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(a)θ is an acute angle, with sinθ = 14. Find the value of cosθ, without using a calculator. Give your answer in surd form. Show your working out.
(a) working
(b)The diagram below shows the graph of y = cos2x for a limited domain. On the diagram, draw and label the graph of y = sin2x, using the same axes and scales. Use the fact that cos2A + sin2A = 1.
Given graph: y = cos²x — draw y = sin²x directly on it
In this question, all lengths are in cm. Áine has 100 sticks. Each stick has a different integer length from 1 to 100, so the lengths are 1 cm, 2 cm, 3 cm, …, 99 cm, 100 cm. Áine makes triangles, using three of these sticks each time.
(a)Áine makes the triangle shown below, with sides of length 12, 20, and x, where x ∈ ℕ. The size of the angle between the sides of length 12 and 20 is 58°, correct to the nearest degree.
(i)Find the area of the triangle, correct to the nearest cm2.
(a)(i)
(ii)Find the value of x ∈ ℕ.
(a)(ii)
(b)Áine also makes the triangle shown below. One side has a length of 72 and another has a length of y, where y ∈ ℕ. The sizes of two of the angles are 34° and 59°, each correct to the nearest degree, as shown. Find the value of y ∈ ℕ.
(b)
(c)Áine makes a number of different triangles with sides of length n, n+5, and n+10, where n ∈ ℕ. The angle between the sides of length n and n+5 is A, as shown in the diagram below. When Áine changes the value of n, this changes the shape of the triangle and the size of the angle A.
(i)A = 90° for one value of n ∈ ℕ. Find this value of n.
(c)(i)
(ii)Estimate the limit of the size of the angle A, as n tends to infinity. Write a sentence to justify your answer. For this part only, assume that n can get extremely big (beyond 100).
(c)(ii)
(d)Áine makes another triangle with three of her sticks, with sides of length 12, 24, and 30. Find how many different (non-congruent) triangles Áine could make that would be similar to this triangle. Show your working out. Remember that Áine only uses three sticks for each triangle, and that the length of each stick is a different whole number from 1 to 100, inclusive.
(d) working
Lee has a solar farm. The number of daylight hours per day in the solar farm over the course of one year can be modelled by the following function, H(t), for 0 ≤ t ≤ 365, t ∈ ℤ:
H(t) = 4·5 sin( 2π365 t ) + 12
Here, H is in hours, t is the number of days after 21st March, and 2π365t is in radians.
(i)Use H(t) to find the number of daylight hours on 28th March. Give your answer correct to 2 decimal places.
(d)(i)
(ii)Using H(t), solve an equation to find the first day (value of t) and the last day (value of t), after 21st March, for which the number of daylight hours is less than 8. Give each answer as a value of t ∈ ℤ, with 0 ≤ t ≤ 365. You do not need to find the actual date for each day.
(d)(ii)
(a)Find all six solutions to the following equation in A, where −360° ≤ A ≤ 720°:
sin A = 12
(a) — six solutions
(b)f(x) is the following function, where x ∈ ℝ is in radians:
f(x) = |4 sin x| − 1
Part of the graph of y = f(x) is shown below. Write down the period and range of f(x).
(b) — period & range
(c)In the triangle ABC, |AC| = 2, |BC| = 3, and |AB| = 4.
Use the Cosine Rule to find the value of tan∠CBA, without using a calculator. Give your answer in the form √nm, where n, m ∈ ℤ. Show all your working out.
(c) working
(b)In the diagram on the right, [BC] represents a flagpole. |AB| = 15 m and |BC| = 17·5 m, as shown. AB is perpendicular to BC. Ally measures the size of ∠CAB, the angle of elevation of the flagpole. She makes a mistake, and measures that |∠CAB| is 52°, which is not correct. Work out the percentage error in Ally’s value for |∠CAB|. Give your answer correct to 1 decimal place.
(b)
(c)Ally is also working out the height of a round tower. She measures the angle of elevation to the top of the tower. She then moves 10 m away from the tower, on horizontal ground, and measures the angle of elevation of the top again. She measures both of these angles of elevation from a height of 1·25 m. She draws the diagram below to show her measurements. Two lengths, x and h, are shown in the diagram.
