IB HL Mathematics Problems, Organized by

Subject

Core: Algebra

1. (May 1999, #3) The second term of an arithmetic sequence is 7. The sumof the first four terms of the arithmetic sequence is 12. Find the first term,a, and the common difference d, of the sequence.

2. (May 1999, #5) Let z = x+yi. Find the values of x and y if (1−i)z = 1−3i.

3. (May 1999, #15) On the diagram below, draw the locus of the point P (x, y),representing the complex number z = x+yi, given that |z−4−3i| = |z−2+i|.

4. (May 1999, #16) Given that

(1 + x)5(1 + ax)6 ≡ 1 + bx + 10x2 + · · ·+ a6x11,

find the values of a, b ∈ Z∗.

5. (Nov 2000, #7) Find the sum of the positive terms of the arithmetic sequence85, 78, 71, . . . .

6. (Nov 2000, #12) The coefficient of x in the expansion of

(x +

1

ax2

)7

is 73.

Find the possible values of a.

7. (Nov 2000, #15) The sum of an infinite geometric sequence is 1312, and the

sum of the first three terms is 13. Find the first term.

8. (Nov 2000, #18) If z is a complex number and |z + 16| = 4|z + 1|, find thevalue of |z|.

9. (May 2001, #7) The nth term, un, of a geometric sequence is given byun = 3(4)n+1, n ∈ Z+.

(a) Find the common ratio r.

(b) Hence, or otherwise, find Sn, the sum of the first n terms of this se-quence.

10. (May 2001, #14) Given that z = (b + i)2, where b is real and positive, findthe exact value of b when arg z = 60◦.

11. (Nov 2001, #2) The complex number z satisfies i(z + 2) = 1 − 2z, wherei =

√−1. Write z in the form z = a + bi, where a and b are real numbers.

12. (Nov 2001, #4) Consider the infinite geometric series

1 +

(2x

3

)+

(2x

3

)2

+

(2x

3

)3

+ · · · .

(a) For what values of x does the series converge?

(b) Find the sum of the series if x = 1.2

13. (Nov 2001, #19) A sample of radioactive material decays at a rate whichis proportional to the amount of material present in the sample. Find thehalf-life of the material if 50 grams decay to 48 grams in 10 years.

14. (May 2002, #1) Consider the arithmetic series 2 + 5 + 8 + · · · .(a) Find an expression for Sn, the sum of the first n terms.

(b) Find the value of n for which Sn = 1365.

15. (May 2002, #3) (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.

16. (Nov 2002, #3) Find the coefficient of x3 in the binomial expansion of(1− 1

2x

)8

.

17. (Nov 2002, #6) Find50∑

r=1

ln(2r), giving the answer in the form a ln 2, where

a ∈ Q.

18. (May 2003, #1) A geometric sequence has all positive terms. The sum ofthe first two terms is 15 and the sum to infinity is 27. Find the value of

(a) the common ratio;

(b) the first term.

19. (May 2003, #11) The complex number z satisfies the equation

√z =

2

1− i+ 1− 4i.

Express z in the form x + iy where x, y ∈ Z.

20. (May 2003, #12) Find the exact value of x satisfying the equation

(3x)(42x+1) = 6x+2.

Give your answer in the formln a

ln bwhere a, b ∈ Z.

21. (Nov 2003, #5) Consider the equation 2(p + iq) = q − ip− 2(1− i), wherep and q are both real numbers. Find p and q.

22. (Nov 2003, #9) The first four terms of an arithmetic sequence are 2, a −b, 2a + b + 7, and a− 3b, where a and b are constants. Find a and b.

23. (Nov 2003, #10) Solve log163√

100− x2 =1

2.

24. (Nov 2003, #19) Solve 2(5x+1

)= 1 +

3

5x, giving the answer in the form

a + log5 b, where a, b ∈ Z.

25. (May 2004, T1 #4) A geometric series has a negative common ratio. Thesum of the first two terms is 6. The sum to infinity is 8. Find the commonratio and the first term.

26. (May 2004, T2 #4) The three terms a, 1, b are in arithmetic progression.The three terms 1, a, b are in geometric progression. Find the value of aand of b given that a 6= b.

27. (May 2004, T2 #6) Let the complex number z be given by

z = 1 +i

i−√

3.

Express z in the form a + bi, giving the exact values of the real constantsa, b.

6

28. (Nov 2004, #3) The sum of the first n terms of a series is given by

Sn = 2n2 − n, where n ∈ Z+.

(a) Find the first three terms of the series.

(b) Find an expression for the nth term of the series, giving your answer interms of n.

29. (Nov 2004, #4) Given that (a− i)(2− bi) = 7− i, find the value of a and ofb, where a, b ∈ Z.

30. (Nov 2004, #8) Find the expansion of (2 + x)5, giving your answer in as-cending powers of x.

(b) By letting x = 0.01 or otherwise, find the exact value of 2.015.

31. (Nov 2004, #13) Given that z ∈ C, solve the equation z3 − 8i = 0, givingyour answers in the form z = r(cos θ + i sin θ).

Core: Functions and Equations

1. (May 1999, #1) When the function f(x) = 6x4 + 11x3 − 22x2 + ax + 6 isdivided by (x + 1) the remainder is −20. Find the value of a.

