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mathematics
pearson edexcel a level mathematics
Pearson Edexcel A Level Mathematics Pure Mathematics Year 2 1st Edition Greg Attwood, Jack Barraclough, Ian Bettison, David Goldberg, Alistair Macpherson, Joe Petran - Solutions
Find the value of dy/dx at the point where x = 1 on the curve with the equation y=x√10x + 6
a. Find f'(x). f(x) = e2x − ln (x2) + 4, x > 0The curve with equation y = f(x) has a gradient of 2 at point P. The x-coordinate of P is a.b. Show that a(e2a − 1) = 2.
Curve C has equation x = (arc cos y)2. Show that dy dx √1 - cos²√x 2√x
The curve C has equation y = f(x), whereThe point P is a stationary point on C.a. Calculate the x-coordinate of P. The point Q on C has x-coordinate 1.b. Find an equation for the normal to C at Q. 1 f(x) = 3 ln x + x > 0
The curve C has equation y = 2ex + 3x2 + 2. Find the equation of the normal to C at the point where the curve intercepts the y-axis. Give your answer in the form ax + by + c = 0 where a, band c are integers to be found.
A curve C has equation y = 5 sin 3x + 2 cos 3x, -π < x < πa. Show that the point P (0, 2) lies on C.b. Find an equation of the normal to the curve C at P.
The point P lies on the curve with equation y = 2(34x). The x-coordinate of P is 1. Find an equation of the normal to the curve at the point P in the form y = ax + b, where a and b are constants to be found in exact form.
The curve C has equation y = e2x cos x.a. Show that the turning points on C occur when tan x = 2.b. Find an equation of the tangent to C at the point where x = 0.
Given that x = cosec 5y,a. Find dy/dx in terms of y.b. Hence find dy/dx in terms if x.
The curve C is given by the equationswhere t is a parameter.At A, t = 2. The line l is the normal to C at A.a. Find dy/dx in terms of t.b. Hence find an equation of l. 8 x = 4 – 3, 1 = 2 4t = t > 0
A curve has equation f(x) = (x3 − 2x)e−x.a. Find f'(x). The normal to C at the origin O intersects C again at P.b. Show that the x-coordinate of P is the solution to the equation 2x2 = ex + 4.
Given that x = y2 ln y, y > 0,a. Find dx/dyb. Use your answer to part a to find in terms of e, the value of dy/dx at y = e.
A curve C is given by the equations x = 2 cos t + sin 2t, y = cos t − 2 sin 2t, 0 < t < π where t is a parametera. Find dx/dt and dy/dt in terms of t.b. Find the value of dy/dx at the point P on C where t = π/4.c. Find an equation of the normal to the curve at P.
The curve C is given by the equations x = 2t, y = t2, where t is a parameter. Find an equation of the normal to C at the point P on C where t = 3.
A curve is given by x = 2t + 3, y = t3 − 4t, where t is a parameter. The point A has parameter t = −1 and the line l is the tangent to C at A. The line l also cuts the curve at B.a. Show that an equation for l is 2y + x = 7.b. Find the value of t at B.
A car has value £V at time t years. A model for V assumes that the rate of decrease of V at time t is proportional to V. Form an appropriate differential equation for V.
In a study of the water loss of picked leaves the mass, M grams, of a single leaf was measured at times, t days, after the leaf was picked. It was found that the rate of loss of mass was proportional to the mass M of the leaf. Write down a differential equation for the rate of change of mass of the
This graph shows part of the curve C with parametric equationsP is the point on the curve where t = 2. The line l is the normal to C at P. Find the equation of l. x = (t + 1)², y = t³ +3, t>−1
In a pond the amount of pondweed, P, grows at a rate proportional to the amount of pondweed already present in the pond. Pondweed is also removed by fish eating it at a constant rate of Q per unit of time. Write down a differential equation relating P to t, where t is the time which has elapsed
A circular patch of oil on the surface of some water has radius r and the radius increases over time at a rate inversely proportional to the radius. Write down a differential equation relating r and t, where t is the time which has elapsed since the start of the observation.
A metal bar is heated to a certain temperature, then allowed to cool down and it is noted that, at time t, the rate of loss of temperature is proportional to the difference between the temperature of the metal bar, θ, and the temperature of its surroundings θ0. Write down a differential equation
Find the gradient of the curve with equation 5x2 + 5y2 − 6xy = 13 at the point (1, 2).