(i)Use the Sine rule to show that x = 25·5 m, correct to 1 decimal place.
(c)(i)
(ii)Hence, find the total height of the tower, marked h in the diagram. Give your answer in metres, correct to 1 decimal place.
(c)(ii)
(a)ABCD is a parallelogram. |AB| = 10 cm, |BC| = 13 cm, and |∠ABC| = 110°. Find the area of ABCD, correct to the nearest cm2.
(a)
(b)X is an angle, with 0° ≤ X ≤ 360°, and
cos(2X) = √32
Find all the possible values of X.
(b)
(c)KLM is a triangle where |MK| = 15√3 cm, |ML| = 45 cm, and |∠KLM| = 25°. θ is the angle ∠LKM. Work out the two possible values of θ, for 0° < θ < 180° (use the Sine rule). Give each answer correct to the nearest degree.
(c)
A company makes windscreen wipers. In this question, the rectangle PQRS has a width of 180 cm and a height of 100 cm. M is the midpoint of [PQ], N is the midpoint of [SR], and O ∈ NM. All lengths are given in cm.
(a)In the diagram below, the line segment [AB] shows a type of wiper blade. [AB] rotates around the point O, where O ∈ AB, until it reaches the position [A′B′]. The region that it cleans is ABB′A′, which is the sector OBB′ with the sector OAA′ removed. A, A′, B, and B′ lie on the rectangle PQRS, and N lies on the arc from B to B′. The line segment [RQ] is extended 20 cm to T, as shown. ∠OTR is a right angle.
(i)Show that |OB| = 120 cm.
(a)(i)
(ii)Hence, show that |∠BOT| = 41·4°, correct to 1 decimal place.
(a)(ii)
(iii)Hence, work out the area of ABB′A′. Remember that ABB′A′ is the sector OBB′ with the sector OAA′ removed. Give your answer correct to the nearest cm2.
(a)(iii)
(b)Moving the point O along the line NM changes the size of the wiper blade and the region that it cleans. In the diagram below, [DE] rotates about O, where O ∈ DE, until it reaches [D′E′]. As in part (a), E and E′ lie on the rectangle PQRS. Here, |∠E′OE| = 105°. By setting |OE| = x, use the triangle OEE′ and the Cosine Rule to find the value of |OE|. Give your answer in cm, correct to 1 decimal place.
(b)
(a)Prove that sin(A + B) = sin A cos B + cos A sin B.
(a)
(b)Using the formula in part (a), and without using a calculator, find the value of sin 75°. Give your answer in surd form.
(b)
(c)Find all solutions of the following equation in t, for 0° ≤ t ≤ 360°: sin t = sin(2t)
(c)
Olga is a cyclist.
(a)The diagram shows a road [AB] (not to scale). AC is horizontal and BC is vertical. |BC| = 9 m and |AB| = 70 m. The gradient of the road [AB] is |BC||AC| written as a percentage. Find the gradient of [AB], correct to the nearest percent.
(a)
(b)Olga wants to measure the vertical height of a hill. The point H is at the top of the hill. The points R and P are 20 m apart on horizontal ground, at the bottom. Olga measures the angle of elevation from R to H (17°). Taking O to be the point directly below H that is horizontal with R and P, she also measures |∠OPR| = 87° and |∠ORP| = 88° (shown, not to scale). Work out the distance |OH|, the vertical height of the top of the hill relative to R and P. Give your answer correct to the nearest metre.
(b)
Olga has tests done to measure her lung capacity. When resting, the volume of air, V, in her lungs after t seconds can be modelled by V(t) = 2 − 0·4 cos(π2 t), where V is in litres, t ≥ 0, and π2 t is in radians. The graph of y = V(t) for the first 9 seconds is shown.
(c)Find the values marked a and b on the graph, the minimum and maximum values of V.
(c)
(d)What is the connection between V′(t), the derivative of V, and whether Olga is breathing in or breathing out?
(d)
(e)Use the formula V(t) = 2 − 0·4 cos(π2 t) to find each of the following, when Olga is resting. Give each answer correct to 3 decimal places.
(i)Find the volume of air in Olga's lungs, half a second after t = 0.
