IB HL Mathematics Problems, Organized by Subject Core: Algebra Algebra
1. (May 1999, #3) 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.
2. (May 1999, #5) Let z = x + yi. Find the values of x 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 find the values of a, b Z∗ .
2
6 11
≡ 1 + bx + 10x + · · · + a x
,
∈
5. (Nov 2000, 2000, #7) Find the sum of the positiv p ositivee terms of the arithmetic sequence sequence 85, 78, 71, . . . .
6. (Nov 2000, #12) The coefficient of x in the expansion of x + Find the possible values of a.
1
ax2
7
is 37 .
7. (Nov (Nov 2000, 2000, #15) #15) The sum of an infinite infinite geometri geometricc sequenc sequencee is 1312 , and the sum of the first three terms is 13. Find the first term.
8. (Nov 2000, #18) If z is is a complex number and z + + 16 = 4 z + + 1 , find the value of z .
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| | |
|
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9. (May 2001, #7) The nth term, un , of a geometric sequence is given by un = 3(4)n+1 , n Z+ .
∈
(a) Find the common ratio r.
(b) Hence, or otherwise, find S n, the sum of the first n terms of this sequence.
10. (May 2001, #14) Given that z = (b + i)2, where b is real and positive, find the exact value of b when arg z = = 60◦ .
11. (Nov 2001, #2) The complex number z satisfies i(z + + 2) = 1 2z , where 1. Write z in in the form z = = a + bi, where a and b are real numbers. i =
−
√ −
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 which is proportional proportional to the amount amount of material material present present in the sample. sample. Find Find the half-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 S n, the sum of the first n terms.
···
(b) Find the value of n for which S n = 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 (Nov 2002, 2002, #3) Find Find the coeffici coefficien entt of x3 in the binomial expansion of 8 1 x . 1 2
−
50
17. (Nov 2002, #6) Find a
∈ Q.
ln(2r ), giving the answer in the form a ln 2, where where
r=1
18. (May 2003, #1) A geometri geometricc sequenc sequencee has all positive positive terms. terms. The sum of the 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 satisfies the equation
√ z =
2
− i + 1 − 4i. Express z in in the form x + iy where x, y ∈ Z. 1
20. (May 2003, #12) Find the exact value of x satisfying the equation (3x )(42x+1 ) = 6x+2 . Give your answer in the form
ln a where a, b ln b
∈ Z.
21. (Nov 2003, #5) Consider the equation 2( p + iq ) = q a nd q are are both real numbers. Find p and q . p and
− ip − 2(1 − i), where
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 log16
√ 100 − x 3
2
−
1 = . 2
3 24. (Nov 2003, #19) Solve 2 5x+1 = 1 + x , giving the answer in the form 5 a + log5 b, where a, b Z.
∈
25. (May (May 2004, T1 #4) A geometri geometricc series series has a negativ negativee common common ratio. The sum of the first two terms is 6. The sum to infinity is 8. Find the common ratio 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 geometri geometricc progress progression ion.. Find Find the value value of a and of b given that a = 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 constants a, b.
6 28. (Nov 2004, #3) The sum of the first n terms of a series is given by S n = 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 in terms of n.
29. (Nov 2004, #4) Given that (a b, where a, b Z.
∈
− i)(2 − bi) = 7 − i, find the value of a and of
30. (Nov 2004, #8) Find the expansion of (2 + x)5 , giving your answer in ascending 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 z 3 your answers in the form z = r (cos θ + i sin θ).
∈
− 8i = 0, giving
Core: Functions and Equations
1. (May 1999, #1) When the function f (x) = 6x4 + 11 x3 22x2 + ax + 6 is divided 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). The x-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 y 2 = f (x k ), where 0 < k < n m and indicate the coordinates of the points of intersection of y2 with the xaxis.
−
−
3. (May 1999, #12) Given f (x) = x 2 + x(2 k ) + k 2, find the range of values of k for which f (x) > 0 for all real values of x.
−
4. (Nov 2000, #2) Given functions f : x function (f g )−1 .
