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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
a. Find d/dx (xsin x + cosx).b. Hence find the exact value of 3 0 πC x cos x dx.
A curve has equation y = sin2x − x for 0 < x < 2 π/2. Find the x-coordinates of the stationary points of the curve, and determine the nature of these stationary points on the curve giving your answer correct to 3 significant figures.
The diagram shows part of the curve y = 2 − x2 ln(x + 1). The shaded region R is bounded by the curve and by the lines x = 0, x = 1 and y = 0. Use the trapezium rule with intervals to estimate the area of R giving your correct to 2 decimal places. State, with a reason, whether the
a. By differentiating cosx/sinx, show that if y = cot x then dy/dx = -cosec2x.b. b Hence, show that -T ∙π 6 cosec 2 2xdx = √3 | 6
a. Given thatfind the value of the constant A.b. Hence show that 4x 2x - 1 2+ A 2x - 1
The diagram shows the curve Use the trapezium rule with 3 intervals to estimate the value ofgiving your answer correct to 2 decimal places. State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of ܐ> y = e cos x for 0≤x≤. I.
A curve is such thatGiven that dy/dx when x = 0 and that the curve passes through the point (π/2, -3), find the equation of the curve. d² y dx² = -12 sin 2x - 2 cos x.
a. Find the quotient and remainder when 6x2 + 5x is divided by 2x − 5.b. Hence show that 6x² + 5x 2x - 5 -dx 23 5 22-25 (3)
A curve is such thatGiven that the curve passes through the pointGiven that the curve passes through the point (e, e2 ), find the equation of the curve. dy d.x = 2x + 3 x + e
A curve is such that dy/dx = 6e2x + 2e-x. Given that the curve passes through the point (0, 2), find the equation of the curve.
a. FndHere k is a positive constant.b. Hence find the exact value of k Ső (5e-2x + 2e-3x) dx_
A curve is such that d2y/dx2 = 20e-2x. Given that dy/dx = -8 when x = 0 and that the curve passes through the point (1,5/e2), find the equation of the curve.
The point (π/2, 5) lies on the curve for which dy/dx = 4sin(2x - π/2).a. Find the equation of the curve.b. Find the equation of the normal to the curve at the point where x = π/3.
Find the exact area of the region bounded by the curve y = 1 + e2x−5, the x-axis and the lines x = 1 and x = 3.
The diagram shows part of the curve y = 1 + 3 sin2x + cos 2x. Find the exact value of the area of the shaded region. y X
The diagram shows part of the curve y = 3sin2x + 6sin x and its maximum point M. M y n R X
a. Show that sin3x Ξ 3sin x − 4sin3 x.b. Hence show that 2 sin³ x dx = 5 12
The diagram shows part of the curve y = 2/x+3.Given that the shaded region has area 4, find the value of k. YA 01 k x
a. Show that cos 3x Ξ 4 cos3 x − 3cos x.b. Hence show that " π (4cos x + 2 cos.x) dx 17 6
The diagram shows part of the curve y = cos x + 2 sin x. Find the exact volume of the solid formed when the shaded region is rotated through 360° about the x-axis. 0 0 2 B MAX
a. Show thatb. Hence show that 12 sin² x cos²x = 3 2 (1- cos 4x).
a. Show that d/dx(xex - ex) =xex.b. Use your result from part a to evaluate the area of the shaded region. 3 " IN = [ ।
a. Findb. Show that [4e* (3 + e²x) dx.
The points P(1, −2) and Q(2, k) lie on the curve for which dy/dx =3 - 2/x. The tangents to the curve at the points P and Q intersect at the point R. Find the coordinates of R.
a. Find where a is a positive constant.b. Hence find the value of S (4e-²* + Se-x) dx,
i. Show that (2 sin x + cos x)2 can be written in the formii. Hence find the exact value of 5 2 +2 sin 2.x - - 3 2 cos2.x.
The diagram shows part of the graph of y = cos x. The points(π/4, √2/4) and (π/3, 1/3) lie on the curve.a. Find the exact value ofb. Hence show that y √2 2 1 2 0 y = cos x R|4 E|M 3 X
The diagram shows the graph of y = ex. The points (ln2, 2) and (ln3, 3) lie on the curve.a. Find the value ofb. Hence show that y 3 2 در In 2 y = ex In 3
The diagram shows the curve y = 2ex + 8e−x − 7 and its minimum point M. Find the area of the shaded region. y TAT
The diagram shows the curve sin y = x sin x, for 0 x < x < π. The point Q(1/2π, 1/2π) lies on the curve.i. Show that the normal to the curve at Q passes through the point (π, 0).ii. Find iii. Hence evaluate Q TC X
i. By differentiating cosx/sinx that ifii. By expressing cot2 x in terms of cosec2 x and using the result of part i, show thatiii. Express cos 2x in terms of sin2 x and hence show that 1/1 - cos2x can be expressed as1/2 cosec2x. Hence, using the result of part i, find if y = cotx
i. Using the expansion of cos(3x − x) and cos(3x + x), prove that ii. Hence show that -~ (cos 2x - cos 4x) = sin 3x sin.x.
i. Findii. Without using a calculator, find the exact value ofgiving your answer in the form ln(aeb ), where a and b are integers. 1+ cos4 2x cos² 2x dx.