(e)(i)
(ii)Find the rate at which the volume of air in Olga's lungs is increasing, half a second after t = 0.
(e)(ii)
(f)Olga's breathing is also measured while she is doing gentle exercise. During this time: when she breathes in fully, the volume of air in her lungs is 3·6 litres; when she breathes out fully, the volume is 1·3 litres; and she breathes in and out twice as many times per minute as when resting. Use this to write a formula for E(t), the volume of air in Olga's lungs during this time, t seconds after she has breathed out fully. Give your answer in the form E(t) = a + b cos(ct), where a, b, c, t ∈ ℝ, and ct is in radians.
(f)
Ava is looking at different tilings — different ways of covering a region with shapes.
(a)First, she looks at two tilings: one using identical squares and another using identical regular hexagons. (Note: a regular hexagon can be split into 6 congruent equilateral triangles.) A square tile and a hexagonal tile each have an area of 140 cm2.
(i)Work out the length of the side of the square tile, correct to 1 decimal place.
(a)(i)
(ii)The hexagonal tile has sides of length x cm, where x ∈ ℝ. Work out the value of x, correct to 1 decimal place.
(a)(ii)
(b)Next, Ava looks at more complicated tilings. The tiling below is made up of two shapes: an arrowhead (ABED) and a quadrilateral (EBCD). The point E lies on the line AC, and both shapes are symmetrical about AC (diagrams not to scale).
(i)|AD| = 8 cm, |AE| = 6 cm, and |ED| = 4 cm. As shown, α = ∠DAE. Show that α = cos−1(78).
(b)(i)
(ii)In the diagram |∠ADC| = |∠ABC| = 90°. Use this, and part (b)(i), to work out the total area of the quadrilateral ABCD, correct to 2 decimal places.
(b)(ii)
(a)(i) Prove that tan(A − B) = tan A − tan B1 + tan A tan B.
(a)(i)
(ii)Write tan 15° in the form √a − 1√a + 1, where a ∈ ℕ.
(a)(ii)
(b)The triangle ABC is shown below. |AC| = |BC| and |∠ACB| = 45°. |AB| = 10√(2 − √2), as shown. Find the length |AC|.
(b)
Oscar takes measurements of two adjacent triangular fields, Field 1 (ABC) and Field 2 (BDC), as shown (not to scale). B lies on the line AD. |AB| = 30 m, |BD| = 10 m, |AC| = 35 m, and |∠CAD| = 50°. (The angle ABC is not a right angle.)
(a)Find the area of Field 1 and, hence, find the area of Field 2. Give each answer correct to the nearest m2.
(a)
(b)Find the length of the perimeter of Field 1. Give your answer correct to the nearest metre.
(b)
Oscar is watching an airplane, P, fly directly over his head at point O. The x-axis is the horizontal ground and the y-axis runs vertically up from Oscar. P flies at a constant height of 10 km.
(c)Sound travels at roughly 343 metres per second in air.
(i)Use Diagram 1 to show that it takes 41 seconds for the sound the airplane makes at P1 to reach Oscar, correct to the nearest second.
(c)(i)
(ii)The airplane P flies at a constant speed of 255 m/s. By the time Oscar hears the sound made at P1, the airplane has flown on to P2 (Diagram 2). Work out the size of the angle marked θ, correct to 1 decimal place.
(c)(ii)
(d)P3 and P4 are two other points on the flightpath. By the time Oscar hears the sound made at P3, the airplane has flown on to P4 (not to scale). P3 and P4 are both a distance of d km from the y-axis.
(i)Explain briefly why the following equation holds: √(100 + d2)0·343 = 2d0·255
(d)(i)
(ii)Solve the equation above to find the value of d, correct to 1 decimal place.
(d)(ii)
(a)(i) Prove that cos 2A = cos2 A − sin2 A.
(a)(i)
(ii)sin θ2 = 1√5, where 0 ≤ θ ≤ π. Use the formula cos 2A = cos2 A − sin2 A to find the value of cos θ.
(a)(ii)
(b)Solve the equation tan(B + 150°) = −√3, for 0° ≤ B ≤ 360°.
