- Find the following indefinite integrals:a.b.c.d.e. f.g.h.i.j.k.l. 1 xx
- Find y in each of the following cases:a. dy/dx = 4x + 2b. dy/dx = 6x2 – 5x – 1c. dy/dx = 3 – 5x3d = (x − 2)(3x + 2)
- Evaluate the following definite integrals:∫21 3x2 dx
- Find the area of each of the shaded regions: YA ア=? 3.
- The sketch shows the curve y = x3 – x.Calculate the area of the shaded region. /y x-x 0.
- Evaluate the following definite integrals:a.b.c.d.e.f.g.h.i.j. 4 1 2 3х +1
- Find f(x) given thata. f′(x) = 5x + 3b. f′(x) = x4 + 2x3 − x + 8c. f′(x) = (x − 4)(x2 + 2)d. f′(x) = (x − 7)2
- Evaluate the following definite integrals:∫41 4x3 dx
- Find the area of each of the shaded regions: YA ソ= 6r -?
- The sketch shows the curve y = x3 – 4x2 + 3x.a. Calculate the area of each shaded region.b. State the area enclosed between the curve and the x-axis. y. y = x3- 4x2. +3x P 3.
- Find the following integrals:a. ∫5 dxb. ∫5x3 dxc. ∫(2x – 5)dxd. ∫(3x2 – 4x + 3)dx
- Evaluate the following definite integrals:∫1-1 6x2 dx
- Find the area of each of the shaded regions: yA y = x³ + 1 1 1
- The sketch shows the curve y = x4 – 2x.a. Find the coordinates of the point A.b. Calculate the area of the shaded region. ly =x-2x A
- Find the following integrals:a. ∫(3 – x)2 dxb. ∫(2x + 1)(x – 3)dxc. ∫(x + 1)2 dxd. ∫(2x – 1)2 dx
- Evaluate the following definite integrals:∫51 4 dx
- Find the area of each of the shaded regions: yA y = x2 - 2x 4.
- The sketch shows the curve y = x3 + x2 – 6x.Work out the area between the curve and the x-axis. y y =x+x2-6x 23 2
- Find the equation of the curve y = f(x) that passes through the specified point for each of the following gradient functions:a. dy/dx = 2x – 3; (2, 4)b. dy/dx = 4 + 3x3 (4, -2)c. dy/dx = 5x – 6;
- Evaluate the following definite integrals:∫42 (x2 + 1) dx
- Find the area of each of the shaded regions: yA y = x3- 3x + 2x 2
- a. Sketch the curve y = x² for −3 < x < 3.b. Shade the area bounded by the curve, the lines x = −1 and x = 2 and the x-axis.c. Find, by integration, the area of the region you have shaded.
- Find the equation of the curve y = f(x) that passes through the specified point for each of the following gradient functions:a. dy/dx = 2√x – 1; (1, 1)b. f'(x) = x – √x; (4, 2)
- Evaluate the following definite integrals:∫3-2 (2x + 5) dx
- Find the area of each of the shaded regions: y = 5x4 -x5 YA
- a. Sketch the curve y = x² – 2x for −1 < x < 3.b. For what values of x does the curve lie below the x-axis?c. Find the area between the curve and the x-axis.
- You are given that dy/dx = 2x + 3.a. Find ∫(2x + 3) dx.b. Find the general solution of the differential equation.c. Find the equation of the curve with gradient function dy/dx and that passes
- Evaluate the following definite integrals:∫52 (4x3 – 2x + 1) dx
- Find the area of each of the shaded regions: yA y = 9-x 23 3
- a. Sketch the curve y = x3 for −3 < x < 3.b. Shade the area between the curve, the x-axis and the line x = 2.c. Find, by integration, the area of the region you have shaded.d. Without any
- The curve C passes through the point (3, 21) and its gradient at any point is given by dy/dx = 3x2 − 4x + 1.a. Find the equation of the curve C.b. Show that the point (−2, −9) lies on the curve.
- Evaluate the following definite integrals:∫65 (x2 – 5) dx
- Find the area of each of the shaded regions: y = 6+x -x2 2
- a. Shade, on a suitable sketch, the region with an area given by∫2-2(x2 + 1) dx.b. Evaluate this integral.
- a. Evaluate ∫41 (2x + 1) dx.b. Interpret this integral on a sketch graph.
- a. Find ∫(4x − 1) dx.b. Find the general solution of the differential equation dy/dx = 4x − 1.c. Find the particular solution that passes through the point (−1, 4).d. Does this curve pass
- Evaluate the following definite integrals:∫31 (x2 – 3x + 1) dx
- Find the area of each of the shaded regions: y = 4x -x 2
- The curve y = f(x) passes through the point (2, −4) and f´(x) = 2 − 3x2. Find the value of f(−1).
