1 k. cos tan? Higher Maths Non Calculator Practice Practice Paper A. 1. A sequence is defined by the recurrence relation u 2u 1, u 3.

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1 Higher Maths Non Calculator Practice Practice Paper A. A sequence is defined b the recurrence relation u u, u. n n What is the value of u?. The line with equation k 9 is parallel to the line with gradient 7. What is the value of k?. A circle has equation 8. What is the radius of this circle?. What is the derivative of with respect to?. Find d. 6. If 7 is written in the form ( p) q, find the value of q. 7. A sequence is generated b the recurrence relation u 8u 6. n n What is the limit of this sequence as n? 8. A circle with centre (, ) passes through the point (, 7). What is the equation of the circle? 9. The vectors p and q with components k p k and q are perpendicular. What is the value of k?. Identif the nature of the roots of the equation 8.. What is the value of 7 cos tan?

2 . Given that log 8 p, find the value of p.. Find ( ) d. K and L are the points with coordinates (,, ) and (,, ) respectivel. If KM KL, find the coordinates of M.. h ( ). 8 For what values of is h ( ) undefined? 6. Here are two statements about the graph with equation a b, shown opposite. () a ; () is alwas increasing Which of these statements are true? a b 7. The diagram shows part of the graph of a cubic. What is the equation of this graph? 8. Given that log log, epress in terms of. 9. If p. ( p q ) 8 and p, find the value of p. q

3 . The diagram shows part of the curve with equation plog ( k). What is the value of p. Triangle PQR has vertices P(, ), Q(7, ) and R(, ), as shown. P Q R (a) (b) (c) Find the equation of the median RM. Find the equation of the altitude AP. Find the coordinates of the point of intersection of RM and AP.. Find the stationar points on the curve given b nature. 9 and determine their. (a) Functions f and g are defined on suitable domains b Find f ( g( )). f g ( ) and ( ) (b) Sketch the curve with equation f ( g( )). π. (a) Show that 6 sin cos sin cos. (b) Epress sin cos in the form ksin( a) where k and a π (c) Hence, or otherwise, solve sin cos, where π. 6 π.

4 Practice Paper B. Given that f ( ), find f ().. Find ( )( ) d.. P and Q have coordinates (,, ) and (,, ). What is the distance between P and Q?. If 8 is epressed in the form ( p) q, what is the value of q?. Here are two statements about the equation () The roots are equal. () The roots are rational. Which of these statements is true?. 6. Find all the values of in the interval for which cos. 7. S is the point with coordinates (,, ), T(,, ) and U(,, 7). Find the ratio in which T divides SU. d 8. Given that sin( ), find. d 9. The angle between the line shown in the diagram and the -ais is. What is the gradient of the line?. Given that log 9, what is the value of a? a π. What is the maimum value of 9 sin?

5 . Find ( ) d.. The graph shown in the diagram has equation of the form cos( p) q. What are the values of p and q? cos( p) q. Given that h ( ) 6, what is the largest possible domain for h?. Vector t has components. u is a unit vector such that u kt, where k. Find the value of k. 6. The diagram shows the graph of f ( ). Sketch the graph of f ( )? 7. The equation of the parabola shown is of the form k( ). What is the value of k? (, ) k( ) 8. Simplif log log ( ). 9. What is the solution to?

6 . If v t and the rate of change of v with respect to t at t k, k is 6, find the value of k.. A circle with equation 6 9 has centre C. (a) Write down the coordinates of the centre C and find the length of the radius of this circle. A second circle with equation ( ) ( 7) 6 has centre C. (b) (i) Find the distance between the centres C and C. (ii) Hence find the minimum distance between the circumferences of the two circles.. A is the point with coordinates (,, ), B(,, ) and C(,, ). (a) Epress AB and AC in component form. (b) Find the size of angle BAC.. Solve sin cos for.

7 . The diagram shows part of the quartic with equation g( ). There are stationar points at, and a. a On separate diagrams sketch the graph of (a) g( ). (b) g( ).. Find the values of for which the function f ( ) is decreasing. 6. P is the point with coordinates (, 6) and Q is(, ). Find the locus of points which are equidistant from both P and Q.

