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mathematics
cambridge international as & a level mathematics probability & statistics
Cambridge International AS & A Level Mathematics Pure Mathematics 2 & 3 Coursebook 1st Edition Sue Pemberton, Julianne Hughes, Julian Gilbey - Solutions
The third term of a geometric progression is 16 and the sixth term is − 1/4.a. Find the common ratio and the first term.b. Find the sum to infinity.
The coefficient of x2 in the expansion of (1 + ax)4 is 30 times the coefficient of x in the expansion of Find the value of a. 1+ ax 3 3
The diagram shows two circles, C1 and C2, touching at the point T. Circle C1 has centre P and radius 8 cm; circle C2 has centre Q and radius 2 cm. Points R and S lie on C1 and C2 respectively, and RS is a tangent to both circles.i. Show that RS = 8 cm.ii. Find angle RPQ in radians correct to 4
Robert buys a car for $8000 in total (including interest). He pays for the car by making monthly payments that are in arithmetic progression. The first payment that he makes is $200 and the debt is fully repaid after 16 payments. Find the fifth payment.
The second term of a geometric progression is 24 and the third term is 12(x + 1).a. Find, in terms of x, the first term of the progression.b. Given that the sum of the first three terms is 76, find the possible values of x.
The first term of an arithmetic progression is 1.75 and the second term is 1.5. The sum of the first n terms is −n. Find the value of n.
a. Find the first three terms, in ascending powers of y, in the expansion of (1 + y)4.b. By replacing y with 5x − 2x2, find the coefficient of x2 in the expansion of (1 + 5x − 2x2)4.
a. Find, in ascending powers of x, the first three terms in the expansion ofb. Hence, obtain the coefficient of x2 in the expansion of 8
The first three terms of a geometric progression are 135, k and 60. Given that all the terms in the progression are positive, find:a. The value of kb. The sum to infinity.
a. Find, in ascending powers of x, the first three terms in the expansion of (2 + x)10.b. By replacing x with 2y − 3y2, find the first three terms in the expansion of (2 + 2y − 3y2)10.
In the diagram, OAB is an isosceles triangle with OA = OB and angle AOB = 2θ radians.Arc PST has centre O and radius r, and the line ASB is a tangent to the arc PST at S.i. Find the total area of the shaded regions in terms of r and θ.ii. In the case whereand r = 6, find the total perimeter of
The sixth term of an arithmetic progression is −3 and the sum of the first ten terms is −10.a. Find the first term and the common difference.b. Given that the nth term of this progression is −59, find the value of n.
The second term of a geometric progression is −1458 and the fifth term is 432. Find:a. The common ratiob. The first termc. The sum to infinity.
Find the power of x that has the greatest coefficient in the expansion of 4 3x*++).
The first three terms of a geometric progression are k + 12 , k and k − 9, respectively.a. Find the value of k.b. Find the sum to infinity.
The sum of the first n terms, Sn, of a particular arithmetic progression is given by Sn = 4n2 + 3n.Find the first term and the common difference.
A company makes a donation to charity each year. The value of the donation increases exponentially by 10% each year. The value of the donation in 2010 was $10 000.a. Find the value of the donation in 2016.b. Find the total value of the donations made during the years 2010 to 2016, inclusive.
An arithmetic progression has first term a and common difference d. The sum of the first 100 terms is 25 times the sum of the first 20 terms.a. Find d in terms of a.b. Write down an expression, in terms of a, for the 50th term.
a. Write down the expansion of (x + y)5.b. Without using a calculator and using your result from part a, find the value of correct to the nearest hundred. (101), 4
Find the first three terms, in ascending powers of x, in the expansion of (2 − 3x)4(1 + 2x)10.
The fourth term of a geometric progression is 48 and the sum to infinity is five times the first term.Find the first term.
i. Prove the identityii. Hence solve the equation sin 0 1 1- cos Ꮎ sinᎾ 1 tane
The sum of the first n terms, Sn, of a particular arithmetic progression is given by Sn = 12n − 2n2.Find the first term and the common difference.
