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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 function f is such that f´(x) = 12x3 + 10x and f (−1) = 1.Find f(x).
Differentiate with respect to x:a. (x + 4)6 b. (2x + 3)8 c. (3 − 4x)5d.e.f. 5(2x − 1)5 g. 2(4 − 7x)4h.i. (x2 + 3)5 j. (2 − x2 )8 k. (x2 + 4x)3l. 2 [+x
Find the equation of the tangent to each curve at the given point.a. y = x2 − 3x + 2 at the point (3, 2)b. y = (2x − 5)4 at the point (2, 1)c. d. y = x3-5 x at the point (-1, 6)
Find the gradient of the curveat the point where x = 2. y = 8 4x - 5
Find f´´(x) for each of the following functions.a.b.c.d.e.f. f(x) = = 5 X 1/w 2x5
Differentiate with respect to x:a. x5 b. x9 c. x−4 d. 1/xe. 8f.g. x3 × x2h. x5/x2 2
Find the equation of the normal to each curve at the given point.a. y = 3x3 + x2 − 4x + 1 at the point (0, 1)b.c. y = (5 − 2x)3 at the point (3, −1)d. 3 √√x+1 at the point (-2, -3)
Given that y = 4√x, show that 4x2 d² y dx² dy - 4x = y. dx
Given that y = 4x − (2x − 1)4, find dy dx and d² y d.x2
Differentiate with respect to x:a.b.c. d.e. f. g. h. √x-5
The normal to the curveat the point (3, 6) meets the x-axis at P and the y-axis at Q.Find the midpoint of PQ. y = || 6 √x - 2
A curve passes through the point A(4, 2) and has equation a. Find the equation of the tangent to the curve at the point A. [5]b. Find the equation of the normal to the curve at the point A. y 2 (x-3)²
Find the gradient of the curve at the point where the curve crosses the x-axis. y 5x − 10 x²
A curve passes through the point P(5, 1) and has equationa. Show that the equation of the normal to the curve at the point P is 5x + 2y = 27.The normal meets the curve again at the point Q.b. i. Find the coordinates of Q.ii. Find the midpoint of PQ. y = 3 - 10 ४ ४
Find the coordinates of the points on the curve y = x3 − 3x − 8 where the gradient is 9.
The diagram shows part of the curvewhich crosses the x-axis at A and the y-axis at B.The normal to the curve at A crosses the y-axis at C.i. Show that the equation of the line AC is 9x + 4y = 27.ii. Find the length of BC. y = 2- 18 2x + 3
A curve has equation and passes through the points A(1, −1) and B(4, 11). At each of the points C and D on the curve, the tangent is parallel to AB. Find the equation of the perpendicular bisector of CD. y= y = 3x 4 X
A curve has equation y = x5 − 8x3 + 16x. The normal at the point P(1, 9) and the tangent at the point Q(−1, −9) intersect at the point R.Find the coordinates of R.
A curve has equationand passes through the points A(2, 12) anda. Find the coordinates of the points C and D. Give your answer in exact form.b. Find the equation of the perpendicular bisector of CD. 12 y = 3x + = X
A curve has equation y = x3 + 2x2 − 4x + 6.a. Show that dy/dx = 0 when 2 and when x = -2 and when x = 2/3.b. Find the value of d2y/dx2 when x = -2 and when x = 2/3.
A curve has equation y = 2(x − 1)3 + 2. The normal at the point P(4, 4) and the normal at the point Q(9, 18) intersect at the point R.a. Find the coordinates of R.b. Find the area of triangle PQR.
The curve y = x2 − 4x − 5 and the line y = 1 − 3x meet at the points A and B.a. Find the coordinates of the points A and B.b. Find the gradient of the curve at each of the points A and B.
The curve has gradient 16 when x = 1 and gradient −8 when x = −1.Find the value of a and the value of b. y = ax + b 2 X²
A curve has equationand passes through the points P(−1, 1). Find the equation of the tangent to the curve at P and find the angle that this tangent makes with the x-axis. y = 5 2 - 3x
The gradient of the curve y = ax2 + bx at the point (3, −3) is 5. Find the value of a and the value of b.
