Questions and Answers of Pearson Edexcel A Level Mathematics

Prospectors are drilling for oil. The cost of drilling to a depth of 50 m is £500. To drill a further 50 m costs £640 and, hence, the total cost of drilling to a depth of 100 m is £1140. Each
The first three terms of an arithmetic sequence are 5p, 20 and 3p, where p is a constant. Find the 20th term in the sequence.
Prove that the sum of the first n odd numbers is n2.
In a convergent geometric series the common ratio is r and the first term is 2. Given that S∞ = 16 × S3,a. Find the value of the common ratio, giving your answer to 4 significant figures.b. Find
Find the least value of n for which Σ(4r – 3) > 2000. - r=1
Each year, for 40 years, Anne will pay money into a savings scheme. In the first year she pays in £500. Her payments then increase by £50 each year, so that she pays in £550 in the second year,
A geometric sequence has first term 200 and a common ratio p where p > 0. The 6th term of the sequence is 40.a. Show that p satisfies the equation 5 log p + log 5 = 0.b. Hence or otherwise, find
Given thata. Show that (3k + 58)(k − 13) = 0b. Hence find the value of k. k Σ(8 + 3r) = 377, p=l
The first three terms in an arithmetic sequence are −8, k2, 17k … Find two possible values of k.
The fifth term of an arithmetic series is 33. The tenth term is 68. The sum of the first n terms is 2225.a. Show that 7n2 + 3n − 4450 = 0.b. Hence find the value of n.
The first term of a geometric series is 30. The sum to infinity of the series is 240.a. Show that the common ratio, r, is 7/8.b. Find to 3 significant figures, the difference between the 4th and 5th
A virus is spreading such that the number of people infected increases by 4% a day. Initially 100 people were diagnosed with the virus. How many days will it be before 1000 are infected?
A geometric sequence has first term 4 and fourth term 108. Find the smallest value of k for which the kth term in this sequence exceeds 500 000.
The sum of the first two terms of an arithmetic series is 47. The thirtieth term of this series is −62. Find:a. The first term of the series and the common difference.b. The sum of the first 60
An arithmetic series is given by (k + 1) + (2k + 3) + (3k + 5) + … + 303a. Find the number of terms in the series in terms of k.b. Show that the sum of the series is given by 152k + 46208/k + 2.c.
An arithmetic sequence has first term k2 and common difference k, where k > 0. The fifth term of the sequence is 41. Find the value of k, giving your answer in the form p + q √5, where p and q
A geometric series has first term a and common ratio r. The second term of the series is 11/8 and the sum to infinity of the series is 8.a. Show that 64r2 − 64r + 15 = 0.b. Find the two
I invest £A in the bank at a rate of interest of 3.5% per annum. How long will it be before I double my money?
a. Calculate the sum of all the multiples of 3 from 3 to 99 inclusive, 3 + 6 + 9 + …+ 99b. In the arithmetic series 4p + 8p + 12p +…+ 400 where p is a positive integer and a factor of 100,i.
a. Find the sum of the integers which are divisible by 3 and lie between 1 and 400. b. Hence, or otherwise, find the sum of the integers, from 1 to 400 inclusive, which are not divisible by 3.
The fish in a particular area of the North Sea are being reduced by 6% each year due to overfishing. How long will it be before the fish stocks are halved?
A geometric series is given by 1 + 3x + 9x2 +… The series is convergent.a. Write down the range of possible values of x. Given thatb. Calculate the value of x. Σ(x)"-1 = 2 2 = r=1
A polygon has 10 sides. The lengths of the sides, starting with the shortest, form an arithmetic series. The perimeter of the polygon is 675 cm and the length of the longest side is twice that of the
The man who invented the game of chess was asked to name his reward. He asked for 1 grain of corn to be placed on the first square of his chessboard, 2 on the second, 4 on the third and so on until
Prove that the sum of the first 2n multiples of 4 is 4n(2n + 1).
