Higher Maths. Calculator Practice. Practice Paper A. 1. K is the point (3, 2, 3), L(5, 0,7) and M(7, 3, 1). Write down the components of KL and KM.

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1 Higher Maths Calculator Practice Practice Paper A. K is the point (,, ), L(5,,7) and M(7,, ). Write down the components of KL and KM. Calculate the size of angle LKM.. (i) Show that ( ) is a factor of f ( ). (ii) Hence factorise f( ) full. Solve ( ) ( ).. Find the equation of the tangent to the parabola with equation 6 at the point (, 6). Show that this line is also a tangent to the circle with equation In the right-angled triangle shown in Diagram, tan a. Find the eact values of (i) cos a ; (ii) cos a. In the right-angled triangle shown in Diagram, tan b. Find the eact value of sin( a b).

2 5. Solve log ( ) log ( 5), The diagram below shows part of the graph of p qsin r. 4 p qsin r. Write down the values of p, q and r. The graph of p qsin r. has a minimum turning point at A and a maimum turning point at B. B A p qsin r. Calculate the shaded area in the diagram above.

3 7. Cobalt-6 is used in food irradiation and decas to Nickel-6, which is a stable substance. kt Cobalt-6 decas according to the law m m e, where m is the initial mass t of Cobalt-6 present and m is the mass remaining after t ears. t The time taken for half the mass of Cobalt-6 to deca to Nickel-6 is 5 ears. Find the value of k, giving our answer correct to significant figures. In a sample of Cobalt-6 what percentage has decaed to Nickel-6 after ears? 8. A rectangular park measures metres b metres. A path connecting the two entrances, at opposite corners of the park, is to be laid through the park as shown. Entrance Park metres metres metres metres Entrance The cost per metre of laing the path through the park is twice the cost, per metre, of laing the path along the perimeter. Show that the total cost of laing this path can be modelled b C ( ) Find the value of which would minimise the cost of laing the path.

4 Practice Paper B. A sequence is defined b the recurrence relation u 4u 6, u. n n Determine the values of u, u and u. Wh does this sequence have a limit as n? (c) A second sequence, generated b v pv 4, has the same limit as the sequence in. Find the value of p. n n. A function f is defined on the set of real numbers b f ( ) 4 7. Show that ( ) is a factor of f( ), and hence factorise f( ) full. The graph shown has equation of the form S 4 T Calculate the shaded area labelled S. (c) Find the total shaded area.. D has coordinates (7,,) and F is (,, 5). Find the coordinates of E which divides DF in the ratio :. G has coordinates (6,, 5). Show that EG is perpendicular to DF.

5 4. P, Q and R have coordinates ( 4, 6), (8, ) and (, 8) respectivel. Show that PQ is perpendicular to QR. Hence find the equation of the circle which passes through P, Q and R. 5. Two functions f and g are defined on the set of real numbers b Find (i) f ( g( )); (ii) g( f ( ))., where k f ( ) k and g( ) k Find the value of k for which f( g( )) g( f( )) has equal roots. 6. A closed wooden bo, in the shape of a cuboid, is constructed from a sheet of wood of area 6 cm. The base of the bo measures cm b cm. The height of the bo is h cm. cm h cm cm Assuming the thickness of the sides of the bo are negligible, show that the volume (in cubic centimetres) of the bo is given b 4 V( ) (i) Calculate the value of for which this volume is a maimum. (ii) Find the maimum volume of the bo.