Solvea.b.c.d.|3x − 5| = x + 1 e. x +|x + 4 |= 8 f. 8 − |1 − 2x| = x 2x + 1 x-21 = 5
Solvea. |4x − 3| = 7b. |1 − 2x|= 5c.d.e.f.|2x + 7| = 3x 3x-2 5 = 4
Given that cosA = - 4/5 and sinB = -8/17 and that A and B lie in the same quadrant, find the value of:a. sin(A + B)b. cos(A + B)c. tan(A − B)
Find the quotient and remainder for each of the following.a. (x3 + 2x2 + 5x − 4) ÷ (x + 1) b. (6x3 + 7x2 − 6) ÷ (x − 2)c. (8x3 + x2 − 2x + 1) ÷ (2x − 1) d. (2x3 − 9x2 − 9) ÷ (3 − x)e. (x3 + 5x2 − 3x − 1) ÷ (x2 + 2x + 1) f. (5x4 − 2x2 − 13x + 8) ÷ (x2 + 1)
Use the factor theorem to show that 2x − 5 is a factor of 2x3 − 7x2 + 9x − 10.
Solve,a. |2x + 1| = |x|b. |3 − 2x|= |3x|c. |2x − 5| = |1 − x|d. |3x + 5| = |1 + 2x|e. |x − 5| = 2|x + 1|f. 3|2x - 1| = la x - 3
Solvea. |x2 − 2| = 7 b. |5 − x2|= 3 − x c. |x2 + 2x| = x + 2d. |x2 − 3| = 2x + 1 e. |2x2 − 5x|= 4 − x f. |x2 − 7x + 6|= 6 − x
Express 8 and 0.25 as powers of 2 and hence simplify. log, 8 log, 0.25
Solve, giving your answers correct to 3 significant figures.a. 5x = 18 b. 2x = 35 c. 32x = 8 d. 2x+1 = 25e. 32x−5 = 20 f. 3x = 2x+1 g. 5x+3 = 74 - 3xh. 41−3x = 3x−2i. 2x = 2(5x) j. 4x = 7(3x) k. 2x+1 = 3(5x) l. 5(2x−3 ) = 4(3x+4 )
Write as a single logarithm.a. log2 7 + log2 11b. log6 20 - log6 4c. 3log5 2 − log5 4d. 2 log3 8 − 5 log3 2e.1 + 2 log2 3f. 2 log4 2
a. Show that 2x+1 + 6(2x−1) = 12 can be written as 2(2x ) + 3(2x ) = 12.b. Hence solve the equation 2x+1 + 6(2x−1) = 12, giving your answer correct to 3 significant figures.
Solve, giving your answers correct to 3 significant figures.a. 2x+2 + 2x = 220 b. 3x+1 = 3x−1 + 32 c. 2x+1 + 5(2x−1) = 24d. 5x + 53 = 5x+2e. 4x-1 = 4x - 43f. 3x+1 - 2(3x-2) = 5
Without using a calculator, find the exact value of each of the following:a. sin20° cos 70° + cos 20° sin70° b. sin172° cos 37° − cos172° sin37°c. cos 25° cos 35° − sin25° sin35° d. cos 99° cos 69° + sin99° sin69°e. f. tan 25° + tan 20⁰ 1-tan 25° tan 20⁰
Solve, giving your answers correct to 3 significant figures.a. 4x + 2x − 12 = 0 b. 3(16x ) − 10(4x ) + 3 = 0c. 4x + 15 = 4(2x+1) d. 2(9x ) = 3x+1 + 27
Given that log4 x = p and log4 y = q, express in terms of x and/or y:a. log4(64 p)b. log4(- 2/q)c. pqd. log4 P2 - log4 (4√q)
Find the set of values of x satisfying the inequality 3|x − 1|< |2x + 1|.
Expand and simplify cos(x + 30°).
