Pearson Edexcel A Level Mathematics Pure Mathematics Year 2 1st Edition Greg Attwood, Jack Barraclough, Ian Bettison, David Goldberg, Alistair Macpherson, Joe Petran - Solutions

The function t is defined by t: x ↦5 − 2xSolve the equation t2(x) − (t(x))2 = 0.
Find the values of the constants P, Q and R. -15x + 21 = (x-2)(x + 1)(x - 5) P x-2 + Q x + 1 + R X-5
A student attempts to solve the equation |3x + 4| = x. The student writes the following working:Explain the error made by the student. 3x + 4 = x 4 = 2x x = -2 or -(3x + 4) = x -3x 4 = x -4 = 4x x = -1 Solutions are x = -2 and x = -1. -
Show that q(x) can we written in the formfind the values of the constants A, B, C and D. = (x)b 8x³+1 4x² - 4x + 1
The functions p and q are defined bya. Find qp(x) and state its range.b. Find the value of qp(7).c. Solve qp(x) = −126. p: x ln(x + 3), x ER, x>-3 q: xe³x - 1, XER
Find the values of the constants A, B, C, D and E in the following identity: 3x4 4x³8x² + 16x − 2 = (Ax² + Bx + C)(x² − 3) + Dx + E -
Given thatfind the values of the constants A, B, C and D. 6x³ - 7x² + 3 3x² + 11x - 10 = Ax + B + C 3x - 5 + D x + 2
The function p is defined bya. Sketch y = p(x).b. Find the values of a, to 2 decimal places, such that p(a) = 50. p(x) = e-, -5≤ x < 0 \x³+4, 0≤x≤4
Given f(x) = e5x and g(x) = 4 ln x, find in its simplest form:a. gf(x)b. fg(x)
Use proof by contradiction to prove the statement: ‘There are no integer solutions to the equation x2 − y2 = 2'
Express the following as partial fractions: a 6x² + 7x - 3 X³ - X b 8x + 9 10x² + 3x - 4
Solve |3x − 5| = 11 − x.
The functions s and t are defined bys(x) = 2x, x ∈ ℝt(x) = x + 3, x ∈ ℝa. Find an expression for st(x).b. Find an expression for ts(x).c. Solve st(x) = ts(x), leaving your answer in the form Ina/Inb
By factorising the denominator, express the following as partial fractions: a 4x² + 17x11 x² + 3x - 4 b x4 - 4x³ + 9x² - 17x + 12 x³ - 4x² + 4x
a. Prove by contradiction that if n2 is a multiple of 3, n is a multiple of 3.b. Hence prove by contradiction that √3 is an irrational number.
Show that g(x) can be written in the formand find the values of p, q, r, s and t. g(x) = x4+3x²-4 x² + 1
The function s is defined bya. Sketch y = s(x).b. Find the value(s) of a such that s(a) = 43.c. Solve s(x) = x. s(x) = x²-6, x
Express the following as partial fractions: a 2x2 12x - 26 (x + 1)(x-2)(x + 5) b -10x² - 8x + 2 x(2x + 1)(3x - 2) с -5x² 19x - 32 (x + 1)(x + 2)(x - 5)
a. On the same diagram, sketch the graphs y = −2x and y =|1/2 x - 2|.b. Solve the equation - 2x = |1/2 x - 2|.
Expressas a single fraction in its simplest form. 6x + 1 5x+3 x-5 x²-3x - 10 +
Use proof by contradiction to show that there exist no integers a and b for which 21a + 14b = 1.
Find the values of the constants A, B and C. 10x² - 10x + 17 (2x + 1)(x - 3)2 = A 2x + 1 + B x-3 + с : (x — 3)2,.X > 3 -
a. Show thatb. Hence differentiate f(x) and find f'(4). f(x): = 2x² + 13x + 6 2
Given thatthe values of the constants A, B and C. 2x² - 1 x² + 2x + 1 = A + B C + x + 1 ' (x + 1)2
Given f(x)a. Prove thatb. Find an expression for f3 (x) . f(x): || 1 [ + x , X # -1
Show thatcan be written in the formwhere A, B and C are constants to be found. 39x2 + 2x + 59 (x + 5)(3x − 1)²
Prove that the sum of the first 50 natural numbers is 1275.
Show that the sum of the first 2n natural numbers is n(2n + 1).
Given that the geometric series 1 − 2x + 4x2 − 8x3 +… is convergent,a. Find the range of possible values of x.b. Find an expression for S∞ in terms of x.
Find in terms of k: a Σ4(-2)" r=| b Σ(100 – 21) rel c Σ(7 – 2) r=10
The first three terms of a geometric sequence are given by 8 − x, 2x, and x2 respectively where x > 0.a. Show that x3 − 4x2 = 0.b. Find the value of the 20th term.c. State, with a reason, whether 4096 is a term in the sequence.
