
Sarah has a total of 1800 g of the chemical available. Show that N, the greatest number of
experiments possible, satisfies the inequality
1.3N ⩽ 91,
and use logarithms to calculate the value of N.
[6]
,3. a. The first term of a geometric progression is 50 and the common ratio is 0.8. Use logarithms
to find the smallest value of k such that the value of the kth term is less than 0.15.
[4]
b. In a different geometric progression, the second term is −3 and the sum to infinity is 4.
Show that there is only one possible value of the common ratio and hence find the first
term.
[8]
4. A geometric progression has first term 3 and second term − 6.
i. State the value of the common ratio.
[1]
ii. Find the value of the eleventh term.
[2]
iii. Find the sum of the first twenty terms.
[2]
5. An arithmetic progression u1, u2, u3, … is defined by u1 = 5 and un+1 = un + 1.5 for n ⩾ 1.
i. Given that uk = 140, find the value of k.
[3]
A geometric progression w1, w2, w3, … is defined by wn = 120 ×(0.9) n−1 for n ⩾ 1.
ii. Find the sum of the first 16 terms of this geometric progression, giving your answer
correct to 3 significant figures.
[2]
iii. Use an algebraic method to find the smallest value of N such that
[6]
, 6. Business A made a £5000 profit during its first year.
In each subsequent year, the profit increased by £1500 so that the profit was £6500 during
the second year, £8000 during the third year and so on.
Business B made a £5000 profit during its first year.
In each subsequent year, the profit was 90% of the previous year’s profit.
(a) Find an expression for the total profit made by business A during the first n years.
[2]
Give your answer in its simplest form.
(b) Find an expression for the total profit made by business B during the first n years.
[3]
Give your answer in its simplest form.
(c) Find how many years it will take for the total profit of business A to reach £385
000. [3]
(d) Comment on the profits made by each business in the long term. [2]
7. In this question you must show detailed reasoning.
It is given that the geometric series
is convergent.
(a) Find the set of possible values of x, giving your answer in set notation. [5]
(b) Given that the sum to infinity of the series is , find the value of x. [3]
8(i). The seventh term of a geometric progression is equal to twice the fifth term. The sum of
