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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

Write each of the following in the form x + iy, where x and y are real values.a. (1 + 3i)(2 − i) b. (4 − 5i)(4 + 5i)c. (7 − 3i)2d. (3 − i)4e.f.g.h. 17-i 3+i
Relative to the origin O, the position vectors of the points A and B are given bya. Find a vector equation of the line AB.b. The line AB is perpendicular to the line L with vector equation:i. Find
At a point with coordinates (x, y) the gradient of a particular curve is directly proportional to xy2. At a particular point P with coordinates (1, 3) it is known that the gradient of this curve is
Write each of the following in the form x + iy, where x and y are real values.a. z1 + z2*b. z1* + z2c. z1z2d. z1/z2z1 = 5 − 3i and z2 = 1 + 2i
a. Express 5 - 2i/1+3i in the form x + iy where x and y are real numbers.b. Solve w2 − 2w + 26 = 0.c. On a sketch of an Argand diagram, shade the region whose points represent complex numbers
a. Given that z1 = −i is a root of the equation z3 + z2 + z + k = 0, find the value of the constant k.b. Write down the other complex root of the equation and find the third root, stating
Find the value of x and the value of y.a. (x + 2y) + i(3x − y) = 1 + 10i b. (x + y − 4) + 2xi = (5 − y)ic. (x − y) + (2x − y)i = −1
The point P( p, q, −1) lies on the line L with vector equation r = i − j + 2k + λ(i + 8j + k).a. Find the value of each of the constants p and q.b. The position vector of Q, relative to the
Find the roots of the equation 2z2 + z + 3 = 0, giving your answer in modulus argument form.
On a single Argand diagram, sketch the loci |z| = 4 and |z + 2| = |z − 4|. Hence determine complex numbers that satisfy both loci, giving your answers in Cartesian form.
a. Show thatis a solution of the equation 5z2 − 2z + 5 = 0.b. Write down the other solution of the equation. N || 1 5 + 2√√6. 5
i. Use the trapezium rule with two intervals to estimate the value ofgiving your answer correct to 2 decimal places.ii. Find 1 So 6 +²2ex dx, 0
The complex number z is given by z = (√3) + i.i. Find the modulus and argument of z.ii. The complex conjugate of z is denoted by z*. Showing your working, express in the form x + iy, where x and y
The variables x and θ are related by the differential equation.where 0 < θ < 1/2 π,When θ = 1/12π, x = 0. Solve the differential equation, obtaining an expression for x in terms of θ, and
It is given that z − 3 is a factor of z3 − 3z2 + 25z − 75.Solve z3 − 3z2 + 25z −75 = 0.
On an Argand diagram, sketch the locus |z| =|z + 8i|. Find the Cartesian equation of this locus.
Find the quadratic equations that have the following roots.a.b.c.d. α = -7i ß= 7i
PQRS is a parallelogram. The vertices, P,Q and R have position vectors, relative to an origin O,a. Find OSvector.b. Find the lengths of the sides of the parallelogram.c. Find the interior angles of
a. Find the exact values of the modulus and argument of z.b. Given thatWrite z/w in the form reiθ , where > 0 and − π < θ < π.z = 4/3 − 4i w = 2√2 π COS +isin 12 I 12
Find the value of x and the value of y, when x and y are real and positive.(x + iy)2 = 55 + 48i
Sketch the locus z − (3 − 6i) = 3 on an Argand diagram. Write down the Cartesian equation of this locus.
The complex number u is defined byi. Without using a calculator and showing your working, express u in the form x + iy, where x and y are real.ii. Sketch an Argand diagram showing the locus of the
It is given that 2z + 1 is a factor of 2z3 − 11z2 + 14z + 10.Solve 2z3 − 11z2 + 14z + 10 = 0.
In a certain country the government charges tax on each litre of petrol sold to motorists. The revenue per year is R million dollars when the rate of tax is x dollars per litre. The variation of R
Find the square root of:a. 24 − 10ib. 7 + (6√2)ic. 5/4 - i√2)id. 7 - 24ie. -4 + (2√5)if. Hele
For complex numbers z satisfying |z − 8 − 16i| = 2√5, find the least possible value of |z| and the greatest possible value of |z|.
