Questions and Answers of Edexcel AS And A Level Mathematics

The surface area, A cm2, of an expanding sphere of radius r cm is given by A = 4πr2. Find the rate of change of the area with respect to the radius at the instant when the radius is 6 cm.
Calculate the x-coordinates of the points on the curve with equation y = 7x2 − x3 at which the gradient is equal to 16.
G is the point with coordinates (4, 16) on the curve with equation y = x2.a. Find the gradients of the chords joining the point G to the points with coordinates:i. (5, 25) ii. (4.5,
The point A with coordinates (−1, 4) lies on the curve with equation y = x3 − 5x.The point B also lies on the curve and has x-coordinate (−1 + h).a. Show that the gradient of the line segment
Given that 2y2 − x3 = 0 and y > 0, find dy/dx.
The diagram shows a square and four congruent right-angled triangles.Use the diagram to prove that a2 + b2 = c2. b a C
Prove that *^^ + x^= Ã^ − x^ x-y
f(x) = 4x3 + 4x2 − 11x − 6a. Use the factor theorem to show that (x + 2) is a factor of f(x).b. Factorise f(x) completely.c. Write down all the solutions of the equation 4x3 + 4x2 − 11x − 6 =
Prove that A(3, 1), B(1, 2) and C (2, 4) are the vertices of a right-angled triangle.
A circle has equation (x − 1)2 + y2 = k, where k > 0.The straight line L with equation y = ax cuts the circle at two distinct points. Prove that k> 9² 1 + a²
f(x) = 10x3 + 43x2 – 2x − 10Find the remainder when f(x) is divided by (5x + 4).
a. Show that (x − 2) is a factor of 9x4 – 18x3 – x2 + 2x.b. Hence, find four real solutions to the equation 9x4 − 18x3 − x2 + 2x = 0.
Prove that quadrilateral A(1, 1), B(2, 4), C(6, 5) and D(5, 2) is a parallelogram.
Use completing the square to prove that n2 − 8n + 20 is positive for all values of n.
Prove that (x - 1)(x² + x) = x²(x² - 1)
f(x) = 3x3 – 14x2 – 47x – 14a. Find the remainder when f(x) is divided by (x − 3).b. Given that (x + 2) is a factor of f(x), factorise f(x) completely.
Prove that quadrilateral A(2, 1), B(5, 2), C (4, −1) and D(1, −2) is a rhombus.
Prove that A(−5, 2), B(−3, −4) and C (3, −2) are the vertices of an isosceles right-angled triangle.
Prove that the quadrilateral A(1, 1), B(3, 2), C (4, 0) and D(2, −1) is a square.
a. Find the remainder when x3 + 6x2 + 5x – 12 is divided byi. x − 2,ii. x + 3.b. Hence, or otherwise, find all the solutions to the equation x3 + 6x2 + 5x – 12 = 0.
A student is trying to prove that 1 + x22. The student writes:a. Identify the error made in the proof.b. Provide a counter-example to show that the statement is not true. (1 + x)² = 1 + 2x + x². So
Prove that the sum of two consecutive positive odd numbers less than ten gives an even number.
Find the coefficient of x3 in the binomial expansion of: a (3 + x)5 e (1 + x) 10 i (1-x)6 b (1 + 2x)5 f (3-2x)6 (3 + x)² j c (1-x)6 g (1 + x)20 8 k (2-x) ³ d (3x + 2)³ h (4-3x)7 1 (5+ 1 x)³
a. Find the first four terms of the binomial expansion, in ascending powers of x, ofb. By substituting an appropriate value for x, find an approximate value for 0.996. -755) 6. (1-10
f(x) = 2x3 + 3x2 – 8x + 3a. Show that f(x) = (2x – 1)(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
Prove that the line 4y − 3x + 26 = 0 is a tangent to the circle (x + 4)2 + ( y − 3)2 = 100.
The 16th row of Pascal’s triangle is shown below.a. Find the next two values in the row.b. Hence find the coefficient of x3 in the expansion of (1 + 2x)15. 1 15 105
Prove that the statement ‘n2 − n + 3 is a prime number for all values of n’ is untrue.
f(x) = 12x3 + 5x2 + 2x – 1a. Show that (4x – 1) is a factor of f(x) and write f(x) in the form (4x – 1)(ax2 + bx + c).b. Hence, show that the equation 12x3 + 5x2 + 2x – 1 = 0 has exactly 1
a. Write down the first four terms of the binomial expansion ofb. By substituting an appropriate value for x, find an approximate value for 2.110. (2+ 5, 10
The equation x2 – kx + k = 0, where k is a positive constant, has two equal roots. Prove that k = 4.
Given thatwrite down the value of a. (45)= 45! 17!a!'
Prove that the distance between opposite edges of a regular hexagon of side length √3 is a rational value.
a. Prove that the difference of the squares of two consecutive even numbers is always divisible by 4.b. Is this statement true for odd numbers? Give a reason for your answer.