2. (May 1999, #7) The diagram below shows the graph of y1 = f(x). Thex-axis is a tangent to f(x) at x = m and f(x) crosses the x-axis at x = n.

On the same diagram sketch the graph of y2 = f(x−k), where 0 < k < n−mand indicate the coordinates of the points of intersection of y2 with the x-axis.

3. (May 1999, #12) Given f(x) = x2 + x(2− k) + k2, find the range of valuesof k for which f(x) > 0 for all real values of x.

4. (Nov 2000, #2) Given functions f : x 7→ x + 1 and g : x 7→ x3, find thefunction (f ◦ g)−1.

5. (Nov 2000, #10) Find the real number k for which 1 + ki, (i =√−1), is a

zero of the polynomial z2 + kz + 5.

6. (Nov 2000, #20) The following graph is that part of the graph of y = f(x)for which f(x) > 0.

Sketch, on the axes provided below, the graph of y2 = f(x) for −2 ≤ x ≤ 2.

7. (May 2001, #5) Let f : x 7→√

1

x2− 2. Find

(a) the set of real values of x for which f is real and finite;

(b) the range of f .

8. (May 2001, #10) (z + 2i) is a factor of 2z3 − 3z2 + 8z − 12. Find the othertwo factors.

9. (May 2001, #17) An astronaut on the moon throws a ball vertically upwards.The height, s metres, of the ball, after t seconds, is given by the equations = 40t + 0.5at2, where a is a constant. If the ball reaches its maximumheight when t = 25, find the value of a.

10. (May 2001, #18) The equation kx2 − 3x + (k + 2) = 0 has two distinct realroots. Find the set of possible values of k.

11. (May 2001, #19) The diagram shows the graph of the functions y1 and y2.

On the same axes sketch the graph ofy1

y2

. Indicate clearly where the x-

intercepts and asymptotes occur.

12. (Nov 2001, #3) The polynomial f(x) = x3 + 3x2 + ax + b leaves the sameremainder when divided by (x − 2) as when divided by (x + 1). Find thevalue of a.

13. (Nov 2001, #5) The function f : x 7→ 2x + 1

x− 1, x ∈ R, x 6= 1. Find the

inverse function f−1, clearly stating its domain.

14. (Nov 2001, #11) Find the values of x for which |5− 3x| ≤ |x + 1|.

15. May 2002, #15 The one-one function f is defined on the domain x > 0 by

f(x) =2x− 1

x + 2.

(a) State the range, A, of f .

(b) Obtain an expression for f−1(x), for x ∈ A.

16. (May 2002, #16) Find the set of values of x for which (ex−2)(ex−3) ≤ 2ex.

17. (Nov 2002, #1) When the polynomial x4 + ax + 3 is divided by (x− 1), theremainder is 8. Find the value of a.

18. (Nov 2002, #4) Find the equations of all the asymptotes of the graph of

y =x2 − 5x− 4

x2 − 5x + 4.

19. (Nov 2002, #7) The functions f(x) and g(x) are given by f(x) =√

x− 2and g(x) = x2 + x. The function (f ◦ g)(x) is defined for x ∈ R, except forthe interval ]a, b[ .

(a) Calculate the value of a and of b.

(b) Find the range of f ◦ g.

20. (Nov 2002, #9) Solve the inequality x2 − 4 +3

x< 0.

21. (May 2003, #4) The polynomial x3 +ax2−3x+ b is divisible by (x−2) andhas a remainder 6 when divided by (x + 1). Find the value of a and of b.

22. (May 2003, #7) The function f is given by f(x) = 2− x2 − ex.

Write down

(a) the maximum value of f(x);

(b) the two roots of the equation f(x) = 0.

23. (May 2003, #13) Solve the inequality |x− 2| ≥ |2x + 1|.

24. (May 2003, #17) The function f is defined for x ≤ 0 by f(x) =x2 − 1

x2 + 1.

Find an expression for f−1(x).

25. (Nov 2003, #6) The diagram shows the graph of f(x).

(a) On the same diagram, sketch the graph of1

f(x), indicating clearly any

asymptotes.

(b) On the diagram write down the coordinates of the local maximum point,

the local minimum point, the x-intercepts and the y-intercept of1

f(x).

26. (Nov 2003, #13) Consider the equation (1+2k)x2−10x+k−2 = 0, k ∈ R.Find the set of values of k for which the equation has real roots.

27. (Nov 2003, #17) Let f(x) =x + 4

x + 1, x 6= −1, and g(x) =

x− 2

x− 4, x 6= 4.

Find the set of values of x such that f(x) ≤ g(x).

28. (Nov 2004 T1, #1) The polynomial x3 − 2x2 + ax + b has a factor (x − 1)and a remainder 8 when divided by (x + 1). Calculate the value of a and ofb.

29. (May 2004, T1 #3) For 0 ≤ x ≤ 6, find the coordinates of the points ofintersection of the curves

y = x2 cos x and x + 2y = 1.

30. (May 2004, T1 #9) The function f is defined on the domain [−1, 0] by

f : x 7→ 1

1 + x2.

(a) Write down the range of f .