◦
3
→ x + 1 and g : x → x , find the
5. (Nov 2000, #10) Find the real number k for which 1 + ki, (i = zero of the polynomial z 2 + kz + 5.
√ −1), is a
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 y 2 = f (x) for
−2 ≤ x ≤ 2.
7. (May 2001, #5) Let f : x
→ − 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 + 2 i) is a factor of 2z 3 two factors.
2
− 3z + 8z − 12. Find the other
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 equation s = 40t + 0.5at2 , where a is a constant. If the ball reaches its maximum height when t = 25, find the value of a.
10. (May 2001, #18) The equation kx2 3x + (k + 2) = 0 has two distinct real roots. 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 of intercepts and asymptotes occur.
y1 . Indicate clearly where the xy2
12. (Nov 2001, #3) The polynomial f (x) = x3 + 3 x2 + ax + b leaves the same remainder when divided by (x 2) as when divided by (x + 1). Find the value of a.
−
2x + 1 , x 13. (Nov 2001, #5) The function f : x x 1 inverse function f −1, clearly stating its domain.
→ −
∈ R, x = 1.
Find the
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 2x 1 f (x) = . 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
x
x
− 2)(e − 3) ≤ 2e .
17. (Nov 2002, #1) When the polynomial x4 + ax + 3 is divided by (x remainder is 8. Find the value of a.
− 1), the
18. (Nov 2002, #4) Find the equations of all the asymptotes of the graph of x2 5x 4 . y = 2 5x + 4 x
− − −
√ −
19. (Nov 2002, #7) The functions f (x) and g (x) are given by f (x) = x 2 and g (x) = x 2 + x. The function (f g )(x) is defined for x R, except for the 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 + x3 < 0.
21. (May 2003, #4) The polynomial x 3 + 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.
−
−
22. (May 2003, #7) The function f is given by f (x) = 2 Write down
2
x
−x −e .
(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 Find an expression for f −1 (x).
≤
x2 1 0 by f (x) = 2 . x +1
−
25. (Nov 2003, #6) The diagram shows the graph of f (x). 1 (a) On the same diagram, sketch the graph of , indicating clearly any f (x) asymptotes.
(b) On the diagram write down the coordinates of the local maximum point, 1 the local minimum point, the x-intercepts and the y -intercept of . f (x)
26. (Nov 2003, #13) Consider the equation (1 + 2k )x2 10x + k 2 = 0, k Find the set of values of k for which the equation has real roots.
−
−
∈ R.
x + 4 x−2 , x = −1, and g (x) = , x = 4. x + 1 x−4 Find the set of values of x such that f (x) ≤ g (x).
27. (Nov 2003, #17) Let f (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 of b.
29. (May 2004, T1 #3) For 0 ≤ x ≤ 6, find the coordinates of the points of intersection of the curves y = x 2 cos x and
x + 2 y = 1.
30. (May 2004, T1 #9) The function f is defined on the domain [−1, 0] by f : x →
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 ≤ 3. x − 12
32. (May 2004, T2 #1) The polynomial x 2 − 4x +3 is a factor of x 3 + (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 of intersections of the curves y = x sin x
and x + 3 y = 1.
k
34. (May 2004, T2 #10) Let f (x) =
x
− k , x = k, k > 0.
(a) On the diagram below, sketch the graph of f . Label clearly any points of intersection with the axes, and any asymptotes. y
x k
1 (b) On the diagram below, sketch the graph of . Label clearly any points f
of intersection with the axes. y
x k
35. (May 2004, T2 #10) The function f is defined by f : x x 3 . 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) = x 3 (x + 2) is a factor of f (x).
2
− 2x − 5x + k . Find the value of k if
38. (Nov 2004, #7) (a) Find the largest set S of values of x such that the 1 function f (x) = takes real values. 3 x2
√ −
(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 ].
√
− π2 < x < π2 .
