Questions and Answers of Cambridge IGCSE And O Level Additional Mathematics 1st Edition

State the transformations required, in the correct order, to obtain this graphs from the graph of y = tanx. y. 3-- 22.5° 45° 90° 135° 180° 225° 270° 315° 360oX
In the diagram, ABED is a trapezium with right angles at E and D, and CED is a straight line. The lengths of AB and BC are (2√3)d and 2d respectively, and angles BAD and CBE are 60° and 30°
In the triangle LMN, angle M = π/2 and cot N = 1.a. Find the angles L and N.b. Find sec L, cosec L and tan L.c. Show that 1 + tan2 L = sec2 L.
a. For what values of α are sin α, cos α and tan α all positive given that 0° ≤ α ≤ 360°?b. Are there any values of α for which sin α, cos α and tan α are all negative? Explain your
State the transformations required, in the correct order, to obtain this graph from the graph of y = cos x. 37 2π X 2 2 -1- -2- -3- -4- -54
Prove the identity cot θ + tan θ ≡ sec θ cosec θ.
In the diagram, ABC is a triangle in which AB = 6 cm, BC = 4 cm and angle ABC = 2π/3. The line CX is perpendicular to the line ABX.a. Work out the exact length of BX.b. Show that angle CAB =
Malini is 1.5 m tall. At 8 o’clock one evening, her shadow is 6 m long.Given that the angle of elevation of the sun at that moment is α:a. Show that cot ‑α = 4,b. Find the value of α.
Solve the following equations for 0° ≤ θ ≤ 360°.a. cos (θ − 20°) = 1/2b. tan (θ + 10°) = √3/3c. sin (θ + 80°) = √2/2d. tan 2θ = √3e. sin (1/2 ) = 1/2f. cos
This is the graph of y = sec θ.a. Solve the equation secθ = 1 for -2π ≤ θ ≤ 2π.b. What happens if you try to solve sec θ = 0.5? y y = cosx 2n x y = secx
Solve the equation tan θ = sec θ for − 360° ≤ θ ≤ 360°.
a. For what values of α are sin α, cos α and tan α all positive? Give your answers in both degrees and radians.b. Are there any values of α for which sin α, cos α and tan α are all negative?
Solve the following equations for −2π ≤ x ≤ 2π.a. 10 sin x = 1 b. 2 cos x – 1 = 0 c. tan x + 2 = 0d. 5 sin x + 2 = 0 e. cos² x = 1 − sin x
a. Show that 12sin2 x + cos x − 1 = 11 + cos x − 12cos2 x.b. Solve 12sin2 x + cos x − 1 = 0 for 0° ≤ x ≤ 360°.
a. Show that sin2θ + 1 – sinθ − cos2θ = 2sin2θ − sinθ.b. Solve sin2θ + 1 − sinθ − cos2θ = 0 for 0 ≤ θ ≤ 2π.
Calculate:a. 7! b. 9!/7! c. 4! × 6!/7! × 2!
a. Find the values of i. 7P3 ii. 9P4 iii. 10P8b. Find the values of i. 7C3 ii. 9C4 iii. 10C8c. Show that, for the values of n and r in parts a and b, nCr = nPr/r!.
Simplify:a. n!/(n + 1)!b. (n – 2)!/(n -3)!
There are 15 competitors in a camel race. How many ways are there of guessing the first three finishers?
Simplify:a. (n + 2)!/n!b. (n + 1)!/(n – 1)!
A group of 6 computer programmers is to be chosen to work the night shift from a set of 14 programmers. In how many ways can the programmers be chosen if the 6 chosen must include the shift-leader
Write in factorial notation:a. 9 × 8 × 7/6 × 5 × 4b. 14 × 15/5 × 4 × 3 × 2c. (n + 2)(n + 1)n/4 × 3 × 2
Zaid decides to form a band. He needs a bass player, a guitarist, a keyboard player and a drummer. He invites applications and gets 6 bass players, 8 guitarists, 4 keyboard players and 3 drummers.