(b)
The triangle ABC shows the 3 sections of a level triathlon course. Contestants swim 4 km from C to B, cycle from B to A, then run 28 km from A to C. Mary cycles at 25 km/hour, and it takes her 1 hour 12 minutes to cycle from B to A.
(a)Show that the total length of the course is 62 km.
(a)
(b)On average, Mary can run 5·6 times as fast as she can swim. It takes her 4·8 hours to complete the course. Find her average swimming speed in km/h.
(b)
(c)Show that |∠ACB| = 116·5°, correct to 1 decimal place.
(c)
(d)To comply with safety regulations, the region inside the triangular course must be kept clear of people. Find the area of this region. Give your answer in km2, correct to 1 decimal place.
(d)
(e)Find the shortest distance from the point C to the side AB. Give your answer in km, correct to 1 decimal place.
(e)
(f)The course is viewed from a camera tower rising vertically from point A. The top of the tower is T. The angle of elevation of T from B is 0·05°. Find |AT|, the vertical height of the tower. Give your answer correct to the nearest metre.
(f)
(a)An aeroplane flies east from point A for 2 hours at a constant 420 km/h to point B. It then changes direction, heading 20° towards the south at the same speed, until it reaches point C (shown). The direct distance from A to C is 1450 km and |∠BAC| = 8·57°.
(i)Find how long it took to fly from B to C. Give your answer correct to the nearest minute.
(a)(i)
(ii)The average fuel consumption of the plane is 3·8 litres per second and the fuel capacity is 100 000 litres. Show that the plane can complete the journey A to B to C and directly back to A at 420 km/h without refuelling.
(a)(ii)
(b)The voltage V(t) (in Volts) of a certain alternating current is V(t) = 110√2 sin(120πt), where t is in seconds.
(i)Find the period and range of the function V(t).
(b)(i)
(ii)Sketch the function for 0 ≤ t ≤ p, where p is the period. Indicate the period and range on your graph.
(b)(ii)
(iii)Use V(t) to find the voltage when t = 6·67 seconds. Give your answer correct to two decimal places.
(b)(iii)
(iv)Find one value for t where the voltage is 110 Volts. Give your answer in the form ab, where a, b ∈ ℕ.
(b)(iv)
(v)Find the rate of change of the voltage when t = 2 seconds. Give your answer correct to the nearest unit.
(b)(v)
(a)A flagpole [GH], shown in the diagram, is vertical and the ground is inclined at an angle of 5° to the horizontal between E and G. The angles of elevation from E and F to the top of the pole are 35° and 52° respectively. The distance from E to F along the incline is 6 m. Find how far F is from the base of the pole (G) along the incline. Give your answer correct to two decimal places.
(a)
(b)In the diagram the large circle s has centre O and the small circle c has centre D. The circle c touches the circle s at the point C. OA and OB are tangents to c as shown. The radius of c is r. |∠BOA| = 60°. The ratio of the area of s to the area of c is k : 1. Find the value of k.
(b)
(a)Find the two values of θ for which tan θ2 = −1√3, where 0 ≤ θ ≤ 4π.
(a)
(b)The diagram shows OAB, a sector of a circle of radius 7 cm with centre O. In the sector, |∠BOA| = 1·2 radians. The area of the shaded region is 21 cm². Find |BC|. Give your answer correct to 1 decimal place.
(b)
(a)Show that cos 2θ = 1 − 2 sin2 θ (a trigonometric identity).
(a)
(b)Find the cosine of the acute angle between two diagonals of a cube.
(b)
The diagram shows a triangular patch of ground ΔSGH, with |SH| = 58 m, |GH| = 30 m, and |∠GHS| = 68°. The circle is a helicopter pad. It is the incircle of ΔSGH and has centre P.
(a)Find |SG|. Give your answer in metres, correct to 1 decimal place (cosine rule).
(a)
(b)Find |∠HSG|. Give your answer in degrees, correct to 2 decimal places.
(b)
(c)Find the area of ΔSGH. Give your answer in m², correct to 2 decimal places.
(c)
(d)(i)Find the area of ΔHSP, in terms of r, where r is the radius of the helicopter pad.
(d)(i)
(ii)Show that the area of ΔSGH, in terms of r, can be written as 71·2r m².
