Math HL Revision Pack
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Year 12 Mathematics HL Ȃ Revision Pack 1.
When the function f (x) = 6x4 + 11x3 ± 22x2 + ax + 6 is divided by (x + 1) the remainder is ±20. Find the value of a.
2.
The second term of an arithmetic sequence is 7. The sum of the first four terms of the arithmetic sequence is 12. Find the first term, a, and the common difference, d, of the sequence.
3.
Find the coordinates of the point where the line given by the parametric equations x = 2O + 4, y = ±O ± 2, z = 3O + 2, intersects the plane with equation 2x + 3y ± z = 2.
4.
Let z = x + yi. Find the values of x and y if (1 ± i)z = 1 ± 3i.
5.
If 2x2 ± 3y2 = 2, find the two values of
6.
(a)
dy when x = 5. dx
Find a vector perpendicular to the two vectors: &
&
&
OP = i ± 3 j + 2 k
& & & O Q = ±2 i + j ± k
(b)
If OP and O Q are position vectors for the points P and Q , use your answer to part (a), or otherwise, to find the area of the triangle OPQ . )
7.
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.
Pascal Ashkar Ȃ Ǯͳʹ
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Year 12 Mathematics HL Ȃ Revision Pack 8.
Given that (1 + x)5 (1 + ax)6 { 1 + bx + 10x2 + ............... + a6 x11, find the values of a, b
*.
9.
A particle moves along a straight line. When it is a distance s from a fixed point, where s > 1, (3s 2) the velocity v is given by v = . Find the acceleration when s = 2. (2s 1)
10.
The coordinates of the points P, Q, R and S are (4, 1, ±1), (3, 3, 5), (1, 0, 2c), and (1, 1, 2), respectively. (a)
Find the value of c so that the vectors QR and PR are orthogonal. For the remainder of the question, use the value of c found in part (a) for the coordinate of the point R.
11.
(b)
Evaluate PS × PR .
(c)
Find an equation of the line l which passes through the point Q and is parallel to the vector PR .
(d)
Find an equation of the plane S which contains the line l and passes through the point S.
(e)
Find the shortest distance between the point P and the plane S.
The ratio of the fifth term to the twelfth term of a sequence in an arithmetic progression is 6 . 13 If each term of this sequence is positive, and the product of the first term and the third term is 32, find the sum of the first 100 terms of this sequence.
Pascal Ashkar Ȃ Ǯͳʹ
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Year 12 Mathematics HL Ȃ Revision Pack 12.
Let x and y be real numbers, and Z be one of the complex solutions of the equation z3 = 1. Evaluate: (a)
1 + Z + Z2;
(b)
(Z x + Z2y)(Z2x + Z y).
13.
Using mathematical induction, prove that the number 22n ± 3n ± 1 is divisible by 9, for n = 1, 2, ..... .
14.
The roots Į and ȕ of the quadratic equation
x2 ± kx + (k + l) = 0 are such that Į2 + ȕ2 = 13. Find the possible values of the real number k.
15.
& & & & The vector n = 2 i ± j +3 k is normal to a plane which passes through the point (2, 1, 2).
(a)
Find an equation for the plane.
(b)
Find a if the point (a, a ± 1, a ± 2) lies on the plane.
16.
For what values of k is the straight line y = kx + 1 a tangent to the circle with centre (5, 1) and radius 3?
17.
(a)
Evaluate (1 + i)2, where i =
(b)
Prove, by mathematical induction, that (1 + i)4n = (±4)n, where n
(c)
Hence or otherwise, find (1 + i)32.
Pascal Ashkar Ȃ Ǯͳʹ
1 .
*.
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Year 12 Mathematics HL Ȃ Revision Pack
18.
19.
Let z1 =
6 i 2 , and z2 = 1 ± i. 2
(a)
Write z1 and z2 in the form r(cos ș + i sin ș), where r > 0 and ±
(b)
Show that
(c)
Find the value of
ʌ ʌ dșd . 2 2
z1 = cos S + i sin S . 12 12 z2
z1 in the form a + bi, where a and b are to be determined exactly in z2 radical (surd) form. Hence or otherwise find the exact values of cos S and sin S . 12 12
Consider the points A(l, 2, 1), B(0, ±1, 2), C(1, 0, 2), and D(2, ±1, ±6). (a)
Find the vectors AB and BC .
(b)
Calculate AB × BC .
(c)
Hence, or otherwise find the area of triangle ABC.
(d)
Find the equation of the plane P containing the points A, B, and C.