8 Practice Paper C. A sequence is defined b the recurrence relation What is the value of u? u u, u 6 n n. Here are two statements about the line with equation 8. () This line is parallel to a line with gradient () This line cuts the -ais at the point (, 8). Which of these statements is true?.. Functions f and g are defined on suitable domains b Find an epression for f ( g( )). f( ) and g( ).. A curve has equation. What is the gradient of the tangent at the point where?. A circle with centre (, ) passes through the point (, ). What is the equation of the circle? 6. Find d. 7. g ( ) 7. What is the remainder when g ( ) is divided b ( )? R 8. Vectors u and v are shown in the diagram below. QR ST Find PQ in terms of u and v. Q S v u T u P

9 9. P and Q are the points with coordinates (,, ) and (,, ) respectivel. If PR PQ, find the coordinates of R.. What is the eact value of sin cos?. Find cos( ) d.. Given that log log log 8, epress in terms of.. Given that d d sin, find.. If 6 is written in the form p q ( ), what is the value of p?. Solve tan for 6. The diagram shows the graph with equation log ( a ). b (, ) log ( a) b What are the values of a and b? 7. What is the nature of the roots of the quadratic equation? 8. The diagram shows part of the graph of cubic with equation g( ). The graph has turning points at and. Sketch the graph of = g ().

10 9. Solve 8.. The diagram shows part of the curve f ( ). L The curve passes through the points K(, ) and L(, 7). K Which of the following represents the equation of the curve? A B C e D. A function f is defined b f ( ), where is a real number. (a) Find the coordinates of the points where the curve with equation f ( ) crosses the and -aes. (b) Find the stationar points on the curve f ( ) and determine their nature. (c) (i) Sketch the curve f ( ). (ii) Hence solve.. Two sequences are generated b the recurrence relations u v n n u 8 kv n n The two sequences approach the same limit as n. (a) Evaluate this limit. (b) Hence determine the value of k.. Given that sin a and sin b, where π π b, a and find the eact values of : (a) sin( a b); (b) tan( a b).. In the triangle opposite a b units a b Find a.( a b c ) c

11 Practice Paper D. The midpoint of the line joining G(,, 7) to H(,, p) is M( q,, ). What are the values of p and q?. Given that f( ), find f ( ).. If 7 is written in the form ( a) r, find the value of r.. A straight line passes through the points (, ) and (, ). What is the equation of the line?. Functions f and g are defined on the set of real numbers b What is the value of g( f( ))? f g ( ) and ( ) 6. The vectors with components 7 What is the value of t? and t are perpendicular. 7. The diagram shows a right-angled triangle with sides, and. What is the value of cos? 8. Find 6 d 9. For what value of k does the equation k have equal roots?. DE and EF have components and respectivel. Given that D has coordinates (,, ), what are the coordinates of F?

12 7π. What is the maimum value of 9. Find ( ) d. 8 sin?. How man solutions does the equation( 7 cos )(tan 9) have in the interval find f. Given that f( ) sin, 6.. The diagram shows the line ST with equation. The angle between ST and the positive direction of the -ais is Find an epression for A tan B tan C tan D tan 6. What is the value of log? log 8 7. The diagram shows a sketch of the curve with equation k( )( )( a) What are the values of a and k? 8. Here are two statements about the function f ( ). () The largest possible domain is. () The range is f( ). Which of these statements is true? 9. Given that, for f ( ), for, for Sketch a curve to represent f ( )?. If a, find an epression for.

13 . A(, ),B(, ) and C(, 8) are the vertices of triangle ABC shown in the diagram. C A B (a) Write down the equation of the altitude from C. (b) (c) Find the equation of the perpendicular bisector of BC. Find the point of intersection of the lines found in (a) and (b).. P is the point (,, ), Q is (,, ) and R is (7,, ). (a) (b) Show that P, Q and R are collinear. Find the ratio in which Q divides PR.. Find the equation of the tangent to the curve with equation at the point where.. (a) Given that f ( ) and ( ) is a factor of f( ), find a formula for f( ). (b) Hence factorise f( ) full. (c) Solve f( ).

14 b. The graph illustrates the law a. The straight line joins the points (, ) and (, ). log Find the values of a and b. log

15 Practice Paper E. K and L have position vectors and respectivel. What is the magnitude of KL?. If f( ) 7, find f ( ).. Find d. A function f is defined on the set of real numbers b f ( ). Find an epression for f ( f ( )).. Evaluate. sin cos 6. A circle with centre (, ) passes through the point (, ). What is the equation of the circle? 7. f ( ). What is the remainder when f( ) is divided b ( )? 8. The diagram shows the part of the graph of the cubic f ( ). Sketch the graph of f ( )?

16 9. The graph shown in the diagram has equation p sin( q). What are the values of p and q?. A sequence is generated b the recurrence relation u 7 u. n n If u, what is the value of u?. For what value of k does the equation k 6 have equal roots?. Find ( 7) d.. Given that and (), f ( ) 6 f find a formula for f( ) in terms of.. What are the coordinates of the centre of the circle with equation 6 8?. The diagram shows part of the graph of a cubic function. 6 What is the equation of this graph?

