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cambridge international as & a level mathematics probability & statistics

Cambridge International AS And A Level Mathematics Pure Mathematics 1 Coursebook 1st Edition Sue Pemberton, Julian Gilbey - Solutions

The graph of y = ax2 + bx + c is translated by the vectorThe resulting graph is y = 2x2 − 11x + 10. Find the value of a, the value of b and the value of c. 2 -5
a. Find the least value of k for which f is one-one.b. Find an expression for f−1(x).f : x ↦ 2x2 + 12x − 14 for x ∈ ℝ, x ≥ k
Find the range for each of these functions.a. f(x) = x2 − 2 for x ∈ ℝb. f : x ↦ x2 + 3 for −2 ≤ x ≤ 5c. f(x) = 3 − 2x2 for x ≤ 2 d. f(x) = 7 − 3x2 for −1≤ x ≤ 2
Find the range for each of these functions.a. f(x) = (x − 2)2 + 5 for x ≥ 2b.c.d. f(x) = (2x - 1)² - 7 for x ≥ 2
The function f : ↦ x2 −4x + k is defined for the domain x ≥ p , where k and p are constants.i. Express f(x) in the form (x + a)2 + b + k, where a and b are constants.ii. State the range of f in terms of k.iii. State the smallest value of p for which f is one-one.iv. For the value of p found
Determine the sequence of transformations that maps y = f(x) to each of the following functions.a.b. y = −f(x) + 2c. = f(2x − 6) d. y = 2f(x) − 8 - y = f(x) + 3
The diagram shows the function f defined for −1≤ x ≤ 4 , wherei. State the range of f .ii. Copy the diagram and on your copy sketch the graph of y = f−1(x) .iii. Obtain expressions to define the function f−1, giving also the set of values for which each expression is valid. f(x)
a. Find the range of f.b. State, with a reason, whether f has an inverse.f : x ↦ x2 − 6x for x ∈ ℝ
Solve the equation gf(x) = 1. f: xH x+1 2 for x ER g: xH 2x + 3 x-1 for x E R, x = 1
Determine the sequence of transformations that maps:a. The curve y = x3 onto the curveb. The curve y = x3 onto the curvec. The curve y = 1/(x+5)³
Find an expression for ff(x), giving your answer as a single fraction in its simplest form. f(x) = x+1 2x + 5 for x = R, x > 0
The diagram shows the graph of y = f −1(x), wherea. Find an expression for f(x).b. State the domain of f. y= x 5x-1 X
a. State the smallest value of k for which f has an inverse.f(x) = 9 − (x − 3)2 for x ∈ ℝ, k ≤ x ≤ 7b. For this value of k:i. Find an expression for f−1(x)ii. State the domain and range of f−1.
Given that f(x) = √x, write down the equation of the image of f(x) after:a. Reflection in the x-axis, followed by translationfollowed by translationfollowed by a stretch parallel to the x-axis with stretch factor 2.b. Translationfollowed by a stretch parallel to the x-axis with stretch factor 2,
The function f is defined by f(x) = 4x2 − 24x + 11, for x ∈ ℝ.i. Express f(x) in the form a(x − b)2 + c and hence state the coordinates of the vertex of the graph of y = f(x). The function g is defined by g(x) = 4x2 − 24x + 11, for x ≤ 1.ii. State the range of g.iii. Find an expression
Given that gf(−1) = 2 and g−1(7) = 1, find the value of a and the value of b. f(x) = 3x + a for x ER g(x)=b-5x for x ER
Express each function in the form a( x + b)2 + c, where a, b and c are constants and, hence, state the range of each function.a. f(x) = x2 + 6x − 11 for x ∈ ℝ b. f(x) = 3x2 − 10x + 2 for x ∈ ℝ
a. Represent, on a graph, the function:b. Find the range of the function. f(x) = 3-x² for 0≤x≤2 3x7 for 2 ≤x≤4
