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

Solve the following equations:a. 3x2 + 5x = 2 b. (2x − 3)2 = 9 c. (x − 7)2 = 36 d. 2x2 = 8 e. 3x2 = 5f. (x − 3)2 = 13 g. (3x − 1)2 = 11 h. 5x2 − 10x2 = −7 + x + x2i. 6x2 − 7 = 11x j. 4x2 + 17x = 6x − 2x2
Write each of these expressions in the form p(x + q)2 + r, where p, q and r are constants to be found:a. 2x2 + 8x + 1 b. 5x2 − 15x + 3 c. 3x2 + 2x − 1 d. 10 − 16x − 4x2 e. 2x − 8x2 + 10
x2 − 14x + 1 = (x + p)2 + q, where p and q are constants.a. Find the values of p and q.b. Using your answer to part a, or otherwise, show that the solutions to the equation x2 − 14x + 1 = 0 can be written in the form r ± s√3 , where r and s are constants to be found.
2kx − y = 44kx + 3y = −2are two simultaneous equations, where k is a constant.a. Show that y = −2.b. Find an expression for x in terms of the constant k.
The curve and the line given by the equationswhere k is a non-zero constant, intersect at a single point.a. Find the value of k.b. Give the coordinates of the point of intersection of the line and the curve. kx² - xy + (k+1)x= 1 k 2x+y=1
a. By eliminating y from the equationsx + y = 2x2 + xy − y2 = −1show that x2 − 6x + 3 = 0.b. Hence, or otherwise solve the simultaneous equationsx + y = 2x2 + xy − y2 = −1giving x and y in the form a ± b√6, where a and b are integers.
The sketch shows the graphs of the straight lines with equations:y = x + 1, y = 7 – x and x = 1.a. Work out the coordinates of the points of intersection of the functions.b. Write down the set of inequalities that represent the shaded region shown in the sketch. कठफचलता 5- x = 1 3 4
On a coordinate grid, shade the region that satisfies the inequalities:y > x2 – 2 and y ≤ 9 – x2.
A person throws a ball in a sports hall. The height of the ball, h m, can be modelled in relation to the horizontal distance from the point it was thrown from by the quadratic equation:The hall has a sloping ceiling which can be modelled with equationDetermine whether the model predicts that the
Determine the number of points of intersection for these pairs of simultaneous equations. a y = 6x² + 3x - 7 y = 2x + 8 by=4x² 18x + 40 y = 10x -9 - cy=3x² - 2x +4 7x + y + 3 = 0
3x + ky = 8x − 2ky = 5are simultaneous equations where k is a constant.a. Show that x = 3.b. Given that y = 1/2 determine the value of k.
a. By eliminating y from the equationsy = 2 − 4x3x2 + xy + 11 = 0show that x2 − 2x – 11 = 0.b. Hence, or otherwise, solve the simultaneous equationsy = 2 − 4x3x2 + xy + 11 = 0giving your answers in the form a ± b√3, where a and b are integers.
Find the coordinates of the points at which the line with equation y = x − 4 intersects the curve with equation y2 = 2x2 − 17.
The sketch shows the graphs of the curves with equations:y = 2 – 5x – x2, 2x + y = 0 and x + y = 4.Write down the set of inequalities that represent the shaded region shown in the sketch. 8 7- A -9% 5- 3 Nort si9- 23
a. Given that 3x = 9y − 1, show that x = 2y − 2.b. Solve the simultaneous equations:x = 2y − 2x2 = y2 + 7
a. Find the range of values of k for which the equation x2 − kx + (k + 3) = 0 has no real roots.b. Find the range of values of p for which the roots of the equation px2 + px − 2 = 0 are real.
On a coordinate grid, shade the region that satisfies the inequalities: y > (x – 3)2, y + x ≥ 5 and y < x – 1.
Given that x ≠ 3, find the set of values for which 5 x-3
2x − py = 54x + 5y + q = 0are simultaneous equations where p and q are constants. The solution to this pair of simultaneous equations is x = q, y = −1. Find the value of p and the value of q.
