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mathematics
cambridge international as & a level mathematics probability & statistics

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

a. Show that the equation sin(x + 30°) = 5 cos(x − 60°) can be written in the form cot x = − √3.b. Hence solve the equation sin(x + 30°) = 5 cos(x − 60°) for −180°< x <180°.
The polynomial 2x3 − 4x2 + ax + b, where a and b are constants, is denoted by p(x). It is given that when p(x) is divided by (x + 1) the remainder is 4, and that when p(x) is divided by (x − 3)
a. Prove that 2 cosec 2θ tan Ξ sec2θ.b. Hence solve the equation cosec 2θ tanθ = 2 for -π < θ < π.
a. Given that 3sec θ + 4 cosec θ = 2 cosec 2θ, show that 3sinθ + 4 cosθ = 1.b. Express 3sinθ + 4 cosθ in the form Rsin(θ + α) , where R > 0 and 0°< α < 90°.Give the value of α
Solve each equation for 0°< x < 360°.a. cos(x + 30°) = 2 sin xb. cos(x − 60° ) = 3cos xc. sin(30° − x) = 4sin xd. cos(x + 30°) = 2 sin(x + 60°)
i. Express cos x + 3 sin x in the form R cos (x − α), where R > 0 and 0° < α < 90°, giving the exact value of R and the value of α correct to 2 decimal places.ii Hence solve the
a. Prove that cos 4x + 4 cos 2x Ξ 8 cos4 x − 3.b. Hence solve the equation 2cos 4x + 8 cos 2x = 3 for − π < x < π.
Solve each equation for 0° < x <180°.a. 2 tan(60° − x) = tan xb. 2 tan(45° − x) = 3tan xc. sin(x + 60°) = 2 cos(x + 45°)d. sin(x − 45°) = 2 cos(x + 60°)e. tan(x + 45°) = 6tan
It is given that 2 ln(4x − 5) + ln(x + 1) = 3ln3.i. Show that 16x3 − 24x2 − 15x − 2 = 0.ii. By first using the factor theorem, factorise 16x3 − 24x2 − 15x − 2 completely.iii. Hence
a. Show that θ = 18° is a root of the equation sin3θ = cos 2θ.b. Express sin3θ and cos 2θ in terms of sinθ.c. Hence show that sin18° is a root of the equation 4x3 − 2x2 − 3x + 1 = 0.d.
a. Find the maximum and minimum values of 7 sin2θ + 9 cos2θ + 4sinθ cosθ + 2.b. Hence, or otherwise, solve the equation 7 sin2θ+ 9 cos2θ + 4sinθ cosθ = 10 for 0° < θ < 360°.
Solve each equation for 0° < θ < 180°.a. 6sec3θ − 5sec2θ − 8secθ + 3 = 0b. 2 cot3θ + 3cosec2θ − 8 cot = 0
Solve the equation tan(x − 45°) + cot x = 2 for 0° < x < 180°.
Prove that a = 4b cos 2x cos x. B X a А b 3x C
Find the range of values of θ between 0 and 2π for which cos 2θ > cosθ.
Use the expansions of cos(5x + x) and cos(5x − x) to prove that cos 6x + cos 4x Ξ 2 cos5x cos x.
Solve the inequality cos 2θ − 3sinθ - 2 > 0 for 0° < θ < 360°.
The polynomial 3x3 + 2x2 + ax + b, where a and b are constants, is denoted by p(x). It is given that (x − 1) is a factor of p(x), and that when p(x) is divided by (x − 2) the remainder is
Given that sin x + sin y = p and cos x + cos y = q, find an expression for cos(x − y), in terms of p and q.
i. The polynomial x3 + ax2 + bx + 8, where a and b are constants, is denoted by p(x). It is given that when p(x) is divided by (x − 3) the remainder is 14, and that when p(x) is divided by (x
a. Use the expansions of cos(2x + x) and cos(2x − x) , to prove that:cos 3x + cos x + 2 cos 2x cos xb. Solve cos 3x + cos 2x + cos x > 0 for 0°< x < 360°.
