Prelim practice - Higher Maths · PDF fileHigher Mathematics Prelim practice [SQA] 1. Part Marks Level Calc. Content Answer U1 OC1 3 C CR G2 1992 P1 Q13 [SQA] 2. Find the equation

Higher Mathematics

Prelim practice

1.[SQA]

Part Marks Level Calc. Content Answer U1 OC1

3 C CR G2 1992 P1 Q13

2.[SQA] Find the equation of the perpendicular bisector of the line joining A(2,−1) andB(8, 3) . 4


4 C CN G2, G5 1996 P1 Q1

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Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes

Higher Mathematics

3.[SQA]


3 C CN G2, G5, G3 1997 P1 Q1

4.[SQA]


(a) 2 C CN G3 1999 P1 Q2

(b) 3 C CN G8

5.[SQA] Find the equation of the straight line which is parallel to the line with equation2x+ 3y = 5 and which passes through the point (2,−1) . 3


3 C CN G3, G2 2x+ 3y = 1 2001 P1 Q1

•1 ss: express in standard form•2 ic: interpret gradient•3 ic: state equation of straight line

•1 y = − 23 x+ 53 stated or implied by •2

•2 mline = − 23 stated or implied by •3•3 y− (−1) = − 23(x− 2)

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Higher Mathematics

6.[SQA] A quadrilateral has vertices A(−1, 8) , B(7, 12) , C(8, 5) and D(2,−3) as shown inthe diagram.

A

B

E

D

C

O x

y

(a) Find the equation of diagonal BD. 2

(b) The equation of diagonal AC is x+ 3y = 23.

Find the coordinates of E, the point of intersection of the diagonals. 3

(c) (i) Find the equation of the perpendicular bisector of AB.

(ii) Show that this line passes through E. 5


(a) 2 C CN G3, G2 y− 12 = 3(x− 7) 2011 P1 Q21

(b) 3 C CN G8 E(5, 6)

(ci) 4 C CN G7 y− 10 = −2(x− 3)(cii) 1 C CN A6 proof

•1 pd: find gradient of BD•2 ic: state equation of BD

•3 ss: start solution of simultaneouseqs

•4 pd: solve for one variable•5 pd: solve for second variable

•6 ss: know and find midpoint of AB•7 pd: find gradient of AB•8 ic: interpret perpendicular gradient•9 ic: state equation of perp. bisector•10 ic: justification of point on line

•1 155 or equiv.•2 y− (−3) = 3(x− 2)

•3 3x− y = 9 and x+ 3y = 23•4 x = 5 or y = 6•5 y = 6 or x = 5

•6 (3, 10)•7 48 or equiv.•8 − 84 or equiv•9 y− 10 = −2(x− 3)•10 when x = 5, y = −2× 5+ 16 = 6

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Higher Mathematics

7.[SQA]


(a) 6 C NC G3, G5, G8 1992 P1 Q2

(b) 2 C NC G8

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Higher Mathematics

8.[SQA] Triangle PQR has vertex P on the x -axis,as shown in the diagram.

Q and R are the points (4, 6) and (8,−2)respectively.

The equation of PQ is 6x− 7y+ 18 = 0.

(a) State the coordinates of P. 1

(b) Find the equation of the altitude ofthe triangle from P. 3

(c) The altitude from P meets the lineQR at T. Find the coordinates of T. 4

6 – 7 + 18 = 0 T

P

Q(4, 6)

R(8, –2)

Ox

x

y

y


(a) 1 C CN G4 P(−3, 0) 2009 P1 Q21

(b) 3 C CN G7 y = 12(x+ 3)

(c) 4 C CN G8 T(5, 4)

•1 ic: interpret x-intercept

•2 pd: find gradient (of QR)•3 ss: know and use m1m2 = −1•4 ic: state equ. of altitude

•5 ic: state equ. of line (QR)•6 ss: prepare to solve sim. equ.•7 pd: solve for x•8 pd: solve for y

•1 P = (−3, 0)

•2 mQR = −2•3 malt. = 1

2

•4 y− 0 = 12(x+ 3)

•5 y+ 2 = −2(x− 8)•6 x− 2y = −3 and 2x+ y = 14•7 x = 5•8 y = 4

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Higher Mathematics

9.[SQA] Triangle ABC has vertices A(−1, 6) ,B(−3,−2) and C(5, 2) .

