Questions and Answers of Cambridge International AS & A Level Further Mathematics

Prove, by mathematical induction, that 32n - 1 is divisible by 8 for all n ≥ 1.
The sequence a1, a2, a3, ... is such that a1 > 5 and an+1 = 4an/5 + 5/an for every positive integer n.Prove by mathematical induction that an > 5 for every positive integer n.Prove also that an
Prove, by mathematical induction, that 3(3n+2) + 4 is divisible by 13 for n ≥ 0.
Prove by mathematical induction thatHence, determine an expression for 1). %3D 3 r=1
A function is given as y = cos2x + 2sin 2x.a. Find the second derivative.b. Using mathematical induction, show that = (-1)"22"[cos 2.x + 2 sin 2x] for values n>1. dx" 2n
The cubic equation x3 – 5x2 + 1 = 0 has roots α, β, γ.a. Find the values of S1 and S3.Another cubic equation has roots α2, β2, γ2.b. Find the cubic equation with these roots.c. Hence, or
A function is given as f(n) = (2n + 1)3 + (3n - 2)2 + n + 3. Using mathematical induction, prove that f(n) is always even for n ≥ 0.
The recurrence relation un+1 = 2un - 2 has first term u1 = 10.a. Find the values of u2, u3 and u4b. Prove by mathematical induction that un 4 × 2n + 2.
Using mathematical induction, prove that 8n - 1 is always divisible by 7.
The curve C is given as y = 2 + x/x2 + 5x + 4.a. Write down the asymptotes of C.b. Find dy/dx.c. Determine the number of turning points.d. Sketch the curve C, showing all asymptotes, and stating
Prove, by mathematical induction, that un > 1/2 is true for the relation 2 + 1 where u = 1. Un+1 Un + 2
Prove, by mathematical induction, that n3 - n is divisible by 6 for n ≥ 2.
Three planes are given as:∏1: x- 2y + 3z = 5, ∏2: -x + 4y + z = -5 and ∏3: 2x - 2y + 9z = 9.By writing these three equations in the form Ax = b, state the matrix A and perform row operations on
Using mathematical induction, prove that >(3,2 + r) = n(n+ 1).
The polar curve C is given as r = 2 sin 2θ, for 0 ≤ θ ≤ π/2.a. Sketch the curve C.b. Find the greatest distance of the curve from the y-axis.c. Evaluate the area inside the polar curve C, for
A function is given as f(n) = 34n+3 + 72n+1 + 6. Show by using proof by induction that f(n) is always divisible by 8 for n ≥ 0.
A matrix is given asa. Determine the values of A2, A3 and A4.b. Prove, by mathematical induction, thatc. State An when n → ∞. 3 A = -1 1
Four points A(2, 3, 1), B(0, 4, -.3), C(2, 2, 0) and D(-1, 0, 1) are given.The line I1 passes through A and B, and the line I2 passes through C and D.a. Find the angle between the lines I1 and I2.b.
Prove by mathematical induction that, for all values n ≥ 1, 7n + 23n - 1 is divisible by 7.
Express un = 1/4n2 – 1 in partial fractions, and hence findIn terms of n.Deduce that the infinite series u1 + u2 + u3 + … is convergent and state the sum to infinity.