(b) Find an expression for f−1(x).

31. (May 2004, T1 #16) Solve the inequality∣∣∣∣x + 12

x− 12

∣∣∣∣ ≤ 3.

32. (May 2004, T2 #1) The polynomial x2−4x+3 is a factor of x3 +(a−4)x2 +(3− 4a)x + 3. Calculate the value of the constant a.

33. (May 2004, T2 #3) For −3 ≤ x ≤ 3, find the coordinates of the points ofintersections of the curves

y = x sin x and x + 3y = 1.

34. (May 2004, T2 #10) Let f(x) =k

x− k, x 6= k, k > 0.

(a) On the diagram below, sketch the graph of f . Label clearly any pointsof intersection with the axes, and any asymptotes.

-

6

kx

y

(b) On the diagram below, sketch the graph of1

f. Label clearly any points

of intersection with the axes.

-

6

kx

y

35. (May 2004, T2 #10) The function f is defined by f : x 7→ x3.

Find an expression for g(x) in terms of x in each of the following cases.

(a) (f ◦ g)(x) = x + 1.

(b) (g ◦ f)(x) = x + 1.

36. (May 2004, T2 #16) Solve the inequality∣∣∣∣x + 9

x− 9

∣∣∣∣ ≤ 2.

37. (Nov 2004, #1) Consider f(x) = x3 − 2x2 − 5x + k. Find the value of k if(x + 2) is a factor of f(x).

38. (Nov 2004, #7) (a) Find the largest set S of values of x such that the

function f(x) =1√

3− x2takes real values.

(b) Find the range of the function f defined on the domain S.

39. (Nov 2004, #15) Find the range of values of m such that for all x

m(x + 1) ≤ x2.

Core: Circular Functions and Trigonometry

1. (Nov 2000, #16) In a triangle ABC, ABC = 30◦, AB = 6 cm, and AC =3√

2 cm. Find the possible lengths of [BC].

2. (May 2001, #2) Solve 2 sin x = tan x, where −π

2< x <

π

2.

3. (Nov 2001, #16) Let θ be the angle between the unit vectors a and b, where0 < θ < π. Express |a− b| in terms of sin 1

2θ.

4. (May 2002, #10) The angle θ satisfies the equation tan θ + cot θ = 3, whereθ is in degrees. Find all the possible values of θ lying in the interval ]0◦, 90◦[.

5. (May 2002, #12) The function f is defined on the domain [0, π] by f(θ) =4 cos θ + 3 sin θ.

(a) Express f(θ) in the form R cos(θ − α) where 0 < α < π2.

(b) Hence, or otherwise, write down the value of θ for which f(θ) takes itsmaximum value.

6. (Nov 2002, #12) Triangle ABC has AB = 8 cm, BC = 6 cm, and BAC =20◦. Find the smallest possible area of 4ABC.

7. (Nov 2002, #2) Find all the values of θ in the interval [0, π] which satisfythe equation cos 2θ = sin2 θ.

8. (May 2003, #8) In the triangle ABC, A = 30◦, BC = 3 and AB = 5. Find

the two possible values of B.

9. (Nov 2003, #1) Consider the points A(1, 2,−4), B(1, 5, 0) and C(6, 5,−12).Find the area of 4ABC.

10. (Nov 2003, #14) Let f(x) = sin

(arcsin

x

4− arccos

3

5

), for −4 ≤ x ≤ 4.

(a) On the grid below, sketch the graph of f(x).

(b) On the sketch, clearly indicate the coordinates of the x-intercept, they-intercept, the minimum point and the endpoints of the curve of f(x).

(c) Solve f(x) = −1

2.

11. (May 2004, T1 #20) The following three-dimensional diagram shows thefour points A, B, C, and D. A, B, and C are in the same horizontal plane,

and AD is vertical. ABC = 45◦, BC = 50 m, ABD = 30◦, ACD = 20◦.

Using the cosine rule in the triangle ABC, or otherwise, find AD.

12. (May 2004, T2 #20) The diagram shows a sector AOB of a circle of radius

1 and centre O, where AOB = θ.

The lines (AB1), (A1B2), (A2B3) are perpendicular to OB. A1B1, A2B2 areall arcs of circles with centre O.

Calculate the sum to infinity of the arc length

AB + A1B1 + A2B2 + A3B3 + · · · .

13. (Nov 2004, #9) The diagram below shows a circle centre O and radiusOA = 5 cm. The angle AOB = 135◦.

Find the area of the shaded region.

14. (Nov 2004, #10) Consider the equation e−x = cos 2x, for 0 ≤ x ≤ 2π.

(a) How many solutions are there to this equation?

(b) Find the solution closest to 2π, giving your answer to four decimalplaces.

Core: Matrices and Geometry

1. (May 1999, #6) Find the value of a for which the following system of equa-tions does not a have a unique solution.

4x− y + 2z = 1

2x + 3y = −6

x− 2y + az =7

2

2. (Nov 2000, #1) Find the values of the real number k for which the determi-

nant of the matrix

[k − 4 3−2 k + 1

]is equal to zero.

3. (Nov 2000, #9) (a) Describe the transformation of the plane whose matrix is

M =

12

−√

32

√3

212

.

