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Grade: 11
Level: HL
Summer Break Revision Worksheet:
Session-2013-2014
CALCULATOR IS PERMITTED
All the answers must be given to 3 sf , angle to 1 dp or the degree of accuracy specified in the
question.
Show a neat sketch of the graph wherever needed with ALL NECESSARY DETAILS ON IT.
Algebra, Polynomials and Functions
1 i) The polynomial x3 + ax2 – 3x + b is divisible by (x – 2) and has a remainder 6 when divided
by (x + 1). Find the value of a and of b.
ii) The equation kx2 – 3x + (k + 2) = 0 has two distinct real roots. Find the values of k.
x 9
x 9
2.
Solve the inequality
3.
Solve the inequality x2 – 4 +
4.
The polynomial x2 – 4x + 3 is a factor of x3 + (a – 4)x2 + (3 – 4a) x + 3. Calculate the value
of the constant a.
5.
Consider the equation (1 + 2k)x2 – 10x + k – 2 = 0, k
which the equation has real roots.
6.
When the function f (x) = 6x4 + 11x3 – 22x2 + ax + 6 is divided by (x + 1) the remainder is –
20. Find the value of a.
7.
When the polynomial x4 + ax + 3 is divided by (x – 1), the remainder is 8. Find the value of
a.
8.
Let f (x) =
x 4
,x
x 1
2
3
< 0.
x
–1 and g (x) =
Find the set of values of x such that f (x)
9.
Solve the equation e 2 x
1
x
2
x–2
,x
x–4
. Find the set of values of k for
4.
g (x).
= 2.
10. Consider the curves C1, C2 with equations
C1 : y = x2 + kx + k, where k
C2 : y = –x2 + 2x – 4.
0 is a constant
Both curves pass through the point P and the tangent at P to one of the curves is also a
tangent at P to the other curve.
(a)
Find the value of k.
(b) Find the coordinates of P.
11. The quadratic function Q is defined by Q(x) = kx2 − (k − 3) x + (k − 8), k
Determine the values of k for which Q (x) = 0 has no real roots.
12. A circle has equation x2 + (y – 2)2 = 1. The line with equation y = kx, where k
tangent to the circle. Find all possible values of k.
, is a
13. When the polynomial P (x) = 4x2 + px2 + qx + 1 is divided by (x – 1) the remainder is
13
–2. When P (x) is divided by (2x – 1) the remainder is
. Find the value of p and of
4
q.
14.
Solve the inequality:
15.
Let f (x) =
f (x)
16
x 4
,x
x 1
3
x2 – 4 +
< 0.
x
–1 and g (x) =
x–2
,x
x–4
4. Find the set of values of x such that
g (x).
Solve the inequality x – 2
2x+ 1 without using calculator.
m(x + 1) x2.
17.
Find the range of values of m such that for all x
18
The polynomial f (x) = x3 + 3x2 + ax + b leaves the same remainder when divided by (x – 2)
as when divided by (x + 1). Find the value of a.
19.
Given f (x) = x2 + x(2 – k) + k2, find the range of values of k for which f (x) > 0 for all real
values of x.
20
Given that x >0, find the solution of the following system of equations:
8x 3
y
21.
22.
3 xy – y = x2 + 9
4
The polynomial p(x) = (ax + b)3 leaves a remainder of –1 when divided by (x + 1), and a
remainder of 27 when divided by (x – 2). Find the values of the real numbers a andb.
23. The roots α and β of another quadratic equation
x2 – kx + (k + l) = 0 are such that α2 + β2 = 13. Find the possible values of the
real number k.
24.Sketch the graph of
.
25. The diagram below shows the graph of y1 = f (x). The x-axis is a tangent to f (x) at x = m and
f (x) crosses the x-axis at x = n.
y
0
y1 = f(x)
m
x
n
On the same diagram sketch the graph of y2 = f (x – k), where 0 < k < n – m and indicate the
coordinates of the points of intersection of y2 with the x-axis.
26. For which values of the real number x is | x + k | = | x | + k, where k is a positive real number?
27. Let f (x) =
4
,x
x 2
2 and g (x) = x − 1.
If h = g ◦ f, find
(a)
h (x);
(2)
(b)
h−1 (x), where h−1 is the inverse of h.
(4)
(Total 6 marks)
28.
The functions f and g are defined as:
2
f (x) = e x , x
0
g (x) =
1
x 3
(a)
,x
3.
Find h (x) where h (x) = g ◦ f (x).
(2)
(b)
State the domain of h−1 (x).
(2)
(c)
Find h−1 (x).
(4)
(Total 8 marks)
29.
The real root of the equation x3 – x + 4 = 0 is –1.796 to three decimal places.
Determine the real root for each of the following.
(a)
(x – 1)3 – (x – 1) + 4 = 0
(2)
(b)
8x3 – 2x + 4 = 0
(3)
(Total 5 marks)
30.
