Edexcel AS And A Level Mathematics Pure Mathematics Year 1/AS 1st Edition Greg Attwood - Solutions

The circle C with equation (x + 5)2 + (y + 2)2 = 125 meets the positive coordinate axes at A(a, 0) and B(0, b).a. Find the values of a and b.b. Find the equation of the line AB.c. Find the area of the triangle OAB, where O is the origin.
The circle C has equation x2 + 6x + y2 − 2y = 7. The lines l1 and l2 are tangents to the circle. They intersect at the point R(0, 6).a. Find the equations of lines l1 and l2, giving your answers in the form y = mx + b.b. Find the points of intersection, P and Q, of the tangents and the circle.c.
The point P(7, −14) lies on the circle with equation x2 + y2 + 6x − 14y = 17. The point Q also lies on the circle such that PQ is a diameter. Find the coordinates of point Q.
The line with equation y = 2x + 5 meets the circle with equation x2 + kx + y2 = 4 at exactly one point. Find two possible values of k.
The circle, centre (p, q) radius 25, meets the x-axis at (−7, 0) and (7, 0), where q > 0.a. Find the values of p and q.b. Find the coordinates of the points where the circle meets the y-axis.
The point A(−3, −7) lies on the circle centre (5, 1). Find the equation of the tangent to the circle at A.
The line segment AB is a chord of a circle centre (2, −1), where A and B are (3, 7) and (−5, 3) respectively. AC is a diameter of the circle. Find the area of △ABC.
Simplify these fractions as far as possible: a b с 3x4 - 21x 3x x² - 2x - 24 x² - 7x + 6 2x² + 7x - 4 2x² + 9x + 4
The line y = −3x + 12 meets the coordinate axes at A and B.a. Find the coordinates of A and B.b. Find the coordinates of the midpoint of AB.c. Find the equation of the circle that passes through A, B and O, where O is the origin.
The points A(−3, −2), B(−6, 0) and C(1, q) lie on the circumference of a circle such that ∠ BAC = 90°.a. Find the value of q.b. Find the equation of the circle.
The points R(−4, 3), S(7, 4) and T(8, −7) lie on the circumference of a circle.a. Show that RT is the diameter of the circle.b. Find the equation of the circle.
Simplify these fractions as far as possible: a d g j m (x+3)(x - 2) (x - 2) x² + 10x + 21 (x + 3) x² + x - 20 x² + 2x - 15 2x² + 7x + 6 (x - 5)(x + 2) 2x² + 3x + 1 x²-x-2 b h k n (x + 4)(3x - 1) (3x - 1) x² + 9x + 20 (x + 4) x² + 3x + 2 x2+5x+4 2x² + 9x - 18 (x + 6)(x + 1) x² + 6x +
The points A(−4, 0), B(4, 8) and C(6, 0) lie on the circumference of circle C.Find the equation of the circle.
The points A(−7, 7), B(1, 9), C(3, 1) and D(−7, 1) lie on a circle.a. Find the equation of the perpendicular bisector of:i. AB ii. CDb. Find the equation of the circle.
Prove that when n is an integer and 1 ≤ n ≤ 6, then m = n + 2 is not divisible by 10.
Prove that X 1 + √2 ==x√2 - x.
Write each polynomial in the form (x ± p)(ax2 + bx + c) by dividing:a. x3 + 6x2 + 8x + 3 by (x + 1) b. x3 + 10x2 + 25x + 4 by (x + 4)c. x3 − x2 + x + 14 by (x + 2) d. x3 + x2 − 7x − 15 by (x − 3)e. x3 − 8x2 + 13x + 10 by (x − 5) f. x3 − 5x2 − 6x − 56 by (x − 7)
Use the factor theorem to show that:a. (x − 1) is a factor of 4x3 − 3x2 − 1 b. (x + 3) is a factor of 5x4 − 45x2 − 6x − 18c. (x − 4) is a factor of −3x3 + 13x2 − 6x + 8.
Prove that n2 − n is an even number for all values of n.
where a, b and c are constants. Work out the values of a, b and c. 6x3 + 3x2 84x 6x2 33x + 42 - ax(x + b) X + C
Prove that every odd integer between 2 and 26 is either prime or the product of two primes.