(b) Find the smallest positive integer n for which Mn =

[1 00 1

].

4. (May 2001, #3) Give a full geometric description of the transformationrepresented by the matrix 4

335

35−4

5

.

5. (May 2001, #9) Find the equation of the line of intersection of the twoplanes −4x + y + z = −2 and 3x− y + 2z = −1

6. (May 2001, #12) Find an equation of the plane containing the two lines

x− 1 =1− y

2= z − 2 and

x + 1

3=

2− y

3=

z + 2

5.

7. (Nov 2001, #6) If A =

[x 44 2

]and B =

[2 y8 4

], find the values of x and

y, given that AB = BA.

8. (Nov 2001, #9) The matrix

1 −2 −31 −k −13

−3 5 k

is singular. Find the values

of k.

9. (Nov 2001, #12) A linear transformation T maps Triangle 1 to Triangle 2,as shown in the diagram.

Find a matrix which represents T .

10. (May 2002, #4) The matrix A is given by

A =

2 1 k1 k −13 4 2

.

Find the values of k for which A is singular.

11. (May 2002, #18) A transformation T of the plane is represented by thematrix

T =

[2 31 2

].

(a) T transforms the point P to the point (8, 5). Find the coordinates ofP .

(b) Find the coordinates of all points which are transformed to themselvesunder T .

12. (Nov 2002, #10) Find an equation for the line of intersection of the followingtwo planes.

x + 2y − 3z = 2

2x + 3y − 5z = 3.

13. (Nov 2002, #19) The transformation M represents a reflection in the line

y = x√

3. The transformation R represents a rotation throughπ

6radians

anticlockwise about the origin. Give a full geometric description of the singletransformation which is equivalent to M followed by R.

14. (May 2003, #5) Given that A =

[3 −2

−3 4

]and I =

[1 00 1

], find the

values of λ for which (A− λI) is a singular matrix.

15. (May 2003, #15) The point A is the foot of the perpendicular from the point(1, 1, 9) to the plane 2x + y − z = 6. Find the coordinates of A.

16. (Nov 2003, #3) The matrices A, B, C and X are all non-singular 3 × 3matrices. Given that A−1XB = C, express X in terms of the other matrices.

17. (Nov 2003, #7) Find the angle between the plane 3x − 2y + 4z = 12 andthe z-axis. Give your answer to the nearest degree.

18. (May 2004, T1 #5) The composite transformation T is defined by a clockwiserotation of 45◦ about the origin followed by a reflection in the line x+y = 0.Calculate the 2× 2 matrix representing T .

19. (May 2004, T2 #5) The linear transformations M and S are represented bythe matrices

M =

0 1−1 0

and S =

−1 00 1

.

Give a full geometric description of the single transformation represented bythe matrix SMS.

20. (Nov 2004, #2) Given that the matrix A =

1 −1 22 p 31 −2 5

is singular, find

the value of p.

21. (Nov 2004, #11) Consider the four points A(1, 4,−1), B(2, 5,−2), C(5, 6, 3),and D(8, 8, 4). Find the point of intersection of the lines (AB) and (CD).

22. (Nov 2004, #19) (a) Find the cartesian equation of the plane that containsthe origin O and the two points A(1, 1, 1) and B(2,−1, 3).

(b) Find the distance from the point C(10, 5, 1) to the plane OAB.

23. (Nov 2004, #20) The following diagram shows the liines x−2y−4 = 0, x+y = 5 and the point P (1, 1). A line is drawn from P to intersect withx− 2y − 4 = 0 at Q, and with x + y = 5 at R, so that P is the midpoint of[QR].

Core: Vectors

1. (May 1999, #4) Find the coordinates of the point where the line given bythe parametric equations x = 2λ + 4, y = −λ− 2, z = 3λ + 2, intersects theplane with equation 2x + 3y − z = 2.

2. (May 1999, #10) (a) Find a vector perpendicular to the two vectors:

−→OP = ~i− 3~j + 2~k−→OQ = −2~i +~j − ~k

(b) If−→OP and

−→OQ 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.

3. (Nov 2000, #11) Let α be the angle between the vectors a and b, where

a = (cos θ)i + (sin θ)j, b = (sin θ)i + (cos θ)j and 0 < θ <π

4. Express α in

terms of θ.

4. (Nov 2001, #15) Point A(3, 0,−2) lies on the line r = 3i−2k+λ(2i−2j+k),where λ is a real parameter. Find the coordinates of one point which is 6units from A, and on the line.

5. (May 2002, #5) Find the angle between the vectors v = i + j + 2k andw = 2i + 3j + k. Give your answer in radians.

6. (May 2002, #8) The vector equations of the lines L1 and L2 are given by

L1 : r = i + j + k + λ(i + 2j + 3k)

L2 : r = i + 4j + 5k + µ(2i + j + 2k).

The two lines intersect at the point P . Find the position vector of P .

7. (Nov 2002, #2) The graph of the function f(x) = 2x3 − 3x2 + x + 1 is

translated to its image, g(x), by the vector

[1−1

]. Write g(x) in the form

g(x) = ax3 + b2 + cx + d.

8. (Nov 2002, #18) Given two non-zero vectors a and b such that |a + b| =|a− b|, find the value of a · b.