17 6. The diagram shows part of the graph of the cubic f ( ). There are stationar points at and. Sketch the graph of f ( )? 7. If 8 is epressed in the form ( p) q, what is the value of q? 8. If log t log, find the value of t. 9. If p find the rate of change of p with respect to when.. Find the solutions for 8?. A line joins the points P(, ) and Q(, 7). Find the equation of the perpendicular bisector of PQ. P Q. Show that the line with equation is a tangent to the circle with equation and find the coordinates of the point of contact of the tangent and circle.. The diagram shows a right-angled triangle with height units, base unit and an angle of p. (a) Find the eact values of: (i) cos p ; (ii) cos p. (b) B writing p p p, find the eact value of cos p. p

18 . A function f is defined b f ( ), where. Find the maimum and minimum values of f.. (a) Epress cos sin in the form k cos( a), where k and a 6. (b) Find: (i) the maimum value of sin cos ; (ii) a value of where this maimum value occurs in the interval 6.

19 PRACTICE PAPER F. If f( ) ( )( ), find f ( ).. Vectors p is given b i j k and q is i j k. What are the components of p q?. A circle has equation 8. What is the radius of this circle?. The line with equation k is parallel to the line with gradient. What is the value of k?. What is the derivative of 6, with respect to? 6. Find d. 7. A circle centre (, ) passes through the point (, ). What is the equation of the circle? 8. What is the value of π 6 sin cos? 9. Determine the coordinates of the stationar point and it s nature for the curve with equation = ( 9). Here are two statements about the equation () The roots are unequal; () The roots are irrational. Which of these statements is true? π. What is the minimum value of 6 cos?

20 d. If sin(7 ), find. d. Find the radius of the circle with equation 8.. g ( ). For what value(s) of is g ( ) undefined?. The diagram shows part of the graph of the cubic f ( ). f ( ) m There are three roots at, and m as shown. There are two stationar points ling between the roots. Sketch the graph of f ( )? 6. The equation of the parabola shown is of the form k( )( ). What is the value of k? k( )( ) (, ) 7. What is the maimum value of ( )( )? 8. Given that a. b and a.( a b ), find a. 9. If log log log epress in terms of Sketch the graph of log?

21 . (a) (i) Show that ( ) is a factor of f ( ) 6. (ii) Hence factorise f( ) full. 6 6 d 6, p, find the value of p. (b) Given that p. (a) Write 6 in the form ( a) b. (b) (i) Sketch the graph of 6. (c) (ii) State the range of values of. Write down the maimum value of. 6. The diagram shows part of the curve with equation log ( a). b Q log ( a) b P The curve passes through the points P(, ) and Q(, ). Find the values of a and b.. (a) Write (b) Find cos in terms of cos. cos d.. In the triangle shown p q and r. r Find p.( p q r ) q p

22 ANSWERS PAPER A units. ( ) c. 8. (9,, 7). 6 c. and Onl statement is correct ( )( )( ) 8. ( ) ( ) real and distinct roots.. (a) (b) (c) (, ). Maimum turning point at (, 8) Minimum turning point at (, ) 6. (a) (b) ( ) or 7 7. (a) Proof (b) sin 6 (c), 6

23 PAPER B. 9.. c. ( ). 9 units. p =, q =. -.,. real and distinct roots. c 6. 7 and : cos( ) 8. log (a) Centre (, ) and radius unit (b) (i) units (ii) units. (a) (b) AB and AC Angle BAC 9 or.,. (a) (b) g( ) g( ) a a. :, 6. 9

24 PAPER C. 9.. Onl statement is correct.. f( g( )). sin( ) c 8 sin... ( ) ( ) 8. 6 cos 6. c 6. a =, b = real and distinct roots 8. u v (, 6, ) 9. or... (a) (, ) and, (b) Maimum turning point at (, ) Minimum turning point at (, ) (c) (i) (ii). (a) Limit : (b) 6 k 7. (a) sin( ab) (b) tan( a b). 8

25 PAPER D. p =, q =... ( 6 ) c tan a =, k = Onl statement is correct (,, ). log a. (a) (b) 9 (c),. (a) Proof e.g. show that QR PQ (b) :.. (a) f ( ) 8 (b) f( ) ( )( )( ) (c) {,, }. a 6 and b

26 PAPER E c. ( 7) f. f( f( )) 6. (, ). -. c ( ) ( ) ( ) 6. ( ) ( ) p =, q = or. 7. Point of contact (, ). (a) (i) (b) (ii). Maimum value, minimum value 7. (a) cos( ) (b) Maimum value 7 at

27 PAPER F cos(7 ) and c 7 6. ( ) ( ) maimum at (9, ) 9.. Both statements are correct.. (a) (i) Show that f ( ) (ii) ( )( )( ) (b). (a) p ( ) (b) (i) (ii) (c). a, b. (a) (cos ) or cos (b) sin c. 8