Edexcel AS And A Level Mathematics Pure Mathematics Year 1/AS 1st Edition Greg Attwood - Solutions

Sketch, on separate sets of axes, the graphs of the following in the interval −360° ≤ θ ≤ 360°. In each case give the periodicity of the function.a. y = sin 1/2 θ b. y = −1/2 cos θ c. y = tan (θ − 90°) d. y = tan 2θ
Show that cos A = 1/8 4 cm B A 6cm 5 cm C
In △ABC, AB = 10 cm, BC = a√3 cm, AC = 5√13 cm and ∠ABC = 150°. Calculate:a. The value of ab. The exact area of △ABC.
In △PQR, PQ = 15 cm, QR = 12 cm and ∠PRQ = 75°. Find the two remaining angles.
A helicopter flies on a bearing of 080° from A to B, where AB = 50 km.It then flies for 60 km to a point C.Given that C is 80 km from A, calculate the bearing of C from A.
In △ABC, AB = x cm, AC = (5 + x) cm and ∠BAC = 150°. Given that the area of the triangle is 3(3/4)cm2a. Show that x satisfies the equation x2 + 5x − 15 = 0.b. Calculate the value of x, giving your answer to 3 significant figures.
In the diagram AD = DB = 5 cm, ∠ABC = 43° and ∠ACB = 72°. Calculate:a. ABb. CD 72° A 5 cm D 5cm 43° B
In the triangle, cos ∠ABC = 5/8a. Calculate the value of x.b. Find the area of triangle ABC. B 6cm 2cm C (x + 1) cm
The graph shows the curve with equation y = sin (x + k), −360° ≤ x ≤ 360°, where k is a constant.a. Find one possible value for k.b. Is there more than one possible answer to part a? Give a reason for your answer. -240° y 1 -60⁰0 -1- 120⁰ 300°
Show that cos P = −1/4 2 cm 4 cm P 3 cm R
In a triangle, the largest side has length 2 cm and one of the other sides has length √2 cm.Given that the area of the triangle is 1 cm2, show that the triangle is right-angled and isosceles.
The distance from the tee, T, to the flag, F, on a particular hole on a golf course is 494 yards. A golfer’s tee shot travels 220 yards and lands at the point S, where ∠STF = 22°.Calculate how far the ball is from the flag.
A park is in the shape of a triangle ABC as shown. A park keeper walks due north from his hut at A until he reaches point B. He then walks on a bearing of 110° to point C.a. Find how far he is from his hut when at point C. Give your answer in km to 3 s.f.b. Work out the bearing of the hut from
The three points A, B and C, with coordinates A(0, 1), B(3, 4) and C (1, 3) respectively, are joined to form a triangle.a. Show that cos ∠ACB = −4/5b. Calculate the area of △ABC.
A zookeeper is building an enclosure for some llamas. The enclosure is in the shape of a quadrilateral as shown. If the length of the diagonal BD is 136 ma. Find the angle between the fences AB and BCb. Find the length of fence AB A 76m 66° D 80m 98° C B
Town B is 6 km, on a bearing of 020°, from town A. Town C is located on a bearing of 055° from town A and on a bearing of 120° from town B. Work out the distance of town C from:a. Town A b. Town B
The longest side of a triangle has length (2x − 1) cm. The other sides have lengths (x − 1) cm and (x + 1) cm. Given that the largest angle is 120°, work outa. The value of x.b. The area of the triangle.
In △ABC, AB = √2 cm, BC = √3 cm and ∠BAC = 60°. Show that ∠ACB = 45° and find AC.
The variation in the depth of water in a rock pool can be modelled using the function y = sin (30t)°, where t is the time in hours and 0 ≤ t ≤ 6.a. Sketch the function for the given interval.b. If t = 0 represents midday, during what times will the rock pool be at least half full?
The diagram shows part of the graph of y = f(x).It crosses the x-axis at A(120°, 0) and B(p, 0).It crosses the y-axis at C (0, q) and has a maximum value at D, as shown.Given that f(x) = sin (x + k), where k > 0, write downa. The value of pb. The coordinates ofc. The smallest value of kd. The
In △ABC, AB = 5 cm, BC = 6 cm and AC = 10 cm. Calculate the size of the smallest angle.