9. (May 2003, #3) Given that a = i + 2j − k, b = −3i + 2j + 2k, andc = 2i− 3j + 4k, find (a× b) · c.

10. (May 2004, T1 #10) The line x− 1 =y + 1

2=

z

3and the plane r · (i + 2j−

k) = 1 interesct at the point P . Find the coordinates of P .

11. (May 2004, T2 #15) Given that a = (i + 2j + k)×(−2i + 3k),

(a) find a;

(b) find the vector projection of a onto the vector −2j + k.

Core: Statistics and Probability

1. (May 1999, #2) A bag contains 2 red balls, 3 blue balls, and 4 green balls.A ball is chosen at random from the bag and is not replaced. A second ballis chosen. Find the probability of choosing one green ball and one blue ballin any order.

2. (May 1999, #8) In a biliingual school there is a class of 21 pupils. In thisclass, 15 of the pupils speak Spanish as their first language and 12 of these15 pupils are Argentine. The other 6 pupils in the class speak English astheir first language and 3 of these 6 pupils are Argentine.

A pupil is selected at random from the class and is found to be Argentine.Find the probability that the pupil speaks Spanish as his/her first language.

3. (May 1999, #17) A biased die with four faces is used in a game. A playerpays 10 counters to roll the die. The table below shows the possible scores onthe die, the probability of each score and the number of counter the playerreceives in return for each score.

Score 1 2 3 4

Probability 12

15

15

110

Number of counters player receives 4 5 15 n

Find the value of n in order for the player to get an expected return of 9counter per roll.

4. (May 1999, #18) A factory has a machine designed to produce 1 kg bags ofsugar. It is found that the average weight of sugar in the bags is 1.02 kg.Assuming that the weights of the bags are normally distributed, find thestandard deviation if 1.7% of the bags weigh below 1 kg. Give your answercorrect to the nearest 0.1 gram.

5. (Nov 2000, #4) The box-and-whisker plots shown represent the heights offemale students and the heights of male students at a certain school.

-

150 160 170 180 190 200 210 Height (cm)

Females

Males

(a) What percentage of female students are shorter than any male students?

(b) What percentage of male students are shorter than some female stu-dents?

(c) From the diagram, estimate the mean height of the male students.

6. (Nov 2000, #6) Given that events A and B are independent with P(A∩B) =0.3 and P(A ∩B′) = 0.3, find P(A ∪B).

7. (Nov 2000, #19) In how many ways can six different coins be divided be-tween two students so that each student receives at least one coin?

8. (May 2001, #6) A machine produces packets of sugar. The weights in gramsof thirty packets chosen at random are shown below.

Weight (g) 29.6 29.7 29.8 29.9 30.0 30.1 30.2 30.3

Frequency 2 3 4 5 7 5 3 1

Find unbiased estimates of

(a) the mean of the population from which this sample is taken;

(b) the variance of the population from which this sample is taken.

9. (May 2001, #11) Given that P(X) =2

3, P(Y |X) =

2

5and P(Y |X ′) = 1

4,

find

(a) P(Y ′);

(b) P(X ′ ∪ Y ′).

10. (May 2001, #13) Z is the standardized normal random variable with mean0 and variance 1. Find the value of a such that P(|Z| ≤ a) = 0.75.

11. (May 2001, #15) X is a binomial random variable, where the number oftrials is 5 and the probability of success of each trial is p. Find the valuesof p if P(X = 4) = 0.12.

12. (Nov 2001, #1) A coin is biased so that when it is tossed the probability ofobtaining heads is 2

3. The coin is tossed 1800 times. Let X be the number

of heads obtained. Find

(a) the mean of X;

(b) the standard deviation of X.

13. (Nov 2001, #8) A continuous random variable X has probability densityfunction

f(x) =

4

π(1 + x2), for 0 ≤ x ≤ 1,

0, elsewhere.

Find E(X).

14. (Nov 2001, #17) How many four-digit numbers are there which contain atleast one digit 3?

15. (Nov 2001, #18) The probability that a man leaves his umbrella in any shophe visits is 1

3. After visiting two shops in succession, he finds that he has

left his umbrella in one of them. What is the probability that he left hisumbrella in the second shop?

16. (May 2002, #7) The probability that it rains during a summer’s day in acertain town is 0.2. In this town, the probability that the daily maximumtemperature exceeds 25 ◦C is 0.3 when it rains and 0.6 when it does not rain.Given that the maximum daily temperature exceeded 25 ◦C on a particularsummer’s day, find the probability that it rained on that day.

17. (May 2002, #9) When John throws a stone at a target, the probability thathe hits the target is 0.4. He throws a stone 6 times.

(a) Find the probability that he hits the target exactly 4 times.

(b) Find the probability that he hits the target for the first time on histhird throw.

18. (May 2002, #11) The weights of a certain species of bird are normally dis-tributed with mean 0.8 kg and standard deviation 0.12 kg. Find the proba-bility that the weight of a randomly chosen bird of the species lies between0.74 kg and 0.95 kg.

19. (May 2002, #14) The 80 applicants for a Sports Science course were requiredto run 800 metres and their times were recorded. The results were used toproduce the following cumulative frequency graph.