A ball is dropped from a height of 10 m. It bounces to a height of 7 m and continues to bounce. Subsequent heights to which it bounces follow a geometric sequence. Find out:a. How high it will bounce
A sequence of numbers is defined, for n ≽ 1, by the recurrence relation un+1 = kun – 4, where k is a constant. Given that u1 = 2:a. Find expressions, in terms of k, for u2 and u3.b. Given also
Richard is doing a sponsored cycle. He plans to cycle 1000 miles over a number of days. He plans to cycle 10 miles on day 1 and increase the distance by 10% a day.a. How long will it take Richard to
A sequence is defined by the recurrence relation an+ 1 1 an a₁ = p
The fifth term of an arithmetic series is 14 and the sum of the first three terms of the series is –3.a. Use algebra to show that the first term of the series is –6 and calculate the common
A sequence a1, a2, a3, … is defined byWhere k is an integer?a. Given that the sequence is increasing for the first 3 terms, show that k > p, where p is an integer to be found.b. Find a4 in terms
A savings scheme is offering a rate of interest o 3.5% per annum for the lifetime of the plan. Alan wants to save up £20 000. He works out that he can afford to save £500 every year, which he will
The fourth term of an arithmetic series is 3k, where k is a constant, and the sum of the first six terms of the series is 7k + 9.a. Show that the first term of the series is 9 – 8k.b. Find an
The first term of a geometric series is 130. The sum to infinity of the series is 650.a. Show that the common ratio, r, is 4/5.b. Find, to 2 decimal places, the difference between the 7th and 8th
The adult population of a town is 25 000 at the beginning of 2012. A model predicts that the adult population of the town will increase by 2% each year, forming a geometric sequence.a. Show that the
Kyle is making some patterns out of squares. He has made 3 rows so far.a. Find an expression, in terms of n, for the number of squares required to make a similar arrangement in the nth row.b. Kyle
A convergent geometric series has first term a and common ratio r. The second term of the series is −3 and the sum to infinity of the series is 6.75.a. Show that 27r2 − 27r − 12 = 0.b. Given
a. Fully factories the expression x4 − 1. b. Hence, or otherwise, write the algebraic fraction x4 - 1/x + 1 in the form (ax + b)(cx2 + dx + e) and find the values of a, b, c, d and e.
The function h has domain −10 ≤ x ≤ 6, and is linear from (−10, 14) to (−4, 2) and from (−4, 2) to (6, 27).a. Sketch y = h(x).b. Write down the range of h(x).c. Find the values of a, such
For each function:i. Sketch the graph of y = f(x)ii. State the range of f(x)iii. State whether f(x) is one-to-one or many to-one. a fix + 3x + 2 for the domain {x>0} b f(x)=x² + 5 for the domain {x
Sketch the graph of each of the following. In each case, write down the coordinates of any points at which the graph meets the coordinate axes. a y = |x-1| e y = 17 - xl g y = |x| by = 12x + 31 f
Given that f(x) can be expressed in the formfind the values of A, B and C. f(x) = 3x² + x + 1 x²(x + 1) , x = 0, x = -1 #
Show that g(x) can we written in the formand find the values of the constants A, B and C. g(x) = x² + 3x - 2 (x+1)(x-3)
Find the values of the constants A, B, C and D. x³ + 2x² + 3x - 4 x + 1 -= Ax² + Bx + C + D x + 1
Write as a single fraction. 1 1 a = + = b 3 2/5 1 c = + = 3 1 d + 4x 8.x 32 X-2 I 1 X f a 5b - 3 26
Find the values of the constants D, E and F such that g(x)= -x - 10x - 5 (x + 1) ( x 1) x * - - x = -1, x 1
Given the functions p(x) = 1 − 3x, q(x) = x/4 and r(x) = (x − 2)2, find:a. pq(−8)b. qr(5)c. rq(6)d. p2(−5)e. pqr(8)
For each of the following mappings:i. State whether the mapping is one-to-one, many-to-one or one-to-many.ii. State whether the mapping could represent a function. a y y O AX b C y. O y. O y 0 X
Given thatfind the values of the constants A, B and C. x² - 10 (x − 2)(x + 1) = A + B x = 2 - + C I + x
Write as a single fraction. a 3 X 2 x + 1 1 d (x + 2) - (x+3) 1 IC b e 2 TIN x-1 T 3x (x + 4)² 3 x + 2 1 x +4 f 4 2x + 1 + 5 2(x + 3) 2 x-1 + 4 3(x - 1)
Write down the negation of each statement.a. All rich people are happy.b. There are no prime numbers between 10 million and 11 million.c. If p and q are prime numbers then (pq + 1) is a prime
Given the functions f(x) = 4x + 1, g(x) = x2 − 4 and h(x) = 1/x, find expressions for the functions:a. fg(x)b. gf(x)c. gh(x)d. fh(x)e. f2(x)
Show that can be written in the form where A and B are constants to be found. -2x - 5 (4 + x)(2-x)
Prove that if q2 is an irrational number then q is an irrational number.