In △ABC, AB = (2 − x) cm, BC = (x + 1) cm and ∠ABC = 120°.a. Show that AC2 = x2 − x + 7.b. Find the value of x for which AC has a minimum value.
A windmill has four identical triangular sails made from wood. If each triangle has sides of length 12 m, 15 m and 20 m, work out the total area of wood needed.
In △ABC, AB = 9.3 cm, BC = 6.2 cm and AC = 12.7 cm. Calculate the size of the largest angle.
In △ABC, AB = x cm, BC = (4 − x) cm, ∠BAC = y and ∠BCA = 30°. Given that sin y = 1/√2, show that x = 4(√2 − 1)
Triangle ABC is such that BC = 5√2 cm, ∠ABC = 30° and ∠BAC = θ, where sin θ = √5/8 Work out the length of AC, giving your answer in the form a √b , where a and b are integers.
Two points, A and B are on level ground. A church tower at point C has an angle of elevation from A of 15° and an angle of elevation from B of 32°. A and B are both on the same side of C, and A, B and C lie on the same straight line. The distance AB = 75 m.Find the height of the church tower.
In △ABC, AB = x cm, BC = 5 cm, AC = (10 − x) cm.a. Show that b. Given that cos ∠ABC = −1/7 , work out the value of x. LABC= 4x - 15 2x
The lengths of the sides of a triangle are in the ratio 2 : 3 : 4. Calculate the size of the largest angle.
The perimeter of △ABC = 15 cm. Given that AB = 7 cm and ∠BAC = 60°, find the lengths of AC and BC and the area of the triangle.
Describe geometrically the transformations which map:a. The graph of y = tan x onto the graph of tan 1/2 xb. The graph of y = tan 1/2 x onto the graph of 3 + tan 1/2 xc. The graph of y = cos x onto the graph of −cos xd. The graph of y = sin (x − 10) onto the graph of sin (x + 10).
In △ABC, AB = (x − 3) cm, BC = (x + 3) cm, AC = 8 cm and ∠BAC = 60°. Use the cosine rule to find the value of x.
In the triangle ABC, AB = 14 cm, BC = 12 cm and CA = 15 cm.a. Find the size of angle C, giving your answer to 3 s.f.b. Find the area of triangle ABC, giving your answer in cm2 to 3 s.f.
a Sketch on the same set of axes, in the interval 0 ≤ x ≤ 180°, the graphs of y = tan (x − 45°) and y = −2 cos x, showing the coordinates of points of intersection with the axes.b. Deduce the number of solutions of the equation tan (x − 45°) + 2 cos x = 0, in the interval 0 ≤ x ≤
In △ABC, AB = x cm, BC = (x − 4) cm, AC = 10 cm and ∠BAC = 60°.Calculate the value of x.
In △ABC, AB = (5 − x) cm, BC = (4 + x) cm, ∠ABC = 120° and AC = y cm.a. Show that y2 = x2 − x + 61.b. Use the method of completing the square to find the minimum value of y2, and give the value of x for which this occurs.
A microchip company models the probability of having no faulty chips on a single production run as:where p is the probability of a single chip being faulty, and n being the total number of chips produced.a. State why the model is restricted to small values of p.b. Given that n = 200, find an
After 5 years, the value of an investment of £500 at an interest rate of X % per annum is given by:Find an approximation for this expression in the form A + BX + CX2, where A, B and C are constants to be found. You can ignore higher powers of X. 500(1+ X 100 5
Show that:a. nC1 = nb. "C₂= n(n-1) 2
The coefficient of x2 in the binomial expansion of (1 + x/2)n, where n is a positive integer, is 7.a. Find the value of n.b. Using the value of n found in part a, find the coefficient of x4.
Given thatwrite down the value of a. 50 13. ♡ 50! 13!a!?
In the binomial expansion of (1 + x)30, the coefficients of x9 and x10 are p and q respectively. Find the value of q/p.
a. Use the binomial theorem to expand (3 + 10x)4 giving each coefficient as an integer.b. Use your expansion, with an appropriate value for x, to find the exact value of 10034. State the value of x which you have used.