i. Given that (x + 2) and (x + 3) are factors of 5x3 + ax2 + b, find the values of the constants a and b.ii. When a and b have these values, factorise 5x3 + ax2 + b completely, and hence solve
Solve the inequality cos 4θ + 3cos 2θ + 1, 0 for 0° < θ < 360°.
i. Find the quotient and remainder when x4 + x3 + 3x2 + 12x + 6 is divided by (x2 − x + 4).ii. It is given that, when x4 + x3 + 3x2 + px + q is divided by (x2 − x + 4), the remainder is
The angle α lies between 0° and 90° and is such that 2 tan2α + sec2α = 5 − 4tan α.i. Show that 3tan2 α + 4tan α − 4 = 0 and hence find the exact value of tan α.ii. It is given that
Use a calculator to evaluate correct to 3 significant figures:a. e3b. e2.7c. e0.8d. e−2
Use the substitution y = 2x to solve the equation 22x + 32 = 12(2x ).
Solve, giving your answers correct to 3 significant figures.a. 52x − 5(5x) + 6 = 0b. 22x + 5 = 6 × 2xc. 32x = 6 × 3x + 7d. 42x + 27 = 12(4x)
Use the substitution u = 2x to solve the equation 22x − 5(2x+1) + 24 = 0.
Given that log8 y = log8 (x-2)-2 log8 x, express y in terms of x.
Given that log3(z-1) - log3 y, express z in terms of y.
The mass, m grams, of a radioactive substance is given by the formula m = m0e−kt, where t is the time in days after the mass was first recorded and m0 and k are constants.The table below shows
Given that log10 (x2 y) = 4 and log10 (x4/y3) = 18, find the value of log10 x and of log10 y.
Solve, giving your answers correct to 3 significant figures.a. ln x = 5b. ln x = −4c. ln(x − 2) = 6d. ln(2x + 1) = −2
The temperature, T °C , of a hot drink, t minutes after it is made, can be modelled by the equation T = 25 + ke−nt, Where k and n are constants. The table below shows experimental values of
For each of the following equations find the value of x/y correct to 3 significant figures.a. 3x = 7yb. 7x = (2.7)y c. 42x = 35y
Solve, giving your answers correct to 3 significant figures.a. ln(3 − x2) = 2 ln x b. 2ln(x + 2) − ln x = ln(2x − 1)c. 2ln( x + 1) = ln( 2x + 3)d. ln(2x + 1) = 2 ln x + ln5e. ln(x + 2) −
Solve the equation x2.5 =20x1.25, giving your answers in exact form.
Express y in terms of x for each of these equations.a. 2ln(y + 1) − ln y = ln(x + y) b. ln(y + 2) − ln y = 1 + 2 ln x
Solve, giving your answers correct to 3 significant figures.a. |4x − 8| = 2b. |2(3x ) − 4| = 8c. 3|2x − 5| = 2x d. |3x + 4| = |3x − 9|e. |5(2x ) + 3| = |5(2x ) − 10| f. |2x+2 + 1|
Solve, giving your answers in exact form.a. e2x + 2ex − 15 = 0 b. e2x − 5ex + 6 = 0c. 6e2x − 13ex − 5 = 0d. ex − 21e-x = 4
Given that 24x+1 × 32−x = 8x × 35−2x, find the value of:a. 6xb. x
Given that the function f is defined as f : x -> 5ex + 2 for x ∈ R , find an expression for f−1(x).
Solve the equation 8(8x - 1 - 1) = 7(4x - 2x+1).
Solve 2 ln(3 − e2x) = 1, giving your answer correct to 3 significant figures.
Solve ln(2x + 1) < ln(x + 4).