Given that g(x) = x2 , write down the equation of the image of g(x) after:a. Translationfollowed by a reflection in the y-axis, followed by translationfollowed by a stretch parallel to the y-axis with stretch factor 3.b. A stretch parallel to the y-axis with stretch factor 3, followed by
The value of k is now given to be 7.i. Express 2x2 − 12x + 13 in the form a(x + b)2 + c, where a, b and c are constants.ii. The function f is defined by f(x) = 2x2 − 12x + 13, for x > k, where k is a constant. It is given that f is a one-one function. State the smallest possible value of
Express each function in the form a − b(x + c)2, where a, b and c are constants and, hence, state the range of each function.a. f(x) = 7 − 8x − x2 for x ∈ ℝ b. f(x) = 2 − 6x − 3x2 for x ∈ ℝ
Express each of the following as a composite function, using only f and/or g.f : x ↦ x2 for x ∈ ℝ g : x ↦ x + 1 for x ∈ ℝa. x ↦ (x + 1)2 b. x ↦ x2 + 1 c. x ↦ x + 2d. x ↦ x4 e. x ↦ x2 + 2x + 2 f. x ↦ x4 + 2x2 + 1
a. Find expressions for f−1(x) and g−1(x).b. Show that the equation f−1(x) = g−1(x) has two real roots. f(x)= 3x1 for x = R g(x) = 3 2x - 4 for xe R, x 2
Find two different ways of describing the combination of transformations that maps the graph of f(x) = √x onto the graphand sketch the graphs of y = f(x) and y = g(x). g(x)=√√-x-2
Find the values of k for which the equation fg(x) = x has two equal roots. f(x)= k 2x for x = R 2 g(x) = ² for x = R₁ x # 0 X
The function f is defined for p ≤ x ≤ q, where p and q are positive constants, by f : x ↦ x2 − 2x − 15.The range of f is given by c ≤ f(x) ≤ d, where c and d are constants.i. Express x2 − 2x − 15 in the form (x + a)2 + b.ii. State the smallest possible value of c. For the case
Show that the equation gf(x) = 0 has no real solutions.f(x) = x2 − 3x for x ∈ ℝ g(x) = 2x + 5 for x ∈ ℝ
The function f : x ↦ x2 + 6x + k, where k is a constant, is defined for x ∈ ℝ. Find the range of f in terms of k.
Functions f and g are defined for x ∈ ℝ byi. Find and simplify expressions for fg(x) and gf(x).ii. Hence find the value of a for which fg(a) = gf(a).iii. Find the value of b (b ≠ a) for which g(b) = b.iv. Find and simplify an expression for f−1g(x). The function h is defined by h : x ↦ x2
The function f is defined by f : x ↦ 2x2 −12x + 7 for x ∈ ℝ i. Express f(x) in the form a(x − b)2 − c.ii. State the range of f.iii. Find the set of values of x for which f(x) < 21. The function g is defined by g : x ↦ 2x + k for x ∈ ℝ.iv. Find the value of the constant k
a. Find an expression for f−1(x).b. Find the domain of f−1.f : x ↦ (2x − 1)3 − 3 for x ∈ ℝ, 1 ≤ x ≤ 3
a. Find an expression for f−1(x).b. Show that if f(x) = f−1(x), then x2 − x − 1 = 0.c. Find the values of x for which f(x) = f−1(x).Give your answer in surd form. f(x) = x-1 for xe R, x≠ 1
Find two different ways of describing the sequence of transformations that maps the graph of y = f(x) onto the graph of y = f(2x + 10).
The function g : x ↦ 5 − ax − 2x2, where a is a constant, is defined for x ∈ ℝ. Find the range of g in terms of a.
Functions f and g are defined byi. Express f(x) in the form a(x + b)2 + c, where a, b and c are constants.ii. State the range of f.iii. State the domain of f−1.iv. Sketch on the same diagram the graphs of y = f(x), y = g(x) and y = f−1(x), making clear the relationship between the graphs.v.