The equation x3 - 4x + 2 = 0 has roots α, β, γ.a. Find the value of -1/α2 + 1/β2 + 1/γ2.b. Show that the matrixis non-singular. 1 а 1 2y 0 B 0 3B/
The curve C has equation y = 5(x – 1)(x + 2)/(x – 2)(x + 3).i. Express y in the form P + Q/x – 2 + R/x + 3.ii. Show that dy/dx = 0 for exactly one value of x and find the corresponding value of
Given that the matrixis non-singular find the matrix B such that BA2 = I. (2 II
Given thathas first term u1 = 3, prove, by mathematical induction, that un < 4 for all n ≥ 1. 4и, + 7 Un+1 и, + 2
Given that x3 - 3x2 + 12 = 0 has roots α, β, γ, find the following values:a. α + β + γ and αβ + αγ + βγb. α2 + β2 + γ2
The quartic equation 5x4 - 3x3 + x - 13 = 0 has roots α, β, γ, δ. Find:a. Σα and Σα2b. Σ1/α
Each of the following quadratic equations has roots α, β. Find the values of α + β and 4.a. x2 + 5x + 9 = 0b. x2 - 4x + 8 = 0c. 2x2 + 3x – 7 = 0
Each of the following cubic equations has roots α, β, γ. Find, for each case, α + β + γ and αβγ.a. x3 + 3x2 – 5 = 0b. 2x3 + 5x2 - 6 = 0c. x3 + 7x – 9 = 0
For each of the following quartic equations, find the values of Σα and Σαβ.a. x4 - 2x3 + 5x2 + 7 = 0b. 2x4 + 5x3 - 3x + 4 = 0c. 3x4 - 2x2 + 9x - 11 =0
The quadratic equation x2 + 5x + 3 = 0 has roots α, β. Find the quadratic equation with roots 3α, 3β.
The roots of the equation x3 + 4x – 1 = 0 are α, β and γ. Use the substitution y = 1/1 + x to show that the equation 6y3 – 7y2 + 3y – 1 = 0 has roots 1/α + 1, 1/β + 2 and 1/γ + 1. 1 For
Given that 3x2 + 4x + 12 = 0 has roots α, β, find:a. α + β and αβb. α2 + β2
The quadratic equation 2x2 - 4x + 7 = 0 has roots α, β.a. Find the quadratic equation with roots α2, β2.b. Find the quadratic equation with roots 2α - 3, 2β - 3.
The roots of the quartic equation x4 - 4x3 + 3x2 – 4x + 1 = 0 are α, β, γ and δ.Find the values ofUsing the substitution y = x + 1, find a quartic equation in y. Solve this quartic equation and
The roots of each of the following cubic equations are α, β, γ. In each case, find the values of S2 and S-1.a. x3 - 2x2 + 5= 0b. 3x3 + 4x – 1 = 0c. x3 + 3x2 + 5x – 7 = 0
A quartic equation is given as x4 + x + 2 = 0. It has roots α, β, γ, δ. State the values of S1 and S-1, and find the value of S2.
Given that 3x2 - 2x + 9 = 0 has roots α, β, find the quadratic equation with roots α + 1/α, β + 1/β.
The cubic equation x3 - x2 - 3x - 10 = 0 has roots α, β, γ.i. Let u = -α + β + γ. Show that u + 2α = 1, and hence find a cubic equation having roots -α + β + γ, α -β + y, α + β –
If a + b = -3 and α2 + β2 = 7, find the value of ab and, hence, write down a quadratic equation with roots α and β.
The cubic equation x3 - x + 7 = 0 has roots α, β, γ. Find the values of Σα and Σα2.
The quartic equation 2x4 + x3 - x + 7 = 0 has roots α, β, γ, δ. Given that S3 = 11/8, and using Sn, find the value of S4.
The quadratic equation x2 - 4.x + 9 = 0 has roots α, β. Find the quadratic that has roots 1/α, 1/β.
If x2 + bx + c = 0 has roots a and A prove that:a. If α = 3β, then b2 = 16/3cb. if α = β, then b2 = 4(c + 1).
Given that 2x3 + 5x2 + 1 = 0 has roots α, β, γ, and that Sn = αn + βn + γn, find the values of S2 and S3.
You are given that x4 - x3 + x + 2 = 0, where the roots are α, β, γ, δ. Find the values of Σα, Σα2 and Σ1/α. Hence, determine the value of Σα3.
Given that 2x3 - 5x + 1 = 0 has roots α, β, γ, find the cubic equation with roots α2, β2, γ2. Hence, find the value of S4.
You are given the quadratic equation px2 + qx - 16 = 0, which has roots a and ft. Given also that α + β = -1/2 and αβ = -8, find the values of p and q.