Find the exact values of:a. sec 60°b. cosec 45° c. cot120° d. sec 300°e. cosec135° f. cot 330° g. sec150° h. cot(−30°)
a. Express 15sin − 8 cos in the form R sin(θ - α),where R > 0 and 0° < α < ,90°.Give the value of α correct to 2 decimal places.b. Hence solve the equation 15sin − 8 cosθ = 10 for 0° < θ <360°.
Prove each of these identities.a. tanA + cot A Ξ 2 cosec 2A b. 1 − tan2 A Ξ cos 2Asec2Ac. tan2A − tanA Ξ tan Asec 2Ad. (cosA + 3sinA)2 Ξ 5 − 4 cos 2A + 3sin2Ae. cot 2A + tanA Ξ 1/2 cosecA secA f. cosec 2A + cot 2A Ξ cot Ag. sec2A cosec2A Ξ 4 cosec22Ah. sin(A + B) sin(A − B)
Prove each of these identities.a.b.c.d.e.f. sin A cos B cos A 2 cos(A - B) sin B sin 2B
Solve the equation 2|3x − 1| = 3x, giving your answers correct to 3 significant figures.
Given that tan x = 4/3 where 0° < x < 90°, find the exact value of:a. sin2xb. cos 2xc. tan2xd. tan3x
a. Express 2 cosθ− 3sinθ in the form Rcos(θ + α), where R . 0 and 0° < α <90°.Give the exact value of R and the value of α correct to 2 decimal places.b. Hence solve the equation 2 cosθ − 3sinθ = 1.3 for 0° < θ < 360°.
Use the fact that 4A = 2 × 2A to show that:a.b. cos 4A + 4 cos 2A Ξ 8 cos4 A − 3 sin 4A sin A = 8 cos³ A-4 cos A
i. Solve the equation |3x + 4| = |3x − 11|.ii. Hence, using logarithms, solve the equation |3 × 2y + 4| = |3 × 2y − 11|, giving the answer correct to 3 significant figures.
Without using a calculator, find the exact value of:a. sin75°b. tan75°c. cos105d. tan(−15°)e. sinπ/12f. cos19π/2g. cot7π/12h. sin7π/12
Given that cos 2x = - 527/625 and that 0°< x < 90°, find the exact value of:a. sin2xb. tan2xc. cos xd. tan x
Solve each equation for 0°< x < 360°.a. sec x = 3b. cot x = 0.8c. cosec x = −3d. 3sec x − 4 = 0
a. Express 15sinθ − 8 cosθ in the form Rsin(θ − α) where R > 0 and 0° < α < 90°.Give the value of α correct to 2 decimal places.b. Hence solve the equation 15sinθ − 8 cosθ = 3 for 0° < θ < 360°.
Given that cos x = 4/5 and that 0°< x < 90°, without using a calculator find the exact value of cos(x − 60°).
Given that cos x = - 3/5 and that 90° < x < 180°, find the exact value of:a. sin2xb. sin 4xc. tan2xd. tan1/2x
Solve each equation for 0 < x < 2π.a. cosec x = 2 b. sec x = −1 c. cot x = 2d. 2cot x + 5 = 0
a. Express 4sinθ − 6 cosθ in the form Rsin(θ − α), where R > 0 and 0° < α < 90°.Give the exact value of R and the value of correct to 2 decimal places.b. Hence solve the equation 4sinθ − 6cosθ = 3 for 0° < θ < 180°.c. Find the greatest and least values of (4sinθ
a. Express cos θ + √3 sinθ in the form Rcos(θ - α), where R > 0 and 0 < α < π/2.Give the exact values of R and α.b. Hence prove that 1 (cose + (3sine) sec²0 -
Prove the identity 8sin2 x cos2 x Ξ 1 − cos 4x.
Given that sinA = 4/5 and sinB = 5/13, where A and B are acute, show that sin(A + B) = 63/65.
Given that 3cos 2x + 17 sin x = 8, find the exact value of sin x.
Solve each equation for 0°< x < 180°.a. cosec 2x = 1.2b. sec 2x = 4c. cot 2x = 1d. 2 sec 2x = 3
a. Express 3sinθ + 4 cos θ in the form Rsin(θ + α ) , where R > 0 and 0° < ,90°.Give the value of α correct to 2 decimal places.b. Hence solve the equation 3sinθ + 4 cosθ = 2 for 0° < θ < 360°.c. Find the least value of 3sinθ + 4 cosθ + 3 as θ varies.
Prove the identity (2 sinA + cosA)2 Ξ 1/2 (4sin2 A - 3cos 2A + 5).