Show that ff(x) = x. f(x) = x + 5 2x - 1 x # 1/1/12 for xe R, x #
Find the values of k for which the equation gf(x) = k has real solutions.f(x) = x2 − 3x for x ∈ ℝ g(x) = 2x − 5 for x ∈ ℝ
a. Express f(x) in the form (x − a)2 − b.b. Find an expression for f−1(x) and state the domain of f−1.f : x ↦ x2 − 10x for x ∈ ℝ, x ≥ 5
If the range of the function f is −4 ≤ f(x) ≤ 5, find the value of a.f(x) = x2 − 2x − 3 for x ∈ ℝ, −a ≤ x ≤ a
Determine which of the following functions are self-inverse functions.a.b.c. f(x) = 1 3-x for x ER, x 3
If the range of the function f is −2 ≤ f(x) ≤ 16, find the possible values of a.f(x) = x2 + x − 4 for x ∈ ℝ, a ≤ x ≤ a + 3
a. Express 2x2 + 4x − 8 in the form a(x + b)2 + c.b. Find the least value of k for which the function is one-one.f(x) = 2x2 + 4x − 8 for x ∈ ℝ, x ≥ k
a. Express f(x) in the form a(x + b)2 + c.f(x) = 2x2 − 8x + 5 for x ∈ ℝ, 0 ≤ x ≤ kb. State the value of k for which the graph of y = f(x) has a line of symmetry.c. For your value of k from part b, find the range of f .
Find the set of values of m for which the line y = mx + 1 intersects the circle (x − 7)2 + (y − 5)2 = 20 at two distinct points.
The coordinates of three of the vertices of a trapezium, ABCD, are A(3, 5), B(−5, 4) and C(1, −5).AD is parallel to BC and angle ADC is 90°.Find the coordinates of D.
a. Find an expression for (fg)−1(x).b. Find expressions for:i. f−1 g−1(x) ii. g−1 f−1(x).c. Comment on your results in part b.Investigate if this is true for other functions. f:x3x-5 for x = R g: x4-2x for x ER
Find the largest possible domain for each function and state the corresponding range.a. f(x) = 3x − 1 b. f(x) = x2 + 2 c. f(x) = 2xd.e.f. f(x) =
a. Find ff(x) and state the domain of this function.b. Show that if f(x) = ff(x) then x2 + x − 2 = 0.c. Find the values of x for which f(x) = ff(x). 2 x + 1 for x ER, x # -1
Functions P, Q, R and S are composed in some way to make a new function, f(x).For each of the following, write f(x) in terms of the functions P, Q, R and/or S, and state the domain and range for each composite function.a. f(x) = x2 + 4x + 3b. f(x) = 2x + 1 c. f(x) = xd.e.f.g. f(x) = x
a. Find the set of values of x for which f(x) ≥ 7.f(x) = x2 − 2x + 4 for x ∈ ℝb. Express x2 − 2x + 4 in the form (x − a)2 + b.c. Write down the range of f .
A line has equation 2x + y = 20 and a curve has equationwhere a is a constant.Find the set of values of a for which the line does not intersect the curve. 18 x-3³ y = a +-
a. Find fg(x).b. Find the range of the function fg(x).f(x) = x2 − 5x for x ∈ ℝg(x) = 2x + 3 for x ∈ ℝ
The diagram shows the curve y = 7√x and the line y = 6x + k, where k is a constant.The curve and the line intersect at the points A and B.i. For the case where k = 2, find the x-coordinates of A and B.ii. Find the value of k for which y = 6x + k is a tangent to the curve y = 7√x. y B y = 6x +
Solve the equation 4 X 17 +1 +18= =-. X²
Find the points of intersection of the line y = x − 3 and the circle (x − 3)2 + (y + 2)2 = 20.
