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Higher Mathematics
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Polynomials and Quadratics
Paper 1 Section A
Each correct answer in this section is worth two marks.
1. A parabola has equation y = −2x2+ 4x + 5.
Which of the following are true?
I. The parabola has a minimum turning point.
II. The parabola has no real roots.
A. Neither I nor II is true
B. Only I is true
C. Only II is true
D. Both I and II are true
Key Outcome Grade Facility Disc. Calculator Content SourceA 2.1 C 0.37 0.4 CN A15, A17 HSN 070
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2. Given p(x) = x2 + x − 6, which of the following are true?
I. (x + 3) is a factor of p(x) .
II. x = 2 is a root of p(x) = 0.
A. Neither I nor II is true
B. Only I is true
C. Only II is true
D. Both I and II are true
Key Outcome Grade Facility Disc. Calculator Content SourceD 2.1 C 0.78 0.67 NC A21 HSN 170
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3. When 2ax3 + (a + 1)x − 6 is divided by x + 2, the remainder is 2.
What is the value of a?
A. 53
B. − 49
C. − 59
D. − 57
Key Outcome Grade Facility Disc. Calculator Content SourceC 2.1 C 0.41 0.77 NC A21 HSN 174
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[END OF PAPER 1 SECTION A]
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Paper 1 Section B4. (a)[SQA] Express f (x) = x2 − 4x + 5 in the form f (x) = (x − a)2 + b . 2
(b) On the same diagram sketch:(i) the graph of y = f (x) ;
(ii) the graph of y = 10 − f (x) . 4
(c) Find the range of values of x for which 10 − f (x) is positive. 1
Part Marks Level Calc. Content Answer U1 OC2(a) 2 C NC A5 a = 2, b = 1 2002 P1 Q7(b) 4 C NC A3 sketch(c) 1 C NC A16, A6 −1 < x < 5
•1 pd: process, e.g. completing thesquare
•2 pd: process, e.g. completing thesquare
•3 ic: interpret minimum•4 ic: interpret y-intercept•5 ss: reflect in x-axis•6 ss: translate parallel to y-axis
•7 ic: interpret graph
•1 a = 2•2 b = 1
•3 any two from:parabola; min. t.p. (2, 1); (0, 5)
•4 the remaining one from above list•5 reflecting in x-axis•6 translating +10 units, parallel to
y-axis
•7 (−1, 5) i.e. −1 < x < 5
5.[SQA] Find the values of x for which the function f (x) = 2x3 − 3x2 − 36x is increasing. 4
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6.[SQA]
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7.[SQA]
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8.[SQA] For what value of k does the equation x2 − 5x + (k + 6) = 0 have equal roots? 3
Part Marks Level Calc. Content Answer U2 OC13 C CN A18 k = 1
4 2001 P1 Q2
•1 ss: know to set disc. to zero•2 ic: substitute a, b and c into
discriminant•3 pd: process equation in k
•1 b2 − 4ac = 0 stated or implied by •2
•2 (−5)2 − 4 × (k + 6)•3 k = 1
4
9.[SQA]
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10.[SQA] Find the values of k for which the equation 2x2 + 4x + k = 0 has real roots. 2
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11.[SQA] For what value of a does the equation ax2+ 20x + 40 = 0 have equal roots? 2
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12.[SQA] Factorise fully 2x3+ 5x2 − 4x − 3. 4
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13.[SQA]
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14.[SQA] One root of the equation 2x3 − 3x2+ px + 30 = 0 is −3.