(e)
Find a set of parametric equations for the line through the point D and perpendicular to the plane P.
(f)
Find the distance from the point D to the plane P.
(g)
Find a unit vector which is perpendicular to the plane P.
(h)
The point E is a reflection of D in the plane P. Find the coordinates of E.
Pascal Ashkar Ȃ Ǯͳʹ
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Year 12 Mathematics HL Ȃ Revision Pack 20.
x 1n x kx, x ! 0 Consider the function fk (x) = ® , where k x 0 ¯0,
(a)
Find the derivative of f k (x), x > 0.
(b)
Find the interval over which f0 (x) is increasing. The graph of the function f k (x) is shown below.
y
0
(c)
(d)
21.
A
x
(i)
Show that the stationary point of f k (x) is at x = ek±1.
(ii)
One x-intercept is at (0, 0). Find the coordinates of the other x-intercept.
Find the equation of the tangent to the curve at A.
Let z1 = a §¨ cos S i sin S ·¸ and z2 = b §¨ cos S i sin S ·¸. 4 4¹ 3 3¹ © © 3
§z · Express ¨ 1 ¸ in the form z = x + yi. © z2 ¹
22.
Find a vector that is normal to the plane containing the lines L1, and L2, whose equations are:
L1: r = i + k + O (2i + j ± 2k) L 2: r = 3 i + 2 j + 2 k + µ ( j + 3 k )
Pascal Ashkar Ȃ Ǯͳʹ
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Year 12 Mathematics HL Ȃ Revision Pack
23.
Find the coordinates of the point which is nearest to the origin on the line
L: x = 1 ± O, y = 2 ± 3O, z = 2.
24.
The plane 6x ± 2y + z = 11 contains the line x ± 1 =
25.
Let f : x esin x. (a)
y 1 2
z 3 . Find l . l
Find f c (x). There is a point of inflexion on the graph of f, for 0 < x < 1.
(b)
26.
Write down, but do not solve, an equation in terms of x, that would allow you to find the value of x at this point of inflexion.
The diagram shows the graph of y = fc (x).
y
\ = I¶([)
x
Indicate, and label clearly, on the graph (a)
the points where y = f (x) has minimum points;
(b)
the points where y = f (x) has maximum points;
(c)
the points where y = f (x) has points of inflexion.
Pascal Ashkar Ȃ Ǯͳʹ
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Year 12 Mathematics HL Ȃ Revision Pack
27.
The area of the triangle shown below is 2.21 cm2. The length of the shortest side is x cm and the other two sides are 3x cm and (x + 3) cm.
x
3x
x+3 (a)
Using the formula for the area of the triangle, write down an expression for sin șin terms of x.
(b)
Using the cosine rule, write down and simplify an expression for cos șin terms of x.
(c)
(i)
Using your answers to parts (a) and (b), show that, § 3x 2 2 x 3 · ¨ ¸ 2x 2 © ¹
(ii)
2
§ · 1 ¨ 4.42 2 ¸ © 3x ¹
2
Hence find (a)
the possible values of x;
(b)
the corresponding values of ș in radians, using your answer to part (b) above.
Pascal Ashkar Ȃ Ǯͳʹ
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Year 12 Mathematics HL Ȃ Revision Pack 28.
The diagram below shows the graphs of y = ±x3 + 3x2 and y = g (x), where g (x) is a polynomial of degree 3.
y y = ±x 3 + x 2
g (x) A
x
0 A¶
(a)
If g (±2) = 0, g (0) = ±4, gc (±2) = 0, and gc (0) = (0) show that g (x) = x3 + 3x2 ± 4. The graph of y = ± x3 + 3x2 is reflected in the y-axis, then translated using the vector § ± 1· ¨¨ ¸¸ to give the graph of y = h (x). © ± 1¹
(b)
Write h (x) in the form h (x) = ax3 + bx2 + cx + d. The graph of y = x3 + 3x2 ± 4 is obtained by applying a composition of two transformations to the graph of y = ±x3 + 3x2.
(c)
State the two transformations whose composition maps the graph of y = ±x3 + 3x2 onto the graph of y = x3 + 3x2 ± 4 and also maps point A onto point Ac.
29.
Prove by mathematical induction that d (xn) = nxn±1, for all positive integer values of n. dx
30.
Given functions f : x x + 1 and g : x x3, find the function (f ° g)±l.
Pascal Ashkar Ȃ Ǯͳʹ
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Year 12 Mathematics HL Ȃ Revision Pack 31.