The diagram shows the curveand the point A(1, 2) which lies on the curve. The tangent to the curve at A cuts the y-axis at B and the normal to the curve at A cuts the x-axis at C.i. Find the equation of the tangent AB and the equation of the normal AC.ii. Find the distance BC.iii. Find the
The curveintersects the x-axis at P. The tangent to the curve at P intersects the y-axis at Q. Find the distance PQ. y = 12 2x - 3 4
The curve y = x(x − 1)(x + 2) crosses the x-axis at the points O(0, 0), A(1, 0)v and B(−2, 0). The normals to the curve at the points A and B meet at the point C. Find the coordinates of the point C.
The gradient of the curve y = x3 + ax2 + bx + 7 at the point (1, 5) is −5. Find the value of a and the value of b.
The equation of a curve is y = 3 + 4x − x2.i. Show that the equation of the normal to the curve at the point (3, 6) is 2y = x + 9.ii. Given that the normal meets the coordinate axes at points A and B, find the coordinates of the mid-point of AB.iii. Find the coordinates of the point at which the
Given that the gradient of the curve y = x3 + ax2 + bx + 3 is zero when x = 1 and when x = 6, find the value of a and the value of b.
The normal to the curve y = 2x2 + kx − 3 at the point (3, −6) is parallel to the line x + 5y = 10.a. Find the value of k.b. Find the coordinates of the point where the normal meets the curve again.
Given that y = 2x3 − 3x2 − 36x + 5, find the range of values of x for which dy/dx < 0.
Given that y = 4x3 + 3x2 − 6x − 9, find the range of values of x for which dy/dx ≥ 0.
An oil pipeline under the sea is leaking oil and a circular patch of oil has formed on the surface of the sea.At midday the radius of the patch of oil is 50m and is increasing at a rate of 3 metres per hour.Find the rate at which the area of the oil is increasing at midday.
A piece of wire, of length 40 cm, is bent to form a sector of a circle with radius rcm and sector angle θ radians, as shown in the diagram. The total area enclosed by the shape is A cm2.a. Express θ in terms of r.b. Show that A = 20r − r2.c. Find the value of r for which there is a stationary
Find the set of values of x for whichis increasing. f(x) = — (5—2x)²³ - + 4x 6
The equation of a curve isFind dy/dx and, hence, explain why the curve does not have a stationary point. y = X -9 X
The diagram shows a rectangular enclosure for keeping animals.There is a fence on three sides of the enclosure and a wall on its fourth side.The total length of the fence is 50m and the area enclosed is Am2.a. Express y in terms of x.b. Show thatc. Find the maximum possible area enclosed and the
A cone has base radius r cm and a fixed height of 30 cm.The radius of the base is increasing at a rate of 0.01cms−1.Find the rate of change of the volume when r = 5.
A curve has equation y = 3x3 + 6x2 + 4x − 5. Show that the gradient of the curve is never negative.
A point with coordinates (x, y) moves along the curve in such a way that the rate of increase of x has the constant value 0.01 units per second. Find the rate of increase of y at the instant when x = 4. y = √1 + 2x
The volume of a spherical balloon is increasing at a constant rate of 40cm3per second.Find the rate of increase of the radius of the balloon when the radius is 15cm.
Find the set of values of x for which each of the following is increasing.a. f(x) = x2 − 8x + 2 b. f(x) = 2x2 − 4x + 7c. f(x) = 5 − 7x − 2x2 d. f(x) = x3 − 12x2 + 2e. f(x) = 2x3 − 15x2 + 24x + 6 f. f(x) = 16 + 16x − x2 − x3
Find the coordinates of the stationary points on each of the following curves and determine the nature of each stationary point. Sketch the graph of each function and use graphing software to check your graphs.a. y = x2 − 4x + 8 b. y = (3 + x)(2 − x)c. y = x3 − 12x + 6 d. y = 10 +
Find the coordinates of the stationary points on each of the following curves and determine the nature of each stationary point.a.b.c.d.e.f. √x - + x^= A 6
A point is moving along the curve y = 3x − 2x3 in such a way that the x-coordinate is increasing at 0.015 units per second. Find the rate at which the y-coordinate is changing when x = 2, stating whether the y-coordinate is increasing or decreasing.