Find the value of p and the other roots. 4
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16.[SQA] Express x4 − x in its fully factorised form. 4
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17.[SQA]
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18.[SQA] Express x3 − 4x2 − 7x + 10 in its fully factorised form. 4
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19.[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
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20.[SQA]
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21.[SQA]
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22.[SQA]
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23.[SQA]
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24.[SQA]
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25.[SQA]
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26.[SQA]
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27.[SQA]
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28.[SQA]
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29.[SQA]
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30.[SQA]
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31.[SQA]
(a) Find the coordinates of the points of intersection of the curves with equationsy = 2x2 and y = 4 − 2x2 . 2
(b) Find the area completely enclosed between these two curves. 3
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32.[SQA] For what range of values of k does the equation x2+ y2
+ 4kx − 2ky − k − 2 = 0represent a circle? 5
Part Marks Level Calc. Content Answer U2 OC45 A NC G9, A17 for all k 2000 P1 Q6
•1 ss: know to examine radius•2 pd: process•3 pd: process•4 ic: interpret quadratic inequation•5 ic: interpret quadratic inequation
•1 g = 2k, f = −k, c = −k − 2stated or implied by •2
•2 r2= 5k2
+ k + 2•3 (real r ⇒) 5k2
+ k + 2 > 0 (accept ≥)•4 use discr. or complete sq. or diff.•5 true for all k
[END OF PAPER 1 SECTION B]
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Paper 21.[SQA]
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2.[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 real roots. 2
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3.[SQA] Show that the roots of the equation (k − 2)x2 − (3k − 2)x + 2k = 0 are real. 4
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4.[SQA]
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5.[SQA]
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6.[SQA]
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7.[SQA] Show that the equation (1 − 2k)x2 − 5kx − 2k = 0 has real roots for all integervalues of k . 5
Part Marks Level Calc. Content Answer U2 OC15 A/B CN A18, A16, 0.1 proof 2002 P2 Q9
•1 ss: know to use discriminant•2 ic: pick out discriminant•3 pd: simplify to quadratic•4 ss: choose to draw table or graph•5 pd: complete proof using disc.≥ 0
•1 discriminant = . . .•2 disc = (−5k)2 − 4(1 − 2k)(−2k)•3 9k2 + 8k•4 e.g. draw a table, graph, complete
the square•5 complete proof and conclusion
relating to disc.≥ 0
8.[SQA] The diagram shows part of the graph of thecurve with equation y = 2x3 − 7x2 + 4x + 4.(a) Find the x -coordinate of the maximum
turning point. 5
(b) Factorise 2x3 − 7x2 + 4x + 4. 3
(c) State the coordinates of the point A andhence find the values of x for which2x3 − 7x2 + 4x + 4 < 0. 2
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A(2, 0)
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Part Marks Level Calc. Content Answer U2 OC1(a) 5 C NC C8 x = 1
3 2002 P2 Q3(b) 3 C NC A21 (x − 2)(2x + 1)(x − 2)
(c) 2 C NC A6 A(− 12 , 0), x < − 1
2
•1 ss: know to differentiate•2 pd: differentiate•3 ss: know to set derivative to zero•4 pd: start solving process of equation•5 pd: complete solving process
•6 ss: strategy for cubic, e.g. synth.division
•7 ic: extract quadratic factor•8 pd: complete the cubic factorisation
•9 ic: interpret the factors•10 ic: interpret the diagram
•1 f ′(x) = . . .•2 6x2 − 14x + 4•3 6x2 − 14x + 4 = 0•4 (3x − 1)(x − 2)•5 x = 1
3
•6· · · 2 −7 4 4
· · · · · · · · ·
· · · · · · · · · 0•7 2x2 − 3x − 2•8 (x − 2)(2x + 1)(x − 2)
•9 A(− 12 , 0)
•10 x < − 12
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9.[SQA] Find p if (x + 3) is a factor of x3 − x2 + px + 15. 3
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10.[SQA] When f (x) = 2x4 − x3 + px2 + qx + 12 is divided by (x − 2) , the remainder is 114.
One factor of f (x) is (x + 1) .