The sum of the first n terms, Sn, of a particular arithmetic progression is given byFind an expression for the nth term. Sm=(5n² - 17n).
The tenth term of an arithmetic progression is 17 and the sum of the first five terms is 190.a. Find the first term of the progression and the common difference.b. Given that the nth term of the progression is −19, find the value of n.
a. Given thatfind the value of p and the value of q.b. Hence, without using a calculator, find the exact value of X X — ² + ₁ x α = ( ² - ²x² ) - ₂ ( ² + ₂x )
The function f is such that f(x) = 2 sin2x − 3cos2x for 0 ≤ x ≤ π.i. Express f(x) in the form a + b cos2x, stating the values of a and b.ii. State the greatest and least values of f(x).iii. Solve the equation f(x) + 1 = 0.
The first four terms, in ascending powers of x, in the expansion of (1 + ax + bx2)7 are 1 − 14x + 91x2 + px3.Find the values of a, b and p.
A geometric progression has first term a and common ratio r. The sum of the first three terms is 3.92 and the sum to infinity is 5. Find the value of a and the value of r.
The first two terms, in ascending powers of x, in the expansion ofFind the values of n , p and q. (1 + x) ( 2 − 4 ) ² are p+qx².
a. The fifth term of an arithmetic progression is 18 and the sum of the first eight terms is 186.Find the first term and the common difference.b. The first term of a geometric progression is 32 and the fourth term is 1/2. Find the sum to infinity of the progression.
The first term of a geometric progression is 1 and the second term is 2 cos x, where Find the set of values of x for which this progression is convergent. 0
A circle is divided into ten sectors. The sizes of the angles of the sectors are in arithmetic progression. The angle of the largest sector is seven times the angle of the smallest sector. Find the angle of the smallest sector.
Let Sn = 1 + 11+ 111 + 1111 + 11111+ … to n terms.Show that - Sn 10+1-10-9n 81
The function f is defined byi. State the range of f.ii. Find the coordinates of the points at which the curve y = f(x) intersects the coordinate axes.iii. Sketch the graph of y = f(x).iv. Obtain an expression for f−1(x), stating both the domain and range of f−1. f: x + 4 sinx - 1 for
a. The seventh term of an arithmetic progression is 19 and the sum of the first twelve terms is 224.Find the fourth term.b. A geometric progression has first term 3 and common ratio r. A second geometric progression has first term 2 and common ratio 1/5r. The two progressions have the same sum to
a. The first two terms of an arithmetic progression are 1 and cos2x respectively. Show that the sum of the first ten terms can be expressed in the form a − b sin2x, where a and b are constants to be found.b. The first two terms of a geometric progression are 1 and 1/3 tan2θ respectively,
y = x +1/xa. Express x3 + 1/x3 in terms of y.b. Express x5 + 1/x5 in terms of y.
An arithmetic sequence has first term a and common difference d. The sum of the first 20 terms is seven times the sum of the first five terms.a. Find d in terms of a. b. Find the 65th term in terms of a.
a. A geometric progression has first term a, common ratio r and sum to infinity S.A second geometric progression has first term 5a, common ratio 3r and sum to infinity 10S.Find the value of r.b. An arithmetic progression has first term −4. The nth term is 8 and the (2n)th term is 20.8.Find the
a. The first term of a geometric progression in which all the terms are positive is 50. The third term is 32.Find the sum to infinity of the progression.b. The first three terms of an arithmetic progression are 2sinx, 3cosx and (sinx + 2cosx) respectively, where x is an acute angle.i. Show that tan
The tenth term in an arithmetic progression is three times the third term. Show that the sum of the first ten terms is eight times the sum of the first three terms.
A television quiz show takes place every day. On day 1 the prize money is $1000. If this is not won the prize money is increased for day 2. The prize money is increased in a similar way every day until it is won. The television company considered the following two different models for increasing
The first term of an arithmetic progression is sin2x and the second term is 1.a. Write down an expression, in terms of sinx, for the fifth term of this progression.b. Show that the sum of the first ten terms of this progression is 10 + 35 cos2x.