Estimate

(a) the median;

(b) the interquartile range.

20. (Nov 2002, #5) An integer is chosen at random from the first one thousandpositive integers. Find the probability that the integer chosen is

(a) a multiple of 4;

(b) a multiple of both 4 and 6.

21. (Nov 2002, #8) Consider the six numbers 2, 3, 6, 9, a, and b. The mean ofthe numbers is 6 and the variance is 10. Find the value of a and b, if a < b.

22. (Nov 2002, #15) The probability density function f(x), of a continuousrandom variable X is defined by

f(x) =

{14x(4− x2), 0 ≤ x ≤ 2,

0 otherwise.

Calculate the median value of X.

23. (May 2003, #6) When a boy plays a game at a fair, the probability that hewins a prize is 0.25. He plays the game 10 times. Let X denote the totalnumber of prizes that he wins. Assuming that the games are independent,find

(a) E(X);

(b) P(X ≤ 2).

24. (May 2003, #9) The independent events A, B are such that P(A) = 0.4 andP(A ∪B) = 0.88. Find

(a) P (B);

(b) the probability that either A occurs or B occurs, but not both.

25. (May 2003, #14) The random variable X is normally distributed and

P(X ≤ 10) = 0.670

P(X ≤ 12) = 0.937.

Find E(X).

26. (May 2003, #19) A teacher drives to school. She records the time taken oneach of 20 randomly chosen days. She finds that

20∑i=1

xi = 626 and20∑i=1

x2i = 19780.8,

where xi denotes the time, in minutes, taken on the ith day.

Calculate an unbiased estimate of

(a) the mean time taken to drive to school;

(b) the variance of the time taken to drive to school.

27. (Nov 2003, #2) The cumulative frequency curve below indicates the amountof time 250 student spend eating lunch.

(a) Estimate the number of students who spend between 20 and 40 minuteseating lunch.

(b) If 20% of the students spend more than x minutes eating lunch, estimatethe value of x.

28. (Nov 2003, #4) A continuous random variable, X, has probability densityfunction

f(x) = sin x, 0 ≤ x ≤ π

2.

Find the median of X.

29. (Nov 2003, #12) On a television channel the news is shown at the sametime each day. The probability that Alice watches the news on a given dayis 0.4. Calculate the probability that on five consecutive days, she watchesthe news on at most three days.

30. (Nov 2003, #18) A committee of four children is chosen from eight children.The two oldest children cannot both be chosen. Find the number of waysthe committee may be chosen.

31. (May 2004, T1 #6) The weights of adult males of a type of dog may beassumed to be normally distributed with mean 25 kg and a standard devia-tion 3 kg. Given that 30% of the weights lie between 25 kg and x kg, wherex > 25, find the value of x.

32. (May 2004, T1 #12) Marian shoots ten arrows at a target. Each arrow hasprobability 0.4 of hitting the target, independently of all other arrows. LetX denote the number of these arrows hitting the target.

(a) Find the mean and standard deviation of X.

(b) Find P (X ≥ 2).

33. (May 2004, T1 #13) A desk has three drawers. Drawer 1 contains three goldcoins, Drawer 2 contains two gold coins and one silver coin, and Drawer 3contains one gold coin and two silver coins. A drawer is chosen at randomand from it a coin is chosen at random.

(a) Find the probability that the chosen coin is gold.

(b) Given that the chosen coin is gold, find the probability that Drawer 3was chosen.

34. (May 2004, T1 #15) The heights of 60 children entering a school were mea-sured. The following cumujlative frequency graph illustrates the data ob-tained.

Estimate

(a) the median height;

(b) the mean height.

35. (May 2004, T2 #7) The following diagram shows the probability densityfunction for the random variable X, which is normally distributed withmean 250 and standard deviation 50.

Find the probability of the shaded region.

36. (May 2004, T2 #13) The discrete random variable X has the followingprobability distribution:

P(X = x) =

k

x, x = 1, 2, 3, 4

0, otherwise.

Calculate

(a) the value of the constant k;

(b) E(X).

37. (May 2004, T2 #14) Robert travels to work by train every weekday fromMonday to Friday. The probability that he catches the 08.00 train on Mon-day is 0.66. The probability that the catches the 08.00 train on any otherweekday is 0.75. A weekday is chosen at random.

(a) Find the probability that he catches the train on that day.

(b) Given that he catches the 08.00 train on that day, find the probabilitythat the chosen day is Monday.

38. (Nov 2004, #6) A fair six-sided die, with sides numbered 1, 1, 2, 3, 4, 5 isthrown. Find the mean and variance of the score.

39. (Nov 2004, #12) A continuous random variable X has probability densityfunction given by

f(x) =

{k(2x− x2) for 0 ≤ x ≤ 2

0 elsewhere.

(a) Find the value of k.

(b) Find P(0.25 ≤ x ≤ 0.5).

Core: Calculus

1. (May 1999, #9) If 2x2 − 3y2 = 2, find the two values ofdy

dxwhen x = 5.

2. (May 1999, #11) Differentiate y = arccos(1 − 2x2) with respect to x, andsimplify your answer.

