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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
Use partial fractions to integrate the following: a 3x + 5 (x + 1)(x + 2) b 3x - 1 (2x + 1)(x - 2) с 2x - 6 (x + 3)(x - 1) 3 (2 + x)(1-x)
Integrate the following with respect to x. a 3 sec² x + 5 + X c 2(sin x cos x + x) 2 x-² e 5ex + 4 cos x 1 1 - g + 2 1 X x-3 i 2 cosecx cotx - sec² x b 5e-4 sin x + 2x³ 2 X d 3 sec x tan x 1 f h - + 2 cosec² x 2x ex+ sin x + cos.x 1 +- cosec² x X jer+
Use the substitutions given to find: a |x1+xde;u=1+ x fsin³x dx; u = COS. e [sec² x tan.x√/1 + tan x dx; u² = 1 + tan.x с b fl + sin x COS X dx; u = sin x 2 dx; u = √x √x(x-4) f [sec+ x dx; ¹x dx; u = tan x a f d
Find general solutions to the following differential equations. Give your answers in the form y = f(x). dy dx c cos²x- a = (1+y)(1-2x) dy d.x = y² sin²x dy dx dy d = = 2ex-y dx b = y tan x
Integrate the following functions. a x² + 4 e2x d (e²x + 1)³ grer b e2x e2x + 1 cos 2x 3 + sin 2x h cos 2x (1 + sin 2x)4 e X (x² + 4)³ sin 2x (3 + cos2x)³ i sec² x tan² x с f
By choosing a suitable method of integration, find: a a f(2x - 3)7 dx d fx ln xdx b fx√4x - 1 dx 4 sin x cos x 4 - 8 sin²x e dx C fsin² x cos.x d.x 1 √3-4x dx f f/3
f(x) = x4 − 21x − 18a. Show that there is a root of the equation f(x) = 0 in the interval [−0.9, −0.8].b. Find the coordinates of any stationary points on the graph y = f(x).c. Given that f(x) = (x − 3)(x3 + ax2 + bx + c), find the values of the constants a, b and c.d. Sketch the
a. On the same axes, sketch the graphs of y = ln x and y = ex − 4.b. Write down the number of roots of the equation ln x = ex − 4.c. Show that the equation ln x = ex − 4 has a root in the interval (1.4, 1.5).
g(x) = ex − 1 + 2x − 15a. Show that the equation g(x) = 0 can be written asThe root of g(x) = 0 is α. The iterative formula xn + 1 = ln (15 − 2xn) + 1, x0 = 3, is used to find a value for α.b. Calculate the values of x1, x2 and x3 to 4 decimal places.c. By choosing a suitable interval, show
f(x) = (105x3 − 128x2 + 49x − 6) cos 2x, where x is in radians. The diagram shows a sketch of y = f(x).a. Calculate f(0.2) and f(0.8).b. Use your answer to part a to make a conclusion about the number of roots of f(x) in the interval 0.2 < x < 0.8.c. Further calculate f(0.3), f(0.4),
f(x) = x4 − 3x3 − 6a. Show that the equation f(x) = 0 can be written as x = 3√px4 + q, where p and q are constants to be found.b. Let x0 = 0. Use the iterative formula together with your values of p and q from part a, to find, to 3 decimal places, the values of x1, x2
y = f(x), where f(x) = x2 sin x − 2x + 1. The points P, Q, and R are roots of the equation. The points A and B are stationary points, with x-coordinates a and b respectively.a. Show that the curve has a root in each of the following intervals:i. [0.6, 0.7]ii. [1.2, 1.3]iii. [2.4, 2.5]b. Explain
a. Show that a root α of the equation f(x) = 0 lies in the interval [1.3, 1.4].b. Differentiate f(x) to find f′(x).c. By taking 1.3 as a first approximation to α, apply the Newton Raphson process once to f(x) to obtain a second approximation to α.Give your answer to 3 decimal places. 0xx ,
Given that y = (1 + ln 4x)3/2 find the value of dy/dx at x = 1/4 e3.
Where x is in radiansa. Show that f(x) = 0 has a solution, α, in the interval 0.4 < x < 0.5.b. Taking 0.4 as a first approximation to α, apply the Newton–Raphson process once to f(x) to obtain a second approximation to α. Give your answer to 3 decimal places.c. Show that x = −1.190 is
2 f(x) = 3 + x2 − x3a. Show that the equation f(x) = 0 has a root, α, in the interval [1.8, 1.9].b. By considering a change of sign of f(x) in a suitable interval, verify that α = 1.864 correct to 3 decimal places.
f(x) = x3 − 6x − 2a. Show that the equation f(x) = 0 can be written in the formand state the values of the integers a and b. f(x) = 0 has one positive root, αThe iterative formula =is used to find an approximate value for α.b. Calculate the values of x1, x2, x3 and x4 to 4 decimal places.c.