Calculate the value(s) of a, b, c and d given that: a p(a) = 16 where p: x + 3x - 2, x ER c r(c) = 34 where r: x + 2(2x) + 2, x ER b q(b) = 17 where q:xx²-3, xER d s(d) = 0 where s: x + x² + x - 6,
f(x) = |7 − 5x| + 3. Write down the values of:a. f(1)b. f(10)c. f(−6)
Find the values of the constants A, B, C and D in the following identity: x²-x²-x-3 x(x - 1) = Ax + B + C X + D x-1
Write as a single fraction. a d ܀ 2 2+2+1 2 3 y²-x² -X + +1 b 7 x²-4 + 3 +2 3 x² + 3x + 2 ܬ ܘܬ x2 1 + 4: + 4 f 2 x² + 6x + 9 +2 x²-x-12 3 x2 + 41 + 3 +1 x2 + 5 + 6
If n2 is odd then n is odd.a. Write down the negation of this statement.b. Prove the original statement contradiction.
The functions f and g are defined byf(x) = 3x − 2, x ∈ ℝg(x) = x2, x ∈ ℝa. Find an expression for fg(x).b. Solve fg(x) = gf(x).
For each function:i. Represent the function on a mapping diagram, writing down the elements in the rangeii. State whether the function is one-to-one or many-to-one. a f(x) = 2x + 1 for the domain {x
The expression A/(x - 4)(x + 8) can be written in partial fractions as 2/(x - 4) + B/(x + 8) where A and B are constants to be found. Find the values of the constants A and B.
g(x) = |x2 − 8x|. Write down the values of:a. g(4)b. g(–5)c. g(8)
Given thatfind the values of m, n and p. 2x² + 4x + 5 I - z.X = m + nx + p I - z.X
Prove the following statements by contradiction.a. There is no greatest even integer.b. If n3 is even then n is even.c. If pq is even then at least one of p and q is even.d. If p + q is odd then at
Show thatand find the values of the constants a and b, where a and b are integers. 2x² 11x 40 x² - 4x - 32 X x² + 8x + 16 6x²3x45 ÷ 8x² + 20x48 a 10x²45x +45 b
Show thatcan be expressed in the form where A, B, C and D are constants to be found. -3x³ 4x² + 19x + 8 x² + 2x - 3
The functions p and q are defined bya. Find an expression for qp(x) in the form ax + b/cx + d.b. Solve qp(x) = 16 1 x-2₁ XER, x #2 p(x) = - q(x) = 3x + 4, X ER x
a. Simplify fullyb. Given that Infind x in terms of e. 4x² - 8x x²-3x-4 x² + 6x + 5 2x² + 10x
Expressas a single fraction in its simplest form. 6x + 1 x² + 2x 15 4 x-3
Given that h(x) can be expressed in the formfind the values of A, B and C. h(x)= = 2x² 12x 26 (x + 1)(x − 2)(x + 5) ³ -x > 2 -
a. Simplify fullyb. Given that find x in terms of e. x² + 2x - 24 2x² + 10x X x² – 3x x² + 3x - 18
Find the values of the constants A, B, C and D in the following identity: 8x3 + 2x² + 5 = (Ax + B)(2x2 + 2) + Cx + D
a. Prove that if ab is an irrational number then at least one of a and b is an irrational number.b. Prove that if a + b is an irrational number then at least one of a and b is an irrational
Show that p(x) can be written in the formwhere A, B and C are constants to be found. p(x) = 4x² + 25 4x² - 25
Express each of the following as a fraction in its simplest form. a 3 X 2 1 + x + 1 + x + 2 b 4 3x - 2 x-2 1 2x + 1 С 3 x-1 + 2 x + 1 + 4 x-3