3. (May 1999, #13) The area of the enclosed region shown in the diagram isdefined by

y ≥ x2 + 2, y ≤ ax + 2, where a > 0.

This region is rotated 360◦ about the x-axis to form a solid of revolution.Find, in terms of a, the volume of this solid of revolution.

4. (May 1999, #14) Using the substitution u = 12x + 1, or otherwise, find the

integral ∫x

√1

2x + 1 dx.

5. (May 1999, #19) When air is released from an inflated ballon it is foundthat the rate of decrease of the volume of the balloon is proportional to thevolume of the balloon. This can be represented by the differential equationdv

dt= −kv, where v is the volume, t is the time, and k is the constant of

proportionality.

(a) If the initial volume of the balloon is v0, find an expression, in terms ofk, for the volume of the balloon at time t.

(b) Find an expression, in terms of k, for the time when the volume isv0

2.

6. (May 1999, #20) A particle moves along a straight line. When it is a distance

s from a fixed point, where s > 1, the velocity v is given by by v =(3s + 2)

(2s− 1).

Find the acceleration when s = 2.

7. (Nov 2000, #3) For the function f : x 7→ x2 ln x, x > 0, find the functionf ′, the derivative of f with respect to x.

8. (Nov 2000, #5) Calculate the area bounded by the graph of y = x sin(x2)and the x-axis, between x = 0 and the smallest positive x-intercept.

9. (Nov 2000, #8) For the function f : x 7→ 12sin 2x + cos x, find the possible

values of sin x for which f ′(x) = 0.

10. (Nov 2000, #13) For what values of m is the line y = mx + 5 a tangent tothe parabola y = 4− x2?

11. (Nov 2000, #14) 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.

12. (Nov 2000, #17) Solve the differential equation xydy

dx= 1 + y2, given that

y = 0 when x = 2.

13. (May 2001, #1) Let f(t) =

(1− 1

2t53

). Find

∫f(t) dt.

14. (May 2001, #4) Find the gradient of the tangent to the curve 3x2 +4y2 = 7at the point where x = 1 and y > 0.

15. (May 2001, #8) Let f : x 7→ sin x

x, π ≤ x ≤ 3π. Find the area encolsed by

the graph of f and the x-axis.

16. (May 2001, #16) Find the general solution of the differential equationdx

dt= kx(5− x), where 0 < x < 5, and k is a constant.

17. (May 2001, #20) The function f is given by f : x 7→ e(1+sin πx), x ≥ 0.

(a) Find f ′(x).

Let xn be the value of x where the (n + 1)th maximum or minimumpoint occurs, n ∈ N (i.e., x0 is the value of x where the first maximumor minimum occirs, x1 is the value of x where the second maximum orminimum occurs, etc.).

(b) Find xn in terms of n.

18. (Nov 2001, #7) 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.

19. (Nov 2001, #10) Consider the function y = tan x− 8 sin x.

(a) Finddy

dx.

(b) Find the value of cos x for whichdy

dx= 0.

20. (Nov 2001, #13) 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 curveagain.

21. (Nov 2001, #14) A point P (x, x2) lies on the curve y = x2. Calculate theminimum distance from the point A

(2,−1

2

)to the point P .

22. (Nov 2001, #20) Find the area enclosed by the curves y =2

1 + x2and

y = ex/3, given that −3 ≤ x ≤ 3.

23. (May 2002, #2) A particle is projected along a straight line path. After t

seconds, it velocity v metres per second is given by v =1

2 + t2.

(a) Find the distance travelled in the first second.

(b) Find an expression for the acceleration at time t.

24. (May 2002, #6) (a) Use integration by parts to find

∫x2 ln x dx.

(b) Evaluate

2∫1

x2 ln x dx.

25. (May 2002, #13) The figure below shows part of the curve y = x3 − 7x2 +14x− 7. The curve crosses the x-axis at the points A, B, and C.

(a) Find the x-coordinate of A.

(b) Find the x-coordinate of B.

(c) Find the area of the shaded region.

26. (May 2002, #17) A curve has equation xy3 + 2x2y = 3. Find the equationof the tangent to this curve at the point (1, 1).

27. (May 2002, #19) A rectangle is drawn so that its lower vertices are on the

x-axis and its upper vertices are on the curve y = e−x2. The area of this

rectangle is denoted by A.

(a) Write down an expression for A in terms of x.

(b) Find the maximum value of A.

28. (May 2002, #20) The diagram below shows the graph of y1 = f(x), 0 ≤x ≤ 4.

On the axes below, sketch the graph of y2 =

x∫0

f(t) dt, marking clearly the

points of inflexion.

-

6

1 2 3 4 x

y

29. (Nov 2002, #11) A particle moves in a straight line with velocity, inmetresper second, at time t second, given by

v(t) = 6t2 − 6t, t ≥ 0.

Calculate the total distance travelled by the particle in the first two secondsof motion.

30. (Nov 2002, #13) Find

∫(θ cos θ − θ) dθ.

31. (Nov 2002, #14) Find the x-coordinate of the point of inflexion on the graphof y = xex, −3 ≤ x ≤ 1.

32. (Nov 2002, #16) Air is pumped into a spherical ball which expands at arate of 8 cm3 per second (8 cm3s−1). Find the exact rate of increase of theradius of the ball when the radius is 2 cm.