Given thatwrite down the value of p. (35)= 35! p!18!'
a. Expand (1 + 2x)12 in ascending powers of x up to and including the term in x3, simplifying each coefficient.b. By substituting a suitable value for x, which must be stated, into your answer to part a, calculate an approximate value of 1.0212.c. Use your calculator, writing down all the digits in
Expandsimplifying the coefficients. (x - 1)³
In the binomial expansion of (2k + x)n, where k is a constant and n is a positive integer, the coefficient of x2 is equal to the coefficient of x3.a. Prove that n = 6k + 2.b. Given also that k = 2/3, expand (2k + x)n in ascending powers of x up to and including the term in x3, giving each
a. Expand (2 + x)6 as a binomial series in ascending powers of x, giving each coefficient as an integer.b. By making suitable substitutions for x in your answer to part a, show that (2 + √3)6 − (2 − √3)6 can be simplified to the form k√3 , stating the value of the integer k.
The coefficient of x2 in the binomial expansion of (2 + kx)8, where k is a positive constant, is 2800.a. Use algebra to calculate the value of k.b. Use your value of k to find the coefficient of x3 in the expansion.
a. Given that (2 + x)5 + (2 − x)5 ≡ A + Bx2 + Cx4, find the value of the constants A, B and C.b. Using the substitution y = x2 and your answers to part a, solve (2 + x)5 + (2 − x)5 = 349.
In the binomial expansion of (2 + px)5, where p is a constant, the coefficient of x3 is 135. Calculate:a. The value of p,b. The value of the coefficient of x4 in the expansion.
ABCD is a square. Angle CED is obtuse. Find the area of the shaded triangle. 10 cm C 8 cm E 40° B D
Consider the function f(x) = sin px, p ∈ ℝ, 0 ≤ x ≤ 360°.The closest point to the origin that the graph of f(x) crosses the x-axis has x-coordinate 36°.a. Determine the value of p and sketch the graph of y = f(x).b. Write down the period of f(x).
Given that angle A is obtuse andshow that cos A 7 11'
The scatter graph shows the height, h cm, and inseam leg measurement, l cm, of six adults. A line of best fit has been added to the scatter graph.a. Use two points on the scatter graph to calculate the gradient of the line.b. Use your answer to part a to write a linear model relating height and
Express in terms of sin θ only:a. cos2 θ b. tan2 θ c. cos θ tan θd. cos θ/tan θe. (cos θ − sin θ)(cos θ + sin θ)
Express the following in terms of trigonometric ratios of acute angles:a. sin 240° b. sin (−80°) c. sin (−200°) d. sin 300° e. sin 460°f. cos 110° g. cos 260° h. cos (−50°) i. cos (−200°) j. cos 545°k. tan 100° l. tan 325° m. tan
The diagram shows a sketch of y = cos x.a. Use your calculator to find the principal solution to the equation cos x = 0.4.b. Use the graph and your answer to part a to find solutions to the equation cos x = ± 0.4 in the range 0 ≤ x ≤ 360°. y 90 180° 270° 360° x
Solve for θ, in the interval 0 ≤ θ ≤ 180°, the following equations. Give your answers to 3 significant figures where they are not exact.a. 4 (sin2 θ − cos θ) = 3 − 2cos θ b. 2sin2 θ = 3(1 − cos θ) c. 4cos2 θ − 5sin θ − 5 = 0
Solve the following equations in the interval given:a. 3 sin 3θ = 2 cos 3θ, 0 ≤ θ ≤ 180°b. 4 sin (θ + 45°) = 5 cos (θ + 45°), 0 ≤ θ ≤ 450°c. 2 sin 2x – 7 cos 2x = 0, 0 ≤ x ≤ 180°d. √3 sin (x – 60°) + cos(x – 60°) = 0, –180° ≤ x ≤ 180°
The diagram shows a sketch of y = tan x.a. Use your calculator to find the principal solution to the equation tan x = −2.b. Use the graph and your answer to part a to find solutions to the equation tan x = −2 in the range 0 ≤ x ≤ 360°. 2- -2 90° 180° 270% 560⁰ x
Solve the following equations for θ, in the interval 0 < θ ≤ 360°:a. sin θ = −1 b. tan θ = √3 c. cos θ = 1/2d. sin θ = sin 15° e. cos θ = −cos 40° f. tan θ = −1g. cos θ = 0h. sin θ = −0.766
Given that sin x cos y = 3 cos x sin y, express tan x in terms of tan y.