2. (May 2001, #2) Solve 2 sin x = tan x, where
3. (Nov 2001, #16) Let θ be the angle between the unit vectors a and b , where 0 < θ < π . Express a b in terms of sin 21 θ.
| − |
4. (May 2002, #10) The angle θ satisfies the equation tan θ +cot θ = 3, where θ 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) = 4cos 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 its maximum value.
6. (Nov 2002, #12) Triangle ABC has AB = 8 cm, BC = 6 cm, and B AC = 20◦ . Find the smallest possible area of ABC .
7. (Nov 2002, #2) Find all the values of θ in the interval [0, π ] which satisfy the 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 ABC .
−
−
3 10. (Nov 2003, #14) Let f (x) = sin arcsin arccos , for 4 5 (a) On the grid below, sketch the graph of f (x). x
−
−4 ≤ x ≤ 4.
(b) On the sketch, clearly indicate the coordinates of the x-intercept, the y -intercept, the minimum point and the endpoints of the curve of f (x).
(c) Solve f (x) =
− 12 .
11. (May 2004, T1 #20) The following three-dimensional diagram shows the four 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◦ , AC D = 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 ), (A1 B2 ), (A2 B3) are perpendicular to OB . A1B1 , A2 B2 are all arcs of circles with centre O .
Calculate the sum to infinity of the arc length AB + A1 B1 + A2 B2 + A3 B3 +
·· · .
13. (Nov 2004, #9) The diagram below shows a circle centre O and radius OA = 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 (a) How many solutions are there to this equation?
≤ x ≤ 2π.
(b) Find the solution closest to 2π , giving your answer to four decimal places.
Core: Matrices and Geometry
1. (May 1999, #6) Find the value of a for which the following system of equations does not a have a unique solution.
4x y + 2 z = 1 2x + 3 y = 6 7 x 2y + az = 2
−
−
−
2. (Nov 2000, #1) Find the values of the real number k for which the determi3 k 4 nant of the matrix is equal to zero. 2 k + 1
− −
3. (Nov 2000, #9) (a) Describe the transformation of the plane whose matrix is √ M =
1 2
−
√
3 2
1 2
3 2
.
(b) Find the smallest positive integer n for which M n =
1 0 . 0 1
4. (May 2001, #3) Give a full geometric description of the transformation represented by the matrix
− 4 3
3 5
3 5
4 5
.
5. (May 2001, #9) Find the equation of the line of intersection of the two planes 4x + y + z = 2 and 3x y + 2 z = 1
−
−
−
−
6. (May 2001, #12) Find an equation of the plane containing the two lines x
− 1 = 1 −2 y = z − 2
7. (Nov 2001, #6) If A =
x 4
4 2
and
and B =
y , given that AB = BA .
8. (Nov 2001, #9) The matrix of k .
x + 1
−
1 1 3
3
2
− y = z + 2 . 3
5
2 y , find the values of x and 8 4
−2 −3 −k −13 5
=
k
is singular. Find the values
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 1 k 3 4
− k
1 2
.
Find the values of k for which A is singular.
11. (May 2002, #18) A transformation T of the plane is represented by the matrix T =
2 3 1 2
.
(a) T transforms the point P to the point (8, 5). Find the coordinates of P .
(b) Find the coordinates of all points which are transformed to themselves under T .
12. (Nov 2002, #10) Find an equation for the line of intersection of the following two planes.
x + 2 y
− 3z 2x + 3 y − 5z
= 2 = 3.
13. (Nov 2002, #19) The transformation M represents a reflection in the line π y = x 3. The transformation R represents a rotation through radians 6 anticlockwise about the origin. Give a full geometric description of the single transformation which is equivalent to M followed by R.
√
3 2 14. (May 2003, #5) Given that A = and I = 3 4 values of λ for which (A λI ) is a singular matrix.
−
−
−
1 0 , find the 0 1
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 3 matrices. Given that A−1 XB = C , express X in terms of the other matrices.
×
17. (Nov 2003, #7) Find the angle between the plane 3x the z -axis. Give your answer to the nearest degree.
