H I G H E R M A T H S. Practice Unit Tests (2010 on) Higher Still Higher Mathematics M A T H E M A T I C S. Contents & Information
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1 M A T H E M A T I C S H I G H E R Higher Still Higher Mathematics M A T H S Practice Unit Tests (00 on) Contents & Information 9 Practice NABS... ( for each unit) Answers New format as per recent SQA changes Pegass Educational Publishing Pegass 00
2 FORMULAE LIST ( UNIT ) The equation g f c 0 represents a circle centre ( g, f ) and radius g f c. The equation ( a) ( b) r represents a circle centre ( a, b ) and radius r. Trigonometric formulae: sin A B sin AcosB cos AsinB cos A B cos AcosB sin AsinB cosa cos A sin A cos A sin A sina sin Acos A Pegass 00
3 FORMULAE LIST ( UNIT ) Scalar Product: a. b a b cos, where is the angle between a and b. or a. b a b a b a b where a a a a and b b b b Trigonometric formulae: sin A B sin AcosB cos AsinB cos A B cos AcosB sin AsinB cosa cos A sin A cos A sin A sina sin Acos A Table of standard derivatives: f ( ) f ( ) sin a a cos a cos a a sin a Table of standard integrals: f ( ) f ( ) d sin a cos a a a cos a sin a C C Pegass 00
4 Practice Assessment () Unit - Mathematics (H) Outcome Marks. A line passes through the points (, 5) and (7, ). Find the equation of this line. (). A line makes an angle of o with the positive direction of the -ais, as shown in Diagram. The scales on the aes are equal. Find the gradient of the line giving our answer correct to significant figures. o o Diagram (). A line L has equation. Write down the gradient of a line which is: (a) parallel to L perpendicular to L () () Pegass 00
5 Outcome. The graph of a cubic function f () is shown in Diagram. On separate diagrams sketch the graphs of : (, 6) (a) f (). () f ( ). o 8 () Diagram 5. The graphs with equations And a are shown in Diagram. (, 6) If the graph with equation a passes through the point (, 6), find the value of a () (,) - o Diagram 6. The graphs of = and its inverse function are shown in Diagram. Write down the equation of the inverse function. () (0, ) o (, 0) Diagram 7. Functions f and g are defined on suitable domains b f ( ) 5 and g ( ). Obtain an epression for f ( g( )). () Pegass 00
6 Outcome 5 8. Given, 0, find d. () d 9. A sketch of the curve with equation 5 6 is shown in Diagram 5. A tangent has been drawn at the point P(, ). 5 6 Find the gradient of the tangent at P. o P(, ) Diagram 5 () 0. A curve has equation 8. Using differentiation, find the coordinates of the stationar points on this curve and determine their nature. (6) Outcome. In a small colon 5% of the eisting insects are eaten b predators each da but during the night 500 insects are hatched. There are U n insects at the start of a particular da. (a) Write down a recurrence relation for U n+, the number of insects at the start of the net da. It is known that the colon cannot survive if there are more than 00 insects. () (i) (ii) Find the limit of the sequence generated b this recurrence relation as n. In the long term, can the colon survive? () Pegass 00
7 Practice Assessment () Unit - Mathematics (H) Outcome Marks. A line passes through the points (, 5) and (, 5). Find the equation of this line. (). A line makes an angle of 65 o with the positive direction of the -ais, as shown in Diagram. The scales on the aes are equal. Find the gradient of the line giving our answer correct to significant figures. o 65 o () Diagram. A line L has equation 5 7. Write down the gradient of a line which is: (a) parallel to L perpendicular to L () () Pegass 00
8 Outcome. The graph of a cubic function f () is shown in Diagram. On separate diagrams sketch the graphs of : (a) f (). (, 5) (0, ) () f ( ). o () Diagram 5. The graphs with equations and a is shown in Diagram. If the graph with equation a passes through the point (, 7), find the value of a (, 7).. (, 9) o Diagram () 5 6. The graphs of 5 and its inverse function are shown in Diagram. Write down the equation of the inverse function. () o (, 0) Diagram (0, ) 7. Functions f and g are defined on suitable domains b f ( ) and ( ) g. Obtain an epression for f ( g( ). () Pegass 00