The diagram shows the graph of y = f(x) for −4 ≤ x ≤ 4.Sketch on separate diagrams, showing the coordinates of any turning points, the graphs of:a. y = f(x) + 5b. y = −2f(x) 4 (-2,21) y 0 B (2,-11)
Calculate the lengths of the sides of the triangle PQR. Use your answers to determine whether or not the triangle is right angled.a. P(−4, 6), Q(6, 1), R(2, 9)b. P(−5, 2), Q(9, 3), R(−2, 8)
Find the equation of the line with:a. Gradient 2 passing through the point (4, 9)b. Gradient −3 passing through the point (1, −4)c. Gradient −2/3 passing through the point (−4, 3).
Find the centre and the radius of each of the following circles.a. x2 + y2 = 16 b. 2x2 + 2y2 = 9c. x2 + (y − 2)2 = 25 d. (x − 5)2 + (y + 3)2 = 4e. (x + 7)2 + y2 = 18 f. 2(x − 3)2 + 2(y + 4)2 = 45g. x2 + y2 − 8x + 20y + 110 = 0 h. 2x2 + 2y2 − 14x − 10y − 163 = 0
Find the equation of each of the following circles.a. Centre (0, 0), radius 8 b. Centre (5, −2), radius 4c. Centre − ( 1, 3), radius √7d. centre G 3 2 radius 5 2
The coordinates of three points are A(−6, 4), B(4, 6) and C(10, 7).a. Find the gradient of AB and the gradient of BC.b. Use your answer to part a to decide whether or not the points A, B and C are collinear.
The line 2x − y + 3 = 0 intersects the circle x2 + y2 − 4x + 6y − 12 = 0 at two points, D and E.Find the length of DE.
The graph of f(x) = ax + b is reflected in the y-axis and then translated by the vectorThe resulting function is g(x) = 1 − 5x. Find the value of a and the value of b. 3
P(1, 6), Q(−2, 1) and R(3, −2). Show that triangle PQR is a right-angled isosceles triangle and calculate the area of the triangle.
Find the equation of the line passing through each pair of points.a. (1, 0 ) and (5, 6)b. (3, −5) and (−2, 4)c. (3, −1) and (−3, −5)
The midpoint of the line segment joining P(−4, 5) and Q(6, 1) is M. The point R has coordinates (−3, −7). Show that RM is perpendicular to PQ.
Show that the line 3x + y = 6 is a tangent to the circle x2 + y2 + 4x + 16y + 28 = 0.
A is the point (a, 3) and B is the point (4, b).The length of the line segment AB is 4√5 units and the gradient is −1/2.Find the possible values of a and b.
The distance between two points, P(a, −1) and Q(−5, a), is 4 √5. Find the two possible values of a.
The curveand the line 3x − 4y + 3 = 0 intersect at the points P and Q.Find the length of PQ. y = 3√x-2
Find the equation of the line:a. Parallel to the line y = 3x − 5, passing through the point (1, 7)b. Parallel to the line x + 2y = 6, passing through the point (4, −6)c. Perpendicular to the line y = 2x − 3, passing through the point (6, 1)d. Perpendicular to the line 2x − 3y = 12, passing
Find the equation of the circle with centre (2, 5) passing through the point (6, 8).
Two vertices of a rectangle, ABCD, are A(−6, −4) and B(4, −8). Find the gradient of CD and the gradient of BC.
Find the equation of the perpendicular bisector of the line segment joining the points:a. (5, 2) and (−3, 6)b. (−2, −5) and (8, 1)c. (−2, −7) and (5, −4).
The distance between two points, P(−3, −2) and Q(b, 2b), is 10. Find the two possible values of b.
The graph of y = (x + 1)2 is transformed by the composition of two transformations to the graph of y = 2(x − 4)2. Find these two transformations.
The graph of y = x2 + 1 is transformed by applying a reflection in the x-axis followed by a translation of Find the equation of the resulting graph in the form y = ax2 + bx + c. 3 7
A diameter of a circle has its end points at A(−6, 8) and B(2, −4).Find the equation of the circle.
