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mathematics
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 diagram shows an open water container in the shape of a triangular prism of length y cm.The vertical cross-section is an isosceles triangle with sides 5xcm, 5xcm and 6xcm.The water container is made from 500 cm2 of sheet metal and has a volume of V cm3.a. Express y in terms of x.b. Show
Find the coordinates of the stationary points on the curve y = x4 − 4x3 + 4x2 + 1 and determine the nature of each of these points. Sketch the graph of the curve.
The curve y = x3 + ax2 + b has a stationary point at (4, −27).a. Find the value of a and the value of b.b. Determine the nature of the stationary point (4, −27).c. Find the coordinates of the other stationary point on the curve and determine the nature of this stationary point.d. Find the
A manufacturing company produces x articles per day. The profit function, P(x), can be modeled by the function P(x) = 2x3 − 81x2 + 840x. Find the range of values of x for which the profit is decreasing.
A point, P(x, y), travels along the curve y = x3 − 5x2 + 5x in such a way that the rate of change of x is constant. Find the values of x at the points where the rate of change of y is double the rate of change of x.
Oil is poured onto a flat surface and a circular patch is formed.The radius of the patch increases at a rate of 2√r cms−1.Find the rate at which the area is increasing when the circumference is 8πcm.
A curve has equation y = x3 + x2 − 5x + 7.a. Find the set of values of x for which the gradient of the curve is less than 3.b. Find the coordinates of the two stationary points on the curve and determine the nature of each stationary point.
The equation of a curve is y = x3 + px2, where p is a positive constant.i. Show that the origin is a stationary point on the curve and find the coordinates of the other stationary point in terms of p.ii. Find the nature of each of the stationary points.Another curve has equation y = x3 + px2 +
Paint is poured onto a flat surface and a circular patch is formed.The area of the patch increases at a rate of 5cm2s−1.a. Find, in terms of π, the radius of the patch after 8 seconds.b. Find, in terms of π, the rate of increase of the radius after 8 seconds
A cylindrical container of radius 8cm and height 25cm is completely filled with water.The water is then poured at a constant rate from the cylinder into an empty inverted cone.The cone has radius 15cm and height 24cm and its axis is vertical.It takes 40 seconds for all of the water to be
a. Show that the equation 3sin2θ + 5sinθcosθ = 2 cos2θ can be written in the form 3tan2θ + 5tanθ − 2 = 0.b. Hence, solve the equation 3sin2θ+ 5sinθcosθ = 2 cos2θ for 0° ≤ θ ≤ 180°.
The curve y = 2x3 + ax2 + bx + 7 has a stationary point at the point (2, −13).a. Find the value of a and the value of b.b. Find the coordinates of the second stationary point on the curve.c. Determine the nature of the two stationary points.d. Find the coordinates of the point on the curve where
Solve each of these equations for 0° ≤ x ≤ 360°.a. 4 cos2x = 1 b. 4tan2x = 9
a. Expand (x2 − 1)4 .b. Find the coefficient of x6 in the expansion of (1 − 2x2)(x2 − 1)4.
By considering the gradient of a suitable sequence of chords, find a value for the gradient of the curve at the given point.a. y = x4 at (1, 1) b. y = x2 − 2x + 3 at (0, 3)c. y = 2√x at (4, 4) d. y = 12/x at (2, 6)
A curve has equation y = 3x3 − 3x2 + x − 7. Show that the gradient of the curve is never negative.
A curve passes through the pointa. Find the equation of the tangent to the curve at the point A.b. Find the equation of the normal to the curve at the point A. 4 (2, 1); A and has equation y = 8 (x + 2)²
Find f´(x) for each of the following.a. f(x) = 2x4b. f(x) = 3x5c. d.e.f. f(x) = −2g.h. f(x)= = 2
The equation of a curve is y = (3 − 5x)3 − 2x. Find dy/dx and d2y/dx2.
Given thatfind the value of f´´(−4). f(x) = 2 √√1-2x
Find the gradient of the curveat the points where the curve crosses the x-axis. 3 x + 2 y = x - -
The tangents to the curve y = 5 − 3x − x2 at the points (−1, 7) and (−4, 1) meet at the point Q.Find the coordinates of Q.
The curve y = x(x − 3)(x − 5) crosses the x-axis at the points O(0, 0), A(3, 0) and B(5, 0).The tangents to the curve at the points A and B meet at the point C.Find the coordinates of the point C.