9 Outcome 6 8. Given, 0, find d. () d A sketch of the curve with equation 0 6 is shown in Diagram 5. A tangent has been drawn at the point P(6, 8). o P(6, 8) Find the gradient of the tangent at P. Diagram 5 () 0. A curve has equation 5. Using differentiation, find the coordinates of the stationar points on this curve and determine their nature. (6) Outcome. In a small rabbit colon one sith of the eisting rabbits are eaten b predators each summer, however during the winter 00 rabbits are born. There are U n rabbits at the start of a particular summer. (a) Write down a recurrence relation for U n+, the number of rabbits at the start of the net summer. It is known that the colon cannot sustain more than 000 rabbits at an one time. () (i) Find the limit of the sequence generated b this recurrence relation as n. (ii) In the long term, can the colon sustain the number of rabbits? () Pegass 00
10 Practice Assessment () Unit - Mathematics (H) Outcome Marks. A line passes through the points (, ) and (, 5). Find the equation of this line. (). A line makes an angle of 5 o with the positive direction of the -ais, as shown in Diagram. The scales on the aes are equal. Find the gradient of the line giving our answer correct to significant figures. o 5 o Diagram (). A line L has equation 9. Write down the gradient of a line which is: (a) parallel to L perpendicular to L () () Pegass 00
11 Outcome. The graph of a function f () is shown in Diagram. (, ) On separate diagrams sketch the graphs of : (a) f (). f ( ). -5 o 5 () () (, ) Diagram 5. The graphs with equations and 5 a is shown in Diagram If the graph with equation a passes through the point (, ), find the value of a (, 5).. (, ) o () Diagram 6. The graphs of and its inverse function are shown in Diagram. Write down the equation of the inverse function. () (0, ) o (, 0) Diagram 7. Functions f and g are defined on suitable domains b f ( ) and g ( ) 5. Obtain an epression for f ( g( )). () Pegass 00
12 Outcome Given, 0, find d. () d 9. A sketch of the curve with equation is shown in Diagram 5. A tangent has been drawn at the point P(, 0).. P(, 0) Find the gradient of the tangent at P. () Diagram 5 o 0. A curve has equation 8. Using differentiation, find the coordinates of the stationar points on this curve and determine their nature. (6) Outcome. For an established ant hill 8% of the worker ants fail to return at the end of each da. However, during the night 60 worker ants are hatched. There are U n worker ants at the start of a particular da. (a) Write down a recurrence relation for U n+, the number of worker ants at the start of the net da. It is known that the colon cannot survive if there are more than 6000 ants. () (i) Find the limit of the sequence generated b this recurrence relation as n. (ii) In the long term, will the colon survive? () Pegass 00
13 Unit - Practice Assessments Answers Practice Assessment Outcome :. m, 7 ( ( 7) or 5 ( ) ). m (a) m m Outcome :. (a) diagram (reflection in -ais) diagram (translated unit left) 5. a 6. log 7. (a) f ( g( )) ( ) 5 d Outcome : m 0. (, 77 ), ma ; (, ), min d Outcome :. (a) U n 075U n 500 (i) L 000, (ii) will survive since 000 < 00. Practice Assessment Outcome :. m, ( 5 ( ) or 5 ( ) ). m. (a) m 5 m 5 Outcome :. (a) diagram (reflection in -ais) diagram (translated units right) 5. a 7 6. log 5 7. (a) f ( g( )) ( ) 6 Outcome : 8. d d m 0. 5, ), ma, (, ), min ( Outcome :. (a) 5 U n 6U n 00 (i) L 800 (ii) will sustain since 800 < 000. Practice Assessment Outcome :. m, ( 5 ( ) or ( ) ). m (a) m m Outcome :. (a) diagram (reflection in -ais) diagram (translated unit down) 5. a 6. log 7. (a) f ( g( )) ( 5) ( 5) 8 55 d 85 Outcome : m 0. ( 6, 5), ma, (, 7), min d Outcome :. (a) U n 09U n 60 (i) L 5750 (ii) will survive since 5750 < Pegass 00
