Complete Pure Mathematics 1 For Cambridge International AS & A Level 2nd Edition Jean Linsky, Brian Western, James Nicholson - Solutions

The first four terms of a geometric progression are 1, 3, 9, 27. Find the smallest number of terms that will give a total greater than 265000.
a) The 15th term of an arithmetic progression is 24. The sum of the first 15 terms is 570. Work out the first three terms.b)The sum of the first N terms of this arithmetic progression is 0.Work out the value of N.
Find the sum of all the positive integers from 1 to 300 that are not divisible by 3.
The sum of the 1st and 2nd term of a geometric progression is 1. The sum of the 4th and 5th terms is 27. Work out the common ratio, r, and the 1st term.
The 1st term of an arithmetic progression is 3x + 2. The last term is 13x + 8 and the common difference is 2. Find, in terms of x,a) The number of termsb) The sum of all the terms.
The 16th term of an arithmetic progression is 53 and the 20th term is 65. Find the sum of all the odd numbers in the sequence that are less than 100.
Each year the value of a company increases by 2%. At the start of 2012, the value of the company was $24 000. Estimate the value of the company at the end of 2025. Write your answer to the nearest $1000.
The first four terms of an arithmetic progression are 2, 6, 10, 14. Find the least number of terms needed so that the sum of the terms is greater than 2000.
The first three terms of a series are x − 3, x + 1 and 3x − 5.a) If the series is an arithmetic progression, find the first three terms.b) i) If the series is a geometric progression,find two possible values of x.ii) Explain why there is no sum toinfinity for either of these values of x.
Three consecutive terms of a geometric progression are x − 1, x² − 1 and x + 1.Find two possible values of x.
Find the sum of the integers between 1 and 500 that are divisible by 6.
The 7th term of an arithmetic progression is 20. The 1st, 3rd and 11th terms of the arithmetic progression are the first three terms of a geometric progression. Find the common ratio, r, of this geometric progression.
A geometric progression has first term a and common ratio r. The 1st, 2nd and 4th terms of the geometric progression form the first three terms of an arithmetic progression.a) Show that (r − 1)(r² + r − 1) = 0.b) Given that the sum to infinity of the geometric progression is finite, find the
In a geometric progression, the sum to infinity is five times the first term.a) Find the common ratio.b) The 2nd term is 12. Find the 1st term.c) Find the percentage difference betweenthe sum of the first ten terms and the sumto infinity, correct to 3 significant figures.
The first term of a geometric progression is 10 and the common ratio is −2.The first term of an arithmetic progression is10 and the common difference is −2.Find the difference between the sums of thefirst ten terms of the two sequences.
The 12th term of an arithmetic progression is equal to three times the 2nd term. The 17th term of the progression is 60. Find the sum of the first 50 terms.
The first three terms of a geometric are 3p, 5p + 5, 8p + 20 progression respectively.a) Show that p² − 10p + 25 = 0.b) Hence show that p = 5 and find the common ratio of this progression.c) Find the sum of the first 10 terms of theprogression.
a) Explain why g is a function and f is not a function.b) Determine whether the function g isone-to-one or many-to-one. f(x) = g(x) = [3x²+2 5x-1 [x³-1 2x when -3 < x < -1 when -1 < x < 4 when -3 < x < -1 when -1 < x < 4
The diagram shows a sketch of the curve f(x), which passes through the origin, O, and the points A(−2, 8) and B(1, −1). On separate diagrams, sketch the graphs ofa) y = 4f(x) b) y = f(−1/2x) c) y = −2f(x + 1)d) y =1+ f(−2x).Mark the new position of the points O, A and B on
The curve y = x² − x + 1 is translated byFind and simplify the equation of the translated curve. 0 3
Given that f(x) = 3√x, for x ∈ R, x ≥ 0 and g(x) = 2x² − 1 for x ∈ R,solve the equation gf(x) = 8.
Find the range of the function defined by h(x) = 5 − 2x − x², x ∈ Rby the method of completing the square.
The function fis defined by f: x ax² + b, where a and b are constants. Given that f(−2) = 2 and f(4) = 14, find thevalues of a and b.