A touring party of cricket players is made up of 6 players from each of India, Pakistan and Sri Lanka and 3 from Bangladesh.a. How many different selections of 11 players can be made for a team?b. In
Factorise: a. 6! + 7! b. n! + (n – 1)!
Write the number 42 using factorials only.
A committee of four is to be selected from ten candidates, five men and five women.a. In how many distinct ways can the committee be chosen?b. Assuming that each candidate is equally likely to be
How many different four-letter arrangements can be formed from the letters P, Q, R and S if letters cannot be repeated?
A committee of four is to be selected from four boys and six girls. The members are selected at random.a. How many different selections are possible?b. What is the probability that the committee will
How many different ways can seven books be arranged in a row on a shelf?
A factory advertises six positions. Nine men and seven women apply.a. How many different selections are possible?b. How many of these include equal numbers of men and women?c. How many of the
There are five drivers in a motoring rally.How many different ways are there for the five drivers to finish?
A small business has 14 staff; 6 men and 8 women. The business is struggling and needs to make four members of staff redundant.a. How many different selections are possible if the four staff are
There are five runners in a 60-metre hurdles race, one from each of the nations Japan, South Korea, Cambodia, Malaysia and Thailand.How many different finishing orders are there?
A football team consists of a goalkeeper, two defense players, four midfield players and four forwards. Three players are chosen to collect a medal at the closing ceremony of a competition.How many
Toben listens to 15 songs from a playlist. If he selects ‘shuffle’ so the songs are played in a random order, in how many different orders could the songs be played?
Find how many different numbers can be made by arranging all nine digits of the number 335 688 999 if:i. There are no restrictionsii. The number made is a multiple of 5.
How many different arrangements are there of the letters in each word?a. ASK b. QUESTION c. SINGAPOREd. GOVERN e. VIETNAM f. MAJORITY
Nimish is going to install 5 new game apps on her phone. She has shortlisted 2 word games, 5 quizzes and 16 saga games. Nimish wants to have at least one of each type of game. How many different
How many arrangements of the letters in the word ARGUMENT are there if:a. There are no restrictions on the order of the lettersb. The first letter is an Ac. The letters A and R must be next to each
A MPV has seven passenger seats – one in the front, and three in each of the other two rows.a. In how many ways can all 8 seats be filled from a party of 12 people, assuming that they can all
Iram has 12 different DVDs of which 7 are films, 3 are music videos and 2 are documentaries.a. How many different arrangements of all 12 DVDs on a shelf are possible if the music videos are all next
A string orchestra consists of 15 violins, 8 violas, 7 cellos and 4 double basses. A chamber orchestra consisting of 8 violins, 4 violas, 2 cellos and 2 double basses is to be chosen from the string
An office car park has 12 parking spaces in a row. There are 9 cars to be parked.a. How many different arrangements are there for parking the 9 cars and leaving 3 empty spaces?b. How many different
Write out the following binomial expressions:a. (1 + x)4 b. (1 + 2x)4 c. (1 + 3x)4
Are the following sequences arithmetic? If so, state the common difference and the seventh term.a. 28, 30, 32, 34, …b. 1, 1, 2, 3, 5, 8, …c. 3, 9, 27, 81, …d. 5, 9, 13, 17, …e. 12, 8, 4, 0,
Are the following sequences geometric? If so, state the common ratio and calculate the seventh term.a. 3, 6, 12, 24, …b. 3, 6, 9, 12, …c. 10, −10, 10, −10, 10, …d. 1, 1, 1, 1, 1, 1, …e.
Write out the following binomial expressions:a. (2 + x)4 b. (3 + x)4 c. (4 + x)4
The first term of an arithmetic sequence is −7 and the common difference is 4.a. Find the eighth term of the sequence.b. The last term of the sequence is 65. How many terms are there in the
A geometric sequence has first term 5 and common ratio 2. The sequence has seven terms.a. Find the last term.b. Find the sum of the terms in the sequence.