Given that e2x + e2y = xy, find dy/dx in terms of x and y.
Find the coordinates of the turning points on the curve y3 + 3xy2 − x3 = 3.
a. If (1 + x)(2 + y) = x2 + y2, find dy/dx in terms of x and y.b. Find the gradient of the curve (1 + x)(2 + y) = x2 + y2 at each of the two points where the curve meets the y-axis.c. Show also that there are two points at which the tangents to this curve are parallel to the y-axis.
A curve has equation 7x2 + 48xy − 7y2 + 75 = 0. A and B are two distinct points on the curve and at each of these points the gradient of the curve is equal to 2/11. Use implicit differentiation to show that the straight line passing through A and B has equation x + 2y = 0.
Given that y = xx, x > 0, y > 0, by taking logarithms show that dy/dx = xx(1 + ln x).
Given that y = (arc sin x)2 show that d²y dx² -(zx - 1) dy dx - -X -2=0
A population P is growing at the rate of 9% each year and at time t years may be approximated by the formula P = P0(1.09)t, t ≥ 0 where P is regarded as a continuous function of t and P0 is the population at time t = 0.a. Find an expression for t in terms of P and P0.b. Find the time T years when
Given that y = x − arctan x prove that d²y dx² dy = 2x(1-dr) 2
Differentiate arcsin x/√1 + x2
A curve C has equation y = ln (sin x), 0 < x < πa. Find the stationary point of the curve C.b. Show that the curve C is concave at all values of x in its given domain.
The mass of a radioactive substance t years after first being observed is modelled by the equation m = 40e−0.244ta. Find the mass of the substance nine months after it was first observed.b. Find dm/dtc. With reference to the model, interpret the significance of the sign of the value of dm/dt
The curve C with equation y = f(x) is shown in the diagram, where The curve has a local minimum at A and a local maximum at B.a. Show that the x-coordinates of A and B satisfy the equation tan 2x = −0.5 and hence find the coordinates of A and B.b. Using your answer to part a, find the
The diagram shows a sketch of the curve with parametric equations a. Find the coordinates of the points A and B.b. The point C has parameter t = π/6. Find the exact coordinates of C. c Find the Cartesian equation of the curve. x = 4 cost, y = 3 sint, 0≤ t < 2π
The diagram shows a sketch of the curve with parametric equationsThe curve is symmetrical about both axes. Copy the diagram and label the points having parameters x = cost, y = sin 2t, 0≤1
A circle has parametric equations x = 4 sin t − 3, y = 4 cos t + 5, 0 ≤ t ≤ 2πa. Find a Cartesian equation of the circle.b. Draw a sketch of the circle.c. Find the exact coordinates of the points of intersection of the circle with the y-axis
The cross-section of a bowl design is given by the O y x following parametric equations a. Find the length of the opening of the bowl.b. Given that the cross-section of the bowl crosses the y-axis at its deepest point, find the depth of the bowl. y 0 X
A curve C has parametric equationsa. Determine the ranges of x and y in the given domain of t.b. Show that the Cartesian equation of C can be written in the formwhere A, b and c are integers to be determined. 3 x = ²/ +2₁ +2, y = 2t-3-t², tER, t #0
The curve C has parametric equations.a. Find a Cartesian equation of C.b. Sketch the curve C on the appropriate domain. x = 8 cost, y =sec²t, ==1
The path of a car on a Ferris wheel at time t minutes is modelled using the parametric equations x = 12 sin t, y = 12 − 12 cos t where x is the horizontal distance in metres of the car from the start of the ride and y is the height in metres above ground level of the car. a. Show that the
The curve C has parametric equations a. Show that the Cartesian equation of C can be written as (x + a)2 + ( y + b)2 = c where a, b and c are integers to be determined.b. Sketch the curve C on the given domain of t.c. Find the length of C. x = 9 cost - 2, y=9 sint + 1, -=1
A particle is moving in the xy-plane such that its O y x position after time t seconds relative to a fixed origin O is given by the parametric equations The diagram shows the path of the particle.a. Find the distance from the origin to the particle at time t = 0.5.b. Find the coordinates of
Find the values of t at the points of intersection of the line 4x − 2y − 15 = 0 with the parabola x = t2, y = 2t and give the coordinates of these points.