− 2y + 4z = 12 and
18. (May 2004, T1 #5) The composite transformation T is defined by a clockwise rotation 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 by the matrices M =
−
0 1 1 0
and S =
−
1 0 0 1
.
Give a full geometric description of the single transformation represented by the matrix SM S .
20. (Nov 2004, #2) Given that the matrix A = the value of p.
1 2 1
−1 p −2
2 3 5
is singular, find
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 contains the 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 with x 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 by the parametric equations x = 2λ + 4, y = λ 2, z = 3λ + 2, intersects the plane with equation 2x + 3y z = 2.
− −
−
2. (May 1999, #10) (a) Find a vector perpendicular to the two vectors:
−→ OP −→ OQ −→
− 3 j + 2 k −2 i+ j − k
= i =
−→
(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 OP Q.
3. (Nov 2000, #11) Let α be the angle between the vectors a and b, where π a = (cos θ )i + (sin θ) j, b = (sin θ )i + (cos θ) j and 0 < θ < . Express α in 4 terms of θ .
4. (Nov 2001, #15) Point A(3, 0, 2) lies on the line r = 3i 2k + λ(2i 2 j + k), where λ is a real parameter. Find the coordinates of one point which is 6 units from A, and on the line.
−
−
−
5. (May 2002, #5) Find the angle between the vectors v = i + j + 2k and w = 2i + 3 j + 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 + 2 j + 3k) L2 : r = i + 4 j + 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 3 x2 + x + 1 is 1 translated to its image, g (x), by the vector . Write g (x) in the form 1 g (x) = ax 3 + 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 + 2 j c = 2i 3 j + 4k, find (a b) c.
−
× ·
10. (May 2004, T1 #10) The line x
|
− k, b = −3i + 2 j + 2k, and
− 1 = y +2 1 = z 3 and the plane r · (i + 2 j −
k) = 1 interesct at the point P . Find the coordinates of P .
11. (May 2004, T2 #15) Given that a = (i + 2 j + k ) (a) find a;
× (−2i + 3k),
(b) find the vector projection of a onto the vector
−2 j + 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 ball is chosen. Find the probability of choosing one green ball and one blue ball in any order.
2. (May 1999, #8) In a biliingual school there is a class of 21 pupils. In this class, 15 of the pupils speak Spanish as their first language and 12 of these 15 pupils are Argentine. The other 6 pupils in the class speak English as their 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 player pays 10 counters to roll the die. The table below shows the possible scores on the die, the probability of each score and the number of counter the player receives in return for each score. Score
1
2
3
4
Probability
1 2
1 5
1 5
1 10
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 9 counter per roll.
4. (May 1999, #18) A factory has a machine designed to produce 1 kg bags of sugar. 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 the standard deviation if 1.7% of the bags weigh below 1 kg. Give your answer correct to the nearest 0.1 gram.
5. (Nov 2000, #4) The box-and-whisker plots shown represent the heights of female students and the heights of male students at a certain school. Females
Males
150
160
170
180
190
200
210
Height (cm)
(a) What percentage of female students are shorter than any male students? (b) What percentage of male students are shorter than some female students? (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 between two students so that each student receives at least one coin?
8. (May 2001, #6) A machine produces packets of sugar. The weights in grams of thirty packets chosen at random are shown below. Weight (g)
29.6
29.7
29.8
Frequency
2
3
4
29.9 30.0 5
30.1
30.2
30.3
5
3
1
7
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.
2 2 9. (May 2001, #11) Given that P(X ) = , P(Y X ) = and P(Y X ) = 41 , 3 5 find (a) P(Y );
|
(b) P(X
|
∪ Y ).
10. (May 2001, #13) Z is the standardized normal random variable with mean 0 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 of trials is 5 and the probability of success of each trial is p. Find the values of p if P(X = 4) = 0.12.
12. (Nov 2001, #1) A coin is biased so that when it is tossed the probability of obtaining heads is 32 . 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 density function f (x) =
Find E(X ).