(d)(ii)
(iii)Find the value of r. Give your answer in metres, correct to 1 decimal place.
(d)(iii)
(e)[ST] is a vertical pole at the point S. The angle of elevation of the top of the pole from the point P is 14°. Find the height of the pole. Give your answer, in metres, correct to 1 decimal place.
(e)
(a)Find all the values of x for which cos(2x) = −√32, where 0° ≤ x ≤ 360°.
(a)
(b)Let cos A = y2, where 0° < A < 90°. Write sin(2A) in terms of y (using the double angle formula).
(b)
In engineering, a crank-and-slider mechanism changes circular motion into back-and-forth straight-line motion. The crank [OD] rotates about the fixed point O. The point C slides back and forth in a horizontal line. [CD] is the rod connecting C to the crank. |OD| = 10 cm and |DC| = 30 cm.
(a)For a particular position with |∠DCO| = 15°, find |∠COD|, correct to the nearest degree (sine rule).
(a)
(b)As D moves in a circle around O, the angle α increases. The distance |CX| is a function of α, written f(α).
(i)Write down the period and range of f.
(b)(i)
(ii)Complete the table for f(α) at α = 0°, 90°, 180°, 270°, 360° (Diagram 1 is α = 0°; f(0°) = 30). Give answers correct to 2 decimal places where appropriate.
(b)(ii)
(iii)Use your table values to draw a rough sketch of f in the domain 0° ≤ α ≤ 360°.
(b)(iii)
(iv)For which of the three positions (Diagrams 1, 2, 3) will a 1 degree change in α cause the greatest change in the position of C? Explain your answer.
(b)(iv)
(c)Another crank-and-slider mechanism has |AB| = 36 cm, |AX| = 31 cm, and |∠BAO| = 10° (with |∠OBA| ≠ 90°). Find r, the length of the crank. Give your answer in cm, correct to the nearest cm.
(c)
Conor's property is bounded by a straight river bank. T is the base of a vertical tree growing near the opposite bank; |TE| is the height of the tree. From C (due west of the tree) the angle of elevation of E, the top, is 60°. From D (15 m due north of C) the angle of elevation of E is 30°. The land on both sides is flat and level.
(a)Use triangle ECT to express |TE| in the form √a |CT| metres, where a ∈ ℕ.
(a)
(b)Show that |TE| may also be expressed as √(225 + |CT|23) metres.
(b)
(c)Hence find |CT|, the distance from the base of the tree to the river bank at Conor's side. Give your answer correct to 1 decimal place.
(c)
(d)Find |TE|, the height of the tree. Give your answer correct to 1 decimal place.
(d)
(e)The tree falls across the river and hits the bank at Conor's side at point F. Find the maximum size of the angle FTC. Give your answer in degrees, correct to 1 decimal place.
(e)
(f)If the tree was equally likely to fall in any direction, find the probability that it would hit the bank at Conor's side. Give your answer as a percentage, correct to 1 decimal place.
(f)
(a)Show that cos 7A + cos Asin 7A − sin A = cot 3A (using the sum-to-product identities).
(a)
(b)Given that cos 2θ = 19, find cos θ in the form ± √ab, where a, b ∈ ℕ (using the double angle formula).
(b)
The height of water in a port was modelled by h(t) = 1·6 + 1·5 cos(π6 t), where t is the number of hours since the last recorded high tide and (π6 t) is in radians. The average height was 1·6 m.
(a)Find the period and range of h(t).
(a)
(b)Find the maximum height of the water in the port.
(b)
(c)Find the rate of change at which the height of the water is changing when t = 2, correct to two decimal places. Explain your answer in the context of the question.
(c)
(d)On a particular day high tide occurred at midnight (t = 0). Complete the table of h(t) values (every 3 hours from midnight to the following midnight) and sketch the graph of h(t).
(d)
(e)Find, from your sketch, the difference in water height between low tide and high tide.
(e)
(f)A fully loaded barge (needs a minimum water level of 2 m) enters the port, unloads (then needs only 1·5 m) and departs. Use your graph to estimate the maximum amount of time the barge can spend in port without resting on the sea-bed.