The function f is defined by f: x ↦ x² + 2x − 1 for 0 ≤ x ≤ 2.a) Express f(x) in the form (x + a)² + b, where a and b are constants.b) State the range of f.c) Find an expression for f−1(x), stating its domain.d)Show that ff−1 = x.e)Sketch on the same diagram the graphs of y = f(x) and
Given that f(x) = √3x, g(x) = x − 5 and h(x) = x², finda) gh(x)b) fg(x)c) hg(−2).
Find the inverse for each of these functions. a) f: x ↦ 3 − x domain: x ∈ Rb) g: x ↦ 6(4x − 1) domain: x ∈ Rc) h: x ↦ 3x² + 2 domain: x ∈ R, x ≥ 0 d) e)f) Ex 2 x+5 domain: x € R, x = -5
a) Sketch the graph of y = f(x) where f(x) = 16 − x².b) On separate diagrams, sketch the graphs ofi) y = f(2x)ii) y = 4f(x) iii) y = f(−4x).Mark on each sketch the coordinates of the points wherethe curve cuts the axes.
Prove that h(x) = 1/x − 1 for x ≠ 1, x ∈ R. hh(x) = x-1 2-x
Explain why f is a function and g is not a function. x²-1 (2x-1 f(x) = { g(x) =< x²+2 2x - 1 when 0≤x≤2 when 2 < x < 4 when 0≤x≤ 2 when 2 < x < 4
Find the range of the function defined by f(x) = 2.4.² x³ + 1 when -3 ≤x≤0 when 0 < x < 2
The functions f and g are defined for x ∈ R by f: x ↦ 10x + x² g: x↦ 2x − 1. Express fg(x) in the form a(x + b)² + c, where a, b and care constants.
For each of the following functions statei) The greatest possible domain for which the function is definedii) The corresponding range of the functioniii) Whether the function is one-to-one or many-to-one.a) f: x ↦ 4x − 5b) g: x ↦√xc) h: x ↦ 1/x²
The functions f and g are defined for x ∈ R by f: x ↦ x² −1 g: x ↦ 3x + 4a) Find and simplify expressions for fg(x) and gf(x).b) Hence find the values of a for whichfg(a) = 4a + gf(a).
a) If f−¹(x) = −5, find the value of x.b) Show that ff−¹(x) = x.f(x) = x/ x +3 , x ∈ R, x ≠ −3.
Given that f(x) = x² + 1, g(x) = x√2 and fg(x) = 1.5, for x ∈ R, find the value of x.
The curve y = x² − 3 is translated byFind and simplify the equation of the translated curve. 0
The functions f and g are defined for x ∈ R byf: x ↦ 5 − 3xg: x ↦ (x − 1)² a) Find the set of values which satisfy gf(x) ≤ 25. b) Sketch on the same diagram the graphs of y = f(x) and y = f−1(x), showing the coordinates of their point of intersection and making clear the
Show that f−¹(−2) - f−¹(2)=43/48.f(x) = 2x − 5/7x + 4
Given that f(x) = gh(x), find the values of x.f(x) = 3 − x, g(x) = x² − 19 and h(x) = x − 2 for x ∈ R.
Find an expression for f−1(x), stating its domain.f(x) = 2 + 1/1 − x , x ∈ R, x ≠ 1.
The functions f and g are defined for x ∈ R by f: x ↦ 4x − 1 g: x ↦ 2x + k. Find the value of k for which fg = gf.
Find the domain. Leave your answer in surd form.f(x) = x² − 8x + 16, x ∈ R, f(x) ≥ 6.
The graph of y = f(x) is transformed to the graph of y = 4 + 3f(x − 2).Describe fully the three single transformations that have been combined to give the resultingtransformation.
The curve y = 3x² + 2x − 8 is reflected in the x-axis.State the equation of the reflected curve in the form y = ax² + bx + c, where a, b and care constants.
a) Find and simplify the equation of the curve 3 + f(x − 1).b) Describe the transformation.f(x) = x(x + 3)
Find f−1(x), stating its domain.f(x) = 1 + 2x/x − 1, x ∈ R, x ≠ 1.
Given that f(x) = 2x − 1, g(x) = x² + 1 and h(x) = 1 − x and x ∈ R,find a) fg(x) b) hf(x) c) hg(−1) d) gf(2).