Find

(a) the equation of the line p , themedian from C of triangle ABC. 3

(b) the equation of the line q , theperpendicular bisector of BC. 4

(c) the coordinates of the point ofintersection of the lines p and q . 1

Ox

yA(−1, 6)

B(−3,−2)

C(5, 2)


(a) 3 C CN G7 y = 2 2002 P2 Q1

(b) 4 C CN G7 y = −2x+ 2

(c) 1 C CN G8 (0, 2)

•1 ss: determine midpoint coordinates•2 pd: determine gradient thro’ 2 pts•3 ic: state equation of straight line

•4 ss: determine midpoint coordinates•5 pd: determine gradient thro’ 2 pts•6 ss: determine gradient perp. to •5•7 ic: state equation of straight line

•8 pd: process intersection

•1 F = midAB = (−2, 2)•2 mFC = 0 stated or implied by •3•3 equ. FC is y = 2

•4 M = midBC = (1, 0)•5 mBC = 1

2•6 m⊥ = −2•7 y− 0 = −2(x− 1)

•8 (0, 2)

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Higher Mathematics

10.[SQA] The diagram shows a sketch of thefunction y = f (x) .

(a) Copy the diagram and on it sketchthe graph of y = f (2x) . 2

(b) On a separate diagram sketch thegraph of y = 1− f (2x) . 3

= f( )

(2, 8)(–4, 8)

O x

x

y

y


(a) 2 B CN A3 sketch 2009 P1 Q23

(b) 3 B CN A3 sketch

•1 ic: scaling parallel to x-axis•2 ic: annotate graph

•3 ss: correct order for refl(x) and trans•4 ic: start to annotate final sketch•5 ic: complete annotation

•1 sketch and one of (0, 0), (1, 8),(−2, 8)

•2 remaining points

•3 reflect in x-axis then verticaltranslation

•4 sketch and one of (0, 1), (1,−7),(−2,−7)

•5 remaining points

11.[SQA]


(a) 2 C NC A3 1991 P1 Q9

(b) 3 C NC A3, C11

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Higher Mathematics

12.[SQA]


(a) 2 C CN A3 1992 P1 Q10

(b) 2 C CN A3

13.[SQA]


3 C NC A3, A1 1993 P1 Q14

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Higher Mathematics

14.[SQA]


3 A/B NC A3, A2 1990 P1 Q17

15.[SQA] f (x) = 3− x and g(x) =3

x, x 6= 0.

(a) Find p(x) where p(x) = f (g(x)) . 2

(b) If q(x) =3

3− x , x 6= 3, find p(q(x)) in its simplest form. 3


(a) 2 C CN A4 3− 3x 2000 P2 Q3

(b) 2 C CN A4 x

(b) 1 A/B CN A4

•1 ic: interpret composite func.•2 pd: process

•3 ic: interpret composite func.•4 pd: process•5 pd: process

•1 f(3x

)stated or implied by •2

•2 3− 3x

•3 p(33−x

)stated or implied by •4

•4 3− 333−x

•5 x

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Higher Mathematics

16.[SQA] On a suitable set of real numbers, functions f and g are defined by f (x) =1

x+ 2

and g(x) =1

x− 2.

Find f(g(x)

)in its simplest form. 3


3 C NC A4 1992 P1 Q6

17.[SQA] The functions f and g , defined on suitable domains, are given by f (x) =1

x2 − 4and g(x) = 2x+ 1.

(a) Find an expression for h(x) where h(x) = g(f (x)

). Give your answer as a

single fraction. 3

(b) State a suitable domain for h . 1


(a) 2 C NC A4 1995 P1 Q11

(a) 1 A/B NC A4

(b) 1 A/B NC A1

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Higher Mathematics

18.[SQA] Functions f and g , defined on suitable domains, are given by f (x) = 2x andg(x) = sin x+ cos x .

Find f(g(x)

)and g

(f (x)

). 4


4 C NC A4 1997 P1 Q3

19.[SQA] Functions f and g are defined by f (x) = 2x+ 3 and g(x) =x2 + 25

x2 − 25 where x ∈ R ,

x 6= ±5.The function h is given by the formula h(x) = g

(f (x)

).