Show that each of these functions has at least one root in the given interval. a. f(x) = x3 − x + 5, −2 < x < −1b. f(x ) = x2 − √x − 10, 3 < x < 4c. f(x) = x3 − 1/x − 2, −0.5 < x < −0.2 d. f(x) = ex − ln x − 5, 1.65 < x < 1.75
f(x) = x2 − 6x + 2a. Show that f(x) = 0 can be written as:b. Starting with x0 = 4, use each iterative formula to find a root of the equation f(x) = 0. Round your answers to 3 decimal places.c. Use the quadratic formula to find the roots to the equation f(x) = 0, leaving your answer in the form a
f(x) = x3 − 2x − 1a. Show that the equation f(x) = 0 has a root, α, in the interval 1 < α < 2.b. Using x0 = 1.5 as a first approximation to α, apply the Newton–Raphson procedure once to f(x) to find a second approximation to α, giving your answer to 3 decimal places.
a. Calculate f(3.9) and f(4.1).b. Explain why the equation f(x) = 0 does not have a root in the interval 3.9 < x < 4.1. The equation f(x) = 0 has a single root, α.c. Use algebra to find the exact value of α. f(x) = 1 4-x +3
f(x) = x2 − 6x + 1a. Show that the equation f(x) = 0 can be written as x = √6x + 1.b. Sketch on the same axes the graphs of y = x and y = √6x - 1.c. Write down the number of roots of f(x).d. d Use your diagram to explain why the iterative formula converges to a root of f(x) when x0
a. On the same axes, sketch the curves of y = p(x) and y = q(x). p(x) = 4 − x2 and q(x) = ex.b. State the number of positive roots and the number of negative roots of the equation x2 + ex − 4 = 0.c. Show that the equation x2 + ex − 4 = 0 can be written in the form x = ±(4 − ex)1/2 The
h(x) = 3 √x − cos x − 1, where x is in radians.a. Show that the equation h(x) = 0 has a root, α, between x = 1.4 and x = 1.5.b. By choosing a suitable interval, show that α = 1.441 is correct to 3 decimal places.
f(x) = 1 − x − cos (x2)a. Show that the equation f(x) = 0 has a root α in the interval 1.4 < α < 1.5.b. Using x0 = 1.4 as a first approximation to α, apply the Newton–Raphson procedure once to f(x) to find a second approximation to α, giving your answer to 3 decimal places.c. By
f(x) = xe−x − x + 2a. Show that the equation f(x) = 0 can be written asf(x) has a root, α, in the interval −2 < x < −1.b. Use the iterative formula with x0 = −1 to find, to 2 decimal places, the values of x1, x2 and x3. x = ln x-21- x = 2.
f(x) = sin x − ln x , x > 0, where x is in radians.a. Show that f(x) = 0 has a root, α, in the interval [2.2, 2.3].b. By considering a change of sign of f(x) in a suitable interval, verify that α = 2.219 correct to 3 decimal places.