Sketch the loci arg(z + 4 + 2i) = 5π/6 and |z - 5i| = 4 on an Argand diagram. Determine whether or not there is a complex number, z, that satisfies both loci.
The variables x and y are related by the differential equationGiven that y = 36 when x = 0, find an expression for y in terms of x. dy 6ye³x d.x 2+e³x.
a. Show that the straight line L with vector equationintersects with the line through the points A and B with coordinates (0, 2, 7) and (7, 1, 27), respectively and find the position vector of this
Find the complex number z satisfying the equation z + 4 = 3iz*.Give your answer in the form x + iy, where x and y are real.
Given that z = 3i is a root, solve z4 − 2z3 + 14z2 − 18z + 45 = 0.
The complex number z is defined byFind, showing all your working.i. An expression for z in the form reiθ, where r > 0 and −π < θ < π,ii. The two square roots of z, giving your answers
On a single Argand diagram, sketch the loci |z + 3 − 2i|= 4 and arg(z + 1)π/4. Hence determine the value of z that satisfies both loci, giving your answer in Cartesian form.
z = 5 + i√3 is a root of a quadratic equation. Find this quadratic equation.
The variables x and y satisfy the differential equationthat y = 2 when x = 1. Solve the differential equation and obtain an expression for y in terms of x in a form not involving logarithms. X dy =
In an electrical circuit, the voltage (volts), current (amperes) and impedance (ohms) are related by the equation voltage = current × impedance.The voltage in a particular circuit is 240V and the
The complex number u is defined byi. Showing all your working, find the modulus of u and show that the argument of u is -1/2π.ii. For complex numbers z satisfying arg(z - u) = 1/4π find the least
Throughout this question the use of a calculator is not permitted.i. The complex numbers u and v satisfy the equations u + 2v = 2i and iu + v = 3.Solve the equations for u and v, giving both answers
With respect to the origin O, the position vectors of the points A, B,C and D are given bya. In the case where ABC is a right angle, find the possible values of the constant m.b. In the case where D
Given that y = 1 when x = 0, solve the differential equationobtaining an expression for y in terms of x. dy dx 4x(3y² +10y + 3),
The variables x and y are related by the differential equationIt is given that y = 2 when x = 0. Solve the differential equation and hence find the value of y when x = 0.5, giving your answer correct
A tank containing water is in the form of a cone with vertex C. The axis is vertical and the semi-vertical angle is 60°, as shown in the diagram. At time t = 0, the tank is full and the depth of
The complex number w is defined by w = 22 + 4i/(2 - i2).i. Without using a calculator, show that w = 2 + 4i.ii. It is given that p is a real number such that 1/4π < arg(w + p) < 3/4π. Find
The complex number 3 − i is denoted by u. Its complex conjugate is denoted by u*.i. On an Argand diagram with origin O, show the points A, B and C representing the complex numbers u, u* and u* −
The line L1 has vector equationThe line L2 has vector equationa. In the case where m = 2, show that L1 and L2 do not intersect.b. Find the value of m in the case where L1 and L2 intersect.c. For
a. Find, in the form r = a + λb, a vector equation of the line AB where the points have coordinates A(2, 5, 7) and B(9, −1, −2).b. Find the obtuse angle between the line AB and a line in
The complex number 1 +(√2)i is denoted by u. The polynomial x4 + x2 + 2x + 6 is denoted by p(x).i. Showing your working, verify that u is a root of the equation p(x) = 0, and write down a
i. Showing all your working and without the use of a calculator, find the roots of the complex number 7 −(6√2)i. Give your answers in the form x + iy, where x and y are real and exact.ii.