For which real values of x is the function h undefined? 4


2 C CN A4, A1 1989 P1 Q19

2 A/B CN A4, A1

20.[SQA]

(a) Express 7− 2x− x2 in the form a− (x+ b)2 and write down the values of aand b . 2

(b) State the maximum value of 7− 2x− x2 and justify your answer. 2


(a) 2 A/B NC A5 1991 P1 Q15

(b) 2 A/B NC A6

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Higher Mathematics

21.[SQA] Express (2x− 1)(2x+ 5) in the form a(x+ b)2 + c . 3


3 C NC A5 1996 P1 Q17

22.[SQA]

(a) Show that the function f (x) = 2x2 + 8x − 3 can be written in the formf (x) = a(x+ b)2 + c where a , b and c are constants. 3

(b) Hence, or otherwise, find the coordinates of the turning point of the functionf . 1


(a) 3 C NC A5 1997 P1 Q9

(b) 1 C NC A6

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Higher Mathematics

23.[SQA]


3 C NC A6 1996 P1 Q3

24.[SQA] The diagram shows a sketch of part of the graphof y = log2(x) .

(a) State the values of a and b . 1

(b) Sketch the graph of y = log2(x+ 1)− 3. 3O x

yy = log2(x)

(a, 0)

(8, b)


(a) 1 A/B CN A7 a = 1, b = 3 2001 P1 Q10

(b) 3 A/B CN A3 sketch

•1 pd: use logp q = 0 ⇒ q = 1 and

evaluate logp pk

•2 ss: use a translation•3 ic: identify one point•4 ic: identify a second point

•1 a = 1 and b = 3

•2 a “log-shaped” graph of the sameorientation

•3 sketch passes through (0,−3)(labelled)

•4 sketch passes through (7, 0)(labelled)

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Higher Mathematics

25.[SQA] A sketch of the graph of y = f (x) where f (x) = x3 − 6x2+ 9x is shown below.The graph has a maximum at A and a minimum at B(3, 0) .

Ox

yA y = f (x)

B(3, 0)

(a) Find the coordinates of the turning point at A. 4

(b) Hence sketch the graph of y = g(x) where g(x) = f (x+ 2) + 4.

Indicate the coordinates of the turning points. There is no need to calculatethe coordinates of the points of intersection with the axes. 2

(c) Write down the range of values of k for which g(x) = k has 3 real roots. 1


(a) 4 C NC C8 A(1, 4) 2000 P1 Q2

(b) 2 C NC A3 sketch (translate 4 up, 2left)

(c) 1 A/B NC A2 4 < k < 8

•1 ss: know to differentiate•2 pd: differentiate correctly•3 ss: know gradient = 0•4 pd: process

•5 ic: interpret transformation•6 ic: interpret transformation

•7 ic: interpret sketch

•1 dydx = . . .

•2 dydx = 3x2 − 12x+ 9•3 3x2 − 12x+ 9 = 0•4 A = (1, 4)

translate f (x) 4 units up, 2 units left

•5 sketch with coord. of A′(−1, 8)•6 sketch with coord. of B′(1, 4)

•7 4 < k < 8 (accept 4 ≤ k ≤ 8)

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Higher Mathematics

26.[SQA]


(a) 5 C CN C8 1991 P2 Q1

(b) 2 C CN A1

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Higher Mathematics

27.[SQA]


(a) 6 C CN C8 1998 P2 Q2

(b) 2 C CN C8

28.[SQA] A curve has equation y = 2x3+ 3x2+ 4x− 5.Prove that this curve has no stationary points. 5


2 C NC C8, C7 1999 P1 Q16

3 A/B NC C8, C7

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Higher Mathematics

29.[SQA] A curve has equation y = x− 16√x, x > 0.

Find the equation of the tangent at the point where x = 4. 6


6 C CN C4, C5 y = 2x− 12 2001 P2 Q2

•1 ic: find corresponding y-coord.•2 ss: express in standard form•3 ss: start to differentiate•4 pd: diff. fractional negative power•5 ss: find gradient of tangent•6 ic: write down equ. of tangent

•1 (4,−4) stated or implied by •6•2 −16x− 12•3 dydx = 1 . . .

•4 . . .+ 8x− 32•5 mx=4 = 2•6 y− (−4) = 2(x− 4)

30.[SQA] A ball is thrown vertically upwards. The height h metres of the ball t seconds afterit is thrown, is given by the formula h = 20t− 5t2 .(a) Find the speed of the ball when it is thrown (i.e. the rate of change of heightwith respect to time of the ball when it is thrown). 3

(b) Find the speed of the ball after 2 seconds.

Explain your answer in terms of the movement of the ball. 2


(a) 1 C NC C6 1995 P1 Q21

(a) 2 A/B NC C6

(b) 2 A/B NC A6

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Higher Mathematics

31.[SQA] Find the x -coordinate of each of the points on the curve y = 2x3 − 3x2 − 12x+ 20at which the tangent is parallel to the x -axis. 4


4 C NC C4 1993 P1 Q4

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Higher Mathematics

32.[SQA]


(a) 6 C CN C4, G3 1999 P2 Q11

(b) 6 A/B CN G8, G1

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Higher Mathematics

33. The parabolas with equations y = 10− x2 and y = 25(10− x2) are shown in the

diagram below.