Let f (x) = ln |x5 ± 3x2|, ±0.5 < x < 2, x z a, x z b; (a, b are values of x for which f (x) is not defined). (a)
(i)
Sketch the graph of f (x), indicating on your sketch the number of zeros of f (x). Show also the position of any asymptotes.
(ii)
Find all the zeros of f (x), (that is, solve f (x) = 0).
(b)
Find the exact values of a and b.
(c)
Find f (x), and indicate clearly where fc (x) is not defined.
(d)
Find the exact value of the x-coordinate of the local maximum of f (x), for 0 < x < 1.5. (You may assume that there is no point of inflexion.)
7
32.
§ · The coefficient of x in the expansion of ¨ x 1 2 ¸ is 7 . Find the possible values of a. 3 ax ¹ ©
33.
For the function f : x x2 1n x, x > 0, find the function fc, the derivative of f with respect to x.
34.
The tangent to the curve y2 ± x3 at the point P(1, 1) meets the x-axis at Q and the y-axis at R. Find the ratio PQ : QR.
35.
Ö C = 30°, AB = 6 cm and AC = 3 2 cm. Find the possible lengths of In a triangle ABC, AB [BC].
36.
If z is a complex number and |z + 16| = 4 |z + l|, find the value of | z|.
Pascal Ashkar Ȃ Ǯͳʹ
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Year 12 Mathematics HL Ȃ Revision Pack 37.
The following graph is that part of the graph of y = f (x) for which f (x) t 0.
y 4 3 2 1 ±
0
±
1
2
x
Sketch, on the axes provided below, the graph of y2 = f (x) for ±2 d x d 2.
y 4 3 2 1 ±
±
0
1
2
x
± ± ± ±
Pascal Ashkar Ȃ Ǯͳʹ
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Year 12 Mathematics HL Ȃ Revision Pack
38.
(a)
Sketch and label the curves
y = x2 for ±2 d x d 2, and y = ± 1 ln x for 0 < x d 2. 2
39.
(b)
Find the x-coordinate of P, the point of intersection of the two curves.
(c)
If the tangents to the curves at P meet the y-axis at Q and R, calculate the area of the triangle PQR.
(d)
Prove that the two tangents at the points where x = a, a > 0, on each curve are always perpendicular.
If u = i +2j + 3k and v = 2i ± j + 2k, show that
(a)
u × v = 7i + 4 j ± 5 k . (b)
40.
41.
Let w = O u ȝv where O and µ are scalars. Show that w is perpendicular to the line of intersection of the planes x + 2y + 3z = 5 and 2x ± y + 2z = 7 for all values of O DQGȝ.
If f (x) = ln(2x ± 1), x !
1 , find 2
(a)
fc (x);
(b)
the value of x where the gradient of f (x) is equal to x.
If f ( x)
x , for x z ±1 and g (x) = (f q f )(x), find x 1
(a)
g ( x)
(b)
(g ° g)(2).
Pascal Ashkar Ȃ Ǯͳʹ
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Year 12 Mathematics HL Ȃ Revision Pack 42.
Find the coordinates of the point of intersection of the line L with the plane P where:
x 3 y ±1 z ±1 2 ±1 2 P : 2x 3y ± z ± 5 L:
43.
Solve, for x, the equation log2 (5x2 ± x ± 2) = 2 + 2 log2 x.
44.
Find the x-coordinate, between ±2 and 0, of the point of inflexion on the graph of the function f : x x 2 e x . Give your answer to 3 decimal places.
45.
For the vectors a = 2i + j ± 2k, b = 2i ±j ± k and c = i + 2j + 2k, show that: (a)
a × b = ±3i ± 2j ± 4k
(b)
( a × b) × c = ±(b c) a
Pascal Ashkar Ȃ Ǯͳʹ
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Year 12 Mathematics HL Ȃ Revision Pack 46.
The diagram shows a sketch of the graph of y = fc (x) for a d x d b.
y
\= I¶([)
a
b
x
On the grid below, which has the same scale on the x-axis, draw a sketch of the graph of y = f (x) for a d x d b, given that f (0) = 0 and f (x) t 0 for all x. On your graph you should clearly indicate any minimum or maximum points, or points of inflexion.
y
a
Pascal Ashkar Ȃ Ǯͳʹ
b
x
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Year 12 Mathematics HL Ȃ Revision Pack
47.
(a)
Sketch the graph of the function
1 C ( x) cos x cos2 x 2 for ±2S d x d 2S
48.
(b)
Prove that the function C (x) is periodic and state its period.
(c)
For what values of x, ±2S d x d 2S, is C (x) a maximum?