The cubic curve y = 8x3 + bx2 + cx + d has two distinct points P and Q, where the gradient is zero.
(a)
Show that b2 > 24c.
(4)
(b)
Given that the coordinates of P and Q are
1
, 12 and
2
3
, 20 , respectively, find
2
the values of b, c and d.
(4)
(Total 8 marks)
Trigonometry
31
Given that tan 2θ =
32.
(a)
(b)
3
, find the possible values of tan θ.
4
If sin (x – α) = k sin (x + α) express tan x in terms of k and α.
Hence find the values of x between 0° and 360° when k =
1
and α = 210°.
2
33. Let α be the angle between the unit vectors a and b, where 0 ≤ α ≤ π.
34.
35.
36.
(a)
Express │a – b│ and │a + b│ in terms of α.
(b)
Hence determine the value of cos α for which │a + b│ = 3│a – b│.
(a)
Show that arctan
(b)
Hence, or otherwise, find the value of arctan (2) + arctan (3).
1
2
arctan
1
3
π
.
4
A triangle has sides of length (n2 + n + 1), (2n + 1) and (n2 – 1) where n > 1.
(a)
Explain why the side (n2 + n + 1) must be the longest side of the triangle.
(b)
Show that the largest angle, θ, of the triangle is 120°.
The lengths of the sides of a triangle ABC are x – 2, x and x + 2. The largest angle is 120°.
(a)
Find the value of x.
(b)
Show that the area of the triangle is
(c)
Find sin A + sin B + sin C giving your answer in the form
15 3
.
4
p q
r
where p, q, r
37. The diagram shows a tangent, (TP), to the circle with centre O and radius r. The size of
ˆ A is θ radians.
PO
(a)
Find the area of triangle AOP in terms of r and θ.
.
38.
39.
(b)
Find the area of triangle POT in terms of r and θ.
(c)
Using your results from part (a) and part (b), show that sin θ < θ < tan θ.
The diagram below shows two concentric circles with centre O and radii 2 cm and 4 cm.
ˆ Q = x,
The points P and Q lie on the larger circle and PO
π
where 0 < x < .
2
(a)
Show that the area of the shaded region is 8 sin x – 2x.
(b)
Find the maximum area of the shaded region.
The radius of the circle with centre C is 7 cm and the
radius of the circle with centre D is 5 cm. If the
length of the chord [AB] is 9 cm, find the area of the
shaded region enclosed by the two arcs AB.
40.
(a)
Sketch the curve f(x) = sin 2x, 0 ≤ x ≤ π.
(2)
(b)
Hence sketch on a separate diagram the graph of g(x) = csc 2x, 0 ≤ x ≤ π, clearly
stating the coordinates of any local maximum or minimum points and the equations of
any asymptotes.
(5)
(c)
Show that tan x + cot x ≡ 2 csc 2x.
(3)
(d)
Hence or otherwise, find the coordinates of the local maximum and local minimum
π
points on the graph of y = tan 2x + cot 2x, 0 ≤ x ≤ .
2
(5)
(e)
Find the solution of the equation csc 2x = 1.5 tan x – 0.5, 0 ≤ x ≤
π
.
2
(6)
(Total 21 marks)
41.
Consider the graphs y = e–x and y = e–x sin 4x, for 0 ≤ x ≤
(a)
5π
.
4
On the same set of axes draw, on graph paper, the graphs, for 0 ≤ x ≤
Use a scale of 1 cm to
5π
.
4
π
on your x-axis and 5 cm to 1 unit on your y-axis.
8
(3)
(b)
Show that the x-intercepts of the graph y = e–x sin 4x are
nπ
, n = 0, 1, 2, 3, 4, 5.
4
(3)
(c)
Find the x-coordinates of the points at which the graph of y = e–x sin 4x meets the
graph of y = e–x. Give your answers in terms of π.
(3)
(d)
(i)
Show that when the graph of y = e–x sin 4x meets the graph of y = e–x, their
gradients are equal.
(ii)
Hence explain why these three meeting points are not local maxima of the
graph y = e–x sin 4x.
(6)
(e)
(i)
Determine the y-coordinates, y1, y2 and y3, where y1 > y2 > y3, of the local
5π
maxima of y = e–x sin 4x for 0 ≤ x ≤
. You do not need to show that they are
4
maximum values, but the values should be simplified.
(ii)
Show that y1, y2 and y3 form a geometric sequence and determine the common
ratio r.
(7)
(Total 22 marks)
42.
A triangle has sides of length (n2 + n + 1), (2n + 1) and (n2 – 1) where n > 1.
(a)
Explain why the side (n2 + n + 1) must be the longest side of the triangle.
(3)
(b)
Show that the largest angle, θ, of the triangle is 120°.
(5)
(Total 8 marks)