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Higher Mathematics
Further Calculus Past Papers Unit 3 Outcome 2
Multiple Choice Questions
Each correct answer in this section is worth two marks.
1. Differentiate 3 cos 2x − π6 with respect to x .
A.
B.
C.
D.
Key
C
−3 sin(2x)
−3 sin(2x − π6 )
−6 sin(2x − π6 )
6 sin(2x − π6 )
Outcome
3.2
Grade
C
Facility
0.68
Disc.
0.23
Calculator
NC
Content
C20
Source
HSN 096
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[END OF MULTIPLE CHOICE QUESTIONS]
Written Questions
2
2. Differentiate sin 2x + √ with respect to x .
x
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1
3
3. Given that f (x) = (5x − 4) 2 , evaluate f 0 (4).
[SQA]
Part
Marks
1
2
Level
C
A/B
Calc.
CN
CN
Content
C21
C21
•1 pd: differentiate power
•2 pd: differentiate 2nd function
•3 pd: evaluate f 0 (x)
Answer
U3 OC2
5
8
2000 P2 Q8
1
•1 12 (5x − 4)− 2
•2 ×5
•3 f 0 (4) = 58
4. Given f (x) = cos2 x − sin2 x , find f 0 (x).
[SQA]
3
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5. Given that f (x) = 5(7 − 2x)3 , find the value of f 0 (4).
[SQA]
4
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3
6. Differentiate 2x 2 + sin2 x with respect to x .
[SQA]
4
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7. Find the derivative, with respect to x , of
[SQA]
1
+ cos 3x .
x3
4
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8. If f (x) = cos2 x −
[SQA]
2
, find f 0 (x).
3x2
4
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Higher Mathematics
√
9. Differentiate 4 x + 3 cos 2x with respect to x .
[SQA]
4
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10. Find
[SQA]
√
dy
given that y = 1 + cos x .
dx
3
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11. Given f (x) = (sin x + 1)2 , find the exact value of f 0 ( π6 ).
[SQA]
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12. Find the equation of the tangent to the curve y = 2 sin(x − π6 ) at the point where
x = π3 .
[SQA]
Part
Marks
4
Level
C
Calc.
CN
Content
C5, C20
•1 pd: find derivative
•2 ss: know derivative at x = . . .
represents grad.
•3 pd: find corresponding y-coordinate
•4 ic: state equation of tangent
13. Find
[SQA]
Z
√
Answer
√
y = 3x + 1 −
4
U3 OC2
π
√
3
2002 P2 Q6
dy
•1 dx = 2 cos(x − π6 )
√
•2 m = 3
•3 y x= π3 = 1
√
•4 y − 1 = 3(x − π3 )
1 + 3x dx and hence find the exact value of
Z
0
1√
1 + 3x dx .
4
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14. Differentiate sin3 x with respect to x .
Z
Hence find
sin2 x cos x dx .
[SQA]
4
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15. Find
[SQA]
Part
Z
1
dx .
(7 − 3x)2
2
Marks
Level
Calc.
Content
2
A/B
CN
C22, C14
•1 pd: integrate function
•2 pd: deal with function of function
16. Evaluate
[SQA]
Z
0
2x + 3
−3
2
Answer
1
+c
3(7 − 3x)
U3 OC2
2000 P2 Q10
1
•1 −1
(7 − 3x)−1
1
•2 × −3
4
dx .
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17.
[SQA]
Z
π
2
cos 2x dx .
3
(b) Draw a sketch and explain your answer.
2
(a) Evaluate
0
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18.
[SQA]
(a) Show that (cos x + sin x)2 = 1 + sin 2x .
Z
(b) Hence find (cos x + sin x)2 dx .
1
3
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19. Find
[SQA]
Z 6x2 − x + cos x dx .
4
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20.
[SQA]
(a) By writing sin 3x as sin(2x + x), show that sin 3x = 3 sin x − 4 sin3 x .
Z
(b) Hence find
sin3 x dx .
4
4
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1
21. (a) Find the derivative of the function f (x) = (8 − x 3 ) 2 , x < 2.
Z
x2
(b) Hence write down
1 dx .
(8 − x3 ) 2
[SQA]
Part
(a)
(b)
Marks
2
1
Level
A/B
A/B
Calc.
CN
CN
Content
C21
C24
2
1
Answer
1
− 32 x2 (8 − x3 )− 2
1
− 23 (8 − x3 ) 2 + c
U3 OC2
2002 P1 Q10
•1 pd: process differentiation
•2 pd: use the chain rule
•1 12 (8 − x3 )− 2
•2 . . . × −3x2
•3 ic:
•3 − 23 f (x) or − 23 (8 − x3 ) 2
1
interpret answer from (a)
1
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π
, 1) and f 0 (x) = cos 2x .
22. The curve y = f (x) passes through the point ( 12
[SQA]
3
Find f (x).
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23. The graph of y = f (x) passes through the point
[SQA]
If f 0 (x) = sin(3x) express y in terms of x .
Part
•1
•2
•3
•4
Marks
4
ss:
pd:
ic:
pd:
Level
A/B
Calc.
NC
Content
C18, C23
π
9,1
.
Answer
y = − 13 cos(3x) +
4
7
6
U3 OC2
2000 P1 Q8
R
•1 y = sin(3x) dx stated or implied by
•2
•2 − 13 cos(3x)
•3 1 = − 13 cos( 3π
9 ) + c or equiv.
7
4
• c=6
know to integrate
integrate
interpret ( π9 , 1)
process
√ dy
= 3 sin(2x) passes through the point 5π
,
3 .
12
dx
Find y in terms of x .
24. A curve for which
[SQA]
Part
•1
•2
•3
•4
Marks
4
pd:
pd:
ss:
pd:
Level
A/B
Calc.
CN
Content
C18, C23
integrate trig function
integrate composite function
use given point to find “c”
evaluate “c”
Answer
√
y = − 32 cos(2x) + 14 3
4
U3 OC2
2001 P2 Q10
R
•1 3 sin(2x) dx stated or implied by •2
•2 − 32 cos(2x)
√
5
•3 3 = − 32 cos(2 × 12
π) + c
√
1
4
• c = 4 3 (≈ 0·4)
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25. A point moves in a straight line such that its acceleration a is given by
1
a = 2(4 − t) 2 , 0 ≤ t ≤ 4. If it starts at rest, find an expression for the velocity
v where a = dv
dt .
[SQA]
Part
Marks
4
Level
C
Calc.
NC
Content
C18, C22
•1 ss: know to integrate acceleration
•2 pd: integrate
•3 ic: use initial conditions with const.
of int.
•4 pd: process solution
Answer
3
V = − 43 (4 − t) 2 +
U3 OC2
32
3
2002 P2 Q8
R
1
•1 V = (2(4 − t) 2 ) dt stated or implied
by •2
3
•2 2 × −13 (4 − t) 2
2
•3 0 = 2 ×
•4 c = 10 23
1
− 23
3
(4 − 0) 2 + c
26.
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27.
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28.
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29.
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[END OF WRITTEN QUESTIONS]
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