(d)
Let x = x0 be the smallest positive value of x for which C (x) = 0. Find an approximate value of x0 which is correct to two significant figures.
(e)
(i)
Prove that C (x) = C (±x) for all x.
(ii)
Let x = x1 be that value of x, S < x < 2S, for which C (x) = 0. Find the value of x1 in terms of x0.
(a)
The function f is defined by
f :x ex ±1 ± x
(b)
(i)
Find the minimum value of f.
(ii)
Prove that ex t 1 + x for all real values of x.
Use the principle of mathematical induction to prove that
1 1 1 (1 1) (1 ) (1 ) ....(1 ) n 1 2 3 n for all integers n t 1. (c)
Use the results of parts (a) and (b) to prove that e
(d)
(1
1 1 1 .... ) 2 3 n
! n.
Find a value of n for which
1 1 1 1 ..... ! 100 2 3 n
Pascal Ashkar Ȃ Ǯͳʹ
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Year 12 Mathematics HL Ȃ Revision Pack 49.
Solve 2 sin x = tan x, where ±
50.
Let f : x
51.
52.
S 2
x
S . 2
1 ± 2 . Find x2
(a)
the set of real values of x for which f is real and finite;
(b)
the range of f.
The nth term, un, of a geometric sequence is given by un = 3(4)n+1, n
+
.
(a)
Find the common ratio r.
(b)
Hence, or otherwise, find Sn, the sum of the first n terms of this sequence.
Find an equation of the plane containing the two lines
x ±1
1± y 2
z ± 2 and
x 1 3
2± y 3
z2 . 5
53.
An astronaut on the moon throws a ball vertically upwards. The height, s metres, of the ball, after t seconds, is given by the equation s = 40t + 0.5 at2, where a is a constant. If the ball reaches its maximum height when t = 25, find the value of a.
54.
The equation kx2 ± 3x + (k + 2) = 0 has two distinct real roots. Find the set of possible values of k.
Pascal Ashkar Ȃ Ǯͳʹ
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Year 12 Mathematics HL Ȃ Revision Pack 55.
The function f is given by f : x e(1 sin Sx ) , x t 0. (a)
Find f c(x).
Let xn be the value of x where the (n + l)th maximum or minimum point occurs, n . ( ie x0 is the value of x where the first maximum or minimum occurs, x1 is the value of x where the second maximum or minimum occurs, etc). (b)
56.
Find xn in terms of n.
The triangle ABC has vertices at the points A(±l, 2, 3), B(±l, 3, 5) and C(0, ±1, 1). (a)
Find the size of the angle T between the vectors AB and AC .
(b)
Hence, or otherwise, find the area of triangle ABC.
Let l1 be the line parallel to AB which passes through D(2, ±1, 0) and l 2 be the line parallel to
AC which passes through E(±l, 1, 1).
(c)
(d)
57.
(i)
Find the equations of the lines l1 and l2.
(ii)
Hence show that l 1 and l 2 do not intersect.
Find the shortest distance between l1 and l2.
Using mathematical induction, prove that values of n.
Pascal Ashkar Ȃ Ǯͳʹ
dn nS · § (cos x ) cos ¨ x ¸, for all positive integer n 2 ¹ dx ©
Page 16
Year 12 Mathematics HL Ȃ Revision Pack 58.
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.
59.
The line y = 16x ± 9 is a tangent to the curve y = 2x3 + ax2 + bx ± 9 at the point (1,7). Find the values of a and b.
60.
Consider the function y = tan x ± 8 sin x.
dy . dx
(a)
Find
(b)
Find the value of cos x for which
dy dx
0.
61.
Find the values of x for which «5 ± 3x « d «x + 1«.
62.
Consider the tangent to the curve y = x3 + 4x2 + x ± 6. (a)
Find the equation of this tangent at the point where x = ±1.
(b)
Find the coordinates of the point where this tangent meets the curve again.
Pascal Ashkar Ȃ Ǯͳʹ
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Year 12 Mathematics HL Ȃ Revision Pack
63.
Let y = sin (kx) ± kx cos (kx), where k is a constant. Show that
64.
dy k 2 x sin ( k x). dx
(a)
Express the complex number 8i in polar form.
(b)
The cube root of 8i which lies in the first quadrant is denoted by z. Express z (i)
in polar form;
(ii)
in cartesian form.
65.
Find the angle between the vectors v = i + j + 2k and w = 2i + 3j + k. Give your answer in radians.
66.
The one-one function f is defined on the domain x > 0 by f (x) =
67.
