Questions and Answers of Edexcel AS And A Level Mathematics

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 +
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.
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
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
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
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
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
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
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.
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
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 –
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
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.
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)
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
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
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
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) =
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.
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
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
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 +
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 −
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
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
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) ≤
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
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 + ≤
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 +
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
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 −
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)
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
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.
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.
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 −

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