g(x) = x5 − 5x − 6a. Show that g(x) = 0 has a root, α, between x = 1 and x = 2.b. Show that the equation g(x) = 0 can be written as x = (px + q)1/r, where p, q and r are integers to be found.The iterative formulais used to find an approximate value for α.c. Calculate the values of x1, x2 and
f(x) = x3 + 5x2 − 2a. Show that f(x) = 0 can be written as: b. Starting with x0 = 10, use the iterative formula in part a (i) to find a root of the equation f(x) = 0. Round your answer to 3 decimal places.c. Starting with x0 = 1, use the iterative formula in part a (iii) to find a different
g(x) = x2 − 3x − 5a. Show that the equation g(x) = 0 can be written as x = √3x + 5.b. Sketch on the same axes the graphs of y = x and y = √3x+ 5.c. Use your diagram to explain why the iterative formula xn + 1 = √3xn + 5 converges to a root of g(x) when x0 = 1. g(x) = 0 can also be
A student observes that the function f(x) = 1/x + 2, x ≠ 0, has a change of sign on the interval [−1, 1]. The student writes:By means of a sketch, or otherwise, explain why the student is incorrect. y = f(x) has a vertical asymptote within this interval so even though there is a change of sign,
f(x) = 5x − 4 sin x − 2, where x is in radians. a. Show that f(x) = 0 has a root, α, between x = 1.1 and x = 1.15.b. Show that f(x) = 0 can be written as x = p sin x + q , where p and q are rational numbers to be found.c. Starting with x0 = 1.1, use the iterative formula xn + 1 = p sin xn
a. Show that f(x) = 0 has a root α in the interval [3.4, 3.5].b. Find f'(x).c. Taking 3.4 as a first approximation to α, apply the Newton–Raphson procedure once to f(x) to obtain a second approximation for α, giving your answer to 3 decimal places. f(x) = ln (3x − 4) – x² + 10, x > 3
f(x) = 3 cos (x2) + x − 2 a. Show that the equation f(x) = 0 can be written asb. Use the iterative formula to find, to 3 decimal places, the values of x1, x2 and x3. c. Given that f(x) = 0 has only one root, α, show that α = 1.1298 correct to 4 decimal places. x =
a. On the same axes, sketch the graphs of b. Write down the number of roots of the equation 1/x = x + 3.c. Show that the positive root of the equation1/x = x + 3 lies in the interval (0.30, 0.31).d. Show that the equation 1/x = x + 3 may be written in the form x2 + 3x − 1 = 0. e. Use
a. Find g'(x).A root α of the equation g(x) = 0 lies in the interval [6.5, 6.7]. b. Taking 6.6 as a first approximation to α, apply the Newton–Raphson process once to g(x) to obtain a second approximation to α. Give your answer to 3 decimal places.c. Given that g(1) = 0, find the exact
f(x) = 4 cot x − 8x + 3, 0 < x < π, where x is in radians.a. Show that there is a root α of f(x) = 0 in the interval [0.8, 0.9].b. Show that the equation f(x) = 0 can be written in the form c. Use the iterative formulato calculate the values of x1, x2 and x3 giving your answers to 4
a. Using the same axes, sketch the graphs of y = e−x and y = x2.b. Explain why the function f(x) = e−x − x2 has only one root.c. Show that the function f(x) = e−x − x2 has a root between x = 0.70 and x = 0.71.
a. Show that the equation f(x) = 0 can be written as x = 1.5 − 0.5e−0.8x. b. Use the iterative formula xn + 1 = 1.5 − 0.5e−0.8x with x0 = 1.3 to obtain x1, x2 and x3. Hence write down one root of f(x) = 0 correct to 3 decimal places.c. Show that the equation f(x) = 0 can be written in
h(x) = sin 2x + e4xa. Show that there is a stationary point, α, of y = h(x) in the interval −0.9 < x < −0.8.b. By considering the change of sign of h'(x) in a suitable interval, verify that α = −0.823 correct to 3 decimal places.
a. On the same axes, sketch the graphs of y = √x and y = 2/3.b. With reference to your sketch, explain why the equation √x = 2/x has exactly one real root.c. Given that f(x) = √x − 2/x , show that the equation f(x) = 0 has a root r, where 1 < r < 2.d. Show that the equation √x = 2/x
a. By writing y = xx in the form ln y = x ln x, show that dy/dx = xx (ln x + 1).b. Show that the function f(x) = xx − 2 has a root, α, in the interval [1.4, 1.6].c. Taking x0 = 1.5 as a first approximation to α, apply the Newton–Raphson procedure once to obtain a second
f(x) = x2 - 4/x + 6x − 10, x ≠ 0.a. Use differentiation to find f'(x). The root, α, of the equation f(x) = 0 lies in the interval [−0.4, −0.3].b. Taking −0.4 as a first approximation to α, apply the Newton–Raphson process once to f(x) to obtain a second approximation to α. Give
f(x) = x2 − 5x − 3a. Show that f(x) = 0 can be written as:b Let x0 = 5. Show that each of the following iterative formulae gives different roots of f(x) = 0. i x = V5x + 3 ii x= = x2 – 3 ยา 5
A curve has parametric equationsExpress t in terms of x, and hence show that a Cartesian equation of the curve is y = x2/2x - 1 X = 1 1 + t' y = 1 (1 + i)(1 - t)' t>1
A curve has parametric equations x = sin t, y = cos 2t, 0 ≤ t < 2π.a. Find a Cartesian equation of the curve. The curve cuts the x-axis at (a, 0) and (b, 0).b Find the values of a and b.