Given thatfind f´´(x) . f'(x) = 3 (2x - 1)8⁹
Find the gradient of the curveat the point where x = 5. y = 15 x² - 2x
Given that f(x) = x3 + 2x2 − 3x − 1, find:a. f(1) b. f´(1) c. f´´(1)
Find dy/dx for each of the following.a. y = 5x2 − x + 1 b. y = 2x3 + 8x − 4 c. y = 7 − 3x + 5x2d. y = (x + 5)(x − 4)e.f.g. h.i. y = (2x² - 3)²
Find the gradient of the curve y = (2x − 3)5 at the point (2, 1).
The equation of a curve is y = 5 − 3x − 2x2.a. Show that the equation of the normal to the curve at the point (−2, 3) is x + 5y = 13.b. Find the coordinates of the point at which the normal meets the curve again.
Find the gradient of the curveat the point where the curve crosses the y-axis. y = 6 2 (x - 1)²
The normal to the curve y = x3 − 5x + 3 at the point (−1, 7) intersects the y-axis at the point P.Find the coordinates of P.
Given that y = x2 − 2x + 5, show that d²y - + (x - 1) / dy dx 4. dx² = 2y.
Find the gradient of the curve y = 5x2 − 8x + 3 at the point where the curve crosses the y-axis.
The equation of a curve isa. Find dy/dxb. Show that the normal to the curve at the point meets the y-axis at the point (0, −3). y = 2x 10 X² - + 8.
The curve passes through the point (2, 1) and has gradient − 3/5 at this point.Find the value of a and the value of b. y = a bx - 1
The normal to the curveat the point (9, 4) meets the x-axis at P and the y-axis at Q.Find the length of PQ, correct to 3 significant figures. y = ע 12
Given that xy = 12, find the value of dy/dx when x = 2.
A curve has equation y = x3 − 6x2 − 15x − 7. Find the range of values of x for which both dy/dx and d2y/dx2 are positive.
Find the coordinates of the point on the curve where the gradient is 0. y = √√(x² - 10x +26)
The equation of a curve is a. Find dy/dx.b. Show that the normal to the curve at the point (2, 13) meets the x-axis at the point (28, 0). y = 5.x + 12 X
The normal to the curve y = 4 − 2√x at the point P(16, −4) meets the x-axis at the point Q.a. Find the equation of the normal PQ.b. Find the coordinates of Q.
A curve has equation y = 2x3 − 21x2 + 60x + 5. Copy and complete the table to show whether are positive (+), negative (−) or zero (0) for the given values of x. dy dx and d²y dx²
Find the gradient of the curve y = (2x − 5)(x + 4) at the point (3, 7) .
The normal to the curve y = 5 √x at the point P(4, 10) meets the x-axis at the point Q.a. Find the equation of the normal PQ.b. Find the coordinates of Q.
The diagram shows part of the graph of y = a + b sin x.State the values of the constants a and b. y 3 y = a + b sinx 2 An -I -π -π 2 2 -1- اسان B 3π 2 2π x
Write down the period of each of these functions.a. y = cos x° b. y = sin2x°c.d. y = 1 + 2 sin3x° e. y = tan(x − 30)° f. y = 5 cos(2x + 45)° y = 3tan-xº
Express 2 sin2x − 7 cos2x + 4 in terms of sinx.
Given that and that θ is acute, find the exact value of:a. sinθb. tanθc. 2sinθcosθd. e.f. cose = 5
Without using a calculator, write down, in degrees, the value of:a. cos−11b.c. tan−1√3d. (sin−1) (−1) e. (tan)−1 (− √3)f. 머글 sin 2
For each of the following diagrams, find the basic angle of θ.a.b.c.d. 7 0 8110⁰ x
a. Show that the equation cosθ + sinθ = 5 cosθ can be written in the form tanθ = k.b. Hence, solve the equation cosθ+ sinθ = 5 cosθ for 0° ≤ θ ≤ 360°.
Write down the amplitude of each of these functions.a. y = sin x° b. y = 5 cos 2x°c.d. y = 2 − 3cos 4x° e. y = 4sin(2x + 60)° f. y = 2 sin(3x + 10)° + 5 y = 7 sinx
Solve each of these equations for 0° ≤ x ≤ 360°.a. tanx = 1.5 b. sinx = 0.4 c. cosx = 0.7 d. sinx = −0.3e. cosx = −0.6 f. tanx = −2 g. 2 cosx − 1 = 0 h. 5sinx + 3 = 0
Find the value of x satisfying the equation sin−1(x − 1) = tan−1(3).