20 people play a game at a school fete. The probability that exactly n people win a prize is modelled aswhere p is the probability of any one person winning.Calculate the probability of:a. 5 people
State the row of Pascal’s triangle that would give the coefficients of each expansion:a. (x + y)3 b. (3x − 7)15 c. (2x + 1/2)n d. (y − 2x)n + 4
Work out:a. 4! b. 9!c. 10!/7! d. 15!/13!
Write down the expansion of the following:a. (1 + x)4 b. (3 + x)4 c. (4 − x)4 d. (x + 2)6 e. (1 + 2x)4 f. (1 − 1/2x)4
The coefficient of x2 in the expansion of (2 + ax)6 is 60. Find two possible values of the constant a.
Write each value a to d from Pascal’s triangle using nCr notation: 1 1 1 6 5 1 a С 1 b 1 2 6 d 1 3 10 1 4 1 5 1 15 6 - 1
Write down the expansion of:a. (x + y)4 b. (p + q)5 c. (a − b)3d. (x + 4)3e. (2x − 3)4 f. (a + 2)5 g. (3x − 4)4 h. (2x − 3y)4
Use the binomial theorem to find the first four terms in the expansion of:a. (1 + x)10 b. (1 − 2x)5 c. (1 + 3x)6 d. (2 − x)8 e. ( 2 − 1/2x)10 f. (3 − x)7
The coefficient of x3 in the expansion of (3 + bx)5 is −720. Find the value of the constant b.
If x is so small that terms of x3 and higher can be ignored, show that: (2 + x)(1 − 3x)5 ≈ 2 − 29x + 165x2
Find the coefficient of x3 in the expansion of:a. (4 + x)4 b. (1 − x)5 c. (3 + 2x)3 d. (4 + 2x)5e. (2 + x)6 f. (4 − 1/2 x)4 g. (x + 2)5 h. (3 − 2x)4
Use the binomial theorem to find the first four terms in the expansion of:a. (2x + y)6 b. (2x + 3y)5 c. (p − q)8d. (3x − y)6 e. (x + 2y)8 f. (2x − 3y)9
The coefficient of x3 in the expansion of (2 + x)(3 − ax)4 is 30. Find two possible values of the constant a.
When (1 − 3/2x)p is expanded in ascending powers of x, the coefficient of x is −24.a. Find the value of p.b. Find the coefficient of x2 in the expansion.c. Find the coefficient of x3 in the
Fully expand the expression (1 + 3x)(1 + 2x)3.
Use the binomial expansion to find the first four terms, in ascending powers of x, of:a. (1 + x)8 b. (1 − 2x)6 c. (1 + x/2)10d. (1 − 3x)5 e. (2 + x)7 f. (3 − 2x)3g. (2 −
When (1 − 2x)p is expanded, the coefficient of x2 is 40. Given that p > 0, use this information to find:a. The value of the constant pb. The coefficient of xc. The coefficient of x3
a. Write down the first four terms in the expansion of (1 + 2x)8.b. By substituting an appropriate value of x (which should be stated), find an approximate value of 1.028.
The sketch shows part of the curve with equation y = x(x2 − 4). Find the area of the shaded region. y = x(x² - 4) X
The graph shows a sketch of part of the curve C with equation y = x(x + 3)(2 − x). The curve C crosses the x-axis at the origin O and at points A and B.a. Write down the x-coordinates of A and B.
The diagram shows the finite region, R, bounded by the curve with equation y = 4x − x2 and the line y = 3. The line cuts the curve at the points A and B.a. Find the coordinates of the points A and
The diagram shows a sketch of the curve with equationThe region R is bounded by the curve, the x-axis and the lines x = 1 and x = 3.Find the area of R. y = 3x + 6 x² - 5, x > 0.
The gradient of a particular curve is given byGiven that the curve passes through the point (9, 0), find an equation of the curve. dy √√x +3 1² d.x ||
Find y when dy/dx is given by the following expressions. In each case simplify your answer. a x²-x--6x-² d 5x10x4 + x-3 b 4x³ + x-x-² ex-3+8x c 4-12x-4 + 2x - ² f 5x4x12x-5
f(x) = − x3  + 4x2  + 11x − 30The graph shows a sketch of part of the curve with equation y = − x3  + 4x2  + 11x − 30.a. Use the factor theorem to show that (x +
The curve C, with equation y = f(x), passes through the point (1, 2) and f′(x) = 2x3 − 1/x2 Find the equation of C in the form y = f(x).
The diagram shows a sketch of part of the curve with equation y = 9 − 3x − 5x2 − x3 and the line with equation y = 4 − 4x.The line cuts the curve at the points A (−1, 8) and B(1, 0).Find
Given that A is a constant and show that there are two possible values for A and find these values. (6√x - A)dx = 4²,
The gradient of a curve is given by f′(x) = x2 − 3x − 2/x2. Given that the curve passes through the point (1, 1), find the equation of the curve in the form y = f(x).
