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mathematics
cambridge international as & a level mathematics probability & statistics

Cambridge International AS And A Level Mathematics Pure Mathematics 1 Coursebook 1st Edition Sue Pemberton, Julian Gilbey - Solutions

Solve each of these equations for 0° < x < 360°.a. sin x cos(x − 60) = 0 b. 5sin2x − 3sinx = 0c. tan2x = 5tanx d. sin2 x + 2 sin x cosx = 0e. 2 sin x cos x = sinx f. sinxtanx = 4sinx
Given that tan 25° = a, express each of the following in terms of a.a. tan205° b. sin25° c. cos65° d. cos245°
a. Prove the identityb. Hence, solve the equation 1 1+ sin 0 + 1 1-sin 0 2 cos² 0
Part of the graph of y = a + b cos cx is shown above.Write down the value of a, the value of b and the value of c. y in 5 4 3. 2. 1 0 60 120 180 240 300 360 x
i. Solve the equation 4sin2x + 8cosx − 7 = 0 for 0° ≤ x ≤ 360°.ii. Hence find the solution of the equation 4 sin² (10) + 8 cos ( 0 ) - 7 = 0 for 0° = 0 = 360°.
a. Express 7 sin2x + 4 cos2x in the form a + b sin2x.b. State the range of the function f(x) = 7 sin2 x + 4 cos2x, for the domain 0 ≤ x ≤ 2π.
Given that cos 77° = b, express each of the following in terms of b.a. sin77° b. tan13° c. sin257° d. cos347°
Given thatwhere A and B are in the same quadrant, fi nd the value of:a. cos A b. tan A c. sin B d. tan B 5 sin A = and cos B = 13 4 +in 5
i. Show that the equation 2 tan2θ sin2θ = 1 can be written in the form 2 sin4θ + sin2θ − 1 = 0.ii. Hence solve the equation 2tan2θsin2θ = 1 for 0° ≤ θ ≤ 360°.
a. Express 4 sinθ − cos2θin the form (sinθ + a)2 + b.b. Hence, state the maximum and minimum values of 4sinθ − cos2θ , for the domain 0 ≤ θ ≤ 2π.
a. Given thatb. Hence, find sinθ and cosθ in terms of a. 1-sin0 2 cose show that 12(1+ sine) cose a
Solve each of these equations for 0° < x < 360°.a. 2 sin2 x + sin x − 1 = 0 b. tan2 x + 2 tan x − 3 = 0c. 3cos2 x − 2 cos x − 1 = 0 d. 2sin2 x − cos x − 1 = 0e. 3cos2 x − 3 = sin x f. cosx + 5 = 6sin2xg. 2cos2 x − sin2x − 2 sin x − 1 = 0 h. 1 + tan x
a. Prove the identityb. Hence, solve the equation sin 0 + 1 tan 8 2 ||| 1 + cose 1- cose
Given thatwhere A and B are in the same quadrant, fi nd the value of:a. sinA b. cosA c. sinB d. tanB tan A = 2 a 3 and cos B -3/4
a. Sketch the graph of y = 2sinx for −π ≤ x ≤ π.The straight line y = kx intersects this curve at the maximum point.b. Find the value of k. Give your answer in terms of .c. State the coordinates of the other points where the line intersects the curve.
i. Prove the identityii. Hence solve the equation sin x tan x 1- cos x =1+_¹___ 1. a COS X
In the table, 0° ≤ θ ≤ 360° and the missing function is from the listWithout using a calculator, copy and complete the table. cose, tan0,- sin 8 and 1 tan 8
Part of the graph of y = a tanbx + c is shown above.The graph passes through the pointFind the value of a, the value of b and the value of c. P(77,8). 4
Solve each of these equations for 0 ≤ x ≤ 2.a. 4tan x = 3cos x b. 2 cos2 x + 5sin x = 4
a. Solve the equationb. Solve the equation sinx − 2cosx = 2(2sinx − 3cosx) for − π ≤ x ≤ π. 3 2 – sin x 1+2sinx 4 for 0 ≤ x ≤ 2π.