33. (Nov 2002, #17) The point B(a, b) is on the curve f(x) = x2 such that B isthe point which is closest to A(6, 0). Calculate the value of a.

34. (Nov 2002, #20) The tangent to the curve y = f(x) at the point P (x, y)meets the x-axis at Q(x− 1, 0). The curve meets the y-axis at R(0, 2). Findthe equation of the curve.

35. (May 2003, #10) A curve has equation x3y2 = 8. Find the equation of thenormal to the curve at the point (2, 1).

36. (May 2003, #16) A particle moves in a straight line. Its velocity v ms−1

after t seconds is given by v = e−√

t sin t. Find the total distance travelledin the time interval [0, 2π].

37. (May 2003, #18) Using the substitution y = 2 − x, or otherwise, find∫ (x

2− x

)dx.

38. (May 2003, #20) The diagram below shows the graph of y1 = f(x).

On the axes below, sketch the graph of y2 = |f ′(x)|.

-

6

x

y2

39. (Nov 2003, #8) Consider the function f(t) = 3 sec2 t + 5t.

(a) Find f ′(t).

(b) Find the exact values of

(i) f(π);

(ii) f ′(π).

40. (Nov 2003, #11) Calculate the area enclosed by the curves y = ln x andy = ex − e, x > 0.

41. (Nov 2003, #15) Consider the equation 2xy2 = x2y + 3.

(a) Find y when x = 1 and y < 0.

(b) Finddy

dxwhen x = 1 and y < 0.

42. (Nov 2003, #16) Let y = e3x sin(πx).

(a) Finddy

dx.

(b) Find the smallest positive value of x for whichdy

dx= 0.

43. (Nov 2003, #20) An airplane is flying at a constant speed at a constantaltitude of 3 km in a straight line that will take it directly over an observerat ground level. At a given instant the observer notes that the angle θ is1

3π radians and is increasing at

1

60radians per second. Find the speed, in

kilometres per hour, at which the airplane is moving towards the observer.

x

3 km

θ

Airplane�

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44. (May 2004, T1 #2) Given thatdy

dx= 2x− sin x and y = 2 when x = 0, find

an expression for y in terms of x.

45. (May 2004, T1 #7) The point P (1, p), where p > 0, lies on the curve2y2 − x3y = 15.

(a) Calculate the value of p.

(b) Calculate the gradient of the tangent to the curve at P .

46. (May 2004, T1 #11) (a) Find

∫ m

0

dx

x2 + 4, giving your answer in terms of

m.

(b) Given that

∫ m

0

dx

x2 + 4=

1

3, calculate the value of m.

47. (May 2004, T1 #14) Find

∫x2ex dx.

48. (May 2004, T1 #17) The function f is defined by f : x 7→ 3x2. Find the

solution of the equation f ′(x) = 2.

49. (May 2004, T1 #18) The figure shows a sector OPQ of a circle of radius r

cm and centre O, where POQ = θ.

The value of r is increasing at the rate of 2 cm per second and the value ofθ is increasing at the rate of 0.1 rad per second. Find the rate of increase of

the area of the sector when r = 3 and θ =π

4.

50. (May 2004, T1 #19) Let f(x) = x3 cos x, 0 ≤ x ≤ π

2.

(a) Find f ′(x).

(b) Find the value of x for which f(x) is a maximum.

(c) Find the x-coordinate of the point of inflexion on the graph of f(x).

51. (May 2004, T2 #2) Given thatdy

dx= ex − 2x and y = 3 when x = 0, find

an expression for y in terms of x.

52. (May 2004, T2 #8) The point P (1, p), where p > 0, lies on the curve 2x2y +3y2 = 16.

(a) Calculate the value of p.

(b) Calculate the gradient of the tangent to the curve at P .

53. (May 2004, T2 #12) (a) Find

∫ m

0

dx

2x + 3, giving your answer in terms of

m.

(b) Given that

∫ m

0

dx

2x + 3= 1, calculate the value of m.

54. (May 2004, T2 #17) The function f is defined by f : x 7→ 3x.

Find the solution of the equation f ′′(x) = 2.

55. (May 2004, T2 #18) Find

∫ln x√

xdx.

56. (May 2004, T2 #18) The following diagram shows an isosceles triangle ABCwith AB = 10 cm and AC = BC. The vertex C is moving in a directionperpendicular to (AB) with speed 2 cm per second.

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A B

C

6

Calculate the rate of increase of the angle CAB at the moment the triangleis equilateral.

57. (Nov 2004, #5) If y = ln(2x− 1), findd2y

dx2.

58. (Nov 2004, #14) Find the total area of the two regions enclosed by the curvey = x3 − 3x2 − 9x + 27 and the line y = x + 3.

59. (Nov 2004, #16) Find the equation of the normal to the curve x3+y3−9xy =0 at the point (2, 4).

60. (Nov 2004, #17) Using the substitution 2x = sin θ, or otherwise, find∫ (√1− 4x2

)dx.

61. (Nov 2004, #18) A closed cylindrical can has a volume of 500 cm3. Theheight of the can is h cm and the radius of the base is r cm.

(a) Find an expresson 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 ofr.