Prove each of these identities.a. cosx tanx ≡ sinxb.c.d.e.f. cos4 x + sin2 x cos2 x ≡ cos2 x 1- cos²x sin x cos x = tan x
Draw a diagram showing the quadrant in which the rotating line OP lies for each of the following angles. On each diagram, indicate clearly the direction of rotation and state the acute angle that the line OP makes with the x-axis.a. 100° b. −100°c. 310° d.−150°e. 400°f.g.h.i.j.
Given thatand that θ is acute, find the exact value of:a. sinθb. cosθc. sin2θ + cos2θd.e. f. tan0 = 2 √5
Without using a calculator, find the exact values of each of the following.a. cos120° b. tan330° c. sin225° d. tan(−300°)e.f.g.h. sin 4T | 3
Without using a calculator, write down, in terms of, the value of:a. sin−1 0 b. tan−11c.d.e.f. '() cos-1
Given that and that θ is acute, find the exact value of:a. cosθb. tanθc. 1 _ sin2θd. e.f. sin 0 = 4
Given thatfind the exact value of:a. sin2θb. tan2θ 0 = cos 3/5
Sketch the graph of each of these functions for 0° ≤ x ≤ 360°.a. y = 2 cos xb.c. y = tan3xd. y = 3cos 2x e. y = 1 + 3cos x f. y = 2 sin3x − 1g. y = sin(x − 45) h. y = 2 cos(x + 60) i. y = tan(x − 90) y = sin x 2
Solve each of the these equations for 0 ≤ x ≤ 2.a. sinx = 0.3 b. cosx = 0.5 c. tanx = 3 d. sinx = −0.7e. tanx = −3 f. cosx = −0.5 g. 4sinx = 3 h. 5tanx + 7 = 0
In each part of this question you are given the basic angle, b, the quadrant in which θ lies and the range in which θ lies. Find the value of θ.a. b = 55°, second quadrant, 0°< θ < 360°b. b = 20°, third quadrant, −180°< θ< 0°c. b = 32°, fourth quadrant,
a. Sketch the graph of each of these functions for 0 ≤ x ≤ 2.i. y = 2 sin xii. iii.b. Write down the coordinates of the turning points for your graph for part a iii. cos (x-7) 2 y = cos
Given that is an acute angle measured in radians and that cos θ = k, find, in terms of k, an expression for:a. sin θb. tan θc. cos(π − θ)
Prove each of these identities.a. (sin x + cos x)2 ≡ 1 + 2 sinxcosx b. 2(1 + cos x) − (1 + cos x)2 ≡ sin2xc. 2 − (sin x + cos x)2 ≡ (sin x − cos x)2 d. (cos2x − 2)2 − 3sin2 x ≡ cos4x + sin2x
Given that sinθ < 0 and tanθ < 0, name the quadrant in which angle θ lies.
Solve each of these equations for 0° ≤ x ≤ 180°.a. cos 2x = 0.6 b. sin3x = 0.8 c. tan2x = 4 d. sin2x = −0.5e. 3cos 2x = 2 f. 5sin2x = −4 g. 4 + 2 tan2x = 0 h. 1 − 5sin2x = 0
a. Show that the equation 8sin2θ + 2 cos2θ − cosθ = 6 can be written in the form 6 cos2θ + cosθ − 2 = 0.b. Hence, solve the equation 8sin2θ + 2 cos2θ − cosθ = 6 for 0° ≤ θ ≤ 360°.
Solve the equation cos−1(8x4 + 14x2 − 16) = π.
