All Matches

Solution Library

Expert Answer

Textbooks

Search Textbook questions, tutors and Books

Oops, something went wrong!

Change your search query and then try again

Result Not Found

Books FREE
Tutors
Study Help
Ask a Question

Search
Sign In
Register

study help
mathematics
cambridge international as & a level mathematics probability & statistics

Questions and Answers of Cambridge International AS & A Level Mathematics Probability & Statistics

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
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
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
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.
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
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)
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 =
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°
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
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
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
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 =
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
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
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
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
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
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
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
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
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

Showing 300 - 400 of 1991

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Last

Get In Touch

About Us
Contact Us
Career
Jobs
FAQ
Campus Ambassador

Company Info

Security
Copyrights
Privacy Policy
Terms & Condition
SolutionInn Fee
Scholarship

Services

Sitemap
Fun
Definitions
Become Tutor
Study Help Categories
Recent Questions
Expert Questions

Copyright © 2024 SolutionInn All Rights Reserved.

Get the SolutionInn - Study Help App