The sum of the digits in the number 67 is 13 (as 6 + 7 = 13).a. Show that the sum of the digits of the integers from 19 to 21 is 15.b. Find the sum of the digits of the integers from 1 to 99.
The first term of a progression is 4x and the second term is x2.i. For the case where the progression is arithmetic with a common difference of 12, find the possible values of x and the corresponding values of the third term.ii. For the case where the progression is geometric with a sum to infinity
a. The third and fourth terms of a geometric progression are 1/3 and 2/9 respectively. Find the sum to infinity of the progression.b. A circle is divided into 5 sectors in such a way that the angles of the sectors are in arithmetic progression. Given that the angle of the largest sector is 4 times
a. In an arithmetic progression the sum of the first ten terms is 400 and the sum of the next ten terms is 1000. Find the common difference and the first term.b. A geometric progression has first term a, common ratio r and sum to infinity 6. A second geometric progression has first term 2a, common
The diagram shows an equilateral triangle, PQR, with side length 5cm. M is the midpoint of the line QR. An arc of a circle, centre P, touches QR at M and meets PQ at X and PR at Y. Find in terms of π and 3:a. The total perimeter of the shaded region.b. The total area of the shaded region. P Y R M
In the diagram, OAB is a sector of a circle with centre O and radius 8cm. Angle BOA is α radians. OAC is a semicircle with diameter OA. The area of the semicircle OAC is twice the area of the sector OAB.i. Find α in terms of π.ii. Find the perimeter of the complete figure in terms of . 0 8 cm a
The diagram shows triangle ABC in which AB is perpendicular to BC. The length of AB is 4cm and angle CAB is α radians. The arc DE with centre A and radius 2 cm meets AC at D and AB at E. Find, in terms of α,i. The area of the shaded region,ii. The perimeter of the shaded region. 2cm arad E 4cm B
Find the discriminant for each equation and, hence, decide if the equation has two distinct roots, two equal roots or no real roots.a. x2 − 12x + 36 = 0 b. x2 + 5x − 36 = 0 c. x2 + 9x + 2 = 0d. 4x2 − 4x +1 = 0e. 2x2 −7x + 8 = 0f. 3x2 + 10x − 2 = 0
Find, in terms of π, the area of a sector of:a. Radius 12 cm and angle π/6 radiansb. Radius 10 cm and angle 2π/5 radiansc. Radius 4.5cm and angle 2π/9 radiansd. Radius 9cm and angle 4π/3 radians
Change these angles to radians, giving your answers in terms of.a. 20° b. 40° c. 25° d. 50° e. 5°f. 150° g. 135° h. 210° i. 225° j. 300°k. 65° l. 540° m. 9° n. 35° o. 600°
Find, in terms of π, the arc length of a sector of:a. Radius 8cm and angle π/4b. Radius 7 cm and angle 3π/7c. Radius 16cm and angle 3π/8d. Radius 24cm and angle 7π/6
Find the area of a sector of:a. Radius 34cm and angle 1.5 radian b. Radius 2.6cm and angle 0.9 radians.
Change these angles to degrees.a. π/2b. π/3c. π/6d. π/12e. 4π/3f. 4π/9g. 3π/10h. 7π/12i. 9π/20j. 9π/2k. 7π/5l. 4π/15m. 5π/4n. 7π/3o. 9π/8
Find the arc length of a sector of:a. Radius 10 cm and angle 1.3 radians b. Radius 3.5cm and angle 0.65 radians.
Find, in radians, the angle of a sector of:a. Radius 4cm and area 9cm2.b. Radius 6cm and area 27cm2.