6 7. Whilst carring out an eperiment a scientist gathered some data. The table shows the data collected b the scientist b The variables and, in the table, are connected b a relationship of the form ae. Find the values of a and b. 8. Solve cos 4sin for

7 Practice Paper C. Given that ( ) is a factor of k 6, find the value of k. Hence, or otherwise, solve k 6.. OABC,DEFG is a rectangular prism as show. z D 7 5 G C E B F 8 A OA is 8 units long, OC is 5 units and OD is 7 units. Write down the coordinates of B and G. Calculate the size of angle BEG.. A circle, centre C, has equation 4. Find the centre C and radius of this circle. (i) Show that the point P(5, ) lies on the circumference of the circle. (ii) Find the equation of radius CP. (c) Find the equation of the chord which passes through (7,) and is perpendicular to radius CP. 4. Solve cos cos 6 for 5. Diagram shows part of the graph with equation 5 8. Calculate the shaded area. Diagram

8 Given that p ( 5 8) d find the total shaded area in diagram. Diagram 6. Find the smallest integer value of c for which has onl one real root. g c ( ) ( )( ) 7. Write sin 5 cos in the form ksin( a), where k and a. Sketch the graph of 4sin 5 cos for 8. For a particular radioactive isotope, the mass of the original isotope remaining, m grams, after time t ears is given b m 8t m e where m is the original mass of the isotope. If the original mass is g, find the mass of the original isotope remaining after ears. The half-life of the isotope is the time taken for half the original mass to deca. Find the half life of this isotope. 9. Find 4 sin 4 d sin. 6

9 Practice Paper D. A sequence is defined b recurrence relation u ku 6, u. n n Given that u 8, find the value of k. (i) Wh does this sequence tend to a limit as n? (ii) Find the value of this limit.. f p q ( ) 4. Given that ( ) is a factor of f( ), and the remainder when f( ) is divided b( ) is 9, find the values of p and q.. Securit guards are watching a parked car, via two CCTV cameras, in a supermarket car park. With reference to a suitable set of aes, the car is at C(5,, ) and the cameras are at positions A(, 6, 4) and B(7, 9, 5) as shown. Calculate the size of angle ACB. 4. Part of the graphs of and 5 are shown opposite. S The curves intersect at the points S and T. Find the coordinates of S and T. T Find the shaded area enclosed between the two curves.

10 5. A circle with centre C has equation 6 5. Write down the coordinates of the centre and calculate the C length of the radius of this circle. A second circle with centre C has a diameter twice that of the circle with centre C. C lies on the circumference of this second circle. The line joining C and C is parallel to the -ais. C C Find the equation of the circle with centre C. 6. A manufacturer of eecutive desks estimates that the weekl cost, in, of making desks is given bc( ) Each eecutive desk sells for. Show that the weekl profit made from making desks is given b P ( ) (i) How man desks would the manufacturer have to make each week in order to maimise his profit? (ii) What would his annual profit be?

11 7. The number of bacteria, b, in a culture after t hours is given b b kt b e where b is the original number of bacteria present. The number of bacteria in a culture increases from 8 to 4 in hours. Find the value of k correct to significant figures. How man bacteria, to the nearest hundred, are present after a further 4 hours? 8. Epress cos 5sin in the form k cos( a), where k and a 9. (i) Hence write cos 5sin in the form Rcos( b), where R and b 9. (ii) Solve cos 5sin 5 in the interval 6.

12 Practice Paper E. A line, l, passes through the points A(, ) and B(5, 4). The line makes an angle of a B with the positive direction on the -ais. A Find the value of a. A second line, l, with equation 4, crosses the line in. B l The angle between the two lines is b, as shown. A a b Find the value of b. l. The rectangular based pramid D,OABC has vertices A(6,, ), B(6, 8, ) and D(, 4, 7). (i) Write down the coordinates of C. (ii) Epress AC and AD in component form. Calculate the size of angle CAD.

13 . (i) Show that ( ) is a factor of 6. (ii) Hence factorise 6 full. The line with equation intersects the curve with equation 6 at the points A, B and C. C B A Find the -coordinates of the points A and C. The area between the curve and the line from A to C is shaded in the diagram below. C B A (c) Calculate the total shaded area shown in the diagram. 7