One pair of solutions for the simultaneous equationsy = kx − 54x2 − xy = 6is (1, p) where k and p are constants.a. Find the values of k and p.b. Find the second pair of solutions for the simultaneous equations.
A swimmer dives into a pool. Her position, p m, underwater can be modelled in relation to her horizontal distance, x m, from the point she entered the water as a quadratic equation p = 1/2x2 – 3x.The position of the bottom of the pool can be modelled by the linear equation p = 0.3x –
Find the set of values of x for which:a. 6x − 7 b. 2x2 − 11x + 5 c.d. Both 6x − 7 2 − 11x + 5 5 V 20 X
Find the coordinates of the points at which the line with equation y = 3x − 1 intersects the curve with equation y2 = xy + 15.
Find the set of values of x that satisfy 6/00 +1 < x2 6 0*x+x
Solve the simultaneous equations:x + 2y = 3x2 − 2y + 4y2 = 18
Find the set of values of x for which x2 − 5x − 14 > 0.
Find the set of values of x for whicha. 2(3x − 1) < 4 – 3x.b. 2x2 – 5x – 3 < 0.c. Both 2(3x – 1) < 4 – 3x and 2x2 – 5x – 3 < 0.
Given the simultaneous equations 2x – y = 1, x2 + 4ky + 5k = 0, where k is a non-zero constanta. Show that x2 + 8kx + k = 0. Given that x2 + 8kx + k = 0 has equal roots,b. Find the value of k.c. For this value of k, find the solution of the simultaneous equations.
a. On a coordinate grid, shade the region that satisfies the inequalities y < x + 4, y + 5x + 3 > 0, y > −1 and x < 2.b. Work out the coordinates of the vertices of the shaded region.c. Which of the vertices lie within the region identified by the inequalities?d. Work out the area of
Give your answers in set notation.a. Solve the inequality 3x − 8 > x + 13.b. Solve the inequality x2 − 5x − 14 > 0.
The equation kx2 – 2kx + 3 = 0, where k is a constant, has no real roots. Prove that k satisfies the inequality 0 ≤ k < 3.
Find the set of values of x for which (x − 1)(x − 4) < 2(x − 4).
a. Use algebra to solve (x − 1)(x + 2) = 18.b. Hence, or otherwise, find the set of values of x for which (x − 1)(x + 2) > 18. Give your answer in set notation.
Find the values of k for which kx2 + 8x + 5 = 0 has real roots.
The equation 2x2 + 4kx − 5k = 0, where k is a constant, has no real roots. Prove that k satisfies the inequality −5/2 < k < 0.
a. Sketch the graphs of y = f(x) = x2 + 2x – 15 and g(x) = 6 − 2x on the same axes.b. Find the coordinates of any points of intersection.c. Write down the set of values of x for which f(x) > g(x).
Find the set of values of x for which the curve with equation y = 2x2 + 3x − 15 is below the line with equation y = 8 + 2x.
On a coordinate grid, shade the region that satisfies the inequalities:y > x2 + 4x – 12 and y < 4 – x2.
a. On a coordinate grid, shade the region that satisfies the inequalities y + x < 6, y < 2x + 9, y > 3 and x > 0.b. Work out the area of the shaded region.
This trapezium has an area of 50 m2. Show that the height of the trapezium is equal to 5(√5 − 1) m. -xm- 2xm (x+10) m-
This shape has an area of 44 m2. Find the value of x. xm -xm- -2xm (x + 3) m
By completing the square, show that the solutions to the equation x2 + 2bx + c = 0 are given by the formula x=-b± √b² - c.
The functions p and q are given by p(x) = x2 − 3x and q(x) = 2x − 6, x ∈ ℝ. Find the two values of x for which p(x) = q(x).
Given that the function f(x) = sx2 + 8x + s has equal roots, find the value of the positive constants.
Solve the equation 5x+3 =v3x+7.
A football stadium has 25 000 seats. The football club know from past experience that they will sell only 10 000 tickets if each ticket costs £30. They also expect to sell 1000 more tickets every time the price goes down by £1.a. The number of tickets sold t can be modelled by the linear equation
Solve the following equations, giving your answers correct to 3 significant figures:a. k2 + 11k − 1 = 0 b. 2t2 − 5t + 1 = 0 c. 10 − x − x2 = 7 d. (3x − 1)2 = 3 − x2
Given that x2 + 3x + 6 = (x + a)2 + b, find the values of the constants a and b.