The diagram shows a sector OAB of a circle with centre O and radius r. Angle AOB is θ radians. The point C on OA is such that BC is perpendicular to OA. The point D is on BC and the circular arc AD has centre C.i. Find AC in terms of r and θ.ii. Find the perimeter of the shaded region ABD
The diagram represents a metal plate OABC, consisting of a sector OAB of a circle with centre O and radius r, together with a triangle OCB which is right-angled at C. Angle AOB = θ radians and OC is perpendicular to OA.i. Find an expression in terms of r and for the perimeter of the plate.ii. For
The diagram shows a sector, POR, of a circle, centre O, with radius 8cm and sector angle π/3 radians. The lines OR and QR are perpendicular and OPQ is a straight line.Find the exact area of the shaded region. R 8 cm رواتيا 3 P
Write each of these angles in radians, correct to 3 significant figures.a. 28° b. 32° c. 47° d. 200° e. 320°
Use your calculator to find:a. sin(0.7) b. tan(1.5) c. cos(0.9)d. e.f. COS π 2
The diagram shows a sector, POQ, of a circle, centre O, with radius 4cm. The length of arc PQ is 7 cm. The lines PX and QX are tangents to the circle at P and Q, respectively.a. Find angle POQ, in radians.b. Find the length of PX.c. Find the area of the shaded region. 4 cm 0 Q P X
Find, in radians, the angle of a sector of:a. Radius 10 cm and arc length 5cm b. Radius 12 cm and arc length 9.6cm.
Find the perimeter of each of these sectors.a.b.c. 1.2 rad 6 cm
The diagram shows a circle with centre A and radius r. Diameters CAD and BAE are perpendicular to each other. A larger circle has centre B and passes through C and D.i. Show that the radius of the larger circle is r√2.ii. Find the area of the shaded region in terms of r. B D E
AOB is a sector of a circle, centre O, with radius 8cm.The length of arc AB is 10 cm. Find:a. Angle AOB, in radians b. The area of the sector AOB.
The diagram shows a sector, AOB, of a circle, centre O, with radius 5cm and sector angle π/3 radians. The lines AP and BP are tangents to the circle at A and B, respectively.a. Find the exact length of AP.b. Find the exact area of the shaded region. 0 لاتي 3 B 5 cm P A
Write each of these angles in degrees, correct to 1 decimal place.a. 1.2 rad b. 0.8rad c. 1.34 rad d. 1.52 rad e. 0.79 rad
Calculate the length of QR. P 1 rad 5 cm R
The High Roller Ferris wheel in the USA has a diameter of 158.5 metres.Calculate the distance travelled by a capsule as the wheel rotates through π/16 radians.
The circle has radius 6cm and centre O.PQ is a tangent to the circle at the point P.QRO is a straight line. Find:a. Angle POQ, in radiansb. The length of QRc. The perimeter of the shaded area. 8 cm R P 6 cm
Robert is told the size of angle BAC in degrees and he is then asked to calculate the length of the line BC. He uses his calculator but forgets that his calculator is in radian mode. Luckily he still manages to obtain the correct answer. Given that angle BAC is between 10° and 15°, use graphing
A piece of wire of length 24 cm is bent to form the perimeter of a sector of a circle of radius r cm.i. Show that the area of the sector, Acm2 , is given by A = 12r − r2.ii. Express A in the form a − (r − b)2, where a and b are constants.iii. Given that r can vary, state the greatest value of
The circle has radius 7 cm and centre O.AB is a chord and angle AOB = 2 radians. Find:a. The length of arc ABb. The length of chord ABc. The perimeter of the shaded segment. A 7 cm 2 rad 0 7 cm B
In the diagram, AOB is a quarter circle with centre O and radius r. The point C lies on the arc AB and the point D lies on OB. The line CD is parallel to AO and angle AOC = θ radians.i. Express the perimeter of the shaded region in terms of r, θ and π.ii. For the case where r = 5 cm and θ =
The diagram shows three touching circles with radii 6cm, 4cm and 2 cm.Find the area of the shaded region. 6 4
