Questions and Answers of Edexcel AS And A Level Mathematics

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
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
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
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
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
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
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.
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
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.
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 +
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
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
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
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
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
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
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
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
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,
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 −
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
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
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
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
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
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
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 ≤
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
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 =
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
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
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 ≤
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
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.
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°
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
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
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.
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
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° ≤
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
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
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
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.
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

Showing 500 - 600 of 1136