Find the values of θ, in the interval 0 ≤ θ ≤ 360°, for which: a sin 40 = 0 d cos 20 = 1/2 b cos 30 -1 1 tan-20= √3 e c tan 20 = 1 f sin (-) = 1 √√2
Without using a calculator, write down the values of:a. sin (−90°) b. sin 450° c. sin 540° d. sin (−450°) e. cos (−180°)f. cos (−270°) g. cos 270° h. cos 810° i. tan 360° j. tan (−180°)
The points A(0, 3), B(k, 5) and C(10, 2k), where k is a constant, lie on the same straight line. Find the two possible values of k.
Without using your calculator, work out the values of:a. cos 270° b. sin 225° c. cos 180° d. tan 240° e. tan 135°
Solve for θ, in the interval −180° ≤ θ ≤ 180°, the following equations. Give your answers to 3 significant figures where they are not exact.a. sin2 2θ = 1 b. tan2 θ = 2 tan θc. cos θ (cos θ − 2) = 1 d. 4sin θ = tan θ
Solve the following equations in the interval given:a. tan (45° − θ ) = −1, 0 ≤ θ ≤ 360° b. 2 sin (θ − 20°) = 1, 0 ≤ θ ≤ 360°c. tan (θ + 75°) = √3 , 0 ≤ θ ≤ 360° d. sin (θ − 10°) = −√3/2 , 0 ≤ θ ≤ 360°e. cos (70° − x) = 0.6, 0 ≤ θ ≤
A farmer has a field in the shape of a quadrilateral as shown.The angle between fences AB and AD is 74°. Find the angle between fences BC and CD. 75m A D 120m 135 m B 60m
a. Sketch on separate sets of axes the graphs of y = cos θ (0 b. Verify that:i. cos α = −cos (180° − α) = −cos (180° + α) = cos (360° − α)ii. tan α = −tan (180° − α) = tan (180° + α) = −tan (360° − α)Data from Question 16The graph below shows y = sin θ, 0 ≤ θ ≤
Given that 2 sin θ = 3 cos θ, find the value of tan θ.
The diagram shows three cargo ships, A, B and C, which are in the same horizontal plane. Ship B is 50 km due north of ship A and ship C is 70 km from ship A. The bearing of C from A is 020°.a. Calculate the distance between ships B and C, in kilometres to 3 s.f.b. Calculate the bearing of ship C
State the quadrant that OP lies in when the angle that OP makes with the positive x-axis is:a. 400° b. 115° c. −210° d. 255° e. −100°
The graph below shows y = sin θ, 0 ≤ θ ≤ 360°, with one value of θ (θ = α) marked on the axis.a. Copy the graph and mark on the θ-axis the positions of 180° − α, 180° + α, and 360° − α.b. Verify that: sin α = sin (180° − α) = −sin (180° + α) = −sin (360° − α).
Solve for θ, in the interval 0 ≤ θ ≤ 360°, the following equations. Give your answers to 3 significant figures where they are not exact.a. 4cos2 θ = 1 b. 2sin2 θ − 1 = 0 c. 3sin2 θ + sin θ = 0d. tan2 θ − 2tan θ − 10 = 0 e. 2cos2 θ − 5cos θ + 2 = 0 f. sin2
Express the following as trigonometric ratios of either 30°, 45° or 60°, and hence find their exact values.a. sin 135° b. sin (−60°) c. sin 330° d. sin 420° e. sin (−300°)f. cos 120° g. cos 300° h. cos 225° i. cos (−210°) j. cos 495°k. tan
Draw diagrams to show the following angles. Mark in the acute angle that OP makes with the x-axis.a. −80° b. 100° c. 200° d. 165° e. −145°f. 225° g. 280° h. 330° i. −160° j. −280°
Find the equation of the line which passes through the points A(−2, 8) and B(4, 6), in the form ax + by + c = 0.