The polynomial 6x3 + x2 + ax − 10, where a is a constant, is denoted by P(x). It is given that when P(x) is divided by (x + 2) the remainder is −12.a. Find the value of a and hence verify that
The polynomials P(x) and Q(x) are defined as:P(x) = x3 + ax2 + b and Q(x) = x3 + bx2 + a.It is given that (x − 2) is a factor of P(x) and that when Q(x) is divided by (x + 1) the remainder is
The polynomial 5x3 − 13x2 + 17x − 7 is denoted by p(x).a. Find the quotient when p(x) is divided by (x − 1), and show that the remainder is 2.b. Hence show that the polynomial 5x3 − 13x2 +
Convert from exponential form to logarithmic form.a. 102 = 100 b. 10x = 200c. 10x = 0.05
Solve the inequalities, giving your answer in terms of base 10 logarithms.a. 2x < 5b. 5x > 7c. (2/3)x > 3d. 0.8x < 0.3
Given that a and b are constants, use logarithms to change each of these non-linear equations into the form Y = mX + c. State what the variables X and Y and the constants m and c represent.a. y =
Solve.a. log5 x + log5 3 = log5 30 b. log3 4x − log3 2 = log3 7c. log2 (2 − x) + log2 9 = 2 log2 8 d. log10 (x - 4) 2 log10 5 + log10 2
Use a calculator to evaluate correct to 3 significant figures:a. ln3 b. ln1.4 c. ln0.9 d. ln 0.15
Solve each of these equations, giving your answers correct to 3 significant figures.a.10x = 52 b.10x = 250c.10x = 0.48
Solve the inequalities, giving your answer in terms of base 10 logarithms.a. 85-x < 10b. 32x + 5 > 20c. 2 x 52x+1 < 3d. 7 x (5/6)3-x > 4
The variables x and y satisfy the equation y = a × xn, where a and n are constants. The graph of ln y against ln x is a straight line passing through the points (0.31, 4.02) and (1.83, 3.22) as
Solve.a. log10 (2x + 9) - log10 5 = 1b. log3 2x - log3 (x - 1) = 2c. log2 (5 - 2x) = 3 + log2 (x + 1)d. log5 (2x - 3) + 2log5 2 = 1+log5 (3 - 2x)
Solve each of these equations, giving your answers correct to 3 significant figures.a. 10x = 52b. 10x = 250c. 10x = 0.48
Simplify.a. 3 log5 2 + 1/2 log5 36 - log512b. 1/2 log3 8+1/2 log3 18 - 1
Without using a calculator, find the value of:a. e1n2b. e1/2n9c. 5eln6d. e-1n1/2
Solve each equation for 0°< x < 360°.a. sec x = 3 b. cot x = 0.8 c. cosec x = −3 d. 3sec x − 4 = 0
The variables x and y satisfy the equation Y = k x en(x−2), where k and n are constants. The graph of ln y against x is a straight line passing through the points (1,1.84) and (7, 4.33) as shown in
Solve 5x2 > 2x . x, giving your answer in terms of logarithms.
Solve.a. log2 x + log2 (x - 1) = log2 20b. log5 (x - 2) + logx (x - 5) = log5 4xc. 2 log3 x - log3 (x - 2) = 2d. 3 + 2 log2 x = log2 (3 − 10x)
Solve.a. eln x = 5 b. ln ex = 15 c. e3 ln x = 64 d. e−ln x = 3
Prove that the solution to the inequality 32x−1 x 21−3x > 5 is lonf log 15 2 1- al∞
Solve each of these equations, giving your answers correct to 3 significant figures.a. log10 x = 1.88 b. log10 x = 2.76c. log10 x = -1.4
Solve.a. logx 40 − logx 5 = 1b. logx 36 − logx 4 = 2c. logx 25 − 2 logx 3 = 2d. 2logx 32 = 3 + 2 logx 4
Solve.a. log3(x + 5) = 2 b. log2 (3x − 1) = 5c. logy (7 − 2x) = 0
Solve, giving your answers correct to 3 significant figures.a. ex = 18b. e2x = 25c. ex+1 = 8d. e2x−3 = 16
In this question you are not allowed to use a calculator.You are given that log10 4 = 0.60206 = correct to 5 decimal places and that 100.206 < 2.a. Find the number of digits in the number 4100.b.
Solve.a. (log2 x)2 - 8 log2 x + 15 = 0b. (log10 x)2 - log10 (x2) = 3c. (log5 x)2 + log5 (x3) = 10d. 4(log2 x)2 - 2 log2 (x2) = 3
The variables x and y satisfy the equation 52y = 32x+1. By taking natural logarithms, show that the graph of ln y against ln x is a straight line, and find the exact value of the gradient of this
Solve the simultaneous equations.a. xy = 81b. 4x = 2yc. log4 (x - y) = 2log4xlog4 (x + y + 9) =0d. log10 x = 2log10 ylog10 (2x - 17y) = 2
Given that log8 y = log8(x - 2) - 2 log8 x, express y in the terms of x.