Write out the following binomial expressions:a. (x + y)4b. (x + 2y)4 c. (x + 3y)4
The first term of an arithmetic sequence is 10, the seventh term is 46 and the last term is 100.a. Find the common difference.b. Find how many terms there are in the sequence.
The first term of a geometric sequence of positive terms is 3 and the fifth term is 768.a. Find the common ratio of the sequence.b. Find the eighth term of the sequence.
Use a non-calculator method to calculate the following binomial coefficients. Check your answers using your calculator’s shortest method. 5. 7 b 3 4 () 7 d. 13 3
There are 30 terms in an arithmetic progression. The first term is −4 and the last term is 141.a. Find the common difference.b. Find the sum of the terms in the progression.
Find the coefficients of the term shown for each expansion:a. x4 in (1 + x)6 b. x5 in (1 + x)7 c. x6 in (1 + x)8
A geometric sequence has first term 1/16 and common ratio 4.a. Find the fifth term.b. Which is the first term of the sequence that exceeds 1000?
The kth term of an arithmetic progression is given byuk = 12 + 4k.a. Write down the first three terms of the progression.b. Calculate the sum of the first 12 terms of this progression.
a. Find how many terms there are in the following geometric sequence: 7, 14, …, 3584.b. Find the sum of the terms in this sequence.
Find the first three terms, in ascending powers of x, in the expansion of (3 + k)5.
Below is an arithmetic progression.118 + 112 + … + 34a. How many terms are there in the progression?b. What is the sum of the terms in the progression?
a. Find how many terms there are in the following geometric sequence: 100, 50, …, 0.390 625.b. Find the sum of the terms in this sequence.
Find the first three terms, in descending powers of x, in the expansion of (3x – 3/x)6.
The fifth term of an arithmetic progression is 32 and the tenth term is 62.a. Find the first term and the common difference.b. The sum of all the terms in this progression is 350. How many terms are
The ninth term of an arithmetic progression is three times the second term, and the first term is 5. The sequence has 20 terms.a. Find the common difference.b. Find the sum of all the terms in the
The first three terms of an infinite geometric progression are 8, 4 and 2.a. State the common ratio of this progression.b. Calculate the sum to infinity of its terms.
Find the coefficients of x3 and x4 for each of the following:a. (1 + x)(1 − x)6b. (1 − x)(1 + x)6
a. Find the sum of all the odd numbers between 150 and 250.b. Find the sum of all the even numbers from 150 to 250 inclusive.c. Find the sum of the terms of the arithmetic sequence with first term
The first three terms of an infinite geometric progression are 0.8, 0.08 and 0.008.a. Write down the common ratio for this progression.b. Find, as a fraction, the sum to infinity of the terms of this
Write down the first four terms, in ascending powers of x, of the following binomial expressions:a. (1 − 2x)6b. (2 − 3x)6c. (3 − 4x)6
The first term of an arithmetic progression is 9000 and the tenth term is 3600.a. Find the sum of the first 20 terms of the progression.b. After how many terms does the sum of the progression become
The first three terms of a geometric sequence are 100, 70 and 49.a. Write down the common ratio of the sequence.b. Which is the position of the first term in the sequence that has a value less than
Find the first four terms, in descending powers of x, of the following binomial expressions:a. (x2 + 1/x)5b. (x2 – 1/x)c. (x3 + 1/x)5d. (x3 – 1/x)5
An arithmetic progression has first term −2 and common difference 7. a. Write down a formula for the nth term of the progression. Which term of the progression equals 110?b. Write down a
A geometric progression has first term 10 and its sum to infinity is 15.a. Find the common ratio.b. Find the sum to infinity if the first term is excluded from the progression.
The first three terms in the expansion of (2 − ax)n in ascending powers of x are 32, −240 and 720. Find the values of a and n.