= 10 – 2

R

ST

Q

P

22

5(10 )= −

O

x

x

x y

y

y

A rectangle PQRS is placed between the two parabolas as shown, so that:

• Q and R lie on the upper parabola.

• RQ and SP are parallel to the x-axis.

• T , the turning point of the lower parabola, lies on SP.

(a) (i) If TP = x units, find an expression for the length of PQ.

(ii) Hence show that the area, A , of rectangle PQRS is given by

A(x) = 12x− 2x3· 3

(b) Find the maximum area of this rectangle. 6


(ai) 2 B CN C11 6− x2 2010 P2 Q5

(aii) 1 B CN C11 2x× (6− x2) = A(x)

(b) 6 C CN C11 max is 8√2

•1 ss: know to and find OT•2 ic: obtain an expression for PQ•3 ic: complete area evaluation

•4 ss: know to and start to differentiate•5 pd: complete differentiation•6 ic: set derivative to zero•7 pd: obtain•8 ss: justify nature of stationary point•9 ic: interpret result and evaluatearea

•1 4•2 10− x2 − 4•3 2x(6− x2) = 12x− 2x3

•4 A′(x) = 12 · · ·•5 12− 6x2•6 12− 6x2 = 0•7

√2

•8 x · · ·√2 · · ·

A′(x) + 0 −•9 Max and 8

√2

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Higher Mathematics

34.[SQA]


(a) 2 C CN C4 1989 P1 Q14

(b) 2 C CN A3

35.[SQA] Differentiate 2√x(x+ 2) with respect to x . 4


4 C NC C1 1998 P1 Q14

36.[SQA] If f (x) = kx3 + 5x− 1 and f ′(1) = 14, find the value of k . 3


3 C NC C1, A6 1994 P1 Q2

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Higher Mathematics

37.[SQA] Functions f and g are given by f (x) = 3x+ 1 and g(x) = x2 − 2.(a) (i) Find p(x) where p(x) = f (g(x)) .

(ii) Find q(x) where q(x) = g( f (x)) . 3

(b) Solve p′(x) = q′(x) . 3


(a) 3 C CN A4 3(x2− 2) + 1, (3x+ 1)2− 2 2009 P2 Q2

(b) 3 C CN C1 x = − 12

•1 ss: substitute for g(x) in f (x)•2 ic: complete•3 ic: sub. and complete for q(x)

•4 ss: simplify•5 pd: differentiate•6 pd: solve

•1 f (x2 − 2)•2 3(x2 − 2) + 1•3 (3x+ 1)2 − 2

•4 p(x) = 3x2 − 5, q(x) = 9x2 + 6x− 1•5 p′(x) = 6x, q′(x) = 18x+ 6•6 x = − 12

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Higher Mathematics

38.[SQA] A function f is defined on the set of real numbers by f (x) = (x− 2)(x2 + 1) .

(a) Find where the graph of y = f (x) cuts:

(i) the x -axis;

(ii) the y-axis. 2

(b) Find the coordinates of the stationary points on the curve with equationy = f (x) and determine their nature. 8

(c) On separate diagrams sketch the graphs of:

(i) y = f (x) ;

(ii) y = − f (x) . 3


(a) 2 CN A6 (2, 0), (0,−2) 2011 P1 Q22

(b) 8 CN C8, C9 max: ( 13 ,− 5027), min:(1,−2)

(ci) 2 CN A8, A7 sketch

(cii) 1 CN A3 reflect in x-axis

•1 ic: interpret x intercept•2 ic: interpret y intercept

•3 ic: write in differentiable form•4 ss: know to and start to differentiate•5 pd: complete derivative and equateto 0

•6 pd: factorise derivative•7 pd: process for x•8 pd: evaluate y-coordinates•9 ic: justify nature of stationarypoints

•10 ic: interpret and state conclusions

•11 ic: curve showing points from (a)and (b) without annotation

•12 ic: cubic curve showing all

intercepts and stationary pointsannotated

•13 ic: curve from (i) reflected in x-axis

•1 (2, 0)•2 (0,−2)

•3 x3 − 2x2 + x− 2•4 3x2 . . .•5 3x2 − 4x+ 1 = 0•6 (3x− 1)(x− 1)•7 13 and 1•8 − 5027 and −2•9 x → 1