4 , for 0 x 1 , π (1 + x2 ) 0, elsewhere.
≤ ≤
14. (Nov 2001, #17) How many four-digit numbers are there which contain at least one digit 3?
15. (Nov 2001, #18) The probability that a man leaves his umbrella in any shop he visits is 31 . 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 his umbrella in the second shop?
16. (May 2002, #7) The probability that it rains during a summer’s day in a certain town is 0.2. In this town, the probability that the daily maximum temperature 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 particular summer’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 that he 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 his third throw.
18. (May 2002, #11) The weights of a certain species of bird are normally distributed with mean 0.8 kg and standard deviation 0.12 kg. Find the probability that the weight of a randomly chosen bird of the species lies between 0.74 kg and 0.95 kg.
19. (May 2002, #14) The 80 applicants for a Sports Science course were required to run 800 metres and their times were recorded. The results were used to produce 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 thousand positive 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 of the 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 continuous random variable X is defined by f (x) =
1 x(4 4
0
2
− x ),
0 x 2 , otherwise.
≤ ≤
Calculate the median value of X .
23. (May 2003, #6) When a boy plays a game at a fair, the probability that he wins a prize is 0.25. He plays the game 10 times. Let X denote the total number 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 and P(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 P(X
≤ 10) ≤ 12)
Find E(X ).
= 0.670 = 0.937.
26. (May 2003, #19) A teacher drives to school. She records the time taken on each of 20 randomly chosen days. She finds that 20
i=1
20
xi = 626 and
x2i = 19780.8,
i=1
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 amount of time 250 student spend eating lunch. (a) Estimate the number of students who spend between 20 and 40 minutes eating lunch.
(b) If 20% of the students spend more than x minutes eating lunch, estimate the value of x.
28. (Nov 2003, #4) A continuous random variable, X , has probability density function f (x) = s i n x, 0
≤ x ≤ π2 .
Find the median of X .
29. (Nov 2003, #12) On a television channel the news is shown at the same time each day. The probability that Alice watches the news on a given day is 0.4. Calculate the probability that on five consecutive days, she watches the 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 ways the committee may be chosen.
31. (May 2004, T1 #6) The weights of adult males of a type of dog may be assumed to be normally distributed with mean 25 kg and a standard deviation 3 kg. Given that 30% of the weights lie between 25 kg and x kg, where x > 25, find the value of x.
32. (May 2004, T1 #12) Marian shoots ten arrows at a target. Each arrow has probability 0.4 of hitting the target, independently of all other arrows. Let X 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 gold coins, Drawer 2 contains two gold coins and one silver coin, and Drawer 3 contains one gold coin and two silver coins. A drawer is chosen at random and 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 3 was chosen.
34. (May 2004, T1 #15) The heights of 60 children entering a school were measured. The following cumujlative frequency graph illustrates the data obtained.
Estimate (a) the median height; (b) the mean height.
35. (May 2004, T2 #7) The following diagram shows the probability density function for the random variable X , which is normally distributed with mean 250 and standard deviation 50.
Find the probability of the shaded region.
36. (May 2004, T2 #13) The discrete random variable X has the following probability distribution: P(X = x ) = Calculate (a) the value of the constant k ;
k , x = 1, 2, 3, 4 x 0, otherwise.
(b) E(X ).
37. (May 2004, T2 #14) Robert travels to work by train every weekday from Monday to Friday. The probability that he catches the 08.00 train on Monday is 0.66. The probability that the catches the 08.00 train on any other weekday 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 probability that the chosen day is Monday.
38. (Nov 2004, #6) A fair six-sided die, with sides numbered 1, 1, 2, 3, 4, 5 is thrown. Find the mean and variance of the score.
39. (Nov 2004, #12) A continuous random variable X has probability density function given by f (x) =
k (2x
0
2
−x )
for 0 x 2 elsewhere.
≤ ≤
(a) Find the value of k . (b) Find P(0.25
≤ x ≤ 0.5).