The sum of two real numbers, x and y, is 9.a. Express y in terms of x.b. i Given that P = x2y, write down an expression for P, in terms of x.ii. Find the maximum value of P.c. i Given that Q = 3x2 + 2y2, write down an expression for Q, in terms of x.ii. Find the minimum value of Q.
A circle has radius r cm and area Acm2.The radius is increasing at a rate of 0.1cms−1.Find the rate of increase of A when r = 4.
A sphere has radius r cm and volume V cm3.The radius is increasing at a rate of Find the rate of increase of the volume when V = 36π. 2π cm s-¹.
A point is moving along the curvein such a way that the x-coordinate is increasing at a constant rateof 0.005 units per second. Find the rate of change of the y-coordinate as the point passes through the point (2, 4). 8 y = 2 x² - 2
Find the set of values of x for which each of the following is decreasing.a. f(x) = 3x2 − 8x + 2 b. f(x) = 10 + 9x − x2c. f(x) = 2x3 − 21x2 + 60x − 5 d. f(x) = x3 − 3x2 − 9x + 5e. f(x) = −40x + 13x2 − x3 f. f(x) = 11 + 24x − 3x2 − x3
A function f is defined as Find an expression for f´´(x) and determine whether f is an increasing function, a decreasing function or neither. f(x) = 4 1 - 2x for x 1.
The diagram shows the curve y = 2x2 and the points X(−2, 0) and P(p, 0) . The point Q lies on the curve and PQ is parallel to the y-axis.i. Express the area, A, of triangle XPQ in terms of p.The point P moves along the x-axis at a constant rate of 0.02 units per second and Q moves along the curve
A point is moving along the curvein such a way that the x-coordinate is increasing at a constant rate of 0.02 units per second. Find the rate of change of the y-coordinate when x = 1. y = 3√x - 5 기념 X
The diagram shows a rectangle, ABCD, where AB = 20 cm and BC = 16 cm.PQRS is a quadrilateral where PB = AS = 2x cm, BQ = x cm and DR = 4xcm.a. Express the area of PQRS in terms of x.b. Given that x can vary, find the value of x for which the area of PQRS is a minimum and find the magnitude of this
A watermelon is assumed to be spherical in shape while it is growing. Its mass, M kg, and radius, r cm, arerelated by the formula M = kr3, where k is a constant. It is also assumed that the radius is increasing at aconstant rate of 0.1 centimetres per day. On a particular day the radius is 10 cm
A curve has equation y = 2x3 − 3x2 − 36x + k.a. Find the x-coordinates of the two stationary points on the curve.b. Hence, find the two values of k for which the curve has a stationary point on the x-axis.
A function f is defined asfor x ≥ 0. Find an expression for f´(x) and determine whether f is an increasing function, a decreasing function or neither. f(x) = 5 (x + 22 2 x + 2
A square has side length x cm and area Acm2.The area is increasing at a constant rate of 0.03cm2s−1.Find the rate of increase of x when A = 25.
The curve y = x3 + ax2 − 9x + 2 has a maximum point at x = −3.a. Find the value of a.b. Find the range of values of x for which the curve is a decreasing function.
A cube has sides of length x cm and volume V cm3.The volume is increasing at a rate of 1.5 cm3s−1.Find the rate of increase of x when V = 8.