The functions f and g are given by f(x) = 2x3 + 30x and g(x) = 17x2, x ∈ ℝ. Find the three values of x for which f(x) = g(x).
Find the range of values of k for which 3x2 − 4x + k = 0 has no real solutions.
Write each of these expressions in the form p(x + q)2 + r, where p, q and r are constants to be found:a. x2 + 12x − 9. b. 5x2 − 40x + 13. c. 8x − 2x2 d 3x2 − (x + 1)2
Write 2 + 0.8x − 0.04x2 in the form A − B(x + C)2, where A, B and C are constants to be determined.
The function f is defined as f(x) = x2 − 2x + 2, x ∈ ℝ.a. Write f(x) in the form (x + p)2 + q, where p and q are constants to be found.b. Hence, or otherwise, explain why f(x) > 0 for all values of x, and find the minimum value of f(x).
The diagram shows a sketch of L1 and L2.L1 has equation 2y + 3x = 6.L2 has the equation x − y = 5.a. Find the coordinates of P, the point of intersection.b. Hence write down the solution to the inequality 2y + 3x > x – y. 0 L₁₂:x=y=5 X L₁: 2y + 3x = 6
The function g(x) = x2 + 3px + (14p − 3), where p is an integer, has two equal roots.a. Find the value of p.b. For this value of p, solve the equation x2 + 3px + (14p − 3) = 0.
In each case:i. Draw the graphs for each pair of equations on the same axesii. Find the coordinates of the point of intersection. a y = 3x - 5 y = 3-x b y = 2x - 7 y = 8-3x c y = 3x + 2 3x + y + 1 = 0
Find the value k for which the equation 5x2 − 2x + k = 0 has exactly one solution.
Find all roots of the following functions:a. f(x) = x6 + 9x3 + 8 b. g(x) = x4 − 12x2 + 32c. h(x) = 27x6 + 26x3 − 1 d. j(x) = 32x10 − 33x5 + 1e. k(x) = x − 7√x + 10 f. m(x) = 2x2/3 + 2x1/3 − 12
a. Find the discriminant of h(x) in terms of k.b. Hence or otherwise, prove that h(x) has two distinct real roots for all values of k.h(x) = 2x2 + (k + 4)x + k, where k is a real constant.
Given that for all values of x: 3x2 + 12x + 5 = p(x + q)2 + ra. Find the values of p, q and r.b. Hence solve the equation 3x2 + 12x + 5 = 0.
The function f is defined as f(x) = 32x − 28(3x) + 27, x ∈ ℝ.a. Write f(x) in the form (3x − a)(3x − b), where a and b are real constants.b. Hence find the two roots of f(x).
The function f is defined as f(x) = 22x − 20(2x) + 64, x ∈ ℝ.a. Write f(x) in the form (2x − a)(2x − b), where a and b are real constants.b. Hence find the two roots of f(x).
Find, as surds, the roots of the equation:2(x + 1)(x − 4) − (x − 2)2 = 0.
Use algebra to solve (x − 1)(x + 2) = 18.
A diver launches herself off a springboard. The height of the diver, in metres, above the pool t seconds after launch can be modelled by the following function: h(t) = 5t − 10t2 + 10, t > 0a. How high is the springboard above the water?b. Use the model to find the time at which the diver hits
For this question, f(x) = 4kx2 + (4k + 2)x + 1, where k is a real constant.a. Find the discriminant of f(x) in terms of k.b. By simplifying your answer to part a or otherwise, prove that f(x) has two distinct real roots for all non-zero values of k.c. Explain why f(x) cannot have two distinct real
Find all of the roots of the function r(x) = x8 − 17x4 + 16.