3 → 1 →f ′(x) + 0 − 0 +

•10 max. at ( 13 − 5027), min. at (1,−2)

•11 sketch•12 sketch•13 reflected sketch

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Higher Mathematics

39.[SQA]


(a) 1 C CN CGD 1996 P2 Q11

(a) 3 A/B CN CGD

(b) 2 C CN C11

(b) 3 A/B CN C11

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Higher Mathematics

40.[SQA]


(a) 3 A/B CR CGD 1998 P2 Q10

(b) 3 C CR C11

(b) 3 A/B CR C11

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Higher Mathematics

41.[SQA] Find the coordinates of the point on the curve y = 2x2− 7x+ 10 where the tangentto the curve makes an angle of 45◦ with the positive direction of the x -axis. 4


4 C NC G2, C4 (2, 4) 2002 P1 Q4

•1 sp: know to diff., and differentiate•2 pd: process gradient from angle•3 ss: equate equivalent expressions•4 pd: solve and complete

•1 dydx = 4x− 7•2 mtang = tan 45◦ = 1•3 4x− 7 = 1•4 (2, 4)

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Higher Mathematics

42. Functions f , g and h are defined on the set of real numbers by

• f (x) = x3− 1

• g(x) = 3x+ 1

• h(x) = 4x− 5.

(a) Find g( f (x)) . 2

(b) Show that g( f (x)) + xh(x) = 3x3+ 4x2 − 5x− 2. 1

(c) (i) Show that (x− 1) is a factor of 3x3 + 4x2− 5x− 2.(ii) Factorise 3x3 + 4x2 − 5x− 2 fully. 5

(d) Hence solve g( f (x)) + xh(x) = 0. 1


(a) 2 C CN A4 3(x3 − 1) + 1 2011 P2 Q2

(b) 1 C CN A6 proof

(c) 5 C CN A21 (x− 1)(3x+ 1)(x+ 2)

(d) 1 C CN A22 −2,− 13 , 1

•1 ic: interpret notation•2 ic: complete process

•3 ic: substitute and complete

•4 ss: know to use x = 1•5 pd: complete evaluation•6 ic: state conclusion•7 ic: find quadratic factor•8 pd: factorise completely

•9 ic: interpret and solve equation in(d)

•1 g(x3 − 1)•2 3(x3 − 1) + 1

•3 3(x3 − 1) + 1+ x(4x− 5)= 3x3 + 4x2 − 5x− 2

•4 evaluating at x = 1...•5 3+ 4− 5− 2 = 0•6 (x− 1) is a factor•7 (x− 1)(2x2 + 7x+ 2)•8 (x− 1)(3x+ 1)(x+ 2)

•9 −2,− 13 , 1

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Higher Mathematics

43.[SQA]

(a) The function f is defined by f (x) = x3 − 2x2− 5x+ 6.

The function g is defined by g(x) = x− 1.Show that f

(g(x)

)= x3 − 5x2+ 2x+ 8. 4

(b) Factorise fully f(g(x)

). 3

(c) The function k is such that k(x) =1

f(g(x)

) .

For what values of x is the function k not defined? 3


(a) 4 C NC A4 1990 P2 Q6

(b) 3 C NC A21

(c) 2 C NC A1

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Higher Mathematics

44.[SQA]


(a) 1 C CN A6 1994 P2 Q9

(b) 2 C CN C4, CGD

(b) 4 A/B CN C4, CGD

(c) 2 A/B CN A17

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Higher Mathematics

45.[SQA]

(i) Write down the condition for the equation ax2 + bx + c = 0 to have no realroots. 1

(ii) Hence or otherwise show that the equation x(x + 1) = 3x − 2 has no realroots. 2


3 C CN A17 1999 P1 Q8

46.[SQA] Given that k is a real number, show that the roots of the equation kx2 + 3x+ 3 = kare always real numbers. 5


1 C NC A17 1991 P1 Q18

4 A/B NC A17

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Higher Mathematics

47.[SQA]


1 C NC A18 1992 P1 Q17

3 A/B NC A18

48. (a)[SQA] Given that x+ 2 is a factor of 2x3 + x2 + kx+ 2, find the value of k . 3

(b) Hence solve the equation 2x3 + x2 + kx+ 2 = 0 when k takes this value. 2


(a) 3 C CN A21 k = −5 2001 P2 Q1

(b) 2 C CN A22 x = −2, 12 , 1

•1 ss: use synth division orf (evaluation)

•2 pd: process•3 pd: process

•4 ss: find a quadratic factor•5 pd: process

•1 f (−2) = 2(−2)3 + · · ·•2 2(−2)3 + (−2)2 − 2k+ 2•3 k = −5

•4 2x2 − 3x + 1 or 2x2 + 3x − 2 orx2 + x− 2

•5 (2x− 1)(x− 1) or (2x− 1)(x+ 2) or(x+ 2)(x− 1)and x = −2, 12 , 1

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Higher Mathematics

49. (a) (i) Show that (x− 1) is a factor of f (x) = 2x3 + x2 − 8x+ 5.