Find the values of p and q . 5
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11.[SQA] Find k if x − 2 is a factor of x3 + kx2 − 4x − 12. 3
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12.[SQA]
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13. (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
Part Marks Level Calc. Content Answer U2 OC1(a) 3 C CN A21 k = −5 2001 P2 Q1(b) 2 C CN A22 x = −2, 1
2 , 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, 1
2 , 1
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14.[SQA]
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15.[SQA] The diagram shows a sketch of thegraph of y = x3 − 3x2 + 2x .(a) Find the equation of the
tangent to this curve at thepoint where x = 1. 5
(b) The tangent at the point (2, 0)has equation y = 2x − 4. Findthe coordinates of the pointwhere this tangent meets thecurve again. 5
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Part Marks Level Calc. Content Answer U2 OC1(a) 5 C CN C5 x + y = 1 2000 P2 Q1(b) 5 C CN A23, A22, A21 (−1,−6)
•1 ss: know to differentiate•2 pd: differentiate correctly•3 ss: know that gradient = f ′(1)•4 ss: know that y-coord = f (1)•5 ic: state equ. of line
•6 ss: equate equations•7 pd: arrange in standard form•8 ss: know how to solve cubic•9 pd: process•10 ic: interpret
•1 y′ = . . .•2 3x2 − 6x + 2•3 y′(1) = −1•4 y(1) = 0•5 y − 0 = −1(x − 1)
•6 2x − 4 = x3 − 3x2 + 2x•7 x3 − 3x2 + 4 = 0
•8· · · 1 −3 0 4
· · · · · · · · ·
· · · · · · · · · · · ·
•9 identify x = −1 from working•10 (−1,−6)
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16.[SQA] The diagram shows a sketch of aparabola passing through (−1, 0) ,(0, p) and (p, 0) .(a) Show that the equation
of the parabola isy = p + (p − 1)x − x2 . 3
(b) For what value of p will the liney = x + p be a tangent to thiscurve? 3
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(−1, 0) (p, 0)
(0, p)
Part Marks Level Calc. Content Answer U2 OC1(a) 3 A/B CN A19 proof 2001 P2 Q11(b) 3 A/B CN A24 2
•1 ss: use a standard form of parabola•2 ss: use 3rd point to determine k•3 pd: complete proof
•4 ss: equate and simplify to zero•5 ss: use discriminant for tangency•6 pd: process
•1 y = k(x + 1)(x − p)•2 k = −1 with justification (i.e.
substitute (0, p))•3 y = −1(x + 1)(x − p) and complete
•4 x2 + 2x − px = 0•5 b2 − 4ac = (2 − p)2 = 0
or (2 − p)2 − 4 × 0 = 0•6 p = 2
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17.[SQA]
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18.[SQA]
(a) On the same diagram, sketch the graphs of y = log10 x and y = 2 − x where0 < x < 5.Write down an approximation for the x -coordinate of the point of intersection. 3
(b) Find the value of this x -coordinate, correct to 2 decimal places. 3
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19.[SQA]
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20.[SQA] The parabola shown crosses the x -axis at(0, 0) and (4, 0) , and has a maximum at(2, 4) .The shaded area is bounded by theparabola, the x -axis and the lines x = 2and x = k .(a) Find the equation of the parabola. 2
(b) Hence show that the shaded area, A ,is given by
A = − 13 k3 + 2k2 − 16
3 . 3
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Part Marks Level Calc. Content Answer U2 OC2(a) 2 C CN A19 y = 4x − x2 2000 P2 Q4(b) 3 C CN C16 proof
•1 ic: state standard form•2 pd: process for x2 coeff.
•3 ss: know to integrate•4 pd: integrate correctly•5 pd: process limits and complete
proof
•1 ax(x − 4)•2 a = −1
•3 ∫ k2 (function from (a))
•4 − 13 x3 + 2x2
•5 − 13 k3 + 2k2 −
(
− 83 + 8
)
21.[SQA]
(a) Write the equation cos 2θ + 8 cos θ + 9 = 0 in terms of cos θ and show that,for cos θ , it has equal roots. 3
(b) Show that there are no real roots for θ . 1
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22.[SQA] Find the possible values of k for which the line x − y = k is a tangent to the circlex2
+ y2= 18. 5
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[END OF PAPER 2]
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