Download 1. If y = , find y′ when x = 4. 2. If f ′(x) = 1 – 2x and f(4) = 7, find f(x

Mathematics
Clyde Valley High School
1. If y =
Department
4x − 1
, find y′ when x = 4.
2 x
2. If f ′(x) = 1 – 2x and f(4) = 7, find f(x).
π
3. Find the values of θ where 0 ≤ θ ≤ 2π for which 6cos( 2θ − )
2
has its maximum value and state this maximum value.
4. Factorize fully 2x3 + 5x2 – 4x – 3.
5. The vertices of a triangle are P(-1,3), Q(5,5) and R(1,-1).
Find the equations of
a) The median from P.
b) The altitude from Q.
6. Solve the equation 4sin2x – 1 = 0, for 0 ≤ x ≤ 360 ° .
7. For the quadratic y = x2 – 7x + 13, complete the square and
state the minimum.
8. Find the equation of the tangent to the curve y = 2 – x – x2 at
the point where x = 1.
9. If f(x) = 1 – 2x2 and g(x) = 3x + 1, express f(g(x)) in the form
ax2 + bx + c.
10. Find m if the equation (2m-1)x2 + (m+1)x + 1 = 0 has equal
roots.
11. State the quotient and the remainder when x4 + 2x3 – 3x -5 is
divided by x – 2.
Mathematics
Clyde Valley High School
Department
12. Find k if x+2 is a factor of x3 + kx2 – x – 2. Hence factorize
fully.
13. Find the equation of the perpendicular bisector of the line
joining the points A(-7,4) and B(3,0).
14. State why the recurrence relation un+1 = 0.72un + 5 with u0 =4
has a limit and find it.
15. Show that the line y = 3x + 10 is a tangent to the circle with
equation x2 + y2 – 8x – 4y – 20 = 0. Find the point of contact.
16. Find the equation of the straight line through the point (5,-2)
and perpendicular to the line with equation 3x – 2y = 5.
17. If P(3,4,1), Q(9,1,-5) and R(11,0,-7) , prove that P, Q and R
are collinear. If M(4,7,1), find the size of angle PMQ.
18. If sin A =
5
4
and cos B = , where A and B are acute, find the
13
5
exact value of cos (A-B).
19. Solve the equation cos2x – cosx + 2 = 0, for 0 ≤ x ≤ 360 ° .
20. A curve has equation y = ax2 + b where a and b are constants.
If the gradient of the tangent to the curve at the point (3,4) on the
curve is 6, find the values of a and b.
21. If f(x) = (x-1)2(x+2) , find the stationary values and determine
their nature.
Mathematics
Clyde Valley High School
Department
22. The area of a rectangle is 12. If one side of the rectangle is x
show that the perimeter of the rectangle is given by P = 2 x +
24
.
x
Find the dimensions of the rectangle of minimum perimeter.
23. Find p if x+3 is a factor of x3 – x2 + px + 15. Hence factorize
fully.
1
24. Evaluate ∫ (x 3 + 1)dx , Draw a diagram to illustrate the area
0
represented by this integral.
25. If f ′(x) = 3x2 and f(2) = 2, find f(x).
26. Show that the roots of the quadratic equation
(k-2)x2 + (2 – 3k)x + 2k = 0
are always real for all values of k.
27. Find the area enclosed between the curves y = 2x2 and
y = 4 – 2x2.
28. The vector ai + b j + k is perpendicular to both
i − j + k and 2i + j + k . Find the values of a and b.
29. Find the equation of the straight line through the point (-1,-2)
and parallel to the line x + 2y – 1 = 0.
30. If sin A =
3
π
where 0 ≤ A ≤ , find the exact value of sin2A.
4
2