Divide 3x3 + 12x2 + 5x + 20 by (x + 4)
Write each polynomial in the form (x ± p)(ax2 + bx + c) by dividing:a. 6x3 + 27x2 + 14x + 8 by (x + 4) b. 4x3 + 9x2 − 3x − 10 by (x + 2)c. 2x3 + 4x2 − 9x − 9 by (x + 3) d. 2x3 − 15x2 + 14x + 24 by (x − 6)e. −5x3 − 27x2 + 23x + 30 by (x + 6) f. −4x3 + 9x2 − 3x + 2
Show that (x − 1) is a factor of x3 + 6x2 + 5x − 12 and hence factorise the expression completely.
Prove that the sum of two consecutive square numbers between 12 to 82 is an odd number.
Simplify (2x3 + 3x + 5)/(x + 1)
Divide:a. x4 + 5x3 + 2x2 − 7x + 2 by (x + 2) b. 4x4 + 14x3 + 3x2 − 14x − 15 by (x + 3)c. −3x4 + 9x3 − 10x2 + x + 14 by (x − 2) d. −5x5 + 7x4 + 2x3 − 7x2 + 10x − 7 by (x − 1)
Show that (x + 1) is a factor of x3 + 3x2 − 33x − 35 and hence factorise the expression completely.
A student is trying to prove that x3 + y33. The student writes:a. Identify the error made in the proof.b. Provide a counter-example to show that the statement is not true. (x + y)² = x3 + 3x²y + 3xy² + y² which is less than x3 + y3 since 3x²y + 3xy² > 0
Prove that (x + √y)(x − √y) ≡ x2 − y.
Prove that all cube numbers are either a multiple of 9 or 1 more or 1 less than a multiple of 9.
Prove that 2 ₂ ( 7 ) − ₂ ( ² + x) = xq + ₂ x - zx
a. Show that (x − 3) is a factor of 2x3 − 2x2 − 17x + 15.b. Hence express 2x3 − 2x2 − 17x + 15 in the form (x − 3)(Ax2 + Bx + C), where the values A, B and C are to be found.
Divide:a. 3x4 + 8x3 − 11x2 + 2x + 8 by (3x + 2) b. 4x4 − 3x3 + 11x2 − x − 1 by (4x + 1)c. 4x4 − 6x3 + 10x2 − 11x − 6 by (2x − 3) d. 6x5 + 13x4 − 4x3 − 9x2 + 21x + 18 by (2x + 3)e. 6x5 − 8x4 + 11x3 + 9x2 − 25x + 7 by (3x − 1) f. 8x5 − 26x4 + 11x3 +
Show that (x − 5) is a factor of x3 − 7x2 + 2x + 40 and hence factorise the expression completely.
Prove that (2x − 1)(x + 6)(x − 5) ≡ 2x3 + x2 − 61x + 30.
Find a counter-example to disprove each of the following statements:a. If n is a positive integer then n4 − n is divisible by 4.b. Integers always have an even number of factors.c. 2n2 − 6n + 1 is positive for all values of n.d. 2n2 − 2n − 4 is a multiple of 3 for all integer values of n.
a. Show that (x − 2) is a factor of x3 + 4x2 − 3x − 18.b. Hence express x3 + 4x2 − 3x − 18 in the form (x − 2)(px + q)2, where the values p and q are to be found.
Prove that the solutions of x2 + 2bx + c = 0 are x = -b ± √b² - C.
Divide:a. x3 + x + 10 by (x + 2) b. 2x3 − 17x + 3 by (x + 3)c. −3x3 + 50x − 8 by (x − 4)
Show that (x − 2) is a factor of 2x3 + 3x2 − 18x + 8 and hence factorise the expression completely.
Prove that 3 2 (x - 1²/7) ² - x X نرا = x³ - 6x + 1/2 - X 8 x3
Factorise completely 2x3 + 3x2 − 18x + 8.
Given that a is a positive real number, prove that: て ≥2 D 1 +D
Divide:a. x3 + x2 − 36 by (x − 3) b. 2x3 + 9x2 + 25 by (x + 5)c. −3x3 + 11x2 − 20 by (x − 2)
Each of these expressions has a factor (x ± p). Find a value of p and hence factorise the expression completely.a. x3 − 10x2 + 19x + 30 b. x3 + x2 − 4x − 4 c. x3 − 4x2 − 11x + 30
Prove that 5 - (x²³ − +/ ) ( x ² + x^²³) = x ² + - +/- )
Prove that for all real values of x(x + 6)2 ≥ 2x + 11
It is claimed that the following inequality is true for all negative numbers x and y:The following proof is offered by a student:a. Explain the error made by the student.b. By use of a counter-example, verify that the inequality is not satisfied if both x and y are negative.c. Prove that this
Find the value of k if (x − 2) is a factor of x3 − 3x2 + kx − 10.