14 Practice Assessment () Unit - Mathematics (H) Outcome Marks. (i) Show that ( ) is a factor of g ( ) 6 5. (ii) Hence factorise g() full. (5). Determine the nature of the roots of the equation 7 0 using the discriminant. () Outcome. Find d, 0 5 (). The curve with equation Diagrams. ( ) is shown in ( ) Calculate the shaded area shown in Diagram. o Diagram (5) 5. The line with equation and the curve with the equation in Diagram. The line and the curve meet at the points where = 0 and =. Calculate the shaded area shown in Diagram. are shown o Diagram (6) Pegass 00
15 Outcome Practice Assessment () Unit - Mathematics (H) 6. Solve the equation sin for 0. () 7. Diagram shows two right-angled triangles. o o Diagram (a) Write down the values of cos and sin. () Show that the eact value of cos ( ) is 80. () 8. (a) Epress sin cos0 cos sin0 in the form sin ( ). () Using the results from (a), solve sin cos0 cos sin0 for () Outcome 9. (a) A circle has radius 7 units and centre C(, 5). Write down the equation of the circle. () A circle has equation 6 0. Write down the coordinates of its centre and the length of its radius. () 0. Show that the line with equation is a tangent to the circle with the equation (5). The point P(, ) lies on the circle with centre C(, ), as shown in Diagram. P(, ) Find the equation of the tangent at P. () C(, ) Pegass 00 Diagram
16 Practice Assessment () Unit - Mathematics (H) Outcome Marks. (i) Show that ( ) is a factor of f ( ) 0. (ii) Hence factorise f () full. (5). Determine the nature of the roots of the equation 0 using the discriminant. () Outcome. Find d, 6 0. The curve with equation Diagram. is shown in () Calculate the shaded area shown in Diagram. o 6 Diagram (5) 5. The curve with equation 6 and the curve with the equation in Diagram. 6 are shown The curves meet at the points where = 0 and. 6 Calculate the shaded area shown in Diagram. (6) 6 o Diagram Pegass 00
17 Outcome 6. Solve the equation tan for 0. () 7. Diagram shows two right-angled triangles o Diagram o (a) Write down the values of sin and cos. () Show that the eact value of sin ( ) is 5. () 8. (a) Epress cos cos0 sin sin0 in the form cos ( ). () Using the results from (a), Outcome solve the equation cos cos0 sin sin0 9 for () 9. (a) A circle has radius units and centre (, ). Write down the equation of the circle. () A circle has equation 8 0. Write down the coordinates of its centre and the length of its radius. () 0. Show that the line with equation 7 is a tangent to the circle with equation (5). The point P(, ) lies on the circle with C(, ), as shown in Diagram. Find the equation of the tangent at P. P(, ) C(, ) () Diagram Pegass 00
18 Practice Assessment () Unit - Mathematics (H) Outcome Marks. (i) Show that ( ) is a factor of f ( ) 5. (ii) Hence factorise f () full. (5). Determine the nature of the roots of the equation 9 0 using the discriminant. () Outcome 7. Find d, 0 (). The curve with equation 9 8 is shown in Diagram. Calculate the shaded area shown in Diagram. 9 8 o 6 Diagram 5. The line with equation and the curve with equation 7 are shown in (5) Diagram. The line and the curve meet at the points where = 0 and = 6. 7 o 6 Diagram Calculate the shaded area shown in Diagram. (6) Pegass 00
19 Outcome 6. Solve the equation cos for 0. () 7. Diagram shows two right-angled triangles.. 5 Diagram (a) Write down the values of o sin and o sin. () Show that the eact value of sin ( ) is 5. () 8. (a) Epress cos cos 5 sin sin 5 in the form cos ( ). () Using the results from (a), solve the equation cos cos 5 sin sin 5 7 for () Outcome 9. (a) A circle has a radius of units and centre (, ). Write down the equation of the circle. () A circle has equation Write down the coordinates of its centre and the length of its radius. () 0. Show that the line with equation 7 is a tangent to the circle with equation (5). The point P(, ) lies on the circle with centre C(, ) as shown in Diagram. C(, ) Find the equation of the tangent at P. () P(, ) Pegass 00 Diagram