Lynn is selling cushions as part of an enterprise project. On her first attempt, she sold 80 cushions at the cost of £15 each. She hopes to sell more cushions next time. Her adviser suggests that she can expect to sell 10 more cushions for every £1 that she lowers the price.a. The number of
Find the set of values of x for which:a. 2x − 3 < 5 b. 5x + 4 ≥ 39c. 6x − 3 > 2x + 7 d. 5x + 6 ≤ −12 − xe. 15 − x > 4 f. 21 − 2x > 8 + 3xg. 1 + x < 25 + 3x h. 7x − 7 < 7 − 7xi. 5 − 0.5x ≥ 1 j. 5x + 4 > 12 − 2x
Find the set of values of x for which:a. x2 − 11x + 24 < 0 b. 12 − x − x2 > 0 c. x2 − 3x − 10 > 0d. x2 + 7x + 12 ≥ 0 e. 7 + 13x − 2x2 > 0 f. 10 + x − 2x2 < 0g. 4x2 − 8x + 3 ≤ 0 h. −2 + 7x − 3x2 < 0 i. x2 − 9 < 0j. 6x2 +
On a coordinate grid, shade the region that satisfies the inequalities:y > x – 2, y < 4x and y ≤ 5 – x.
a. Use graph paper to draw accurately the graphs of 2y = 2x + 11 and y = 2x2 − 3x – 5 on the same axes.b. Use your graph to find the coordinates of the points of intersection.c. Verify your solutions by substitution.
Solve these simultaneous equations by elimination:a. 2x − y = 6 4x + 3y = 22b. 7x + 3y = 16 2x + 9y = 29c. 5x + 2y = 6 3x − 10y = 26d. 2x − y = 12 6x + 2y = 21e. 3x − 2y = −6 6x + 3y = 2f.
For each pair of functions:i Sketch the graphs of y = f(x) and y = g(x) on the same axes.ii Find the coordinates of any points of intersection.iii Write down the solutions to the inequality f(x) ≤ g(x). a f(x) = 3x - 7 g(x) = 13 - 2x d f(x)=3-x² g(x) = 2x - 12 b f(x)=8-5x g(x) = 14 - 3x e
Find the set of values of x for which:a. 2(x − 3) ≥ 0 b. 8(1 − x) > x − 1 c. 3(x + 7) ≤ 8 − xd. 2(x − 3) − (x + 12) e. 1 + 11(2 − x) f. 2(x − 5) ≥ 3(4 − x)g. 12x − 3(x − 3) h. x − 2(5 + 2x) i. x(x − 4) > x2 + 2j x(5 − x) ≥ 3 + x − x2 k. 3x + 2x(x −
Simplify:a. (64x10)1/2b.c. (125x12)1/3d.e.f.g.h. 5x3 - 2x² X5
Solve the simultaneous equations, giving your answers in their simplest surd form: a x - y = 6 xy = 4 b 2x + 3y = 13 x² + y² = 78
Use set notation to describe the set of values of x for which:a. 3(x − 2) > x − 4 and 4x + 12 > 2x + 17b. 2x − 5 23 − xc. 2x − 3 > 2 and 3(x + 2) d. 15 − x 12x + 19e. 3x + ≤ ≥ + 6f. 5x + 3 27g. 4(3x + 7) ≤ 20 or 2(3x - 5) > 7-6x 2
Find the set of values of x for which the curve with equation y = f(x) is below the line with equation y = g(x). a f(x) = 3x² - 2x - 1 g(x) = x + 5 2 d f(x) = , x #0 g(x) = 1 b f(x) = 2x² - 4x + 1 g(x) = 3x - 2 e f(x) = 3 4 .x2 g(x) = -1 X X #0 e f(x) = 5x2x² - 4 g(x) = -2x - 1 f f(x) = 2 x +
Do not use your calculator for this exercise. Simplify: a √28 d √32 √27 3 j√175 + √63 + 2√28 m 3√80 2√20 +5√45 - b √72 /90 e h √20 + √80 k √28-2√63+√7 44 /11 C f 50 √12 2 i 200+√18-√72 1 √80-2√20 + 3√45 o √12+ 3√48 +√75
Solve the simultaneous equations:x + 2y = 3x2 − 4y2 = −33
Find the set of values of x for which:a. x2 < 10 − 3x b. 11 < x2 + 10c. x(3 − 2x) > 1 d. x(x + 11) < 3(1 − x2)
On a coordinate grid, shade the region that satisfies the inequalities:x ≥ −1, y + x < 4, 2x + y ≤ 5 and y > −2.