Prove each of these identities.a. cos2 x − sin2 x ≡ 2 cos2x − 1 b. cos2x − sin2x ≡ 1 − 2 sin2xc. tan2x − sin2x ≡ tan2 x sin2x d. cos4x + sin2x ≡ sin4 x + cos2x
Find the exact value of each of the following.a. sin30°cos 60° b. sin245° c. sin45° + cos 30°d.e.f. sin 60° sin 30°
The function f(x) = 3sin x − 4 is defined for the domaina. Find the range off. b. Find f −1(x) . TC 2
Solve each of these equations for the given domains.a. sin(x − 60°) = 0.5 for 0° ≤ x ≤ 360°b.c. cos(2x + 45°) = 0.8 for 0° ≤ x ≤ 180° d. 3sin(2x − 4) = 2 for 0 < x < πe.f. cos(x + 7) = -0.5 for 0 < x < 2π
Given thatand that θ is obtuse, find the value of:a. cosθ b. tanθ sin0 = 2 5
Prove each of these identities.a.b.c.d.e.f.g.h. cos²x - sin²x cos x − sin x = cos x + sinx
Find the exact value of each of the following.a. b.c.d.e.f. sin cos
Solve the equation 13 sin² 0 2 + cose + cose 2 for 0° 0 180°. ≤ =
a. Show that the equation 4sin4θ + 14 = 19 cos2θ can be written in the form 4x2 + 19x − 5 = 0, where x = sin2θ.b. Hence, solve the equation 4sin4θ + 14 = 19 cos2θ for 0° ≤ θ ≤ 360°.
Solve the equation sin2x = 5 cos2x, for 0° ≤ x ≤ 180°.
Given thatand that 180° ≤ θ ≤ 270°, find the value of:a. sinθb. tanθ cos 0 = √√3
In the table, and the missing function is from the list Without using a calculator, copy and complete the table. 0
The function f(x) = 5 − 2 sin x is defined for the domaina. Find the largest value of p for which f has an inverse.b. For this value of p, find f −1(x) and state the domain of f−1. EIN V/ x = p. VI
The function f(x) = 4 − 2 cos x is defined for the domain 0 ≤ x ≤ π.a. Find the range of f and sketch the graph of y = f(x).b. Explain why f has an inverse and find the equation of this inverse.c. Sketch the graph of y = f −1(x) on your graph for part a.
Prove each of these identities.a.b.c.d.e.f. -| cos x COS X 1 + sin x = tan x
Solve each of these equations for 0° ≤ x ≤ 360°.a. 2 sinx = cos x b. 2 sinx − 3cos x = 0c. 4sinx + 7 cos x = 0 d. 3cos2x − 4sin2x = 0
Given thatand that 180° ≤θ ≤ 360°, find the value of:a. sinθb. cosθ tan 0 = 5 12
a. Show that the equation sinθtanθ = 3 can be written in the form cos2θ + 3 cosθ − 1 = 0.b. Hence, solve the equation sinθtanθ = 3 for 0° ≤ θ ≤ 360°.
a. Show that the equation 5(2 sinθ − cosθ) = 4(sinθ + 2 cosθ) can be written in the form b. Hence, solve the equation 5(2sinθ − cosθ) = 4(sinθ + 2cosθ) for 0° ≤ θ ≤ 360°. tan0 = 13 6
a. On the same diagram, sketch the graphs of y = 2 sin x and y = 2 + cos3x for 0 ≤ x ≤ 2.b. Hence, state the number of solutions, in the interval 0 ≤ x ≤ 2, o f the equation 2sinx = 2 + cos3x.
The functionis defined for the domain 0 ≤ x ≤ 2π.a. Find the range of f . b. Find f−1(x) and state its range. * ( 1 ) - 5 2 f(x) = 4 cos
a. Prove the identityb. Hence, solve the equation sin 0 1+ cose + 1+ cose sin 0 2 sin 0
Solve 4sin(2x + 0.3) − 5 cos(2x + 0.3) = 0 for 0 ≤ x ≤ π.
i. Sketch, on a single diagram, the graphs of y = cos 2θ andii. Write down the number of roots of the equation 2cos2θ − 1 = 0 in the interval 0 ≤ θ≤ 2π.iii. Deduce the number of roots of the equation 2cos2θ − 1 = 0 in the interval 10π ≤ θ ≤ 20π. y = 2 for 0 ≤ 0 < 2n.
Solve the equation 2 cos2x = 5sin x − 1 for 0° ≤ x ≤ 360°.
a. Prove the identityb. Hence, solve the equation cos 1 tan 0 (1 + sine) sin e - 1.
a. On the same diagram, sketch and label the graphs of y = 3sinx and y = cos2x for the interval 0 ≤ x ≤ 2π.b. State the number of solutions of the equation 3sinx = cos2x in the interval 0 ≤ x ≤ 2.
Show that (1 + cosx)2 + (1 − cosx)2 + 2 sin2x has a constant value for all x and state this value.
Part of the graph y = a sinbx + c is shown above.Find the value of a, the value of b and the value of c. YA M 9 8 7 6 5 4 3 2. 1 0 T n E 3r 2π

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