20 Unit - Practice Assessments Answers Practice Assessment Outcome :. proof, g ( ) ( )( )( ). b ac real, distinct and irrational Outcome :. C C 5. units Outcome : (a) 8, 8 sin( 0) 7. (a). units 9 5, 705 cos, sin proof 0 8 Outcome : 9. (a) ( ) ( 5) 9 C(, ), r 0. root,, a tangent. ( ) Practice Assessment Outcome :. proof, f ( ) ( )( )( 5). b ac 0 not real 5 Outcome :. C C units 5 Outcome : (a) 8, 8 cos( 0) 7. (a). 7 units 7, sin, cos proof 5 5 Outcome : 9. (a) ( ) ( ) 6 C(, 7), r 8 0. root, 7, a tangent. ( ) 9 Practice Assessment Outcome :. proof, g ( ) ( )( )( ). b ac 0 equal roots 7 7 Outcome :. C C 5. 6 units 5 7 Outcome : 6., 8. (a) cos( 5) 7. (a). units 75 5, sin, sin proof Outcome : 9. (a) ( ) ( ) C(0, ), r 0. root,, a tangent. Pegass 00
21 Practice Assessment () Unit - Mathematics (H) Outcome Marks. The points A, B and C have coordinates (,, ), (0,, ), ( 6, 7, ) respectivel. (a) Write down the components of AB. () Hence show that A, B and C are collinear. (). The point T divides RS in the ratio : as shown in Diagram. Find the coordinates of T. R(7,, ) Diagram T S(5, 5, ) (). Diagram shows vectors AB and AC where C AB 0 and 6 AC 7 A (a) Find AB. AC B Diagram () Hence find the size of angle BAC. () Outcome. (a) Differentiate cos with respect to. () Given sin, find 5 d. () d 5. Find g () when 5 g ( ) ( ). () 6. (i) Find cos d () (ii) Integrate sin with respect to. () 7. Evaluate ( ) d 0 () Pegass 00
22 Outcome 8. (a) Simplif log log. () Simplif 5log log () 9. Solve e 0. Solve log ( ) () () Outcome. Epress cos sin in the form k cos ( a) where k 0 and 0 a 60. (5) Pegass 00
23 Practice Assessment () Unit - Mathematics (H) Outcome Marks. The points P, Q and R have coordinates (, 6, ), (,, ) and (0,, ) respectivel. (a) Write down the components of PQ. () Hence show that P, Q and R are collinear. (). The point P divides QR in the ratio : as shown in Diagram. Find the coordinates of P. Q(,, ) P Diagram () R(6,, 9). Diagram shows vectors ST and SU where S ST and 5 SU T (a) Find ST. SU. () Hence find the size of angle TSU. U () Outcome. (a) Differentiate sin with respect to. () Given 7 cos, find d. () d 5. Find h () when h( ) 5 ( ). () 6. (a) Find sin d () Integrate cos with respect to. () 7. Evaluate ( ) d () Pegass 00
24 Outcome 8. (a) Simplif log t 5 log t. () Simplif log 6 log 9 () 9. Solve e 0. Solve log ( ) () () Outcome. Epress 5sin cos in the form k cos ( a) where k 0 and 0 a 60. (5) Pegass 00
25 Practice Assessment () Unit - Mathematics (H) Outcome Marks. The points S, T and U have coordinates (, 7, ), (,, 5) and ( 6,, ) respectivel. (a) Write down the components of ST. () Hence show that S, T and U are collinear. (). The point P divides AB in the ratio : as shown in the Diagram. Find the coordinates of P. A (8,, ) P Diagram B (,, 9) (). Diagram shows vectors KL and KM where KL K and KM = 8 5 Diagram L M (a) Find the value of KL. KM. () Hence find the size of angle LKM. () Outcome. (a) Differentiate sin with respect to. () Given cos, find d. () d 5. Find f () when f ( ) ( ). () 6. (a) Find sin d () Integrate cos with respect to. () 7. Evaluate ( ) d () Pegass 00
26 Outcome 8. (a) Simplif log 5 log. () Simplif log log 6 () 9. Solve e 8 0. Solve log 6 ( 5) () () Outcome. Epress sin 7cos in the form k sin ( a) where k 0 and 0 a 60. (5)
27 Unit - Practice Assessments Answers Practice Assessment Outcome :. (a) AB proof. T(,, ). (a) AB. AC 5 7 d Outcome :. (a) sin cos 5. d 5 6. (a) sin C cos C g ( ) 5( ) Outcome : 8. (a) log Outcome :. cos( 56) Practice Assessment Outcome :. (a) PQ proof. P(0,, 7). (a) ST. SU 66 6 d Outcome :. (a) cos 7sin 5. d h ( ) 0( ) 6. (a) cos C sin C 7. Outcome : 8. (a) log t Outcome :. cos( 59 0) Practice Assessment Outcome :. (a) ST 5 proof. P(6,, ). (a) KL. KM d Outcome :. (a) cos sin 5. d 6. (a) cos C sin C f ( ) ( ) 7. units Outcome : 8. (a) log Outcome :. 58 sin( 668)
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