A curve C has parametric equationsa. Show that the Cartesian equation of the curve is given by y = ax(1 − bx2) where a and b are integers to be found.b. State the domain and range of y = f(x) in the given domain of t. 1 X = 3 sint, y=sin 31, 0
A diagram shows a curve C with parametric equationsa. Find a Cartesian equation of the curve in the form y = f(x), and state the domain of f(x).b. Show thatc. Hence determine the range of f(x). x = 3√t, y = t³ - 2t, 0≤t≤2
Find the points of intersection of the parabola x = t2, y = 2t with the circle x2 + y2 − 9x + 4 = 0.
The path of a ski jumper from the point of leaving the ramp to the point of landing is modelled using the parametric equations x = 18t, y = −4.9t2 + 4t + 10, 0 < t < k where x is the horizontal distance in metres from the point of leaving the ramp and y is the height in metres above
Find the coordinates of the point(s) where the following curves meet the x-axis and the y-axis. a x = f² -1, b x = sin 2t, c x = tant, y = cost, 0
The profile of a hill climb in a bike race is modelled by the following parametric equations a. Find the value of t at the highest point of the hill climb.b. Hence find the coordinates of the highest point.c. Find the coordinates when t = 1 and show that at this point, a cyclist will be
Find the coordinates of the point(s) where the following curves meet the x-axis and the y-axis. a x = e' +5, y =lnt, t>0 b x = lnt, y = 1²-64, t>0 c x = e2¹ + 1, y = 2et-1, -1
A curve C has parametric equations x = t3 − t, y = 4 − t2, t ∈ ℝa. Show that the Cartesian equation of C can be written in the form x2 = (a − y)(b − y)2 where a and b are integers to be determined.b. Write down the maximum value of the y-coordinate for any point on this curve.
A computer model for the shape of the path of a rollercoaster is given by the parametric equationsa. Find the coordinates of the point where t = π /6. Given that one unit on the model represents 5 m in real life.b. Find the maximum height of the rollercoaster.c. Find the horizontal distance
A curve has parametric equationsa. Find the Cartesian equation of the curve in the form y2 = f(x).b. Determine the possible values of x and y in the given domain of t. x = tan²t + 5, y = 5 sint, 0
Find the values of t at the points of intersection of the line y = −3x + 2 and the curve with parametric equations x = t2, y = t, and give the coordinates of these points.
Show that the line with equation y = 2x − 5 does not intersect the curve with parametric equations x = 2t, y = 4t(t − 1).
A curve C has parametric equations a. Find the coordinates of the points where the curve intersects the x-axis.b. Show that the curve crosses the line y = 4 wherec. Hence determine the coordinates of points where y = 4 intersects the curve. x = 6 cost, y = 4 sin 2t + 2, 2
Find the values of t at the point of intersection of the line y = x − ln 3 and the curve with parametric equations x = ln (t − 1), y = ln (2t − 5), t > 5/2 , and give the exact coordinates of this point.
A curve C has parametric equations x = tan t,Find a Cartesian equation of C. X = = tant, y=3 sin (t-π), 0
The curve C has parametric equations x = sin t, y = cos t. The straight line l passes through the points A and B where t = π/6 and t = π/2 respectively. Find an equation for the line l in the form ax + by + c = 0.
The curve C has parametric equations x = sin t, y = cos 2t + 1, 0 < t < 2π. Given that the line y = k, where k is a constant, intersects the curve, a. Show that 0 < k < 2 b. Show that if the line y = k is a tangent to the curve, then k = 2.
a. Express 65 cos θ − 20 sin θ in the form R cos (θ + α), where R > 0 and 0 < α < π/2. Give the value of α correct to 4 decimal places. A city wants to build a large circular wheel as a tourist attraction. The height of a tourist on the circular wheel is modelled by the equation H
Find the Cartesian equation of the curves given by the following parametric equations: a x = 2 sint -1, y =5 cost +4, c x= cost, y = 2 cos 2t, 0
A river flows from north to south. The position at time t seconds of a rowing boat crossing the river from west to east is modelled by the parametric equations x = 0.9 t m, y = −3.2 t m where x is the distance travelled east and y is the distance travelled north. Given that the river is 75 m
a. Express 12 sin x + 5 cos x in the form R sin (x + α), where R and α are constants, R > 0 and 0 < α < 90°. Round α to 1 decimal place.A runner’s speed, v in m/s, in an endurance race can be modelled by the equation.where x is the time in minutes since the beginning of the race.b.