Given that f(0) = 3 and thatfind:a. The value of a and the value of bb. The range off.f(x) = a + b sin x for 0 ≤ x ≤ 2π () = 2,
a. Prove the identity cos4θ− sin4θ ≡ 2 cos2θ − 1.b. Hence, solve the equation cos¹0 - sin¹0 = — for 0° ≤ 0 ≤ 360°.
Solve sin2x + 3sinxcosx + 2 cos2x = 0 for 0 ≤ x ≤ 2π.
i. Show thatii. Hence solve the equation sin 0 cose 1 sin + cos sin 0 cose sin² 0 - cos²0
The maximum value of f(x) is 8 and the minimum value is −2.a. Find the value of a and the value of b.b. Sketch the graph of y = f(x).
i. Solve the equation 2cos2θ = 3sinθ, for 0° < θ < 360°.ii. The smallest positive solution of the equation 2 cos2(nθ) = 3sin(nθ), where n is a positive integer, is 10°. State the value of n and hence find the largest solution of this equation in the interval 0° ≤ θ ≤ 360°.
The maximum value of f(x) is 9, the minimum value of f(x) is 1 and the period is 120°.f(x) = a + b sincx for 0° ≤ x ≤ 360°, where a and b are positive constants.Find the value of a, the value of b and the value of c.
Identify whether the following sequences are geometric.If they are geometric, write down the common ratio and the eighth term.a. 2, 4, 8, 14,… b. 7, 21, 63, 189,… c. 81, −27, 9, −3,…d.e. 1, 0.4, 0.16, 0.64,… f. 1, −1, 1, −1,… 247 17.7 ..... すすすす····
The maximum value of f(x) is 7 and the period is 60°.f(x) = A + 5 cosBx for 0° ≤ x ≤ 120°a. Write down the value of A and the value of B.b. Write down the amplitude of f(x).c. Sketch the graph of f(x).
The function i. Solve the equation f(x) = 7, giving your answer correct to 2 decimal places.ii. Sketch the graph of y = f(x).iii. Explain why f has an inverse.iv. Obtain an expression for f−1(x). f:x → 5+3cos(x) is defined for 0≤x≤ 2n.
The graph of y = sin x is reflected in the line x = π and then in the line y = 1.Find the equation of the resulting function.
The graph of y = cos x is reflected in the line x = π/2 and then in the line y = 3.Find the equation of the resulting function.
Use Pascal’s triangle to find the expansions of:a. (x + 2)3 b. (1 − x)4 c. (x + y)3 d. (2 − x)3e. (x − y)4 f. (2x + 3y)3g. (2x − 3)4h. (1². + 3 3 2x-³
Find the highest power of x in the expansion of [(5x4 + 3)⁰ + (1 − 3x³)³(4x² − 5x³)° ]*.
Find the coefficient of x2 in the expansion of 2x+2/2
The first term of a progression is 16 and the second term is 24. Find the sum of the first eight terms given that the progression is:a. Arithmetic b. Geometric
Find the sum to infinity of each of the following geometric series.a.b. 1 + 0.1 + 0.01 + 0.001 +…c. 40 − 20 + 10 − 5 +… d. −64 + 48 − 36 + 27 −… 2 3 +
Without using a calculator, find the value of each of the following.a.b.c.d. 3
The first term in an arithmetic progression is a and the common difference is d.Write down expressions, in terms of a and d , for the seventh term and the 19th term.
The first term of a progression is 20 and the second term is 16.a. Given that the progression is geometric, find the sum to infinity.b. Given that the progression is arithmetic, find the number of terms in the progression if the sum of all the terms is −160.
The first term in a geometric progression is a and the common ratio is r. Write down expressions, in terms of a and r, for the sixth term and the 15th term.
In the expansion of (a + 2x)6, the coefficient of x is equal to the coefficient of x2.Find the value of the constant a.
Express each of the following in terms of n.a.b.c. n 2
In the expansion ofthe coefficient of x2 is zero.Find the value of a. (1 - 1) (5 + x) a
a. Find the first three terms in the expansion ofin descending powers of x.b. Hence, find the coefficient of x2 in the expansion of 3.x. 2 릎) x"
The first four terms of a geometric progression are 1, 0.52, 0.54 and 0.56. Find the sum to infinity.
