Mathematics
Department
Clyde Valley High School
1.
Find
dy
dx
in each example
a) y = (2x+1)(x2-2)
b) y =
2x + 1
x
2.
Solve the equation 3sin2x = 5cosx,
0º < x < 360º
3.
Find the equation of the tangent to the circle
x2 + y2 = 29 at the point (-5,2) on the circle.
4.
Find the equation of the tangent to the curve y =
1
4x
at the point where x = 1.
5.
Find the minimum of the function f(x) = x2 + 5x + 13
by firstly completing the square.
6.
If f(x) = 1 + 2x3 – 3x4, find the stationary values of the
function f and determine their nature.
7.
If f(x) = 2x + 1 and g(x) = 1 – 5x find
a) f(g(x))
b) g(f(x))
Hence solve the equation f(g(x)) - g(f(x)) = 8x + 7
8.
Find the equation of the perpendicular bisector of the line
joining the points A(-1,4) and B(5,0).
Mathematics
Clyde Valley High School
9.
Find the angle which the line
y – 2x – 5 = 0 makes with the
positive direction of the x axis.
10.
The graph shows a curve y = f(x)
On separate diagrams sketch the
graphs of
f(x) + 3
–f(x)
f(x - 4)
2 + f(x - 1)
a)
b)
c)
d)
11. If
Department
dy
= 2x + 1 and y = 5 when x = 2, find y in terms of x.
dx
12. If A and B are acute angles and sinA =
find the exact value of cos(A - B).
13. Find the equation of the tangent to
the curve y = x3 – x + 3 at
x = 1. Find algebraically the
coordinates of the point where
the tangent meets the curve again.
The curve and the tangent are shown
in the diagram below.
4
5
and cosB =
5
13
Mathematics
Department
Clyde Valley High School
12
, find the exact
13
14. If A is an obtuse angle and sinA =
value of cos2A.
15. If f '(x) = 3x2 and f(2) = 3 find f(x).
16. Solve the equation cos2x + 3cosx + 2 = 0, 0º < x < 360º
17. The diagram shows the graph of
the quadratic function
y = x2 – 4x + 3.
Calculate the shaded area.
18. Solve the trig. equation
8cos2x + 10sinx - 9 = 0 , 0º < x < 360º
19. If f(x) = 2x2 and g(x) = 2x + 1 find
a) f(f(-1)) b) g(f(-2))
20. The diagram shows the line y = 1 and
the quadratic y = x2 – 3.
Find the points where they meet and
hence find the shaded area.
21. In the diagram shown, find the
equation of the altitude
from A and the median from B.
A(3,5)
C(-5,1)
B(1,-1)
Mathematics
Department
Clyde Valley High School
22. If
dy
= 1 + 2x + x2 and the curve passes through the
dx
point (4,0) find it’s equation.
23. If A and B are acute angles and sinA =
8
3
and tanB =
4
17
find the exact value of sin(A - B).
24. By completing the square find the maximum value of the
function y = 12 – 2x – x2.
25. Solve the equation 4sin2x – 3 = 0, 0 < x < 2 π
26. Determine the stationary values of the function
y = x3(2 – x)
and the nature of the S.P.s
27. If un+1 = 0.6un + 8, u0 = 3 determine the limit of this
sequence.
2x3 - 11x2 + 17x - 6
28. Factorize fully
29. a) Find m if mx2 + (m-1)x + (7m-2) = 0, has equal roots.
b) Find m if 5x2 - 3mx + 5 = 0 has two real roots.
4
30. Evaluate
a) ∫
1
31
x +1
dx
x
1
b)
∫ (x
2
+ 1) 2 dx
0
If f(x) = (1 - 2x)3, find the value of f '(-2)