Solve, giving your answers in terms of natural logarithms. a. ex >10 b. e5x−2 < 35c. 5 × e2x+3 <1
Given that log3 (z -1) - log3 z = 1 +3 log3 + y, express z in terms of y.
Solve.a. log2 (log3 x) = −1b. log4 23−5x = x2
The diagram shows a metal plate. The plate has a perimeter of 50 cm and consists of a rectangle of width 2rcm and height xcm, and a semicircle of radius r cm.a. Show that the area, Acm2, of the plate
Find the area of the region bounded by the curve y = 2x2 + 1, the line y = 9 and the y-axis. A y = 2x² + 1 X
A curve has equationand A(3, 2) is a point on the curve.a. Find the equation of the normal to the curve at the point A.b. A point P(x, y) moves along the curve in such a way that the y-coordinate is
A curve has equationa. Find dy/dx and d2y/dx2 in terms of x.b. Find the coordinates of the stationary point on the curve and determine its nature.c. Find the volume of the solid formed when the
The equation of a curve is y = 3 + 12x − 2x2.a. Express 3 + 12x − 2x2 in the form a − 2(x + b)2, where a and b are constants to be found.b. Find the coordinates of the stationary point on the
The diagram shows the curveand the line x + 2y = 4 that intersect at the points (4, 0) and (0, 2).a. Find the volume obtained when the shaded region is rotated through 360° about the x-axis.b. Find
A curve is such thatThe curve has a stationary point at P where x = 1. Given that the curve passes through the point (2, 5), find the coordinates of the stationary point P and determine its nature.
The function f : x ↦ 6 − 5cosx is defined for the domain 0 ≤ x ≤ 2π.a. Find the range of f.b. Sketch the graph of y = f(x).c. Solve the equation f(x) = 3. The function g : x ↦ 6 −
A curve is such thatand the point P(3, −4) is a point on the curve. The normal to the curve at P meets the curve again at Q.a. Find the equation of the curve.b. Find the equation of the normal to
a. Differentiate (2x√x − 1)5 with respect to x.b. Hence, find √ 3√x(2x√x – 1)ª dx. -
It is given thatfor x > 0. Show that f is an increasing function. 5 f(x)=2x-3, for x > 0.
The graph of y = x3 − 3 is transformed by applying a translation offollowed by a reflection in the x-axis.Find the equation of the resulting graph in the form y = ax3 + bx2 + cx + d. 2 0
The diagram shows sector OAB of a circle with centre O, radius 6cm and sector angle π/3 radians.The point X lies on the line OA and BX is perpendicular to OA.a. Find the exact area of the shaded
a. Find the first three terms in the expansion of (3 − 2x)7, in ascending powers of x.b. Find the coefficient of x2 in the expansion of (1 + 5x)(3 − 2x)7.
A circle has centre (3, −2) and passes through the point P(5, −6).a. Find the equation of the circle.b. Find the equation of the tangent to the circle at the point P, giving your answer in
a. The sum, Sn, of the first n terms of an arithmetic progression is given by Sn = 11n − 4n2. Find the first term and the common difference.b. The first term of a geometric progression
The diagram shows the curve that intersects the x-axis at A.The normal to the curve at B(4, 3) meets the x-axis at C. Find the area of the shaded region. y = √√√2x+1
A mathematical model for the inside of a bowl is obtained by rotating the curve x2 + y2 = 100 through 360° about the y-axis between y = −8 and y = 0. Each unit of x and y represents 1cm.a. Find
Use integration to prove that the volume, V cm3, of a sphere with radius r cm is given by the formula 4 .3 3 Tr
a. Differentiate (x2 − 3x + 5)6 with respect to x.b. Hence, find [2(2x − 3)(x² – 3x + 5)³ dx.
The diagram shows the curve y = 6x − x2 and the line y = 5. Find the area of the shaded region. I y = 6x-x² 1 I y = 5 X

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