- Evaluate the following definite integrals:∫2-1 (x2 + 3) dx
- Find the area of each of the shaded regions: yA y = x - &r2 + 16x 4
- A curve, C, has stationary points at the points where x = 0 and where x = 2.a. Explain why dy/dx = x2 − 2x is a possible expression for the gradient of C.Give a different possible expression for
- Evaluate the following definite integrals:∫-1-4 (16 – x2) dx
- Evaluate the following definite integrals:∫31 (x + 1)(3 – x) dx
- Evaluate the following definite integrals:∫42 {3x(x + 2)} dx
- Evaluate the following definite integrals:∫1-1 (x + 1)(x – 1)
- Evaluate the following definite integrals:∫2-1 (x + 4x2) dx
- Evaluate the following definite integrals:∫1-1 x(x – 1)(x + 1) dx
- Evaluate the following definite integrals:∫3-1 (x3 + 2) dx
- Evaluate the following definite integrals:∫1-3(9 – x2) dx
- In each of the following cases t ≥ 0. The quantities are given in SI units, so distances are in metres and times in seconds.:i. Find expressions for the velocity and acceleration at time tii. Use
- Find expressions for the velocity, v, and displacement, s, at time t in each of the following cases:a. a = 2 − 6t; when t = 0, v = 1 and s = 0b. a = 4t; when t = 0, v = 4 and s = 3c. a = 12t2 −
- A particle is projected in a straight line from a point O. After t seconds its displacement, s metres, from O is given by s = 3t2 – t3.a. Write expressions for the velocity and acceleration at time
- A particle P sets off from the origin, O, with a velocity of 9 m s−1 and moves along the x-axis.At time t seconds, its acceleration is given by a = (6t− 12) m s−2.a. Find expressions for the
- A ball is thrown upwards and its height, h metres, above ground after t seconds is given by h = 1 + 4t – 5t².a. From what height was the ball projected?b. Write an expression for the velocity of
- A particle P starts from rest at a fixed origin O when t = 0. The acceleration a m s−2 at time t seconds is given by a = 6t − 6.a. Find the velocity of the particle after 1 second.b. Find the
- In the early stages of its motion the height of a rocket, h metres, is given by h = 1/6t4, where t seconds is the time after launch.a. Find expressions for the velocity and acceleration of the rocket
- The speed, v m s−1, of a car during braking is given by v = 30 − 5t, where t seconds is the time since the brakes were applied.a. Sketch a graph of v against t.b. How long does the car take to
- The velocity of a moving object at time t seconds is given by v m s−1, where v = 15t – 2t² – 25.a. Find the times when the object is instantaneously at rest.b. Find the acceleration at these
- A particle P moves in a straight line, starting from rest at the point O. t seconds after leaving O, the acceleration, a m s−2, of P is given by a = 4 + 12t.a. Find an expression for the velocity
- The velocity v m s−1, of a particle P at time t seconds is given by v = t3 − 4t2 + 4t + 2.P moves in a straight line.a. Find an expression for the acceleration, a m s−2, in terms of t.b. Find
- For each curve in question.i. Find dy/dx and the value(s) of x for which dy/dx = 0ii. Classify the point(s) on the curve with these x-valuesiii. Find the corresponding y-value(s)iv. Sketch the
- Express each angle in degrees, rounding your answer to 3 s.f. where necessary:a. 2π/3b. 5π/9c. 3cd. π/7e. 3π/8
- Express each angle in radians, leaving your answer in terms of π if appropriate:a. 120° b. 540° c. 22° d. 150° e. 37.5°
- The table gives information about some sectors of circles.Copy and complete the table. Leave your answers as a multiple of π where appropriate. Radius, Angle at centre Angle at centre Arc length,
- The table gives information about some sectors of circles. Copy and complete the table. Leave your answers as a multiple of π where appropriate. Radius, Angle at centre Arc length, Area, A (cm)
- OAB is a sector of a circle of radius 6 cm. ODC is a sector of a circle radius 10 cm. Angle AOB is π/3 .Express in terms of π:a. The area of ABCDb. The perimeter of ABCD. 元 w/ B 10 cm-
- a. Work out the area of the sector AOB.b. Calculate the area of the triangle AOB.c. Work out the shaded area. A б ст B
- The diagram shows the cross-section of a paperweight. The paperweight is a sphere of radius 5 cm with the bottom cut off to create a circular flat base with diameter 8 cm.a. Calculate the angle ACB
- The perimeter of the sector in the diagram is 6π + 16 cm.Calculate:a. Angle AOBb. The exact area of sector AOBc. The exact area of the triangle AOBd. The exact area of the shaded segment. 8 ст B A
- Prove that sin2θ − cos2θ = 3 − 2sin2θ − 4cos2θ.