Givenfind ∫y dx. y = (x + 1)(2x - 3) √x
Find the following integrals:a. ∫(4x−2 + 3x−1/2)dx b. ∫(6x−2 − x1/2)dx c. ∫(2x−3/2 + x2 − x−1/2)dx
Given that y = 10 at x = 4, find y in terms of x, giving each term in its simplest form. dy 1/² = 3x - ² - 2x√√x, x > 0. d.x
Use calculus to find the value of (2x - 3√x)dx.
Find:a. ∫(8x3 − 6x2 + 5)dx b. ∫(5x + 2) x1/2 dx
Find the following integrals:a. ∫(4x3 − 3x−4 + r)dx b. ∫(x + x−1/2 + x−3/2)dxc. ∫(px4 + 2t + 3x−2)dx
The curve with equation y = f(x) passes through the point (−1, 0). Given that f′(x) = 9x2 + 4x − 3, find f(x).
Find the area of the finite region between the curve with equation y = (3 − x)(1 + x) and thex-axis.
Find the area of the finite region bounded by the curve with equation y = (1 − x)(x + 3) and the line y = x + 3.
Find y given that dy/dx = (2x + 3)2.
Find the following integrals:a. ∫(3t2 − t−2)dt b. ∫(2t2 − 3t−3/2 + 1)dtc. ∫(pt3 + q2 + px3)dt
Find the area of the finite region between the curve with equation y = x(x − 4)2 and the x-axis.
The diagram shows the finite region, R, bounded by the curve with equation y = x(4 + x), the line with equation y = 12 and the y-axis.a. Find the coordinates of the point A where the line meets the
Find the following integrals:a.b. ∫(2x + 3)2 dx c. ∫(2x + 3) √xdx (2x³ + 3) x² -d.x
Given thatcan be written in the form 6x p + 5x q,a. Write down the value of p and the value of q. Given thatand that y = 100 when x = 9,b. Find y in terms of x, simplifying the coefficient of each
Given that dx/dt = (t + 1)2 and that x = 0 when t = 2, find the value of x when t = 3.
Find f(x) given that f′(x) = 3x−2 + 6x1/2 + x − 4.
Evaluategiving your answer in the form a + b√3 , where a and b are integers. 12 2 X dx,
The diagram shows a sketch of part of the curve with equation y = x2 + 1 and the line with equation y = 7 − x.The finite region, R1 is bounded by the line and the curve.The finite region, R2 is
Find the area of the finite region between the curve with equation y = 2x2 − 3x3 and the x-axis.
Find ∫f(x)dx when f(x) is given by the following: a (x + ¹)² b (vx + 2)2 ○(+2)
Given that y1/2 = x1/3 + 3:a. Show that y = x2/3 + Ax1/3 + B, where A and B are constants to be found.b. Hence find ∫y dx.
Given thatcalculate the value of k. 1 X dx = 3,
The displacement of a particle at time t is given by the function f(t), where f(0) = 0.Given that the velocity of the particle is given by f′(t) = 10 − 5t,a. Find f(t)b. Determine the
The finite region R is bounded by the x-axis and the curve with equation y = −x2 + 2x + 3, x > 0.The curve meets the x-axis at points A and B.a. Find the coordinates of point A and point B.b.
Find the value of a and the value of b. a (363 – ab) dx = J - : 2 3.x² + 1 + 14x + c
The diagram shows part of a sketch of the curve with equation y = 2/x2 + x. The points A and B have x-coordinates 1/2 and 2 respectively.Find the area of the finite region between AB and the curve.
The graph shows part of the curve C with equation y = x2(2 − x). The region R, shown shaded, is bounded by C and the x-axis. Use calculus to find the exact area of R. R y = x²(2-x) C 2 X
The diagram shows part of the curve with equation y = 3 √x − √x3 + 4 and the line with equation y = 4 − 1/2 x.a. Verify that the line and the curve cross at the point A(4, 2).b. Find the area
Findgiving each term in its simplest form. [(8.x³ + 6x - 3 d.x,
The line with equation y = 10 − x cuts the curve with y = 2x2 − 5x + 4 at the points A and B, as shown.a. Find the coordinates of A and the coordinates of B. The shaded region R is bounded by the
The diagram shows a sketch of the curve with equationa. Show thatb. At the point B on the curve the tangent to the curve is parallel to the x-axis. Find the coordinates of the point B.c. Find, to 3
The diagram shows part of the curve with equation y = x3 − 6x2 + 9x. The curve touches the x‑axis at A and has a local maximum at B.a. Show that the equation of the curve may be written as y =
The diagram shows the curve with equation y = 5 + 2x − x2 and the line with equation y = 2.The curve and the line intersect at the points A and B.a. Find the x-coordinates of A and B.b. The shaded
a. Find ∫(x1/2 − 4)(x−1/2 − 1)dx.b. Use your answer to part a to evaluategiving your answer as an exact fraction. (x-4)(x-²-1)dx
Consider the function y = 3x1/2 − 4 x−1/2 , x > 0.a. Find dy/dx.b. Find ∫y dxc. Hence show thatwhere A and B are integers to be found. [²³y dx = A + B√3,
Find the value of p and the value of q. P 2 [(₂²/2 + pq) dx = ² + 10x + c

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