The number of birds of a certain species in a forested region is recorded over several years. At time t years, the number of birds is N, where N is treated as a continuous variable.The variation in
u = 5 - 2ia. On an Argand diagram, show the points A, B and C representing the complex numbers u, u* and −u, respectively.b. The points ABCD form a rectangle. Write down the complex number
Solve these equations.a. z2 + 2z + 13 = 0b. z2 + 4z + 5 = 0c. 2z2 − 2z + 5 = 0d. z2 − 6z + 15 = 0e. 3z2 + 8z + 10 = 0f. 2z2 + 5z + 4 = 0
Given that y = 0 when x = 1, solve the differential equationobtaining an expression for y2 in terms of x. dy ху y = y2 + 4, = dx
a. The complex number z is defined as z = k − 6i, where k is a real value.Find and simplify expressions, in terms of k, for zz* and z/z*,your answers in the form x + iy where x and y are real.b.
Solve:a. (z − 5)3 = 8b. (2z + 3)3 = 1/64
The equationhas one real root, denoted by α.a. Find, by calculation, the pair of consecutive integers between whichα lies.b. Show that, if a sequence of values given by the iterative
Differentiate with respect to x.a. x sin x b. 5x cos 3x c. x2 tan x d. x cos3 2xe. 5 / cos 3xf. x / cos xg. tan x / xh. sin x / 2 + cos xi. sin x/ 3x - 1j. 1/sin3 2xk. 3x / sin
Find:a.b. c.d.e.f. [ 5e* (2 + e³x) dx
Differentiate with respect to x.a. xexb. x2e3xc. 5xe−2xd. 2√xexe. e6x/xf. e-2x/√xg. ex - 1/ex + 2h. xe3x + e6x/2i. x2ex - x/ex + 2
a. Show that the equation ln(x + 1) + 2x − 4 = 0 has a root between x = 1 and x = 2.b. Use the iterative formulawith an initial value of = 1.5 x1 to find the value of α correct places. Xn+l 4 -
The equation of a curve is 3x2 + 4xy + y2 = 24. Find the equation of the normal to the curve at the point (1, 3), giving your answer in the form ax + by + c = 0 where a, b and c are integers.
a. By sketching a suitable pair of graphs, show that the equation e2x+1 =14 − x3 has exactly one real root.b. Show by calculation that this root lies between 0.5 and 1.c. Show that this root also
The parametric equations of a curve are x = 3(1 + sin2 t), y = 2 cos3 t. Find dy/dx in terms of t, simplifying your answer as far as possible.
a. By sketching graphs of y = x3 + 5x2 and y = 5 − 2x, determine the number of real roots of the equation x3 + 5x2 + 2x − 5 = 0.b. Verify by calculation that the largest root of x3 + 5x2 + 2x −
The curve y = x3 + 5x − 1 cuts the x-axis at the point (α, 0).a. By sketching the graph of y = x3 and one other suitable graph, deduce that this is the only point where the curve y = x3 + 5x − 1
The equation (x − 0.5) e3x = 1 has a root α.a. Show, by sketching the graph of y = e3x and one other suitable graph, that α is the only root of this equation.b. Use the iterative formula xh+1=
a. By sketching a suitable pair of graphs, show that the equation x3 + 10x = x + 5 has only one root that lies between 0 and 1.b. Use the iterative formulawith a suitable value for x1, to find the
The equation of a curve is x2y + y2 = 6x.i. Show thatii. Find the equation of the tangent to the curve at the point with coordinates (1, 2), giving your answer in the form ax + by + c = 0. dy dx 6-
The sequence of values given by the iterative formulawith initial value x1 = 1, converges to α.a. Use this formula to find α correct to 2 decimal places, showing the result of each iteration to 4
a. On the same axes, sketch the graphs of y = ln(x + 1) and y = 3x − 4.b. Hence deduce the number of roots of the equation ln(x + 1) − 3x + 4 = 0.