a. Express 1.4 sin θ − 5.6 cos θ in the form R sin (θ − α), where R and α are constants, R > 0 and 0 < α < 90°. Round R and α to 3 decimal places.b. Hence find the maximum value of 1.4 sin θ − 5.6 cos θ and the smallest positive value of θ for which this maximum occurs.The
a. Prove that cos4 2θ − sin4 2θ ≡ cos 4θ.b. Hence find, for 0 ≤ θ ≤ 180°, all the solutions of cos4 2θ − sin4 2θ = 1/2.
a. Given that 2 cos θ = 1 + 3 sin θ, show that R cos (θ + α) = 1, where R and α are constants to be found.b. Hence, or otherwise, solve 2 cos θ = 1 + 3 sin θ for 0 ≤ θ ≤ 360°.
a. Express 1.5 sin 2x + 2 cos 2x in the form R sin (2x + α), where R > 0 and 0 < α < π/2 , giving your values of R and α to 3 decimal places where appropriate.b. Express 3 sin x cos x + 4 cos2 x in the form a sin 2x + b cos 2x + c, where a, b and c are constants to be found.c. Hence,
a. Given that sin2 θ/2 = 2 sin θ, show that √17 sin (θ + α) = 1 and state the value of α.b. Hence, or otherα = 14.0° b 0°, 151.9°, 360°ise, solve sin2 ≤ θ ≤ 2 = 2 sin θ for 0 ≤ θ ≤ 360°.
Given that sin x cos y = 1/2 and cos x sin y = 1/3,a. Show that sin (x + y) = 5 sin (x − y). Given also that tan y = k, express in terms of k:b. tan xc. tan 2x
Given that 7 cos 2θ + 24 sin 2θ ≡ R cos (2θ − α), where R > 0 and 0 < α < π < 2 , find:a. The value of R and the value of α, to 2 decimal places.b. The maximum value of 14 cos2 θ + 48 sin θ cos θ.c. Solve the equation 7 cos 2θ + 24 sin 2θ = 12.5, for 0 ≤ θ ≤ 180°,
a. Express sin x − √3 cos x in the form R sin (x − α), with R > 0 and 0 < α < 90°.b. b Hence sketch the graph of y = sin x − √3 cos x, for −360° < x < 360°, giving the coordinates of all points of intersection with the axes.
a. Usingwith an appropriate value of θ.b. Use the result in a to find the exact value of tan 3π/8. tan 20 2 tan 0 1 - tan²0
a. Prove, by counter-example, that the statement sec (A + B) ≡ sec A + sec B, for all A and B is false.b. Prove that tan θ + cot θ ≡ 2 cosec 2θ, θ ± nπ/2, n ∈ Z.
a. Given that α is acute and tan α = 3/4,prove that 3 sin (θ + α) + 4 cos (θ + α) ≡ 5 cos θb. Given that sin x = 0.6 and cos x = −0.8, evaluate cos (x 270°) and cos (x + 540°).
a. Given that 4 sin (x + 70°) = cos (x + 20°), show that tan x = − 3/5 tan 70°.b. Hence solve, for 0 ≤ x ≤ 180°, 4 sin (x + 70°) = cos (x + 20°), giving your answers to 1 decimal place.
a. Given that 2 sin x = cos (x − 60)°, show that tan x = 1/4− √3.b. Hence solve, for 0 ≤ x ≤ 360°, 2 sin x = cos (x − 60°), giving your answers to 1 decimal place.
a. Show that cos 2θ = 5 sin θ may be written in the form a sin2 θ + b sin θ + c = 0, where a, b and c are constants to be found.b. Hence solve, for −π < θ < π, the equation cos 2θ = 5 sin θ.
a. Given that √3 sin 2θ + 2 sin2 θ = 1, show that tan 2θ =1/√3.b. Hence solve, for 0 < θ < π, the equation √3 sin 2θ + 2 sin2 θ = 1.