The cubic equation x3 + ax2 + bx + a = 0 has roots α, β, γ, and the constants a, b are real and positive.a. Find, in terms of a and b, the values of Σα and Σ1/α.b. Given that Σα = Σ1/α,
The quartic polynomial x4 + ax2 + bx + 1 = 0 has roots α, β, γ, δ. Given that S2 = S-1, find 33 in terms of a.
The cubic equation x3 + 3x2 - 1 = 0 has roots α, β, γ. Show that the cubic equation with roots α + 2/α, β + 2/β, γ + 2/γ is y3 – 3y2 – 9y + 3 = 0. Hence, determine the values of:a. (α +
The quadratic equation x2 + 2x - 6 = 0 has roots α and β. Find the values of (α – β)2 and 1/α2 + 1/β2.
The cubic equation x3 - x + 3 = 0 has roots α, β, γ.a. Using the relation Sn, = αn + βn + γn, or otherwise, find the value of S4.b. By considering S1 and S4, determine the value of α3(β + γ)
The polynomial 3x4 + 2x3 + 7x2 + 4 = 0 has roots α, β, γ δ, where Sn = αn + βn + γn + δn.a. Find the values of S1 and S2.b. Find the values of S3 and S4.c. Are there any complex roots? Give a
A quartic equation, 2x4 - x3 - 6 = 0, has roots α, β, γ, δ. Show that the quartic equation with roots α3, β3, γ3, δ3 is 8y4 - y3 - 18y2 - 108y - 216 = 0. Hence, find the values of S6 and S-3.
A quadratic equation has roots α and β. Given that 1/α + 1/β = 1/2 and α2 + β2 = 12, find two possible quadratic equations that satisfy these values.
A cubic polynomial is given as 2x3 - x2 + x - 5 = 0, having roots α, β, γ.a. Show that 2Sn+3 + Sn+1 – 5Sn = 0.b. Find the value of S-2.
For the polynomial x4 + ax3 + bx2 + c = 0, with roots α, β, γ and δ it is given that α + β + γ + δ = 2, αβγδ = 1 and α2 + β2 + γ2 + δ2 = 0. Find the values of the coefficients a, b
The cubic equation x3 - x + 4 = 0 has roots α, β, γ. Find the cubic equation that has roots α2, β2, γ2. Hence, or otherwise, determine the values of S6, S8 and S10.
The quadratic equation 3x2 + 2x - 4 = 0 has roots a and P. Find the values of S1, S2 and S-1.
The cubic equation px3 + qx2 + r = 0 has roots α, β, γ. Find, in terms of p, q, r:a. S1b. S2c. S3
The roots of the quartic x4 - 2x3 + x2 - 4 = 0 are α, β, γ, δ. Show that S4 = 9S3.
For each curve given, find the equations of the asymptotes. a.b. 3x +x+3 x+ 1
You are given the quadratic equation 4x2 - x + 6 = 0 which has roots α and β.
The equation x3 + px2 + qx + r = 0 is such that S1 = 0, S2 = - 2 and = S-1 = 1/5. Find the values of the constants p, q, r.
For each of the following, determine the equations of the asymptotes of the curves. a.b.c. 2х + 3 y = х2 + 3x + 2
For the curve y = 2x -3/x + 2 , determine the values of x such that y > 3.
Determine the number of solutions for each of the following equations.a.b. 1저2-21지- 3 = 0 |x| – 2
A curve is given as f(x) = x2 – x – 5/x – 3.Find the equations of the asymptotes of the curve.Sketch the curve-Hence, or otherwise, sketch the curve 1/f(x). State the equations of the vertical
Write y = 2x2/x2 - 5x – 6 in partial fractions. Hence, state the equations of the asymptotes.
Find the number of turning points for each of the following curves. a. x2 – 5/x + 3b. y = x2 + 5x – 4/2x - 1
For the curve y = x2/x + 3, find the range of values that y can have.
The curve C has equation y = 2x2 + 5x – 1/x + 2.Find the equations of the asymptotes of C.Show that dy/dx > 2 at all points on C.Sketch C.
Determine the number of turning points for each of the following curves. a.b. ソ= (x + 1)(x - 3)
A curve is given as y = 6x2 + x -6/2x -1. Write the curve in the form y = Ax + B + C /2x – 1 and state the questions of the asymptotes.