a. Show that g(x) can be written in the form ax2 + bx + c, where a, b and c are constants to be found. b. Hence differentiate g(x) and find g'(−2). g(x) = 4x3-9x² - 9x 32x + 24 x² –
a. On the same axes, sketch the graphs of y = g(x) and y = h(x).b. Hence solve the equation 3 g(x)=4-24 and and h(x) = 5
Given that, forwhere D, E and F are constants. Find the values of D, E and F. x < -1, -10x²8x + 2 x(2x + 1)(3x - 2) D = + X E 2x + 1 + F 3x-2
The following mappings f and g are defined on all the real numbers bya. Explain why f(x) is a function and g(x) is not.b. Sketch y = f(x).c. Find the values of:i. f(3)ii. f(10)d. Find solution of
Show thatcan be written in the formand find the values of the constants D and E. 16x 1 (3x + 2)(2x - 1)
a. On the same diagram, sketch the graphs of y = − |3x + 4| and y = 2x − 9.b. Solve the inequality − |3x + 4| < 2x − 9.
The function f is defined by f(x) = ax3 + bx − 5 where a and b are constants to be found. Given that f(1) = −4 and f(2) = 9, find the values of the constants a and b.
Find the values of the constants A, B and C. 7x² + 2x - 2 x2(x + 1) = A + В X2 + с x+1
The function g has domain −5 ≤ x ≤ 14 and is linear from (−5, −8) to (0, 12) and from (0, 12) to (14, 5). A sketch of the graph of y = g(x) is shown in the diagram.a. Write down the range
Show that h(x) can be written in the formwhere D, E and F are constants to be found. D + x + 5 E F + (3x - 1) (3x - 1)²
Solve the inequality |2x + 9| < 14 − x.
The equationhas exactly one solution.a. Find the value of k.b. State the solution to the equation. 16 - x| = √x + k 2
The function h is defined by h(x) = x2 − 6x + 20 and has domain x ≥ a. Given that h(x) is a one-to-one function find the smallest possible value of the constant a.
Show thatcan be put in the formFind the values of the constants A, B, C and D. 4x36x² + 8x - 5 2x + 1
Find the values of the constants A, B, C and D in the following identity: x³ − 6x² + 11x + 2 = (x − 2)(Ax² + Bx + C) + D
Find the values of the constants A, B, C, D and E. x4 x² - 2x + 1 D x-1 - Ax² + Bx + C +- E (x - 1)²
Show that where A, B, C and D are constants to be found. x4 +2 x² - 1 = Ax² + Bx + C + D x² - 1
Show that h(x) can be written in the formwhere A, B and C are constants to be found. A + B C + x + 3x - 1
Given that f(x) = 2x3 + 9x2 + 10x + 3a. Show that -3 is a root of f(x)b. Express 10/f(x) as partial fractions.
Show that f(x) can be written in the form A/x + B/x - 1 where A and B are constants to be found. f(x): = x-3 x(x - 1)
Prove by contradiction that 3√2 is irrational.
This student has attempted to use proof by contradiction to show that there is no least positive rational number:a. Identify the error in the student’s proof. b. Prove by contradiction that there
The function g is defined by g(x) = cx + d where c and d are constants to be found. Given g(3) = 10 and g(8) = 12 find the values of c and d.
Show that h(x) can be written asand find the values of A, B, C, D and E. h(x) = x4 + 2x² - 3x + 8 x² + x - 2

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