Higher Homework 6
Section 1 Straight Line
1.
a) Find the equation of l1 the perpendicular bisector of the line joining P(3, -3) to Q(-1, 9).
b) Find the equation of l2 which is parallel to PQ and passes through R(1, -2).
c) Find the point of intersection of l1 and l2.
d) Hence find the shortest distance between PQ and l2.
2.
Calculate the size of the angle  :
3x  2 y  10  0
y

x
Section 2:Functions
3.
4.
If h  x  
a) h  m  x  
1
3
and m  x    3 , find
3 x
x
b) h-1(x)
f ( x)  2 x, g ( x)  x 2 , h( x)  sin x , k ( x)  1  x 2 .
a) Find expressions for: (i) f ( g ( x))
(ii) k (h( x))
𝜋
b) If p(x) = 2 h(x - 3 ), sketch p(x) for 0 < x < 2π
Section 3: Trigonometry
5.
Solve (a)
sin 2 x  sin x  0 (0  x  360)
6.
3 sin x  cos x   0 (0  x  2)
3
A
7.
C
4
Given that (sin A  cos B)2  (cos A  sin B)2  3, find two values for angle ( A  B) between 0 and 2.
Section 4: Differentiation(no calculator)
8.
(iv) k (k ( x))
Calculate the exact value of sin x.
B
x
3
(b)
(iii) h( f ( x ))
Differentiate each of the following with respect to x:
(a)
y  2 x3  4 x 2
(b)
y  x2 
(d)
y
(e)
y
x
1
x
1
x
(c)
y  x( x  2) 2
(f)
y
1
2 x3
9.
Find the equation of the tangent to: y  4 x3  2 x at the point where x = 3
10.
In each of the following, find f (x ) :
(a) f ( x) 
11.
12.
x2  1
x
(b)
f ( x) 
1 x
x
f ( x) 
(c)
x2  2x
x
f ( x)  x 2  3x . Calculate the rate of change of f ( x) when x = 10.
Shown above is a sketch graph of the function y  25  x 2 .
The graph crosses the x-axis at the points A and B, as shown.
The point C on the graph has x coordinate 3.
Calculate the gradient of the tangent at (i) A (ii) B (iii) C.
° C
A
Section 5 Revision for Unit Assessment on Trigonometry(R1.2 and E1.2)
O
B
13. Solve √2 sin 2x° = 1, for 0 ≤ x ≤ 180.
14. Solve 3sin 2p° + sin p° = 0 for 0 ≤ p ≤ 180
15. Given 2cosx° + 3 sin x° = √13 cos(x – 56.3)°, solve 2cos x° + 3sin x° = 3, for 0 < x < 90.
16. Given the diagram shown find the exact value of sin( x + y).
x
y
5
2
3
17. Show that (1 + 5sinx)(1 – 5sinx) = 25 cos2x – 24
18. Express 3sin x° - 4cos x° in the form k sin(x + a)° where k>0 and 0 ≤ a ≤ 360.
Section 6: Revision for Unit assessment on Differentiation (R1.3 A1.4Q9)
3
19. Find f’(x) given that f(x) = 𝑥 5 − 4√𝑥 where x > 0.
20. A bicycle rider travelling along a horizontal road applies the brakes and the rider’s position in metres at any
time t seconds is given by s(t) = 3t – 0.75t2.
What is the rider’s velocity a) at the point the brakes are applied
b) after 2 seconds of braking.
21. Differentiate the function f(x) = 2 sinx with respect to x.
22. A curve has equation y = 5x2 – 7x + 1. Find the equation of the tangent to curve at the point where x = 2.
23. The surface area of a cuboid is known to be 600 cm2.
a) Show that the volume of a square based cuboid can be expressed as V = 150x – ½ x3, where x is the length
of the base of the cuboid.
b) Find the value of x which maximizes the volume of the cuboid.