- In the triangle PQR, PQ = 29 cm, QR = 21 cm and PR = 20 cm.a. Show that the triangle is right-angled.b. Write down the values of sin Q, cos Q and tan Q, leaving your answers as fractions.c. Use your
- Write each value in exact form. Do not use a calculator.a. i. sin 30° ii. cos 30° iii. tan 30°b. i. cosec 30°ii. sec 30°iii. cot 30°
- a. Sketch the curve y = cos x for 0° ≤ x ≤ 360°.b. Solve the equation cos x = 0.5 for 0° ≤ x ≤ 360°, and illustrate the two roots on your sketch.c. State two other roots of cos x = 0.5,
- For each transformation (i) to (iv):a. Sketch the graph of y = sin x and on the same axes sketch its image under the transformation.b. State the amplitude and period of the transformed graph.c. What
- Prove that 4cos2 θ + 5sin2 θ = sin2 θ + 4.
- Without using a calculator, show that cos² 60°cos² 45° = cos² 30°
- Prove that = 1 + 1/tan2 θ = 1/sin2 θ.
- Without using a calculator, show that cos 60°sin 30° + sin 60°cos 30° = 1
- Write each value in exact form. Do not use a calculator.a. i. sin 45°ii. cos 45°iii. tan 45°b. i. cosec 45°ii. sec 45°iii. cot 45°
- a. Sketch the curve of y = sin x for −2π ≤ x ≤ 2π.b. Solve the equation sin x = 0.6 for −2π ≤ x ≤ 2π, and illustrate all the roots on your sketch.c. Sketch the curve y = cos x for
- a. Apply each set of transformations to the graph of y = cos x.b. Sketch the graph of y = cos x and the transformed curve on the same axes.c. State the amplitude and period of the transformed
- Write each value in exact form. Do not use a calculator.a. i. sin π/3ii cos π/3iii. tan π/3b. i. cosec π/3ii. sec π/3iii. cot π/3
- Solve the following equations for 0° ≤ x ≤ 2π.a. tan x = √3b. sin x = 0.5c. cos x = - √3/2d. tan x = 1/√3e. cos x = −0.7 f. cos x = 0.3g. sin x = -1/3h. sin x = −1
- a. Apply these transformations to the graph of y = sin x.b. Sketch the graph of y = sin x and the transformed curve on the same axes.c. State the amplitude and period of the transformed graph.d. What
- In the triangle ABC, angle A = 90° and sec B = 2.a. Work out the size of angles B and C.b. Find tan B.c. Show that 1 + tan2 B = sec2 B.
- Prove that 1 – (sin θ – con θ)2/sin θ cos θ = 2.
- Without using a calculator, show that 3cos² π/3 = sin² π/3.
- Write the following as integers, fractions, or using square roots. You should not need your calculator.a. sin 45°b. cos 60°c. tan 45°d. sin 120° e. cos 150° f. tan 180°g. sin
- State the transformations needed, in the correct order, to transform the first graph to the second graph.a. y = tan x, y = 3 tan 2xb. y = tan x, y = 2 tan x + 1c. y = tan x, y = 2 tan (x − 180°)d.
- Solve the equation cosec2 θ + cot² θ = 2 for 0° < x < 180°.
- In the diagram, AB = 12 cm, angle BAC = 30°, angle BCD = 60° and angle BDC = 90°.a. Calculate the length of BD.b. Show that AC = 4 √3 cm. 12 cm 30° 60° A
- In the triangle ABC, angle A = 90° and cosec B = 2.a. Work out the size of angles B and C.AC = 2 unitsb. Work out the lengths of AB and BC.
- In this question all the angles are in the interval −180° to 180°.Give all answers correct to one decimal place.a. Given that cos α < 0 and sin α = 0.5, find α.b. Given that tan β = 0.3587
- State the transformations required, in the correct order, to obtain this graph from the graph of y = sin x. y A 3- 2- 17 -90° 90° 270° 360° 180° * -1-
- Show that cosec A/ cosec A – sin A = sec2 A.
- In the diagram, OA = 1 cm, angle AOB = angle BOC = angle COD = π/4 and angle OAB = angle OBC = angle OCD = π/2.a. Find the length of OD, giving your answer in the form a√2.b. Show that the
- Given that sin θ = 3/4 and θ is acute, find the values of sec θ and cot θ.
- Solve the equation sec2 θ = 4 for 0 ≤ θ ≤ 2π.
- a. Draw a sketch of the graph y = sin x and use it to demonstrate why sin x = sin (180° − x).b. By referring to the graphs of y = cos x and y = tan x, state whether the following are true or

Copyright © 2024 SolutionInn All Rights Reserved.