(ii) Hence factorise f (x) fully. 5

(b) Solve 2x3 + x2− 8x+ 5 = 0. 1

(c) The line with equation y = 2x − 3 is a tangent to the curve with equationy = 2x3 + x2 − 6x+ 2 at the point G.

Find the coordinates of G. 5

(d) This tangent meets the curve again at the point H.

Write down the coordinates of H. 1


(a) 5 C CN A21 (x− 1)(x− 1)(2x+ 5) 2010 P1 Q22

(b) 1 C CN A22 x = 1,− 52(c) 5 C CN A23 (1,−1)(d) 1 C CN A23 (− 52 ,−8)

•1 ss: know to use x = 1•2 ic: complete evaluation•3 ic: state conclusion•4 pd: find quadratic factor•5 pd: factorise completely

•6 ic: state solutions

•7 ss: set ycurve = yline•8 ic: express in standard form•9 ss: compare with (a) or factorise•10 ic: identify xG•11 pd: evaluate yG

•12 pd: state solution

•1 evaluating at x = 1...•2 2+ 1− 8+ 5 = 0•3 (x− 1) is a factor•4 (x− 1)(2x2 + 3x− 5)•5 (x− 1)(x− 1)(2x+ 5)

•6 x = 1 and x = − 52

•7 2x3 + x2 − 6x+ 2 = 2x− 3•8 2x3 + x2 − 8x+ 5 = 0•9 (x− 1)(x− 1)(2x+ 5) = 0•10 x = 1•11 y = −1

•12 (− 52 ,−8)

50.[SQA] Express x3 − 4x2 − 7x+ 10 in its fully factorised form. 4


4 C NC A21 1998 P1 Q2

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Higher Mathematics

51.[SQA]


(a) 4 C CN G5, G3 1991 P2 Q2

(b) 6 C CN G10, G1

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Higher Mathematics

52.[SQA]


(a) 3 C CN G5, G3 1993 P2 Q3

(b) 5 C CN G10

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Higher Mathematics

53.[SQA]

(a) Show that the point P(5, 10) lies on circle C1 with equation(x+ 1)2 + (y− 2)2 = 100. 1

(b) PQ is a diameter of this circle asshown in the diagram. Find theequation of the tangent at Q. 5P(5, 10)

Q

O x

y

(c) Two circles, C2 and C3 , touch circle C1 at Q.

The radius of each of these circles is twice the radius of circle C1 .

Find the equations of circles C2 and C3 . 4


(a) 1 C CN A6 proof 2009 P2 Q4

(b) 5 C CN G11 3x+ 4y+ 45 = 0

(c) 4 A NC G15 (x− 5)2 + (y− 10)2 = 400,(x+ 19)2+(y+ 22)2 = 400

•1 pd: substitute

•2 ic: find centre•3 ss: use mid-point result for Q•4 ss: know to, and find gradient ofradius

•5 ic: find gradient of tangent•6 ic: state equation of tangent

•7 ic: state radius•8 ss: know how to find centre•9 ic: state equation of one circle•10 ic: state equation of the other circle

•1 (5+ 1)2 + (10− 2)2 = 100

•2 centre = (−1, 2)•3 Q = (−7,−6)•4 mrad = 8

6

•5 mtgt = − 34•6 y− (−6) = − 34(x− (−7))

•7 radius = 20•8 centre = (5, 10)•9 (x− 5)2 + (y− 10)2 = 400•10 (x+ 19)2 + (y+ 22)2 = 400

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Higher Mathematics

54.[SQA] Find the equation of the circle which has P(−2,−1) and Q(4, 5) as the end pointsof a diameter. 3


3 C CN G10 1995 P1 Q9

55.[SQA]


(a) 3 C CN G12 1994 P1 Q8

(b) 3 C CN G10

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Higher Mathematics

56. (a) (i) Show that the line with equation y = 3− x is a tangent to the circle withequation x2 + y2 + 14x+ 4y− 19 = 0.