The functions f and g are defined for x ∈ R byFind a) fg(x)b) gf(x) c) ff(−2).Give your answer as a single fraction in its simplest form. f: x- 1 x-3 g: x1+x.
a) Express f(x) in the form a(x + b)² + c.b) Find the range.f(x) = 3x² + 6x − 18, x ∈ R
a) Sketch the graph of the function defined byb) Find the range. f(x) = 5 - 2x x² +1 when x < 2 when x > 2
The functions f and g are defined by f: x4x − 2 g: x (x + 1)²a) Find gf(x).b) Solve the equation fg(x) = 14.
The diagram shows a sketch of the curve f(x). The curve has a horizontal asymptote with equation y = 1 and a vertical asymptote with equation x = 0. On separate diagrams, sketch the graphs ofa) y = f(2x)b) y = −3f(x)c) y = 2f(x − 3)d) y=4 −f(1/2x)e) y = 3 + 4f(x)f) y = 4f(−1/2x).Mark on
The points P(−3, 5) and Q(−2, −8) lie on the curve with equation y = f(x). Find the coordinates of P and Q after the curve has been transformed by the following transformations:a) f(−x) b) f(x) c) f(−x + 1) d) −f(x) −5
The graph of y = f(x) is transformed to the graph of y = −f(2x).Describe fully the two single transformations that have beencombined to give the resulting transformation.
The diagram shows a sketch of the curve f(x), which passes through the points A(0, 3) and B(2, −1).Sketch the graphs of a) y = −f(x) b) y = f(−x)c) y = −f(x + 4)d) y = 3−f(−x).In each case, mark the new position of the points A and B,writing down their coordinates. A(0,
The graph of y = f(x) is transformed to the graph of y = f(x + 5). Describe the transformation.
The function f is defined by f: x ↦1/x + 4, x ∈ R, x ≠ −4. Evaluate f−1(−3).
a) Sketch the graph of y = g(x), where g(x) = 2x.b) On separate diagrams, sketch the graphs ofi) y = g(x + 2) ii) y = g(x) − 1iii) y = g(x) + 4Mark on each sketch, where possible, the coordinates of the points where the curve cuts the axes and state the equations of any asymptotes.
Given that f(x) = 2x² + 1 and g(x) = 3 − x, and x ∈ R, find a) fg(−2) b) gf(x).
By sketching their graphs or otherwise, find the range of these functions given the domain. State whether each function is a one-to-one function or a many-to-one function.a) f: x ↦ 2x + 1, x ∈ Rb) g: x ↦ x² − 4, x ∈ R, −3 < x < 3c) h: x ↦ −x³, x ∈ R, −2 < x ≤ 1d) f:
Find the range of these functions. a) f(x) = x + 3, −2 < x ≤ 5b)c)d) g(x) = 7√x+1, 3 < x≤ 8
Find the real roots of the equation. 8 4 X 5 + x² = 3.
The graph of y = f(x) is transformed to the graph of y = f(−x) + 7.Describe the transformation.
a) Sketch the graph of y = f(x), where f(x) = (x + 1)(x − 3).b) On separate diagrams, sketch the graphs ofi) y = f(x) + 1ii) y = f(x + 3)iii) y = f(x − 1).Mark on each sketch, where possible, the coordinates of the pointswhere the curve cuts the axes.
Find an expression for f−1(x) for each of the following functions.a) f(x) = 2x + 3, x ∈ Rb) f(x) = x² − 2, x ∈ R x ≥ 0c) f(x) = 1 − 4x, x ∈ Rd) f(x) = x − 3/2, x ∈ Re) f(x) = 1/x, x ∈ R, x ≠ 0f) f(x) = 2(x − 1), x ∈ R
The functions f and g are defined for x ∈ R by f : x ↦ 5x − 1 g: x ↦ x².a) fg(x)b) ff(x) c) gf(-3)d) gg(1).
Find the range of these functions.a) f(x)=x − 2, x ≤ 3b) g(x) = 2x2, −3 ≤ x ≤ 1c) h(x) = x³ + 4, x > −2d) f(x) = 1/x, x ≥ 1e) g(x) = 7x + 1, −1 ≤ x ≤ 3f) h(x) = x4, x ∈ R
a) Sketch the graph of y = f(x), where f(x) = (x + 2)(x − 2).b) On separate diagrams, sketch the graphs of i) y = f(−x) ii) y = −f(x).Mark on each sketch, where possible, the coordinatesof the points where the curve cuts the axes.