a. Prove that for any positive numbers p and q:b. Show, by means of a counter-example, that this inequality does not hold when p and q are both negative. p+q>√4pq
Show that x3 + 2x2 − 5x − 10 = (x + 2)(x2 − 5).
i. Fully factorise the right-hand side of each equation.ii. Sketch the graph of each equation.a. y = 2x3 + 5x2 − 4x − 3 b. y = 2x3 − 17x2 + 38x − 15 c. y = 3x3 + 8x2 + 3x − 2d. y = 6x3 + 11x2 − 3x − 2 e. y = 4x3 − 12x2 − 7x + 30
f(x) = 2x2 + px + q. Given that f(−3) = 0, and f(4) = 21:a. Find the value of p and qb. Factorise f(x).
Find the remainder when:a. x3 + 4x2 − 3x + 2 is divided by (x + 5) b. 3x3 − 20x2 + 10x + 5 is divided by (x − 6)c. −2x3 + 3x2 + 12x + 20 is divided by (x − 4)
Given that (x − 1) is a factor of 5x3 − 9x2 + 2x + a, find the value of a.
h(x) = x3 + 4x2 + rx + s. Given h(−1) = 0, and h(2) = 30:a. Find the values of r and sb. Factorise h(x).
Show that when 3x3 − 2x2 + 4 is divided by (x − 1) the remainder is 5.
Given that (x + 3) is a factor of 6x3 − bx2 + 18, find the value of b.
Use completing the square to prove that 3n2 − 4n + 10 is positive for all values of n.
g(x) = 2x3 + 9x2 − 6x − 5.a. Factorise g(x).b. Solve g(x) = 0.
Show that when 3x4 − 8x3 + 10x2 − 3x − 25 is divided by (x + 1) the remainder is −1.
Given that (x − 1) and (x + 1) are factors of px3 + qx2 − 3x − 7, find the values of p and q.
Use completing the square to prove that −n2 − 2n − 3 is negative for all values of n.
a. Show that (x − 2) is a factor of f(x) = x3 + x2 − 5x − 2.b. Hence, or otherwise, find the exact solutions of the equation f(x) = 0.
Show that (x + 4) is a factor of 5x3 − 73x + 28.
Given that (x + 1) and (x − 2) are factors of cx3 + dx2 – 9x – 10, find the values of c and d.
Prove that x2 + 8x + 20 > 4 for all values of x.
Given that −1 is a root of the equation 2x3 − 5x2 − 4x + 3, find the two positive roots.
Simplify (3x3 − 8x − 8)/(x − 2).
Given that (x + 2) and (x − 3) are factors of gx3 + hx2 – 14x + 24, find the values of g and h.
The equation kx2 + 5kx + 3 = 0, where k is a constant, has no real roots. Prove that k satisfies the inequality 0 ≤ k < 12/25.
f(x) = x3 – 2x2 – 19x + 20a. Show that (x + 4) is a factor of f(x).b. Hence, or otherwise, find all the solutions to the equation x3 – 2x2 – 19x + 20 = 0.
Divide x3 − 1 by (x − 1).
f(x) = 3x3 – 12x2 + 6x − 24a. Use the factor theorem to show that (x − 4) is a factor of f(x).b. Hence, show that 4 is the only real root of the equation f(x) = 0.
The equation px2 – 5x − 6 = 0, where p is a constant, has two distinct real roots. Prove that p satisfies the inequality p > – 25/24.
f(x) = 6x3 + 17x2 – 5x − 6a. Show that f(x) = (3x – 2)(ax2 + bx + c), where a, b and c are constants to be found.b. Hence factorise f(x) completely.c. Write down all the real roots of the equation f(x) = 0.
Divide x4 − 16 by (x + 2).
Given thata. Sketch the graph of y = f(x) – 2 and state the equations of the asymptotes.b. Find the coordinates of the point where the curve y = f(x) – 2 cuts a coordinate axis.c. Sketch the graph of y = f(x + 3).d. State the equations of the asymptotes and the coordinates of the point where
a. On the same axes sketch the curves with equations y = (x – 2)(x + 2)2 and y = –x2 – 8.b. Find the coordinates of the points of intersection.
a. Sketch the graph of y = x3 – 5x2 + 6x, marking clearly the points of intersection with the axes.b. Hence sketch y = (x – 2)3 – 5(x – 2)2 + 6(x – 2).