In △ABC, AB = 5 cm and AC = 4 cm, ∠ABC = (θ − 30°) and ∠ACB = (θ + 30°). Using the sine rule, show that tan θ = 3 √3.
The lines l1 and l2 , with equations y = 2x and 3y = x − 1 respectively, are drawn on the same set of axes. Given that the scales are the same on both axes and that the angles l1 and l2 make with the positive x-axis are A and B respectively,a. Write down the value of tan A and the value of tan
Given that sin x = 1/√5 where x is acute and that cos (x − y) = sin y, show that tan y = √5 + 1/2.
a. Use the double-angle formulae to prove that 1 - cosx/1 + cos2x ≡ tan2 x.b. Hence find, for −π ≤ x ≤ π, all the solutions of 1 - cosx/1 + cos2x = 3, leaving your answers in terms of π.
Find a Cartesian equation for each of these parametric equations, giving your answer in the form y = f(x). In each case find the domain and range of f(x). a x = t-2, y = ² +1,₁ -4 2 b x= 5-t, y=f²-1, tER e x = ₁1₂₁ y = ²²₂ 1>2 t², t-2'
The position of a small plane coming into land at time t minutes after it has started its descent is modelled by the parametric equations x = 80t, y = −9.1t + 3000, 0 ≤ t ≤ 329 where x is the horizontal distance travelled (in metres) and y is the vertical distance travelled (in metres) from
A curve is given by the parametric equations Copy and complete the table and draw a graph of the curve for −5 ≤ t ≤ 5. 5 x = 2t, y = 7² t = 0
Find the coordinates of the point(s) where the following curves meet the x-axis. a x = 5+t, y =6-t c x= t², y=(1 - t)(t + 3) 2t y=t-9, 1 + t' e x = t#-1 b x= 2t + 1, y = 2t - 6 1 d ==—, y=(t-1)(2t-1), t#0
A circle has parametric equations x = sin t − 5, y = cos t + 2a. Find a Cartesian equation of the circle.b. Write down the radius and the coordinates of the center of the circle.c. Write down a suitable domain of t that defines one full revolution around the circle.
For each of these parametric curves:i. Find a Cartesian equation for the curve in the form y = f(x) giving the domain on which the curve is definedii. Find the range of f(x). a x = 2ln (5 t), y=t²-5, t
A curve is given by the parametric equationsCopy and complete the table and draw a graph of the curve for −4 < t < 4. x = t², y= 5
Find the coordinates of the point(s) where the following curves meet the y-axis. a x = 2t, y = 1²-5 cx = 1² +21-3, y = t(t-1) t-1 21 1+1' 1²+1' e x=- y = t = -1 1 b x= 3t-4, y = d x = 27-1³, y = 1 1 #0 t#1
A ball is kicked from the ground with an initial speed of 20 m s−1 at an angle of 30°. Its position after t seconds can be described using the following parametric equationsa. Find the horizontal distance travelled by the ball when it hits the ground. A player wants to head the ball when it is
A circle has parametric equations x = 4 sin t + 3, y = 4 cos t − 1. Find the radius and the coordinates of the center of the circle.
A curve C is defined by the parametric equations x = √t, y = t(9 − t), 0 ≤ t ≤ 5. a. Find a Cartesian equation of the curve in the form y = f(x), and determine the domain and range of f(x).b. Sketch C showing clearly the coordinates of any turning points, endpoints and intersections
A curve is given by parametric equations Copy and complete the table and draw a graph of the curve for the given domain of t. x = tant +1, y = sint, 4 3
A curve is given by the parametric equation x = cos t − 2, y = sin t + 3, −π < t < π. Sketch the curve.
The path of a dolphin leaping out of the water can be modelled with the following parametric equations x = 2t m, y = −4.9t2 + 10t m where x is the horizontal distance from the point the dolphin jumps out of the water, y is the height above sea level of the dolphin and t is the time in seconds
A curve has parametric equations x = 4at2, y = a(2t − 1), where a is a constant. The curve passes through the point (4, 0). Find the value of a.
Find the coordinates of the point of intersection of the line with parametric equations x = 3t + 2, y = 1 − t and the line y + x = 2
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