Core: Calculus
1. (May 1999, #9) If 2x2
− 3y
2
= 2, find the two values of
2. (May 1999, #11) Differentiate y = arccos(1 simplify your answer.
2
dy when x = 5. dx
− 2x ) with respect to x, and
3. (May 1999, #13) The area of the enclosed region shown in the diagram is defined by
≥ x
y
2
+ 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 = 21 x + 1, or otherwise, find the integral
x
1 x + 1 dx. 2
5. (May 1999, #19) When air is released from an inflated ballon it is found that the rate of decrease of the volume of the balloon is proportional to the volume of the balloon. This can be represented by the differential equation dv = 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 v 0 , find an expression, in terms of k , for the volume of the balloon at time t. (b) Find an expression, in terms of k , for the time when the volume is
v0
2
.
6. (May 1999, #20) A particle moves along a straight line. When it is a distance (3s + 2) . s from a fixed point, where s > 1, the velocity v is given by by v = (2s 1) Find the acceleration when s = 2.
−
7. (Nov 2000, #3) For the function f : x f , the derivative of f with respect to x.
2
→ x
ln x, x > 0, find the function
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 values of sin x for which f (x) = 0.
→
1 sin 2x + 2
cos x, find the possible
10. (Nov 2000, #13) For what values of m is the line y = mx + 5 a tangent to the parabola y = 4 x2 ?
−
11. (Nov 2000, #14) The tangent to the curve y 2 = x3 at the point P (1, 1) meets the x-axis at Q and the y -axis at R. Find the ratio P Q : QR .
dy
12. (Nov 2000, #17) Solve the differential equation xy = 1 + y 2 , given that dx y = 0 when x = 2.
−
13. (May 2001, #1) Let f (t) = 1
1
2t
5
. Find
f (t) dt.
3
14. (May 2001, #4) Find the gradient of the tangent to the curve 3x2 + 4y 2 = 7 at the point where x = 1 and y > 0.
sin x , π 15. (May 2001, #8) Let f : x x the graph of f and the x-axis.
→
≤ x ≤ 3π. Find the area encolsed by
16. (May 2001, #16) Find the general solution of the differential equation dx = kx(5 dt
− x), where 0 < x < 5, and k is a constant.
17. (May 2001, #20) The function f is given by f : x (a) Find f (x).
→ e
(1+sin πx)
, x
≥ 0.
Let xn be the value of x where the (n + 1)th maximum or minimum point occurs, n N (i.e., x0 is the value of x where the first maximum or minimum occirs, x1 is the value of x where the second maximum or minimum 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 (a) Find
dy . dx
(b) Find the value of cos x for which
− 8sin x.
dy = 0. dx
20. (Nov 2001, #13) Consider the tangent to the curve y = x 3 + 4x2 + x (a) find the equation of this tangent at the point where x = 1.
−
− 6.
(b) Find the coordinates of the point where this tangent meets the curve again.
21. (Nov 2001, #14) A point P (x, x2) lies on the curve y = x2. Calculate the minimum distance from the point A 2, 12 to the point P .
−
22. (Nov 2001, #20) Find the area enclosed by the curves y = y = e x/3 , given that
−3 ≤ x ≤ 3.
2 and 1 + x2
23. (May 2002, #2) A particle is projected along a straight line path. After t 1 seconds, it velocity v metres per second is given by v = . 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
2
(b) Evaluate
x2 ln x dx.
x2 ln x dx.
1
25. (May 2002, #13) The figure below shows part of the curve y = x3 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.
− 7x
2
+
26. (May 2002, #17) A curve has equation xy 3 + 2 x2y = 3. Find the equation of 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−x . The area of this rectangle is denoted by A. (a) Write down an expression for A in terms of x. 2
(b) Find the maximum value of A.
28. (May 2002, #20) The diagram below shows the graph of y1 = f (x), 0 x 4.
≤
≤
x
On the axes below, sketch the graph of y2 =
f (t) dt, marking clearly the
0
points of inflexion. y
1
2
3
4
x
29. (Nov 2002, #11) A particle moves in a straight line with velocity, inmetres per 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 seconds of 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 graph of y = xe x , 3 x 1 .