The curve C has parametric equationsa. Show that the curve C is part of a straight line.b. Find the length of this line segment. X= 2-3t 1+t' y = 3+2 0≤ t ≤4 1+1'
A curve C has parametric equations x = t2 − 2, y = 2t, 0 ≤ t ≤ 2a. Find the Cartesian equation of C in the form y = f(x).b. State the domain and range of y = f(x) in the given domain of t.c. Sketch the curve in the given domain of t.
A curve C has parametric equations x = 2 cos t, y = 2 sin t − 5, 0 < t < πa. Show that the curve C forms part of a circle.b. Sketch the curve in the given domain of t.c. Find the length of the curve in the given domain of t.
The curve C has parametric equations x = t − 2, y = t3 − 2t2, t ∈ ℝa. Find a Cartesian equation of C in the form y = f(x).b. Sketch the curve C.
Show that the line with equation y = 4x + 20 is a tangent to the curve with parametric equations x = t − 3, y = 4 − t2.
The diagram shows the curve C with parametric equation x = 1 + 2t, y = 4t − 1. The curve crosses the y-axis and the x-axis at points A and B respectively.a. Find the coordinates of A and B. The line l intersects the curve at points A and B. b. Find the equation of l in the form ax + by + c =
The diagram shows the curve C with parametric equations The curve crosses the y-axis and the x-axis at points A and B respectively. The line l intersects the curve at points A and B. Find the equation of l in the form ax + by + c = 0. x = ln t - In 70 2 y = sint, 0 < t < 2π
The curve C has parametric equations x = 2 ln t, y = t2 − 1, t > 0.a. Find the coordinates of the point where the line x = 5 intersects the curve. Give your answer as exact values.b. Given that the line y = k intersects the curve, find the range of values for k.
Find dy/dx for each of the following: a y = 4e7x 4(1)* e y = 4 b f y = 3x y = ln (2x³) X c y = - (2) g y = ³x - e-3x d y = ln 5x (1 + ex)² h_y= ex
Differentiate:a. x(1 + 3x)5b. 2x(1 + 3x2)3c. x3(2x + 6)4d. 3x2(5x − 1)−1
a. Given that f(x) = cos x, show that b. Hence prove that f'(x) = −sin x. hosh-1) 1 cosx. h f'(x) = lim sin h h sin x
Differentiate with respect to x:a. ln x2 b. x2 sin 3x
Find the f'(x) given that a f(x) = 34x f(x) = 2x c f(x) = 24x + 4²x d f(x): = 27x + 8x 42.x
Differentiate:a. y = 2 cos xb. y = 2 sin1/2x c. y = sin 8xd. y = 6 sin 2/3x
a. Given that 2y = x − sin x cos x, 0 < x < 2π, show that dy/dx = sin2 x.b. Find the coordinates of the points of inflection of the curve. a sin x X b ln , x > 0 1 x² +9
Differentiatea. e−2x(2x − 1)5b. sin 2x cos 3xc. ex sin xd. sin (5x) ln (cos x)
Find the function f'(x) where f(x) is: a (sec.x) 2 e sec³ x b √cotx f cot³ x c cosec² x d tan² x
Find the gradient of the curve y = (e2x − e−2x)2 at the point where x = ln 3.
Find the stationary points of the curve C with the equation y = (x − 2)2(2x + 3).
Find f'(x) where f(x) is: a x² sec 3x In x tan x e b f tan 2x X etan x COS X с tan x d ex sec 3x
Given that y =1/(4x + 1)2 find the value of dy/dx at (1/4, 1/4).
a. Given that f(x) is increasing on the interval [−k, k], find the largest possible value of k.b. Find the exact coordinates of the points of inflection of f(x). f(x) = X x² + 2² XER
Find the equation of the tangent to the curve y = 2x + 2−x at the point (2, 17/4).
Find fy/dx given that:a. y = sin 2x + cos 3xb. y = 2 cos 4x − 4cosx + 2cos7xc. y = x2 + 4 cos 3xd. y = 1 + 2x sin 5x/x
A curve C has equation y = (5 − 2x)3. Find the tangent to the curve at the point P with x-coordinate 1.
A curve has equation y = x − sin 3x. Find the stationary points of the curve in the interval 0 ≤ x ≤ π.