The line 2y − x = 12 intersects the circle x2 + y2 − 10x − 12y + 36 = 0 at the points A and B.a. Find the coordinates of the points A and B.b. Find the equation of the perpendicular bisector of AB.c. The perpendicular bisector of AB intersects the circle at the points P and Q.Find the exact
The coordinates of three points are A(5, 8), B(k, 5) and C(−k, 4).Find the value of k if A, B and C are collinear.
The line ax − 2y = 30 passes through the points A(10, 10) and B(b, 10b), where a and b are constants.a. Find the values of a and b.b. Find the coordinates of the midpoint of AB.c. Find the equation of the perpendicular bisector of the line AB.
The line l1 passes through the points P(−10, 1) and Q(2, 10). The line l2 is parallel to l1 and passes through the point (4, −1). The point R lies on l2, such that QR is perpendicular to l2. Find the coordinates of R.
The point (−2, −3) is the midpoint of the line segment joining P(−6, −5) and Q(a, b). Find the value of a and the value of b.
The diagram shows the graph of y = f(x) for −3 ≤ x ≤ 3.Sketch the graph of y = 2 − f(x). wy 2- 1 1- -2- 2 y = f(x) W
Sketch the circle (x − 3)2 + (y + 2)2 = 9.
Show that the circles x2 + y2 = 25 and x2 + y2 − 24x − 18y + 125 = 0 touch each other.Find the coordinates of the point where they touch.
The vertices of triangle ABC are A(−9, 2k − 8), B(6, k) and C(k, 12).Find the two possible values of k if angle ABC is 90°.
The line with gradient −2 passing through the point P(3t, 2t) intersects the x-axis at A and the y-axis at B.i. Find the area of triangle AOB in terms of t.The line through P perpendicular to AB intersects the x-axis at C.ii. Show that the mid-point of PC lies on the line y = x.
P is the point (−4, 2) and Q is the point (5, −4). A line, l , is drawn through P and perpendicular to PQ to meet the y-axis at the point R.a. Find the equation of the line l .b. Find the coordinates of the point R.c. Find the area of triangle PQR.
Three of the vertices of a parallelogram, ABCD, are A(−7, 3), B(−3, −11) and C(3, −5).a. Find the midpoint of AC.b. Find the coordinates of D.c. Find the length of the diagonals AC and BD.
Find the equation of the circle that touches the x-axis and whose centre is (6, −5).
Two circles have the following properties:● The x-axis is a common tangent to the circles● The point (8, 2) lies on both circles● The centre of each circle lies on the line x + 2y = 22.a. Find the equation of each circle.b. Prove that the line 4x + 3y = 88 is a common tangent to these circles.
A is the point (0, 8) and B is the point (8, 6).Find the point C on the y-axis such that angle ABC is 90°.
The curveand the line x − 2y + 6 = 0 intersect at the points A and B.a. Find the coordinates of these two points.b. Find the perpendicular bisector of the line AB. y=x+2. 4 X
The point P is the reflection of the point (−7, 5) in the line 5x − 3y = 18.Find the coordinates of P. You must show all your working.
The linewhere a and b are positive constants, meets the x-axis at P and the y-axis at Q.The gradient of the line PQ is 2/5 and the length of the line PQ is 2√29.Find the value of a and the value of b. x a y b ||. 1.
The line l1 has equation 3x − 2y = 12 and the line l2 has equation y = 15 − 2x.The lines l1 and l2 intersect at the point A.a. Find the coordinates of A.b. Find the equation of the line through A that is perpendicular to the line l1.
The point P(k, 2k) is equidistant from A(8, 11) and B(1, 12). Find the value of k.
The function f is such that f(x) = x2 − 5x + 5 for x ∈ ℝ.a. Find the set of values of x for which f(x) ≤ x. b. The line y = mx − 11 is a tangent to the curve y = f(x).Find the two possible values of m.
The points P(1, −2) and Q(7, 1) lie on the circumference of a circle.Show that the centre of the circle lies on the line 4x + 2y = 15.

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