Solve these simultaneous equations by substitution:a. x + 3y = 11 4x − 7y = 6b. 4x − 3y = 40 2x + y = 5c. 3x − y = 7 10x + 3y = −2d. 2y = 2x − 3 3y = x − 1
a. On the same axes sketch the curve with equation x2 + y = 9 and the line with equation 2x + y = 6.b. Find the coordinates of the points of intersection.c. Verify your solutions by substitution.
Given the simultaneous equationsx − 2y = 13xy − y2 = 8a. Show that 5y2 + 3y − 8 = 0.b. Hence find the pairs (x, y) for which the simultaneous equations are satisfied.
Expand and simplify if possible: a √3(2+√3) d (2-√2)(3+√5) g (5-√3)(1-√3) b √5(3-√3) e (2-√3)(3-√7) h (4+√3)(2-√3) c √2(4-√5) f (4+√5)(2+√5) i (7-√11)(2+√11)
Use set notation to describe the set of values of x for which:a. x2 − 7x + 10 < 0 and 3x + 5 < 17 b. x2 − x − 6 > 0 and 10 − 2x < 5c. 4x2 − 3x − 1 < 0 and 4(x + 2) < 15 − (x + 7) d. 2x2 − x − 1 < 0 and 14 < 3x − 2e. x2 − x − 12 > 0 and 3x
On a coordinate grid, shade the region that satisfies the inequalities: y > (3 – x)(2 + x) and y + x ≥ 3.
Solve the simultaneous equations:x + y = 3x2 − 3y = 1
The diagram shows a rectangle with a square cut out. The rectangle has length 3x − y + 4 and width x + 7. The square has length x − 2. Find an expanded and simplified expression for the shaded area. x-2 3x-y +4 x+7
a. On the same axes sketch the curve with equation y = (x – 2)2 and the line with equation y = 3x – 2.b. Find the coordinates of the point of intersection.
Simplify these fractions: d 6x4 + 10x6 2x 8r3+5x 2x b 3x5-x7 X 7x7 +5x² 5x С f 2x4-4x2 4x 9x5 - 5x³ 3.x
Simplify:a. 1/√5b. 1/√11c. 1/√2d. √3/√15e. √12/√48f. √5/80g. √12/√156h. √7/√63
Simplify:a. y3 × y5 b. 3x2 × 2x5 c. (4x2)3 ÷ 2x5 d. 4b2 × 3b3 × b4
Simplify:a. x3 ÷ x−2 b. x5 ÷ x7 c. x3/2 × x5/2d. (x2)3/2 e. (x3)5/3 f. 3x0.5 × 4x−0.5g. 9x2/3 ÷ 3 x1/6 h. 5x7/5 ÷ x2/5 i. 3x4 × 2x−5j. √x × 3√x k. (√x)3 × (√x)4l. (3√x)2/√x
Simplify giving your answer in the form p + q√5, where p and q are rational numbers. 3-2√5 √5-1
Expand and simplify if possible:a. (x + 4)(x + 7) b. (x − 3)(x + 2) c. (x − 2)2d. (x − y)(2x + 3) e. (x + 3y)(4x − y) f. (2x − 4y)(3x + y)g. (2x − 3)(x − 4) h. (3x + 2y)2 i. (2x + 8y)(2x + 3)j. (x + 5)(2x + 3y − 5) k. (x − 1)(3x − 4y −
Expand and simplify if possible:a. (x + 3)(x − 5) b. (2x − 7)(3x + 1) c. (2x + 5)(3x − y + 2)
Expand and simplify if possible:a. 9(x − 2) b. x(x + 9) c. −3y(4 − 3y)d. x(y + 5) e. −x(3x + 5) f. −5x(4x + 1)g. (4x + 5)x h. −3y(5 − 2y2) i.−2x(5x − 4)j. (3x − 5)x2 k. 3(x + 2) + (x − 7) l. 5x − 6 − (3x − 2)m. 4(c + 3d2) − 3(2c

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

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