Find the number of terms and the sum of each of these arithmetic series.a. 13 + 17 + 21 +…+ 97 b. 152 + 149 + 146 +…+ 50
The first, second and third terms of a geometric progression are the first, fourth and tenth terms, respectively, of an arithmetic progression. Given that the first term in each progression is 12 and the common ratio of the geometric progression is r, where r ≠1, find:a. The value of rb. The
Find the sum of each of these arithmetic series.a. 5 + 12 + 19 +… (17 terms) b. 4 + 1 + (−2) +… (38 terms)c.d. −x − 5x − 9x −… (40 terms) +++ (20 terms)
The first term of a geometric progression is 270 and the fourth term is 80. Find the common ratio.
Find the value of A, the value of B and the value of C.(3 + x)5 + (3 − x)5 ≡ A + Bx2 + Cx4
Find, in ascending powers of x, the first three terms in each of the following expansions.a. (1 + 2x)8b. (1− 3x)10c.d. (1 + x2)12e. f. (2 − x)13 g. (2 + x2)8h. 7 (1+²)
The first term of a geometric progression is 8 and the second term is 6. Find the sum to infinity.
The first term of an arithmetic progression is 15 and the sum of the first 20 terms is 1630.Find the common difference.
The first term of a geometric progression is 50 and the second term is −30. Find the fourth term.
A geometric progression has eight terms. The first term is 256 and the common ratio is 1/2.An arithmetic progression has 51 terms and common difference 1/2.The sum of all the terms in the geometric progression is equal to the sum of all the terms in the arithmetic progression. Find the first term
The coefficient of x2 in the expansion of (3 + ax)4 is 216.Find the possible values of the constant a.
a. Find the first three terms when (1 − 2x)5 is expanded, in ascending powers of x.b. In the expansion of (3 + ax)(1 − 2x)5, the coefficient of x2 is zero.Find the value of a.
The first term of a geometric progression is 270 and the fourth term is 80. Find the common ratio and the sum to infinity.
In an arithmetic progression, the first term is −27, the 16th term is 78 and the last term is 169.a. Find the common difference and the number of terms.b. Find the sum of the terms in this progression.
Find the coefficient of x3 in each of the following expansions.a. (1 − x)9b. (1 + 3x)12c. d. X 2+ 4 7
The first, second and third terms of a geometric progression are the first, sixth and ninth terms, respectively, of an arithmetic progression. Given that the first term in each progression is 100 and the common ratio of the geometric progression is r, where r ≠ 1, find: a. The value of rb.
Find the coefficient of x5 in the expansion of हने 3 + 2 x2
a. Expand (2 + x)4.b. Use your answer to part a to express (2 + √3)4 in the form a + b√3 .
In the expansion of (2 + ax)7, where a is a constant, the coefficient of x is −2240.Find the coefficient of x2.
The first term of a geometric progression is 50 and the second term is −40.a. Find the fourth term.b. Find the sum to infinity.
a. Write the recurring decimal 0.5̇7̇ as the sum of a geometric progression.b. Use your answer to part a to show that 0.5̇7̇ can be written as19/33
The first two terms in an arithmetic progression are 146 and 139. The last term is −43.Find the sum of all the terms in this progression.
The sum of the second and third terms in a geometric progression is 84. The second term is 16 less than the first term. Given that all the terms in the progression are positive, find the first term.
The first term of an arithmetic progression is 16 and the sum of the first 20 terms is 1080.a. Find the common difference of this progression.The first, third and nth terms of this arithmetic progression are the first, second and third terms, respectively, of a geometric progression.b. Find the
a. Expand (1 + x)3.b. Use your answer to part a to express:i. (1 + √5 )3 in the form a + b√5ii. (1 − √5 )3 in the form c + d√5.c. Use your answers to part b to simplify (1 + √5)3 + (1 − √5)3.
The first three terms of a geometric progression are 3k + 14, k + 14 and k, respectively.All the terms in the progression are positive.a. Find the value of k.b. Find the sum to infinity.