By writing down the first four terms or otherwise, find the recurrence formula that defines the following sequences: = 2n1 a Un= due= n+1 2 bun= eun = n² = 3m + 2 CU = n + 2 f u = 3" − 1
For each series:i. Find the number of terms in the seriesii. Write the series using sigma notation. a 7+13 + 19 +...+ 157 b 1 3 + 2 15 + 4 75 + ⠀ + 64 46875 c 8-1-10-19... 127
Find how many terms of the following series are needed to make the given sums.a. 5 + 8 + 11 + 14 + … = 670b. 3 + 8 + 13 + 18 + … = 1575c. 64 + 62 + 60 + … = 0d. 34 + 30 + 26 + 22 + … = 112
Suggest possible recurrence relationships for the following sequences. (Remember to state the first term.)a. 3, 5, 7, 9, …b. 20, 17, 14, 11, …c. 1, 2, 4, 8, …d. 100, 25, 6.25, 1.5625, …e. 1, −1, 1, −1, 1, … f. 3, 7, 15, 31, …g. 0, 1, 2, 5, 26, …h. 26, 14, 8, 5, 3.5, …
For each series:i. Write out every term in the seriesii. Hence find the value of the sum. 5 a Σ(3 + 1) r=1 b Σ32 r=l Σsin(9070) r=1 2(3) 8 d Σ2 V=5
Express the following as partial fractions: a d f 6x-2 (x-2)(x + 3) 2x - 13 (2x + 1)(x-3) 7 - 3x x²-3x - 4 b e 2x + 11 (x + 1)(x + 4) 6x + 6 x²-9 8-x x² + 4x с h -7x - 12 2x(x-4) 2x - 14 x² + 2x - 15
Prove by contradiction that √1/2 is an irrational number.
Select the statement that is the negation of ‘All multiples of three are even’.A. All multiples of three are odd.B. At least one multiple of three is odd.C. No multiples of three are even.
For each of the following functions:i. Draw the mapping diagramii. State if the function is one-to-one or many-to-oneiii. Find the range of the function a f(x) = 5x3, domain {x = 3, 4, 5, 6} b g(x)=x²-3, ch(x) = domain {x = -3, -2, -1, 0, 1, 2, 3} domain {x= -1, 0, 1} 7 4 - 3x²
The function f(x) is defined by f : x ↦ x2 + 6x − 4, x ∈ ℝ, x > a, for some constant a.a. State the least value of a for which f−1 exists.b. Given that a = 0, find f−1, stating its domain.
The function n(x) is defined bya. Find n(−3) and n(3).b. Solve the equation n(x) = 50. n(x) = f5-x, x≤0 x², x>0
The diagram shows the graph of f(x). The points A(4, −3) and B(9, 3) are turning points on the graphSketch on separate diagrams, the graphs ofa. y = f(2x) + 1b. y = |f(x)|c. y = −f(x − 2)Indicate on each diagram the coordinates of any turning points on your sketch. y O y = f(x) A (4, -3) B
The function f is defined by f : x ↦ |2x − a|, {x ∈ ℝ}, where a is a positive constant.a. Sketch the graph of y = f(x), showing the coordinates of the points where the graph cuts the axes.b. On a separate diagram, sketch the graph of y = f(2x), showing the coordinates of the points where
The function p is defined by p: x ↦ −2|x + 4| + 10 The diagram shows a sketch of the graph.a. State the range of p.b. Give a reason why p −1 does not exist.c. Solve the inequality p(x) > −4.d. State the range of values of k for which the equation p(x) = -1/√x+ x + k has no solutions.
The function f has domain −5 x 7 and is linear from (−5, 6) to (−3, −2) and from (−3, −2) to (7, 18). The diagram shows a sketch of the function.a. Write down the range of f.b. Find ff(−3).c. Sketch the graph of y = |f(x)|, marking the points at which the graph meets or cuts the
The diagram shows part of the curve with equation y = f(x), where f(x) = x2 − 7x + 5 ln x + 8, x > 0. The points A and B are the stationary points of the curve.a. Using calculus and showing your working, find the coordinates of the points A and B.b. Sketch the curve with equation y = −3f(x
a. Sketch the graph of y = |2x + a|, a > 0, showing the coordinates of the points where the graph meets the coordinate axes.b. On the same axes, sketch the graph of y = 1/x.c. Explain how your graphs show that there is only one solution of the equation x|2x + a| − 1 = 0d. Find, using algebra,
a. Sketch the graph of y = |x − 2a|, where a is a positive constant. Show the coordinates of the points where the graph meets the axes.b. Using algebra solve, for x in terms of a, |x − 2a| = 1/3x.c. On a separate diagram, sketch the graph of y = a − |x − 2a|, where a is a positive constant.

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