(ii) Find the coordinates of the points of contact, P. 5

(b) Relative to a suitable set of coordinate axes, the diagram below shows thecircle from (a) and a second smaller circle with centre C.

P

C

The line y = 3− x is a common tangent at the point P.The radius of the larger circle is three times the radius of the smaller circle.

Find the equation of the smaller circle. 6


(ai) 4 C CN G13 proof 2010 P2 Q3

(aii) 1 C CN G12 P(−1, 4)(b) 6 B CN G9, G15 (x− 1)2 + (y− 6)2 = 8

•1 ss: substitute•2 pd: express in standard form•3 ic: start proof•4 ic: complete proof•5 pd: coordinates of P

•6 ic: state centre of larger circle•7 ss: find radius of larger circle•8 pd: find radius of smaller circle•9 ss: strategy for finding centre•10 ic: interpret centre of smaller circle•11 ic: state equation

•1 x2+(3− x)2+ 14x+ 4(3− x)− 19 = 0•2 2x2 + 4x+ 2 = 0•3 2(x+ 1)(x+ 1)•4 equal roots so line is a tangent•5 x = −1, y = 4

•6 (−7,−2)•7

√72

•8√8

•9 e.g. “Stepping out”•10 (1, 6)•11 (x− 1)2 + (y− 6)2 = 8

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Higher Mathematics

57.[SQA] Explain why the equation x2 + y2 + 2x+ 3y+ 5 = 0 does not represent a circle. 2


2 C CN G9 1993 P1 Q18

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Higher Mathematics

58.[SQA]


(a) 4 C CN G9, G5 1999 P2 Q2

(b) 1 C CN A6

(c) 1 C CN CGD

(d) 2 C CN G10

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Higher Mathematics

59.[SQA] ABCD is a quadrilateral with vertices A(4,−1, 3) , B(8, 3,−1) , C(0, 4, 4) andD(−4, 0, 8) .(a) Find the coordinates of M, the midpoint of AB. 1

(b) Find the coordinates of the point T, which divides CM in the ratio 2 : 1. 3

(c) Show that B, T and D are collinear and find the ratio in which T divides BD. 4


(a) 1 C CN G6, G25 1989 P2 Q2

(b) 3 C CN G25

(c) 4 C CN G23, G25

60.[SQA]


5 C CN G9, G25 1997 P1 Q12

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Higher Mathematics

61.[SQA]


(a) 2 C CN G16 1998 P1 Q3

(b) 1 C CN G26

(c) 1 C CN G16

62.[SQA]


(a) 2 C CN G16 1998 P1 Q5

(b) 2 C CN G16

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Higher Mathematics

63.[SQA]


(a) 3 C CR G16 1994 P2 Q3

(b) 1 C CR G25

(c) 4 C CR G28

(d) 2 C CR CGD

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Higher Mathematics

64.[SQA] The vector ai + b j + k is perpendicular to both the vectors i − j + k and−2i + j + k .Find the values of a and b . 3


3 C CN G18 a = 2, b = 3 1990 P1 Q12

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Higher Mathematics

65.[SQA]


(a) 2 C CN G20 1999 P2 Q3

(b) 2 C CN G20

(c) 5 C CN G28

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Higher Mathematics

66.[SQA] A cuboid measuring 11 cm by 5 cm by 7 cm is placed centrally on top of anothercuboid measuring 17 cm by 9 cm by 8 cm.

Coordinates axes are taken as shown.

O

x

y

5 7

89

11

17

z

A

BC

(a) The point A has coordinates (0, 9, 8) and C has coordinates (17, 0, 8) .

Write down the coordinates of B. 1

(b) Calculate the size of angle ABC. 6


(a) 1 C CN G22 B(3, 2, 15) 2000 P2 Q9

(b) 6 C CR G28 92·5◦

•1 ic: interpret 3-d representation

•2 ss: know to use scalar product•3 pd: process vectors•4 pd: process vectors•5 pd: process lengths•6 pd: process scalar product•7 pd: evaluate scalar product

•1 B= (3, 2, 15) treat

3215

as bad form

•2 cosAB̂C =−→BA.

−→BC

|−→BA||−→BC|

•3 −→BA =

−37−7

•4 −→BC =

14−2−7

•5 |−→BA| =√107, |−→BC| =

√249

•6 −→BA.

−→BC = −7

•7 AB̂C = 92·5◦

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Higher Mathematics

67.[SQA]


(a) 2 C CN G26 1999 P1 Q17

(b) 4 A/B CN G29, G30

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Higher Mathematics

68.[SQA] Vectors p , q and r are representedon the diagram shown where angleADC = 30◦ .