Find the term independent of x in the expansion ofdddf 3x² 23
The first term of a geometric progression is 150 and the sum to infinity is 200. Find the common ratio and the sum of the first four terms.
Find the coefficient of x4 in the expansion of (2x + 1)12.
The first two terms in an arithmetic progression are 2 and 9. The last term in the progression is the only number that is greater than 150. Find the sum of all the terms in the progression.
Three consecutive terms of a geometric progression are x, 4 and x + 6. Find the possible values of x.
The first term of a progression is 2x and the second term is x2.a. For the case where the progression is arithmetic with a common difference of 15, find the two possible values of x and corresponding values of the third term.b. For the case where the progression is geometric with a third term of
Expand (1 + x)(2 + 3x)4.
The sum of the 1st and 2nd terms of a geometric progression is 50 and the sum of the 2nd and 3rd terms is 30. Find the sum to infinity.
The second term of a geometric progression is 4.5 and the sum to infinity is 18. Find the common ratio and the first term.
Find the term in x5 in the expansion of (5 − 2x)8.
Find the coefficient of x2 in the expansion of (3x-2). 3x X
The first term of an arithmetic progression is 15 and the last term is 27. The sum of the first five terms is 79. Find the number of terms in this progression.
Find the sum of the first eight terms of each of these geometric series.a. 3 + 6 + 12 + 24 +… b. 128 + 64 + 32 + 16 +…c. 1 − 2 + 4 − 8 +… d. 243 + 162 + 108 + 72 +…
a. Find the first three terms in the expansion of (x − 3x2)8, in descending powers of x.b. Find the coefficient of x15 in the expansion of (1 − x)(x − 3x2)8.
i. Show that cos4x ≡ 1 − 2 sin2x + sin4x.ii. Hence, or otherwise, solve the equation 8sin4x + cos4x = 2 cos2x for 0° < x < 360°.
Write the recurring decimal … 0.315151515 as a fraction.
Find the coefficient of x8y5 in the expansion of (x − 2y)13.
The diagram shows a metal plate consisting of a rectangle with sides xcm and rcm and two identical sectors of a circle of radius rcm. The perimeter of the plate is 100 cm.a. Show that the area, Acm2, of the plate is given by A = 50r − r2.b. Express 50r − r2 in the form a − (r − b)2, where a
Find the sum of all the integers between 100 and 300 that are multiples of 7.
The first four terms of a geometric progression are 0.5, 1, 2 and 4. Find the smallest number of terms that will give a sum greater than 1000 000.
a. Find the first three terms in the expansion of (1 + px)8, in ascending powers of x.b. Given that the coefficient of x2 in the expansion of (1 − 2x)(1 + px)8 is 204, find the possible values of p.
A sector of a circle, radius r cm, has a perimeter of 60 cm.a. Show that the area, Acm2, of the sector is given by A = 30r − r2.b. Express 30r − r2 in the form a − (r − b)2, where a and b are constants.Given that r can vary:c. find the value of r at which A is a maximumd. find this
The second term of a geometric progression is 9 and the fourth term is 4. Given that the common ratio is positive, find:a. The common ratio and the first termb. The sum to infinity
Find, in ascending powers of x, the first three terms of each of the following expansions.a. (1− x )(2 + x)7b. (1+ 2x)(1 − 3x)10c. (1+ +x X 2
The diagram shows a running track. The track has a perimeter of 400m and consists of two straight sections of length lm and two semicircular sections of radius rm.a. Show that the area, Am2, of the region enclosed by the track is given by A = 400r − πr2.b. Express 400r − πr2 in the
The first term of an arithmetic progression is 2 and the 11th term is 17. The sum of all the terms in the progression is 500. Find the number of terms in the progression.
A ball is thrown vertically upwards from the ground. The ball rises to a height of 8m and then falls and bounces. After each bounce it rises to 3/4 of the height of the previous bounce.a. Write down an expression for the height that the ball rises after the nth impact with the ground.b. Find the
a. Find the first three terms, in ascending powers of x, in the expansion of:i. (1 + 2x)5ii. (3 − x)5 b. Find the coefficient of x2 in the expansion of [(1+ 2x)(3 − x)]5.

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