The curve C1 has equation y = −a/x2 where a is a positive constant. The curve C2 has the equation y = x2 (3x + b) where b is a positive constant.a. Sketch C1 and C2 on the same set of axes, showing clearly the coordinates of any point where the curves touch or cross the axes.b. Using your sketch
x2 − 8x − 29 ≡ (x + a)2 + b, where a and b are constants.a. Find the value of a and the value of b.b. Hence, or otherwise, show that the roots of x2 − 8x − 29 = 0 are c ± d√5, where c and d are integers.
a. Sketch the graphs of y = x2 + 1 and 2y = x – 1.b. Explain why there are no real solutions to the equation 2x2 – x + 3 = 0.c. Work out the range of values of a such that the graphs of y = x2 + a and 2y = x – 1 have two points of intersection.
a. Sketch the graph of y = x2(x – 3)(x + 2), marking clearly the points of intersection with the axes.b. Hence sketch y = (x + 2)2(x – 1)(x + 4).
a. Solve the simultaneous equations:b. Hence, or otherwise, find the set of values of x for which: 2x2 − 3x − 16 > 5 − 2x. y + 2x = 5 2x² 3x - y = 16.
a. Factorise completely x3 − 6x2 + 9x.b. Sketch the curve of y = x3 − 6x2 + 9x showing clearly the coordinates of the points where the curve touches or crosses the axes.c. The point with coordinates (−4, 0) lies on the curve with equation y = (x − k)3 − 6(x − k)2 + 9(x − k) where k is
The functions f and g are defined as f(x) = x(x − 2) and g(x) = x + 5, x ∈ ℝ. Given that f(a) = g(a) and a > 0, find the value of a to three significant figures.
a. Sketch the graphs of y = x2(x – 1)(x + 1) and y = 1/3x3 + 1.b. Find the number of real solutions to the equation 3x2(x – 1)(x + 1) = x3 + 3.
a. Sketch the graph of y = x3 + 4x2 + 4x.b. The point with coordinates (–1, 0) lies on the curve with equation y = (x + a)3 + 4(x + a)2 + 4(x + a) where a is a constant. Find the two possible values of a.
Sketch on separate axes the graphs of:f(x) = x(x − 2)2a. y = f(x)b. y = f(x + 3)Show on each sketch the coordinates of the points where each graph crosses or meets the axes.
An athlete launches a shot put from shoulder height. The height of the shot put, in metres, above the ground t seconds after launch, can be modelled by the following function: h(t) = 1.7 + 10t − 5t2 t > 0a. Give the physical meaning of the constant term 1.7 in the context of the model.b. Use
Find the set of values for which 6 x + 5 < 2, x = -5.
a. Sketch the graph of y = x(x + 1)(x + 3)2.b. Find the possible values of b such that the point (2, 0) lies on the curve with equation y = (x + b)(x + b + 1)(x + b + 3)2.
Given that f(x) = x2 − 6x + 18, x ≥ 0,a. Express f(x) in the form (x − a)2 + b, where a and b are integers. The curve C with equation y = f(x), x ≥ 0, meets the y-axis at P and has a minimum point at Q.b. Sketch the graph of C, showing the coordinates of P and Q. The line y = 41 meets
The figure shows a sketch of the curve with equation y = f(x). The curve crosses the x-axis at the points (2, 0) and (4, 0). The minimum point on the curve is P(3, −2). In separate diagrams, sketch the curves with equationa. y = −f(x)b. y = f(2x)On each diagram, give the coordinates of the
The function h(x) = x2 + 2 √2 x + k has equal roots.a. Find the value of k.b. Sketch the graph of y = h(x), clearly labelling any intersections with the coordinate axes.
The figure shows a sketch of the curve with equation y = f(x). The curve passes through the points (0, 3) and (4, 0) and touches the x-axis at the point (1, 0). On separate diagrams, sketch the curves with equationsa. y = f(x + 1)b. y = 2f(x)c. y = f(1/2x)On each diagram, show clearly the
The function g(x) is defined as g(x) = x9 − 7x6 − 8x3, x ∈ ℝ.a. Write g(x) in the form x3(x3 + a)(x3 + b), where a and b are integers.b. Hence find the three roots of g(x).
Given that x2 + 10x + 36 (x + a)2 + b, where a and b are constants,a. Find the value of a and the value of b.b. Hence show that the equation x2 + 10x + 36 = 0 has no real roots. The equation x2 + 10x + k = 0 has equal roots.c. Find the value of k.d. For this value of k, sketch the graph of y

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Edexcel AS And A Level Mathematics Pure Mathematics Year 1/AS 1st Edition Greg Attwood - Solutions

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