The diagram shows an L-shape. All measurements are in centimeters.a) Write down two equations in x and y.b)Solve the equations simultaneously and hencefind the perimeter of the L-shape. 9 6 y ส 2x X
Find the set of values of k for which the line y + 4 = kx intersects the curve y = x² at two distinct points.
The diagram shows a trapezium. All measurements are in centimeters. The area of the trapezium is 60 cm². The perimeter of the trapezium is 36 cm. Find the value of x and the value of y. 2y 2x+3->> 5X+3 5x
The solid cuboid has a volume of 54cm³ and a total surface area of 96 cm² and x > y.a) Use this information to write down two equations in x and y.b) Solve the equations simultaneously to find the value of x andthe value of y. xcm y cm 2 cm
The equation x² + 8x − k(x + 8) = 0 has equalroots. Find the value of k.
The diagram shows a rectangle. The area of the rectangle is 3.5 m². The perimeter of the rectangle is 7.8 m. Find the dimensions of the rectangle. x metres y metres
A square garden of side x meters is surrounded by a path of width 1 meter. The area of the garden is the same as the area of the path. Find the value of x. Leave your answer in surd form. +1m: X 1m 1m-
Rectangle A has a height of (x + 1) meters and an area of 4 m². Rectangle B has a height of (2x − 1) metersand an area of 6 m².The sum of the widths of the two rectanglesis less than 8 meters.Find the set of values for x.
Show that there are no real solutions to the simultaneous equations y = 1 + 2x − x² and y = 1/2x + 5.
a) Express 3x² + 12x + 5 in the form p(x + q)² + r.b) Find the minimum value of 3x² + 12x + 5.
Solve these simultaneous equation.3x = 1 + 2y3x² − 2y²+ 5 = 0
Solve the simultaneous equations 3x + y = 8 and 3x² + y² = 28.
Solve these simultaneous equation.y = 2x + 1x² − 2xy + y² = 1
Solve the simultaneous equations x + y = 3 and x² + 2xy = 5.
Solve these simultaneous equations.y = 1 + 2xy² = 2x² + x
Solve the simultaneous equations x + y = 0 and 2x² + y² = 6.
Solve these simultaneous equations.3x + 4y = 152xy = 9
Find the maximum area of the triangle on the right. State the value of x when this occurs. (2x + 1) cm (4- x) cm
Solve the simultaneous equations x + y = 1 and x² − y² = 5.
Solve these simultaneous equations.x² + y² = 82x = y + 2
The equation x² + px + q = 0, where p and q are constants, has roots −1 and 4.a) Find the values of p and q.b)Using these values of p and q, find the valueof r, where r is a constant, and the equationx² + px + q + r = 0 has equal roots.
Solve these simultaneous equations.y = 1/2(1 − x2)y = x − 1
A rectangle has a width of x cm. The perimeter of the rectangle is 32 cm. Find the maximum area of the rectangle. xcm
The quadratic equation x² + kx + 36 = 0 has two different real roots. Find the set of possible values of k.
Faisal is x years old. Faisal has a brother called Omar. The sum of the two boys' ages is 20 years.a) Express the product of their ages in the form y = a(x − b)² + c.b)How old must Faisal be to make the product of theirages a maximum?
Solve these simultaneous equations.y = 6x + 5y =12x² − 5x
Work out whether each of these quadratic equations has two distinct roots, equal roots or no real roots.a) x² + x + 1 = 0b) 4x² + 12x + 9 = 0 c) 3x² + 4x + 1 = 0
Find the relationship between p and q, if the equation px² + 3qx + 9 = 0 has equal roots.
Solve these simultaneous equations.x − 2y = 8xy = 24
A right-angled triangle has a width of x cm. The length of the hypotenuse is 10 cm. The perimeter of the triangle is 24 cm. Find the maximum area of the triangle. x cm 10 cm
Solve x(x − 1)−3(x − 3) +3(x − 2) = x + 6.
The equation px² + qx + r = 0, where p, q and r are constants, has roots −1/2 and 3/4. Find the smallest possible integer values of p, q and r.

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Complete Pure Mathematics 1 For Cambridge International AS & A Level 2nd Edition Jean Linsky, Brian Western, James Nicholson - Solutions

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