(f)
(a)Prove that tan(A + B) = tan A + tan B1 − tan A tan B.
(a)
(b)Find all the values of x for which sin(3x) = √32, 0 ≤ x ≤ 360, x in degrees.
(b)
(a)Joan is playing golf. She is 150 m from the centre of a circular green of diameter 30 m. The diagram shows the range of directions in which Joan can hit the ball so that it could land on the green. Find α, the measure of the angle of this range of directions. Give your answer, in degrees, correct to one decimal place.
(a)
(b)At the next hole, Joan, at T, attempts to hit the ball in the direction of the hole H. Her shot is off target and the ball lands at A, a distance of 190 m from T, where ∠ATH = 18°. |TH| is 385 m. Find |AH|, the distance from the ball to the hole, correct to the nearest metre.
(b)
(c)At another hole, where the ground is not level, Joan hits the ball from K. The ball lands at B. The height of the ball, in metres, above the horizontal line OB is given by h = −6t2 + 22t + 8, where t is the time in seconds after the ball is struck.
(i)Find the height of K above OB.
(c)(i)
(ii)The horizontal speed of the ball over the straight distance [OB] is a constant 38 m s−1. Find the angle of elevation of K from B, correct to the nearest degree.
(c)(ii)
(d)At a later hole, Joan's first shot lands at the point G, on ground sloping downwards. A vertical tree [CE], 25 m high, stands between G and the hole. The distance |GC| is also 25 m. The angle of elevation at G to the top of the tree E is θ, where θ = tan−1(12). The height of the top of the tree above the horizontal GD is h metres and |GD| = d metres.
(i)Write d and CD in terms of h.
(d)(i)
(ii)Hence, or otherwise, find h.
(d)(ii)
The lengths of the sides of a flat triangular field ACB are |AB| = 120 m, |BC| = 134 m and |AC| = 150 m.
(a)(i)Find |∠CBA|. Give your answer, in degrees, correct to two decimal places.
(a)(i)
(ii)Find the area of the triangle ACB correct to the nearest whole number.
(a)(ii)
(b)A vertical mast, [DE], is fixed at the circumcentre, D, of the triangle. The mast is held in place by three taut cables [EA], [EB] and [EC]. Explain why the three cables are equal in length.
(b)
(a)Prove that cos 2A = cos2A − sin2A.
(a)
(b)The diagram shows part of the circular end of a running track with three running lanes. The centre of each of the circular boundaries of the lanes is at O. Kate runs in the middle of lane 1, from A to B. Helen runs in the middle of lane 2, from C to D. Helen runs 3 m further than Kate. |∠AOB| = |∠COD| = θ radians. If each lane is 1·2 m wide, find θ.
(b)
The graph below shows the voltage, V, in an electric circuit as a function of time, t. The voltage is given by V = 311 sin(100πt), where V is in volts and t is in seconds.
(a)(i)Write down the range of the function.
(a)(i)
(ii)How many complete periods are there in one second?
(a)(ii)
(b)(i)A table gives the voltage, correct to the nearest whole number, at 12 equally spaced intervals from t1 to t12 over one complete period (with t6 = 0·01, t12 = 0·02). Given V1 = 156, V2 = 269, V3 = 311, use the properties of the function to complete the table of all twelve values.
(b)(i)
(ii)Using a calculator, or otherwise, calculate the standard deviation, σ, of the twelve values of V in the table, correct to the nearest whole number.
(b)(ii)
(c)(i)The standard deviation, σ, of closely spaced values of any function of the form V = a sin(bt), over 1 full period, is given by kVmax, where k is a constant that does not depend on a or b, and Vmax is the maximum value of the function. Use V = 311 sin(100πt) to find an approximate value for k correct to three decimal places.
(c)(i)
(ii)Using your answer in part (c)(i), or otherwise, find the value of b required so that V = a sin(bt) has 60 complete periods in one second, and the approximate value of a so that it has a standard deviation of 110 volts.
(c)(ii)
Questions reproduced from State Examinations Commission Leaving Certificate examination papers,
© State Examinations Commission (examinations.ie). Gathered here for study use.
© State Examinations Commission (examinations.ie). Gathered here for study use.