Given that sinA = 12/13 and sin B = 3/5, where A is obtuse and B is acute, find the value of:a. sin(A + B)b. cos(A − B)c. tan(A + B)
Solve each equation for 0° < θ < 360°.a. 2 sin2θ = cosθb. 2 cos2θ + 3 = 4 cosθc. 2 cos 2θ + 1 = sinθ
Solve each equation for the given domains.a. cosec(x − 30°) = 2 for 0° < x < 360°b. sec(2x + 60°) = −1.5 for 0° < x < 180°c. cot (x+π/4) = 2 for 0 < x < 2πd. 2 cosec(2x -1) = 3 for - π < x < π
Use the expansions of cos(3x − x) and cos(3x + x) to prove the identity:cos 2x − cos 4x Ξ 2 sin3x sin x
The variables x and y satisfy the equation xny = C, where n and C are constants.When x = 1.10, y = 5.20, and when x = 3.20, y = 1.05.i. Find the values of n and C.ii. Explain why the graph of ln y against ln x is a straight line.
Solve each equation for 0° < θ <180°.a. 2 sin2θ tanθ = 1b. 3cos 2θ + cosθ = 2c. 2 cosec 2θ + 2 tanθ = 3secθd. 2 tan2θ = 3cotθe. tan2θ = 4 cotθf. cot 2θ+ cotθ = 3
Solve each equation for 0° < θ <360°.a. cosec2 x = 4 b. sec2 x = 9c. 9 cot2 1/2x = 4d. sec x = cos xe. cosec x = sec xf. 2 tan x = 3cosec x
Given that tanA = t and that tan(A − B) = 2, find tanB in terms of t.
a. Express cosθ − sinθ in the form Rcos(θ +α ), where R > 0 and < α < π/2.Give the exact values of R and α.b. Show that one solution of the equationand find the other solution in the interval 0 < θ < 2π.c. State the set of values of k for which the equation cosθ− sinθ
Given that 3ex + 8e−x = 14, find the possible values of ex and hence solve the equation 3ex + 8e−x = 14 correct to 3 significant figures.
Express cos2 2x in terms of cos 4x.
Solve each equation for 0° < θ < 360°.a. 2 tan2 θ − 1 = secθb. 3 cosec2 = 13 −a cotθc. 2 cot2 θ− cosecθ = 13 d. cosec + cotθ = 2 sinθe. tan2 θ + 3secθ = 0f. √3 sec2θ = cosecθ
If cos(A − B) = 3cos(A + B), find the exact value of tan A tan B.
The angles θ and Ø lie between 0° and 180°, and are such that tan(θ − Ø) = 3 and tanθ + tanØ = 1. Find the possible values of θ and Ø.
a. Use the expansions of cos(2x + x) to show that cos 3x Ξ 4 cos3 x − 3cos x.b. Express sin3x in terms of sin x.
Solve each equation for 0° < θ <180°.a. secθ = 3cosθ − tanθb. sec22θ = 3tanθ +1c. sec4 θ + 2tan2θd. 2 cot2 2θ + 7 cosec 2θ = 2
a. Given that 8 + cosec2 = 6 cot θ, find the value of tanθ.b. Hence find the exact value of tan(θ + 45°).
i. Solve the equation |4x − 1|= |x − 3|ii. Hence solve the equation |4x+1- 1| = |4y - 3| correct to 3 significant figures.
Solve tan2θ + 2 tanθ = 3cotθ for 0° < θ <180°.
Solve each equation for 0 < θ < 2π.a. tan2θ + 3sec θ + 3 = 0b. 3cot2θ + 7 cosecθ = 2
a. Prove that tanb. Hence find the exact value of tan 0 + cot 0 = 2 sin 20
Prove each of these identities.a. b. c.d.e.f.g.h. 1 tanx + cot = sin x cos x
a. Express√5 cosθ + 2 sinθ in the form Rcos(θ − α), where R > 0 and 0°< α < 90°.Give the value of α correct to 2 decimal places.b. Find the smallest positive angle θ that satisfies the equation √5 cosθ + 2 sinθ = 3.c. Find the smallest positive angle θ that satisfies the
a. Express sinθ − 3cosθ in the form Rsin(θ − α ), where R > 0 and 0° < α < 90°.Give the exact value of R and the value of α correct to 2 decimal places.b. Hence solve the equation sinθ − 3cosθ = −2 for 0°< θ <360°.c. Find the greatest possible value of 1+ sin2θ
a. Given that x is acute and that 2 sec2 x + 7 tan x = 17, find the exact value of tan x.b. Hence find the exact value of tan(225° − x).
The polynomial 4x3 + ax2 + 9x + 9, where a is a constant, is denoted by p(x). It is given that when p(x) is divided by (2x − 1) the remainder is 10.i. Find the value of a and hence verify that (x − 3) is a factor of p(x).ii. When a has this value, solve the equation p(x) = 0.
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