PSfrag replacements
O
x
y
Higher Mathematics
Further Calculus
Paper 1 Section A
Each correct answer in this section is worth two marks.
1. Differentiate 3 cos 2x −
respect to x .
A.
−3 sin(2x)
B.
−3 sin(2x − π6 )
C.
−6 sin(2x − π6 )
D.
6 sin(2x − π6 )
π
6
with
[END OF PAPER 1 SECTION A]
Paper 1 Section B
2.
[SQA]
frag replacements
O
x
y
[SQA]
replacements
O
x
y
3. The graph of y = f (x) passes through the point
π
9,1
.
4
If f 0 (x) = sin(3x) express y in terms of x .
hsn.uk.net
Page 1
c SQA
Questions marked ‘[SQA]’ c
All others Higher Still Notes
PSfrag replacements
O
x
y
Higher Mathematics
[SQA]
2
4. Differentiate sin 2x + √ with respect to x .
x
4
[SQA]
5. Given f (x) = cos2 x − sin2 x , find f 0 (x).
3
[SQA]
6. Given that f (x) = 5(7 − 2x)3 , find the value of f 0 (4).
4
[SQA]
7. Differentiate 2x 2 + sin2 x with respect to x .
[SQA]
8. Find the derivative, with respect to x , of
[SQA]
9. If f (x) = cos2 x −
3
4
1
+ cos 3x .
x3
4
2
, find f 0 (x).
3x2
4
√
10. Differentiate 4 x + 3 cos 2x with respect to x .
[SQA]
4
11. Differentiate sin3 x with respect to x .
Z
Hence find
sin2 x cos x dx .
[SQA]
4
√
dy
given that y = 1 + cos x .
dx
3
[SQA]
12. Find
[SQA]
13. Given f (x) = (sin x + 1)2 , find the exact value of f 0 ( π6 ).
[SQA]
replacements
O
x
y
14. Find
Z √
1 + 3x dx and hence find the exact value of
hsn.uk.net
Page 2
Z
0
1√
3
1 + 3x dx .
c SQA
Questions marked ‘[SQA]’ c
All others Higher Still Notes
4
PSfrag replacements
O
x
y
Higher Mathematics
15. Evaluate
[SQA]
Z
0
2x + 3
−3
2
4
dx .
16.
[SQA]
frag replacements
O
x
y
17.
[SQA]
Z
π
2
cos 2x dx .
3
(b) Draw a sketch and explain your answer.
2
(a) Show that (cos x + sin x)2 = 1 + sin 2x .
Z
(b) Hence find (cos x + sin x)2 dx .
1
(a) Evaluate
0
18.
[SQA]
Z 3
6x2 − x + cos x dx .
4
[SQA]
19. Find
[SQA]
π
20. The curve y = f (x) passes through the point ( 12
, 1) and f 0 (x) = cos 2x .
3
Find f (x).
21.
[SQA]
replacements
O
x
y
(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 .
hsn.uk.net
Page 3
c SQA
Questions marked ‘[SQA]’ c
All others Higher Still Notes
4
4
PSfrag replacements
O
x
y
Higher Mathematics
22.
[SQA]
frag replacements
O
x
y
23.
[SQA]
frag replacements
O
x
y
1
24. (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]
[END OF PAPER 1 SECTION B]
replacements
O
x
y
hsn.uk.net
Page 4
c SQA
Questions marked ‘[SQA]’ c Higher Still Notes
All others 2
1
PSfrag replacements
O
x
y
Higher Mathematics
Paper 2
[SQA]
[SQA]
1. Find the equation of the tangent to the curve y = 2 sin(x − π6 ) at the point where
x = π3 .
4
2. 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 .
4
√ dy
,
3 .
= 3 sin(2x) passes through the point 5π
12
dx
Find y in terms of x .
3. A curve for which
[SQA]
1
[SQA]
4. Given that f (x) = (5x − 4) 2 , evaluate f 0 (4).
[SQA]
5.
3
frag replacements
replacements
O
x
y
O
x
y
hsn.uk.net
Page 5
4
c SQA
Questions marked ‘[SQA]’ c
All others Higher Still Notes
PSfrag replacements
O
x
y
Higher Mathematics
[SQA]
6. Find
[SQA]
7.
Z
1
dx .
(7 − 3x)2
2
frag replacements
O
x
y
[END OF PAPER 2]
replacements
O
x
y
hsn.uk.net
Page 6
c SQA
Questions marked ‘[SQA]’ c Higher Still Notes
All others