Luca’s starting salary in a company is $45 000. During the time he stays with the company, it increases by $1800 each year.a. What is his salary in his sixth year?b. How many years has Luca been
A jogger is training for a 5 km charity run. He starts with a run of 400 m, then increases the distance he runs in training by 100 m each day.a. How many days does it take the jogger to reach a
The first four terms in an infinite geometric series are 216, 72, 24, 8.a. What is the common ratio r?b. Write down an expression for the nth term of the series.c. Find the sum of the first n terms
A tank is filled with 10 litres of water. Half the water is removed and replaced with anti-freeze and then thoroughly mixed. Half this mixture is then removed and replaced with anti-freeze. The
A piece of string 20 m long is to be cut into pieces such that the lengths of the pieces form an arithmetic sequence. a. If the lengths of the longest and shortest pieces are 2 m and 50 cm
A pendulum is set swinging. Its first oscillation is through an angle of 20°, and each followingoscillation is through 95% of the angle of the one before it.a. After how many swings is the
The ninth term of an arithmetic progression is 95 and the sum of the first four terms is −10.a. Find the first term of the progression and the common difference. The nth term of the progression is
A ball is thrown vertically upwards from the ground. It rises to a height of 15 m and then falls and bounces. After each bounce it rises vertically to 5/8 of the height from which it fell.a. Find the
The fourth term of a geometric progression is 36 and the eighth term is 576. All the terms are positive.a. Find the common ratio.b. Find the first term.c. Find the sum of the first ten terms.
a. Simplify (1 + t)3 − (1 − t)3.b. Show that x3 − y3 = (x − y)(x2 + xy + y2).c. Substitute x = 1 + t and y = 1 − t in the result in part b and show that your answer is the same as that for
a. Solve the simultaneous equations ar = 12, ar5 = 3072 (there are two possible answers).b. In each case, find the sum of the first ten terms of the geometric progression with first term a and common
Following knee surgery, Adankwo has to do squats as part of her physiotherapy programme. Each day she must do 4 more squats than the day before. On the eighth day she did 31 squats. Calculate how
The first three terms of an arithmetic sequence, a, a + d and a + 2d, are the same as the first three terms, a, ar, ar², of a geometric sequence (a ≠ 0). Show that this is only possible if r = 1
a, b and c are three consecutive terms in a sequence.a. Prove that if the sequence is an arithmetic progression then a + c = 2b.b. Prove that if the sequence is a geometric progression then ac = b².
Sketch each of the following vectors and find their moduli:a. 3i + 4j b. 3i − 4j c. 7i d. −7ie. 5i + 3j f. 2i − 7j g. 6i − 6j h. i + j
A(−1, 4), B(2, 7) and C(5, 0) form the vertices of a triangle.a. Draw the triangle on graph paper. Using your diagram, write the vectors representing the sides AB, BC and AC as column vectors.b.
Express the following vectors in component form:a.b.c.d. 4 a 2 5 X -1 4. 3.
The coordinates of points P, Q, R and S are (–1, –2), (–2, 1), (1, 2) and (2, –1) respectively. The origin is the point O. a. Mark the points on a grid. Use equal scales on the two
Simplify the following:a. (2i + 3j) − (3i − 2j) b. 3(2i + 3j) − 2(3i − 2j)
For each of the following vectorsi. Draw a diagram ii. Find its magnitude. a -3 b 4
Draw diagrams to illustrate each of the following vectors:a. 2i b. 3j c. 2i + 3j d. 2i – 3j
Given that a = 3i + 4j, b = 2i − 3j and c = −i + 5j, find the following vectors:a. a + b + c b. a + b − c c. a − b + cd. 2a + b + 3c e. a − 2b + 3c f. 2(a + b) − 3(b
Write the vectors joining each pair of pointsi. In the form ai + bjii. As a column vectors.a. (1, 4) to (3, 7)b. (1, 3) to (2, −4)c. (0, 0) to (3, 5)d. (−3, 7) to (7, −3)e. (−4, 2) to (0,
A, B, C and D have coordinates (−3, −4), (0, 2), (5, 6) and (2, 0) respectively.a. Draw the quadrilateral ABCD on graph paper.b. Write down the position vectors of the points A, B, C and D.c.

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