A point, P, travels along the curvein such a way that the x-coordinate of P is increasing at a constant rate of 0.5 units per second. Find the rate at which the y-coordinate of P is changing when P is at the point (1, 4). y = 3x + 1 X
The diagram shows the graph of 3x + 2y = 30. OPQR is a rectangle with area Acm2.The point O is the origin, P lies on the x-axis, R lies on the y-axis and Q has coordinates (x, y) and lies on the line 3x + 2y = 30.a. Show that b. Given that x can vary, find the stationary value of A and
PQRS is a rectangle with base length 2p units and area Aunits2.The points P and Q lie on the x-axis and the points R and S lie on the curve y = 9 − x2.a. Express QR in terms of p.b. Show that A = 2 p (9 − p2) .c. Find the value of p for which A has a stationary value.d. Find this stationary
A farmer divides a rectangular piece of land into 8 equal-sized rectangular sheep pens as shown in the diagram.Each sheep pen measures xm by ym and is fully enclosed by metal fencing. The farmer uses 480m of fencing.i. Show that the total area of land used for the sheep pens, Am2, is given by A =
Show thatis an increasing function. f(x) = x² - 4 X
A point is moving along the curvein such a way that the x-coordinate is increasing at a constant rate of 0.02 units per second. Find the rate at which the y-coordinate is changing when x = 2, stating whether the y-coordinate is increasing or decreasing y= 2 X + 5x
The diagram shows a 24cm by 15cm sheet of metal with a square of side xcm removed from each corner.The metal is then folded to make an open rectangular box of depth x cm and volume Vcm3.a. Show that V = 4x3 − 78x2 + 360x.b. Find the stationary value of V and the value of x for which this
A point moves along the curveAs it passes through the point P, the x-coordinate is increasing at a rate of 0.125 units per second and the y-coordinate is increasing at a rate of 0.08 units per second. Find the possible x-coordinates of P. y = 8 7-2x
The variables x, y and z can take only positive values and are such that z = 3x + 2y and xy = 600.i. Show thatii. Find the stationary value of z and determine its nature. z = 3x + 1200 x
The curve y = 2x3 + ax2 + bx − 30 has a stationary point when x = 3.The curve passes through the point (4, 2).a. Find the value of a and the value of b.b. Find the coordinates of the other stationary point on the curve and determine the nature of this point.
A solid metal cuboid has dimensions xcm by xcm by 4xcm.The cuboid is heated and the volume increases at a rate of 0.15 cm3s−1.Find the rate of increase of x when x = 2.
A function f is defined as f(x) = (2x + 5)2 − 3 for x ≥ 0. Find an expression for f´(x) and explain why f is an increasing function.
The curve y = 2x3 + ax2 + bx − 30 has no stationary points.Show that a2 < 6b.
The diagram shows the dimensions in metres of an L-shaped garden. The perimeter of the garden is 48m.i. Find an expression for y in terms of x.ii. Given that the area of the garden is Am2, show that A = 48x − 8x2.iii. Given that x can vary, find the maximum area of the garden, showing that this
A closed circular cylinder has radius r cm and surface area Acm2, where Given that the radius of the cylinder is increasing at a rate of 0.25 cms−1, find the rate of change of A when r = 10. A = 2πι2 + 400π r
The volume of the solid cuboid shown in the diagram is 576cm3 and the surface area is Acm2.a. Express y in terms of x.b. Show thatc. Find the maximum value of A and state the dimensions of the cuboid for which this occurs. 2xcm y cm x cm
A piece of wire, of length 100 cm, is cut into two pieces.One piece is bent to make a square of side x cm and the other is bent to make a circle of radius r cm.The total area enclosed by the two shapes is Acm2.a. Express r in terms of x.b. Show thatc. Find the value of x for which A has a
The diagram shows a piece of wire, of length 2m, is bent to form the shape PQRST.PQST is a rectangle and QRS is a semicircle with diameter SQ.PT = xm and PQ = ST = ym.The total area enclosed by the shape is Am2.a. Express y in terms of x.b. Show that c. Find dA/dx and d2A/dx2d. Find the value