(a)
State the range, A, of f.
(b)
Obtain an expression for f ±1(x), for x A.
2x ± 1 . x2
A curve has equation xy3 + 2x2y = 3. Find the equation of the tangent to this curve at the point (1, 1).
Pascal Ashkar Ȃ Ǯͳʹ
Page 18
Year 12 Mathematics HL Ȃ Revision Pack 68.
The function f is defined by
f (x) = (a)
(b)
(c)
69.
x2 ± x 1 x2 x 1
(i)
Find an expression for fc (x), simplifying your answer.
(ii)
The tangents to the curve of f (x) at points A and B are parallel to the x-axis. Find the coordinates of A and of B.
(i)
Sketch the graph of y = fc (x).
(ii)
Find the x-coordinates of the three points of inflexion on the graph of f.
Find the range of (i)
f;
(ii)
the composite function f ° f.
Ö C = 37.8q, AB = 8.75 and BC = 6. Consider triangle ABC with BA
Find AC.
70.
71.
72.
z1 = 1 i 3
m
and z2 = 1 i . n
(a)
Find the modulus and argument of z1 and z2 in terms of m and n, respectively.
(b)
H ence, find the smallest positive integers m and n such that z1 = z2.
Find the gradient of the tangent to the curve x3 y2 FRV ʌ\ DWWKHSRLQW í
§ 4 · ¨ ¸ A ray of light comLQJIURPWKHSRLQW í LVWUDYHOOLQJLQWKHGLUHFWLRQRIYHFWRU ¨ 1 ¸ and ¨ 2¸ © ¹ PHHWVWKHSODQHʌ x + y + 2z í Find the angle that the ray of light makes with the plane.
Pascal Ashkar Ȃ Ǯͳʹ
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Year 12 Mathematics HL Ȃ Revision Pack 73.
'HWHUPLQHWKHILUVWWKUHHWHUPVLQWKHH[SDQVLRQRI íx)5 (1+ x)7 in ascending powers of x.
74.
A curve has equation x3 y2 = 8. Find the equation of the normal to the curve at the point (2, 1).
75.
Consider the equation 2xy2 = x2y + 3.
76.
(a)
Find y when x = 1 and y < 0.
(b)
Find
dy when x = 1 and y < 0. dx
André wants to get from point A located in the sea to point Y located on a straight stretch of beach. P is the point on the beach nearest to A such that AP = 2 km and PY = 2 km. He does this by swimming in a straight line to a point Q located on the beach and then running to Y.
When André swims he covers 1 km in 5 5 minutes. When he runs he covers 1 km in 5 minutes. (a)
If PQ = x km, 0 d x ILQGDQH[SUHVVLRQIRUWKHWLPH T minutes taken by André to reach point Y.
(b)
Show that
(c)
(i)
Solve
(ii)
Use the value of x found in part (c) (i) to determine the time, T minutes, taken for André to reach point Y.
(iii)
Show that
dT dx
dT dx
5 5x
x2 4
5.
0.
d 2T dx 2
20 5
x
2
4
3 2
and hence show that the time found in part (c) (ii) is a
minimum.
Pascal Ashkar Ȃ Ǯͳʹ
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Year 12 Mathematics HL Ȃ Revision Pack 2
77.
78.
If f (x) = x ± 3 x 3 , x ! 0, (a)
find the x-coordinate of the point P where f ƍ x) = 0;
(b)
determine whether P is a maximum or minimum point.
The function f is defined by f (x) =
ln x
x3
, x t1.
(a)
Find f ƍ x) and f ƍƍ x), simplifying your answers.
(b)
(i)
Find the exact value of the x-coordinate of the maximum point and justify that this is a maximum.
(ii)
Solve f ƍƍ x) = 0, and show that at this value of x, there is a point of inflexion on the graph of f.
(iii)
Sketch the graph of f, indicating the maximum point and the point of inflexion.
The region enclosed by the x-axis, the graph of f and the line x = 3 is denoted by R.
x3 ± x2 ± 3x + 4 has a local maximum point at P and a local minimum point at 3 Q. Determine the equation of the straight line passing through P and Q, in the form ax + by + c = 0, where a, b, c .
79.
The curve y =
80.
A closed cylindrical can has a volume of 500 cm3. The height of the can is h cm and the radius of the base is r cm. (a)
Find an expression for the total surface area A of the can, in terms of r.
(b)
Given that there is a minimum value of A for r > 0, find this value of r.
Pascal Ashkar Ȃ Ǯͳʹ
Page 21
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