The equation cos−1 3x = 1 − x has a root α.a. Show, by calculation, that α is between π/15 and π/12.b. Show that the given equation can be rearranged into the formwith a suitable starting
Given thatfind the values of A, B,C and D. x³ + x²-7 x-3 Ax² + Bx + C + D x-3'
Given thatfind the values of A, B,C, D and E. x4 + 5x² - 1 x + 1 Ax² + Bx² + Cx + D + E x+1'
Express the following proper fractions as partial fractions.a.b.c.d.e.f. 2x (x + 2)²
Find the first 5 terms, in ascending powers of x, in the expansion of 2 + 3x V1−5x2
a. Expressin partial fractions.b. Hence obtain the expansion ofin ascending powers of x, up to and including the term in x3. 7x² + 4x + 4 (1-x)(2x² +1)
The equation In x2 - 6/2 = x - 5 has a root α, such that p < α <q, where p and q are consecutive integers.a. Show that α also satisfies the equation x = ln x + 2.b. Find the value of p and
Express the following proper fractions as partial fractions.a.b.c.d. 2x²-3x+2 x(x² + 1)
Express the following improper fractions as partial fractions.a.b.c.d. 2x² + 3x + 4 (x - 1)(x + 2)
Expand 3√1 − (6x) in ascending powers of x up to and including the term in x3, simplifying the coefficients.
a. Expressin partial fractions.b. Hence obtain the expansion ofin ascending powers of x, up to and including the term in x2. 19-7x - 6x² (2x + 1)(2-3x)
Investigate whether the seriesis a telescoping series. If it is a telescoping series, find an expression for the sum of the first n terms and the sum to infinity. 1 1 xxa+zxdxaxxs 4x536 + 1x2x 3 2x 3
Expand (2 - x)(1 + 3x)1/2 in ascending powers of x, up to and including the term in x2, simplifying the coefficients.
Given thatfind the values of A, B,C and D. 9x3-11x² + 8x - 4 x²(3x - 2) = B C A++ + x X D 3x - 2
Leti. Express f(x) in partial fractions.ii. Hence obtain the expansion of f(x) in ascending powers of x, up to and including the term in x2. f(x) = 2x² - 7x-1 (x-2)(x² + 3)
Express 7x2 - 3x +21/(x +2)(x2 - 5) in the partial fractions.
when is (3 - 2x) expanded the coefficient of x2 is −15. Find the two possible values of a.
The first 3 terms in the expansion of (1 + ax)n are 1 − 24x + 384x2.a. Find the value of a and the value of n.b. Hence find the term in x3.
Express 4x2 - 5x + 3/(x+ 2)(2x -1) in partial fractions.
Expandin ascending powers of x, up to and including the term in x3. 1 + 2x V 1 - x
i. Expand 1/√(1 - 4x) in ascending powers of x, up to and including the term in x2, simplifying the coefficients.ii. Hence find the coefficient of x2 in the expansion of 1 + 2x √√(4-16x)
i. Prove the identity tan2θ − tanθ Ξ tanθ sec 2θ.ii. Hence show that pfr tan 0 sec 20 de 2 In 3
The diagram shows the curve y = x3 ln x and its minimum point M.i Find the exact coordinates of M. ii Find the exact area of the shaded region bounded by the curve, the x-axis and the line x =
The diagram shows the curveThe x-coordinate of the maximum point is and the shaded region is enclosed by the curve and the lines x = α and y = 0.i. Show that α = 2/3ii. Find the exact value of the
By first expressing 4x2 + 5x + 3/2x2 + 5x + 3 in partial fractions, show that 0 4x² + 5x + 3 2x² + 5x+2 -dx = 8 - In 9.
Leta. Using the substitution x = sinθ, show thatb. Hence show that I = 1 4x² 2 √1-x .d.x.
Expressin partial fractions.ii. Hence obtain the expansion ofin ascending powers of x, up to and including the term in x2. 4 + 12x + x² (3 - x)(1+2x)²
The diagram shows the curveand its maximum point M. The shaded region R is enclosed by the curve, the x-axis and the lines x = 1 and x = p.i. Find the exact value of the x-coordinate of M.ii.
When (1 + ax)−2, where a is a positive constant, is expanded in ascending powers of x, the coefficients of x and x3 are equal.i. Find the exact value of a.ii. When a has this value, obtain the
The diagram shows the curve y = x2e2−x and its maximum point M.i. Show that the x-coordinate of M is 2.ii. Find the exact value of 2 JO x²e²-x dx.
Leta. Express f(x) in partial fractions.b. Hence obtain the expansion of f(x) in ascending powers of x, up to and including the term in x3. f(x) = 5x – 2 (x - 1)(2x² - 1)

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