− ≤ ≤
32. (Nov 2002, #16) Air is pumped into a spherical ball which expands at a rate of 8 cm3 per second (8 cm3 s−1 ). Find the exact rate of increase of the radius of the ball when the radius is 2 cm.
33. (Nov 2002, #17) The point B (a, b) is on the curve f (x) = x 2 such that B is the 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). Find the equation of the curve.
−
35. (May 2003, #10) A curve has equation x3 y 2 = 8. Find the equation of the normal to the curve at the point (2, 1).
−1 36. (May 2003, #16) A particle moves √ in a straight line. Its velocity v ms after t seconds is given by v = e− t sin t. Find the total distance travelled in the time interval [0, 2π].
37. (May 2003, #18) Using the substitution y = 2
x
2
−x
− x,
dx.
or otherwise, find
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) .
|
|
y2
x
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 and y = e x e, x > 0.
−
41. (Nov 2003, #15) Consider the equation 2xy 2 = x 2 y + 3. (a) Find y when x = 1 and y < 0.
(b) Find
dy when x = 1 and y < 0. dx
42. (Nov 2003, #16) Let y = e 3x sin(πx ). (a) Find
dy . dx
(b) Find the smallest positive value of x for which
dy = 0. dx
43. (Nov 2003, #20) An airplane is flying at a constant speed at a constant altitude of 3 km in a straight line that will take it directly over an observer at ground level. At a given instant the observer notes that the angle θ is 1 1 radians per second. Find the speed, in π radians and is increasing at 3 60 kilometres per hour, at which the airplane is moving towards the observer. x 3 km θ
dy
44. (May 2004, T1 #2) Given that = 2x dx an expression for y in terms of x.
Airplane
− sin x and y = 2 when x = 0, find
45. (May 2004, T1 #7) The point P (1, p), where p > 0, lies on the curve 2y 2 x3 y = 15. (a) Calculate the value of p.
−
(b) Calculate the gradient of the tangent to the curve at P .
m
46. (May 2004, T1 #11) (a) Find m.
m
(b) Given that
0
0
dx x2 + 4
, giving your answer in terms of
1 = , calculate the value of m. 3 x2 + 4 dx
47. (May 2004, T1 #14) Find
x2 ex dx.
48. (May 2004, T1 #17) The function f is defined by f : x solution of the equation f (x) = 2.
→ 3
x2
. Find the
49. (May 2004, T1 #18) The figure shows a sector OP Q of a circle of radius r cm and centre O , where P OQ = θ .
The value of r is increasing at the rate of 2 cm per second and the value of θ is increasing at the rate of 0.1 rad per second. Find the rate of increase of π the area of the sector when r = 3 and θ = . 4
50. (May 2004, T1 #19) Let f (x) = x 3 cos x, 0 (a) Find f (x).
≤ x ≤ π2 .
(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).
dy
51. (May 2004, T2 #2) Given that = ex dx an expression for y in terms of x.
− 2x and y = 3 when x = 0, find
52. (May 2004, T2 #8) The point P (1, p), where p > 0, lies on the curve 2x2y + 3y 2 = 16. (a) Calculate the value of p. (b) Calculate the gradient of the tangent to the curve at P .
m
53. (May 2004, T2 #12) (a) Find m.
m
(b) Given that
0
0
dx , giving your answer in terms of 2x + 3
dx = 1, calculate the value of m. 2x + 3
54. (May 2004, T2 #17) The function f is defined by f : x Find the solution of the equation f (x) = 2.
55. (May 2004, T2 #18) Find
√
ln x x
x
→ 3 .
dx.
56. (May 2004, T2 #18) The following diagram shows an isosceles triangle ABC with AB = 10 cm and AC = BC . The vertex C is moving in a direction perpendicular to (AB ) with speed 2 cm per second.
C
B A Calculate the rate of increase of the angle C AB at the moment the triangle is equilateral.
