Questions and Answers of Pearson Edexcel A Level Mathematics

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
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
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
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
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
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
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
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
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
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
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
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
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 −
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
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
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
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
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 −
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
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
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
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
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
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
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
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
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
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
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
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,
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
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
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
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
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
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
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
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
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 –
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.
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 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
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
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
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
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
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
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
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

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