It is also given that |p | = 4 and |q | = 3.

(a) Evaluate p .(q + r ) and r .(p − q) . 6

(b) Find |q + r | and |p − q | . 4

A

pD

30 °

B

r

q

C


(a) 6 B CN G29, G26 6√3, 94 2009 P2 Q7

(b) 2 A CR G21, G30 |q+ r| = 3√32

(b) 2 B CR G21, G30 |p− q| =√

(4− 3√32 )2 + ( 32 )

2

•1 ss: use distributive law•2 ic: interpret scalar product•3 pd: processing scalar product•4 ic: interpret perpendicularity•5 ic: interpret scalar product•6 pd: complete processing

•7 ic: interpret vectors on a 2-Ddiagram

•8 pd: evaluate magnitude of vectorsum

•9 ic: interpret vectors on a 2-Ddiagram

•10 pd: evaluate magnitude of vectordifference

•1 p.q+ p.r•2 4× 3 cos 30◦•3 6

√3(≈ 10·4)

•4 p.r = 0•5 −|r| × 3 cos 120◦•6 r = 3

2 and94

•7 q+ r ≡ from D to the proj. of A ontoDC

•8 |q+ r| = 3√32

•9 p− q =−→AC

•10 |p− q| =√

(4− 3√32 )2 + ( 32)

2(≈ 2·05)

69.[SQA] Differentiate sin 2x+2√xwith respect to x . 4


2 C NC C3 1989 P1 Q10

2 A/B NC C20

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Higher Mathematics

70.[SQA] Given that f (x) = (5x− 4) 12 , evaluate f ′(4) . 3


1 C CN C21 58 2000 P2 Q8

2 A/B CN C21

•1 pd: differentiate power•2 pd: differentiate 2nd function•3 pd: evaluate f ′(x)

•1 12(5x− 4)−12

•2 ×5•3 f ′(4) = 5

8

71.[SQA] Given f (x) = cos2 x− sin2 x , find f ′(x) . 3


1 C NC C21 1999 P1 Q19

2 A/B NC C21, C20

72.[SQA] Differentiate 2x32 + sin2 x with respect to x . 4


1 C NC C21 1992 P1 Q11

3 A/B NC C21

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Higher Mathematics

73.[SQA] If f (x) = cos2 x− 2

3x2, find f ′(x) . 4


2 C NC C21, C1 1990 P1 Q19

2 A/B NC C21, C1

74.[SQA] Given f (x) = (sin x+ 1)2 , find the exact value of f ′(π

6 ) . 3


3 A/B NC C21, C20, T2 1998 P1 Q16

75. Functions f and g are defined on suitable domains by f (x) = cos x andg(x) = x+ π

6 .

What is the value of f(g

(π

6

))?

A. 12 + π

6

B.√32 + π

6

C.√32

D. 12 2

Key Outcome Grade Facility Disc. Calculator Content Source

D 1.2 C 0 0 NC A4, T3 2010 P1 Q11

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Higher Mathematics

76. What is the gradient of the tangent to the curve with equation y = cos 2x at thepoint where x = π

4 ?

A. −2

B. −1

C. 0

D. 2 2


A 3.2 C 0.39 0.52 NC C4, C20, T3 HSN 127

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Higher Mathematics

77. The diagram shows the graph with equation of the form y = a cos bx for0 ≤ x ≤ 2π .

2

π 2π

–2

O x

y

What is the equation of this graph?

A. y = 2 cos 3x

B. y = 2 cos 2x

C. y = 3 cos 2x

D. y = 4 cos 3x 2


A 1.2 C 0 0 CN T4 2010 P1 Q4

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Higher Mathematics

78. The diagram shows the curve with equation of the form y = cos(x+ a) + bfor0 ≤ x ≤ 2π .

–2

2π

π

6

7

6

π

Ox

y

What is the equation of this curve?

A. y = cos(x− π

6

)− 1

B. y = cos(x− π

6

)+ 1

C. y = cos(x+ π

6

)− 1

D. y = cos(x+ π

6

)+ 1 2


A 1.2 C 0 0 NC T4, T1 2012 P1 Q9

[END OF QUESTIONS]

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Prelim practice - Higher Maths · PDF fileHigher Mathematics Prelim practice [SQA] 1. Part Marks Level Calc. Content Answer U1 OC1 3 C CR G2 1992 P1 Q13 [SQA] 2. Find the equation