A series of sand dunes has a cross-section which can be modelled using a sine curve of the form y = sin (60x)° where x is the length of the series of dunes in metres.a. Draw the graph of y = sin (60x)° for 0 ≤ x ≤ 24°.b. Write down the number of sand dunes in this model. c. Give one
Write each of the following as a trigonometric ratio of an acute angle:a. cos 237° b. sin 312° c. tan 190°
The line l passes through the point (9, −4) and has gradient 1/3. Find an equation for l, in the form ax + by + c = 0, where a, b and c are integers.
Solve the following equations for x, giving your answers to 3 significant figures where appropriate, in the intervals indicated: a sin x = -- -180° ≤x≤ 540⁰ √3 2 c cos x = -0.809, -180° ≤ x ≤ 180° √√3 e tan x = - 0≤x≤720° 2 3 b 2 sin x = -0.3, -180° ≤ x ≤ 180° d cos x
a. Show that the equation cos2 x = 2 – sin x can be written as sin2x – sin x + 1 = 0.b. Hence show that the equation cos2 x = 2 – sin x has no solutions.
a. Sketch the graphs of y = 2 sin x and y = cos x on the same set of axes (0 ≤ x ≤ 360°).b. Write down how many solutions there are in the given range for the equation 2 sin x = cos x.c. Solve the equation 2 sin x = cos x algebraically, giving your answers in exact form.
Given that tan θ = − √3 and that θ is reflex, find the exact value of: a. sin θ b. cos θ
The circle C has centre (−3, 8) and passes through the point (0, 9). Find an equation for C.
Prove that, for all values of θ:a. (1 + sin θ)2 + cos2 θ ≡ 2(1 + sin θ) b. cos4 θ + sin2 θ ≡ sin4 θ + cos2 θ
The equation tan kx = −1/√3, where k is a constant and k > 0, has a solution at x = 60°a. Find a possible value of k.b. State, with justification, whether this is the only such possible value of k.
Show that the equation 2 cos2 x + cos x – 6 = 0 has no solutions.
Given that sin θ = 2/3 and that θ is obtuse, find the exact value of: a. cos θ b. tan θ
Using the identities sin2 A + cos2 A ≡ 1 and/orprove that: tan A = sin A cos A (cos A = 0), #
The perpendicular bisector of the line segment joining (5, 8) and (7, −4) crosses the x-axis at the point Q. Find the coordinates of Q.
a. Given that 2(sin x + 2 cos x) = sin x + 5 cos x, find the exact value of tan x.b. Given that sin x cos y + 3 cos x sin y = 2 sin x sin y − 4 cos x cos y, express tan y in terms of tan x.
Find, for 0 ≤ x ≤ 360°, all the solutions of sin2 x + 1 = 7/2 cos2 x giving each solution to one decimal place.
a. Given that 4 sin x = 3 cos x, write down the value of tan x.b. Solve, for 0 ≤ θ ≤ 360°, 4 sin 2θ = 3 cos 2θ giving your answers to 1 decimal place.
Find, without using your calculator, the values of:a. sin θ and cos θ, given that tan θ = 5/12 and θ is acute.b. sin θ and cos θ, given that cos θ = −3/5 and θ is obtuse.c. cos θ and tan θ, given that sin θ = −7/25 and 270° < θ < 360°.
Given that θ is an acute angle, express in terms of cos θ or tan θ:a. cos (180° − θ) b. cos (180° + θ) c. cos (−θ) d. cos (−(180° − θ))e. cos (θ − 360°) f. cos (θ − 540°) g. tan (−θ) h. tan (180° − θ)i. tan (180° + θ) j. tan
The lines y = 2x and 5y + x − 33 = 0 intersect at the point P. Find the distance of the point from the origin O, giving your answer as a surd in its simplest form.
Simplify the following expressions:a. cos4 θ − sin4 θ b. sin2 3θ − sin2 3θ cos2 3θc. cos4 θ + 2 sin2 θ cos2 θ + sin4 θ

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Edexcel AS And A Level Mathematics Pure Mathematics Year 1/AS 1st Edition Greg Attwood - Solutions

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