14 4. Solve cos sin for 5. A new 4 hour anti-biotic is being tested on a patient in hospital. It is know, that over a 4 hour period, the amount of anti-biotic remaining in the bloodstream is reduced b 8%. On the first da of the trial, an initial 5 mg dose is given to a patient at 7 a.m. After 4 hours and just prior to the second dose being given, how much anti-biotic remains in the patient s bloodstream? The patient is then given a further 5 mg dose at 7 a.m. and at this time each subsequent morning thereafter. A recurrence relation of the form u au b can be used to model this course of treatment. n n Write down the values of a and b. It is also known that more than 5 mg of the drug in the bloodstream results in unpleasant side effects. (c) Is it safe to administer this anti-biotic over an etended period of time? 6. The diagram shows part of the graph of cos( ). Find the equation of the tangent at the point T, where Solve log ( ) log ( ),. 8. A circle has the following properties: The -ais and the line are tangents to the circle. The circle passes through the points (,) and (,8). The centre lies in the first quadrant. Find the equation of this circle.

15 Practice Paper F. Find the equation of the tangent to the curve at the point where. Show that this line is also a tangent to the circle with equation and state the coordinates of the point of contact.. The diagram opposite shows a rhombus ABCD. AC and BD are diagonals of the A rhombus. Diagonal AC has equation. B D is point with coordinates ( 4, ). E E is the point of intersection of the diagonals. D Find the equation of diagonal BD. C Hence find the coordinates of E.. Given that p i j k and q i j k, find the angle between p and q. 4. If 7 cos, find the eact value of : 5 cos. tan. 5. A sequence is defined b the recurrence relation u au b, u. n n If u 5, write down the value of b. Given that u, find the possible values of a. This sequence tends to a limit as n. (c) Find the limit of the sequence.

16 6. Epress 4cos 7sin in the form ksin( a), where k and a 6. Hence solve 4cos 7sin for A new fish farm is being established consisting of a number of rectangular enclosures. Each enclosure is made up from eight identical rectangular cages. Each cage measures metres b metres. The total length of edging around the top of all of the caging is 48 metres. Show that the total surface area, in square metres, of the top of the eight cages is given b A( ). (i) Find the value of which maimises this surface area. (ii) Hence find the dimensions of each enclosure. 8. The diagram shows the graphs of cos( ) and 7 5cos( ) for 6. The two graphs intersect at T, which has coordinates ( p, q ). Find the eact value of cos p. Determine the value of p.

17 ANSWERS PAPER A. 4 KL and KM or 94 radians. (i) Show that f () (ii) f( ) ( )( )( ),,. 7 Proof 4. (i) cos a (ii) 5 cos a 5 sin( ab) p, q and r k 9 4 % 8. Proof

18 PAPER B. u 6, u 84 and u (c) 5 p (or 6). Show that f () ; f( ) ( )( 5)( ) (c) 65 (or 5 46) 89 (or 48 6) 6. E(5,, ) Show that EG. DF 4. Show that m m PQ QR ( ) ( 7) 5. (i) f ( g( )) k (ii) g( f ( )) 4 4k k k k 6. Proof (i) cm (ii) Maimum volume : 94 8 cm 7. 7 a e b, 8. 86, 6 5

19 PAPER C. k 5,,. B(8, 5, ) and G(, 5, 7) 7 (or 58 rads). Centre : (, ) Radius : 5 units (i) Proof (ii) 4 (c) 47 4., square units (or 5 75) c 7. sin( 84) g 85 ears 9. 4

20 PAPER D. k (c) 9. p, q or radians 4. S(, ) and T(, ) 9 square units 5. C (, ) and radius 5 units ( ) ( ) 6. Proof (i) 4 (ii) cos( 68 ) (i) 9 cos( 68 ) (ii) 5, 56 8, 5, 6 8

21 PAPER E. a 6 6 b. (i) C(, 8, ) (ii) 6 AC 8 and AD or 95 radians. (i) Proof (ii) ( )( 5)( ) A : C : 5 (c) 8 square units , 94, mg u u 5 n n (c) 5 5 safe to administer long term ( 6) ( )

22 PAPER F. (7, ). 4 E(, ). 8 8 or 899 radians b 5 (c) 6 or 5 5 a sin( 9 7) 7 9, 5 7. Proof (i) 4 (ii) 4 m m 8. cos p 4 4