The diagram shows a semicircle with radius 10 cm and centre O. Angle BOC = θradians. The perimeter of sector AOC is twice the perimeter of sector BOC.a. Show thatb. Find the perimeter of triangle ABC. A 10 cm 0 с B
The diagram shows a metal plate made by fixing together two pieces, OABCD (shaded) and OAED (unshaded). The piece OABCD is a minor sector of a circle with centre O and radius 2r. The piece OAED is a major sector of a circle with centre O and radius r. Angle AOD is α radians. Simplifying your
ABCD is a rectangle with AB = 5 cm and BC = 24 cm.O is the midpoint of BC.OAED is a sector of a circle, centre O. Find:a. The length of AOb. Angle AOD, in radiansc. The perimeter of the shaded region. A B E 0 D C
In the diagram, AB is an arc of a circle with centre O and radius 4cm.Angle AOB is α radians. The point D on OB is such that AD is perpendicular to OB.The arc DC, with centre O, meets OA at C.i. Find an expression in terms of α for the perimeter of the shaded region ABDC.ii. For the case
The diagram shows a semicircle, centre O, with radius 8cm.FH is the arc of a circle, centre E. Find the area of:a. Triangle EOF b. Sector FOGc. Sector FEH d. The shaded region. E 2 rad 0 HG
The diagram shows two circles with radius r cm.The centre of each circle lies on the circumference of the other circle.Find, in terms of r, the exact area of the shaded region. @
The diagram shows a sector, EOG, of a circle, centre O, with radius r cm.The line GF is a tangent to the circle at G, and E is the midpoint of OF.a. The perimeter of the shaded region is P cm.Show thatb. The area of the shaded region is Acm2.Show that rcm G E F
The diagram shows a circle with radius 1cm, centre O.Triangle AOB is right angled and its hypotenuse AB is a tangent to the circle at P.Angle BAO = x radians.a. Find an expression for the length of AB in terms of tan x.b. Find the value of x for which the two shaded areas are equal. A x 0 P B
The diagram shows a sector, AOB, of a circle, centre O, with radius Rcm and sector angle π/3 radians.An inner circle of radius r cm touches the three sides of the sector.a. Show that R = 3r.b. Show that 0 π ان B A
The diagram shows the cross-section of two cylindrical metal rods of radii x cm and y cm. A thin band, of length P cm, holds the two rods tightly together.
Simplify each of the following.a. (x3 + 5x2 + 5x − 2) ÷ (x + 2) b. (x3 − 8x2 + 22x − 21) ÷ (x − 3)c. (3x3 − 7x2 + 6x − 2) ÷ (x − 1) d. (2x3 − 3x2 + 8x + 5) ÷ (2x + 1)e. (5x3 − 13x2 + 10x − 8) ÷ (2 − x) f. (6x4 + 13x − 19) ÷ (1 − x)
Find the remainder when:a. 6x3 + 3x2 − 5x + 2 is divided by x − 1b. x3 + x2 − 11x + 12 is divided by x + 4c. x4 + 2x3 − 5x2 − 2x + 8 is divided by x + 1d. 4x3 − x2 − 18x + 1 is divided by 2x − 1
The graphs of y = |x − 1| and y = 2 |x − 4| are shown on the grid. y 8- 71 6- 5- 4- 3 2- - 1 2 3 y = 2|x-41 4 5 6 7 y = |x-1| 89 x
Use the factor theorem to show that x + 1 is a factor of x4 − 3x3 − 4x2 + 5x + 5.
Solve the equation |2x − 3| = |5x + 1|.
a. When x3 − 3x2 + ax − 7 is divided by x + 2, the remainder is −37. Find the value of a.b. When 9x3 + bx − 5 is divided by 3x + 2, the remainder is −13. Find the value of
Solve the inequality |5x − 3| > 7.
a. On the same axes, sketch the graphs of y = |2x − 1| and y = 4 − |x − 1|.b. Solve the inequality |2x − 1|> 4 − |x − 1|.
a. Complete the table of values for y = |x − 3| + 2.b. Draw the graph of y = |x − 3| + 2 for 0 < x < 6.c. Describe the transformation that maps the graph of y = |x| onto the graph of y = |x − 3|+ 2. X 0 5 J' 1 2 3 نرا 3 4 5 الا
Use algebraic division to show these two important factorisations.a. x3 − y3 = (x − y)(x2 + xy + y2 ) b. x3 + y3 = (x + y)(x2 − xy + y2 )
Given that x + 4 is a factor of x3 + ax2 − 29x + 12, find the value of a.
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