A curve has the equation y = e2x − ln x. Show that the equation of the tangent at the point with x-coordinate 1 is y = (2e2 − 1)x − e2 + 1.
a. Given that ax ≡ ekx, where a and k are constants, a > 0 and x ∈ R, prove that k = ln a.b. Hence, using the derivative of ekx, prove that when y = 2x dy/dx = 2x ln 2c. Hence deduce that the gradient of the curve with equation y = 2x at the point (2, 4) is ln 16.
Find the value of dy/dx at the point (8, 2) on the curve with equation 3y2 − 2y = x.
Assuming standard results for sin x and cos x, prove that:a. The derivative of arc cos x is −a/√1 - x2b. The derivative of arc tan x is 1/1 - x2.
A student is attempting to differentiate ln kx. The student writes: Explain the mistake made by the student and state the correct derivative. y = In kx, so dy -= k Inkx dx
Find dy/dx for the following curves, giving your answers in terms of y.a. x = y2 + yb. x = ey + 4y/c. x = sin 2yd. 4x = ln y + y3
The function f is defined for positive real values of x by f(x) = 12 ln x + x3/2a. Find the set of values of x for which f(x) is an increasing function of x.b. Find the coordinates of the point of inflection of the function f.
A curve C has equation y = x2 cos (x2). Find the equation of the tangent to the curve C at the point in the form ax + by + c = 0 where a, b and c are exact constants. P να πν π 2 8
Find the gradient of the curve y = 2 sin 4x − 4 cos 2x at the point where x = π/2.
A particular radioactive isotope has an activity, R millicuries at time t days, given by the equation R = 200 × 0.9t. Find the value of dR/dt, when t = 8.
Show that if y = sec x then dy/dx = sec x tan x.
Given that a curve has equation y = cos2 x + sin x, 0 < x < 2π, find the coordinates of the stationary points of the curve.
Given that y = 3x2(5x − 3)3, show thatwhere n, A, B and C are constants to be determined. dy dx = Ax (5x - 3)" (Bx + C)
A curve has the equation y = 2 sin 2x + cos 2x. Find the stationary points of the curve in the interval 0 ≤ x ≤ π.
The population of Cambridge was 37 000 in 1900, and was about 109 000 in 2000. Given that the population, P, at a time t years after 1900 can be modelled using the equation P = P0kt,a. Find the values of P0 and kb. Evaluate dP/dt in the year 2000c. Interpret your answer to part b in the context of
Show that if y = cot x then dy/dx = −cosec2 x.
The maximum point on the curve with equation y = x √sin x, 0 < x < π, is the point A. Show that the x-coordinate of point A satisfies the equation 2 tan x + x = 0.
Find the value of dy/dx at the point (5/2, 4) on the curve with equation y² + y² = x.
A curve has the equation y = sin 5x + cos 3x. Find the equation of the tangent to the curve at the point (π, −1).
A curve C has equation y = (x + 3)2 e3x.a. Find dy/dx, using the product rule for differentiation.b, Find the gradient of C at the point where x = 2.
a. Find f'(x).b. By evaluating f'(6) and f'(7), show that the curve with equation y = f(x) has a stationary point at x = p, where 6 < p < 7.f(x) = e0.5x − x2, x ∈ R
A curve has the equation y = 2x2 − sin x. Show that the equation of the normal to the curve at the point with x-coordinate π is x + (4π + 1)y − π(8π2 + 2π + 1) = 0.
a. Differentiate ey = x with respect to y. b. Hence, prove that if y = ln x, then dy/dx = 1/x.
Prove that the derivative of akx is ax k ln a. You may assume that the derivative of ekx is kekx.
Prove, from first principles, that the derivative of sin x is cos x. Prove, from first principles, that the derivative of sin x is cos x. h→0, sin h h 1 and cosh - 1 h 0.
a. Use calculus to find the coordinates of the turning points on the graph of y = f(x).f(x) = e2x sin 2x, 0 < x < πb. Show that f''(x) = 8e2x cos 2x.c. Hence, or otherwise, determine which turning point is a maximum and which is a minimum.d. Find the points of inflection of f(x).
Given that the curve C has equationa. Show that the value of dy/dx whenb. Find the equation of the normal to the curve C at x = √3/2. y = arctan 2x X
The curve C has equation x = 4 cos 2y. a. Show that the point Q (2, π/2) lies on C.b. Show that dy/dx = - 1/ 4√3 at Q.c. Find an equation of the normal to C at Q. Give your answer in the form ax + bx + c = 0, where a, b and c are exact constants.
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