The diagram shows a window made from a rectangle with base 2rm and height hm and a semicircle of radius rm. The perimeter of the window is 5m and the area is Am2.a. Express h in terms of r.b. Show that c. Find dA/dr and d2A/dr2d. Find the value for r for which there is a stationary value of
The curvehas a stationary point at (2, 12).a. Find the value of a and the value of b.b Determine the nature of the stationary point (2, 12). y = ax + . x
The diagram shows a metal plate consisting of a rectangle with sides x cm and y cm and a quarter-circle of radius x cm. The perimeter of the plate is 60 cm.i. Express y in terms of x.ii. Show that the area of the plate, Acm2, is given by A = 30x − x2.Given that x can vary,iii. Find the value of x
It is given that Show that f is a decreasing function. 2 f(x) = -x²¹ for x>0
A curve has equationwhere k is a positive constant. Find, in terms of k, the values of x for which the curve has stationary points and determine the nature of each stationary point. y = 1+ 2x + k² 2x - 3'
A point, P, travels along the curvein such a way that at time t minutes the x-coordinate of P is increasing at a constant rate of 0.012 units per minute. Find the rate at which the y-coordinate of P is changing when P is at the point (1, −1). y = √√/2x² - 3
The curvehas a stationary point at (3, 5).a. Find the value of a and the value of b.b. Determine the nature of the stationary point (3, 5).c. Find the range of values of x for which is decreasing. y = x² + ² + b x
Find the coefficient of x3 in the expansions of:a. (x + 3)4 b. (1 + x)5 c. (3 − x)5 d. (4 + x)4e. (x − 2)5f. (2x − 1)4 g. (4 +3)4h. x (5-2) 4
The inside lane of a school running track consists of two straight sections each of length x metres, and two semicircular sections each of radius r metres, as shown in the diagram. The straight sections are perpendicular to the diameters of the semicircular sections. The perimeter of the inside
The points A(0, 0), B(0.5, 0.75), C(0.8, 1.44), D(0.95, 1.8525), E(0.99, 1.9701) and F(1, 2) lie on the curve y = f(x).a. Copy and complete the table to show the gradients of the chords CF, DF and EF.b. Use the values in the table to predict the value of dy/dx when x = 1.
Find dx2y/dx2 for each of the following functions.a. y = x2 + 8x − 4 b. y = 5x3 − 7x2 + 5c. d. y = (2x − 3)4e.f.g.h. y = 2x2(5 − 3x + x2)i. .x 12/²³ - 7 = 1² 6
Differentiatewith respect to x. 3x5 - 7 4x
Find dx2y/dx2 for each of the following functions.a. y = x2 + 8x − 4 b. y = 5x3 − 7x2 + 5c. d. y = (2x − 3)4e.f.g.h. y = 2x2(5 − 3x + x2)i. .x 12/²³ - 7 = 1² 6
Water is poured into the hemispherical bowl of radius 5cm at a rate of 3πcm3s−1.After t seconds, the volume of water in the bowl, V cm3, is given bywhere hcm is the height of the water in the bowl.a. Find the rate of change of h when h = 1.b. Find the rate of change of h when h = 3. V =
A curve has equationi. Find dy/dx and d2y/dx2ii Find the coordinates of the stationary points and state, with a reason, the nature of eachstationary point. x 1∞ y = =+ 2x.
The diagram shows a right circular cone with radius 10 cm and height 30 cm.The cone is initially completely filled with water.Water leaks out of the cone through a small hole at the vertex at a rate of 4cm3s−1.a. Show that the volume of water in the cone, Vcm3, when the height of the water is h
A solid cylinder has radius r cm and height h cm.The volume of this cylinder is 432 cm3 and the surface area is Acm2.a. Express h in terms of r.b. Show thatc. Find the value for r for which there is a stationary value of A.d. Determine the magnitude and nature of this stationary value. A = 2πr²
The diagram shows a right circular cone of base radius r cm and height h cm cut from a solid sphere of radius 10 cm. The volume of the cone is V cm3.a. Express r in terms of h.b. Show thatc. Find the value for h for which there is a stationary value of V.d. Determine the magnitude and nature of
The diagram shows a solid formed by joining a hemisphere of radius r cm to a cylinder of radius r cm and height h cm. The surface area of the solid is 320 πcm2 and the volume is V cm3.a. Express h in terms of r.b. Show thatc. Find the exact value of r such that V is a maximum. 1 1 h
Differentiate with respect to x:a.b.c. d.e.f.g.h. 1 x + 2

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