Given that y = x2 - 2x — 4/3x — 2 show that the range of the curve is y ∈ R.
Determine the equations of the asymptotes of the curve in the following cases. a. y = 1/x2 – 4b. y = 1/x2 + 2x + 5
The curve C has equation y = 2x2 – 3x – 2/ x2 – 2x + 1. State the equations of the asymptotes of C.Show that y ≤ 25/12 at all points of C.Find the coordinates of any stationary points of
The curve, C, is given as y = 2x + 4/x + 1. Find the equations of the asymptotes for C.
The curve, C, is denoted as y = x2 + 3/x – 2. Find the equations of the asymptotes and, hence, sketch C.
For the curve y = x – 1/x + 2, determine the values of x that satisfy y < 5.
Given that f(x) = x2 - x - 6, sketch the curve y = 1/f(x), showing all the asymptotes and points of intersection.
Sketch the curve y = 3x + 2/x -2, showing all points of intersection with the coordinate axes.
The equation of a curve is given as y = x2 + λx/x + 1, where A is a constant.a. Given that one of the asymptotes is y = x + 2, find the value of A.b. State the other asymptote and sketch the curve.
The curve, C, is given as y = x2 + x – 3/x – 3.a. Find the range of values of y that the curve cannot have.b. Determine the exact coordinates of the turning points.
Sketch the curve y2 = 7 - 3x, showing any points of intersection with the coordinate axes.
A curve is given as y = x – 1/x — 2.a. Write this in the form y = A + x - 2b. State the equations of the asymptotes.c. Sketch the curve, showing points of intersection with the coordinate axes.
A curve is given as y = x2 – 2x + 1/x - 4a. Find the equations of the asymptotes.b. Show that one of the turning points is (1, 0) and determine the other turning point.c. Sketch the curve.
The curve y = 3x - 2 - 4x2/x – 2 – 3x2 has a horizontal asymptote at y = k.a. State the value of k.b. Show that there are no vertical asymptotes.c. Determine the range for y.
Given that f(x) = x/x2 – x – 2, sketch the following curves. Show any asymptotes in each case.a. y = |f(x)|b. y = f(|x|)
An equation is given as y = x2 – x + 1/3 – x.a. Show that the curve can be written in the form ,y = αx + β + γ/3 - x, stating the values of α, β and γ.b. Show that there are two turning
The curve, C, is given as y = 1 - 3x/ x2 + 3x - 10. Determine the values of x that satisfy y < 2.
A curve has equation y = 3 – x/x2 – 1.a. Write down the equations of the asymptotes.b. Find the x-coordinates of any stationary points.c. Sketch the curve.
The curve, C, is given as y = x2 – ax + b/2x – 1. Given that one of the asymptotes is y = 1/2x + 5/4 and that one of the points of intersection is (0, 4):a. Find the values of a and bb. Determine
The equation of a curve is y = x/x2 + x – 2.a. Write down the equations of the asymptotes.b. Show that y ∈ R.c. Determine the values of x that satisfy y > 1.
You are given that f(x) x2 + 2x -5/x + 3.a. Show that f(x) has no turning points.b. Show also that the asymptotes are y = x - 1 and x = - 3.c. Sketch the curve If(x)I, showing all the asymptotes and
Sketch the curve.The curve, C, is written as y = 2x/(x - 1)(x + 3)a. Show that C has no turning points.b. Sketch the curve C. Show on your sketch all the points of intersection with the coordinate
The curve y = x2 – 5/(x – 1)(x + 3) has two vertical asymptotes and one horizontal asymptote.a. Find the equations of these asymptotes.b. Determine the number of turning points.c. Sketch the
Find, in terms of n, an expression for 3 r=2 (- 1)r
Find the sum of the first n terms of the series 1/1 × 3 + 1/2 × 4 + 1/3 × 5 + … and deduce the sum to infinity.
Show thatCan be written as n/5n + 25. 1 (r+4)(r+5)

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