Questions and Answers of Edexcel AS And A Level Mathematics

The diagram shows ΔABC. Calculate the area of ΔABC. 4.3 cm, A 40° D B 3.5 cm -8.6cm- C
Find the gradient of the curve with equation y = f(x) at the point A where:a. f(x) = x(x + 1) and A is at (0, 0)b.c.d. f(x) 2x - 6 42 and A is at (3, 0)
The points A, B and C have coordinates (3, −1), (4, 5) and (−2, 6) respectively, and O is the origin. Find, in terms of i and j:a. i the position vectors of A, B and Cb. Find, in surd form: ii
Differentiate:a. 2√x b. 3/x2 c. 1/3x3 d. 1/3x3(x − 2) 2 e +√x x3 i 2x³ + 3x √x f √√x + 2x j x(x² - x + 2) 2x + 3 k 3x²(x² + 2x) h 3x² - 6 1 (3x − 2)(4x + ¹)
Find the coordinates of the point on the curve with equation y = x2 + 5x − 4 where the gradient is 3.
For f(x) = 12 − 4x + 2x2, find the equations of the tangent and the normal at the point where x = −1 on the curve with equation y = f(x).
By considering the gradient on either side of the stationary point on the curve y = x3 − 3x2 + 3x, show that this point is a point of inflection. Sketch the curve y = x3 − 3x2 + 3x.
The displacement, s metres, of a car from a fixed point at time t seconds is given by s = t2 + 8t. Find the rate of change of the displacement with respect to time at the instant when t = 5.
Given thatfind dy/dx. y = 3√x - 4 x > 0,
Find the x-coordinates of the two points on the curve with equation y = x3 − 11x + 1 where the gradient is 1. Find the corresponding y-coordinates.
Prove, from first principles, that the derivative of 6x is 6.
Find the gradients of the curve y = x2 − 5x + 10 at the points A and B where the curve meets the line y = 4.
f(x ) = (12/p√x) + x , where p is a real constant and x > 0. Given that f'(2) = 3, find p, giving your answer in the form a√2 where a is a rational number.
The point P with x-coordinate 1/2 lies on the curve with equation y = 2x2.The normal to the curve at P intersects the curve at points P and Q.Find the coordinates of Q.
Find the maximum value and hence the range of values for the function f(x) = 27 − 2x4.
A rectangular garden is fenced on three sides, and the house forms the fourth side of the rectangle.a. Given that the total length of the fence is 80 m, show that the area, A, of the garden is given
The function f is defined by f(x) = x + 9/x, x ∈ R, x ≠ 0.a. Find f'(x).b. Solve f'(x) = 0.
Prove, from first principles, that the derivative of 4x2 is 8x.
Find the gradients of the curve y = 2x2 at the points C and D where the curve meets the line y = x + 3.
f(x) = (2 − x)9a. Find the first 3 terms, in ascending powers of x, of the binomial expansion of f(x), giving each term in its simplest form.b. If x is small, so that x2 and higher powers can be
f(x) = x4 + 3x3 − 5x2 − 3x + 1a. Find the coordinates of the stationary points of f(x), and determine the nature of each.b. Sketch the graph of y = f(x).
A closed cylinder has total surface area equal to 600p.a. Show that the volume, V cm3, of this cylinder is given by the formula V = 300πr − πr3, where r cm is the radius of the cylinder.b. Find
Given that a. Find the value of x and the value of y when dy/dx = 0.b. Show that the value of y which you found in part a is a minimum. + ?x=d 48 , x > 0
The shape shown is a wire frame in the form of a large rectangle split by parallel lengths of wire into 12 smaller equal-sized rectangles.a. Given that the total length of wire used to complete the
A sector of a circle has area 100 cm2.a. Show that the perimeter of this sector is given by the formulab. Find the minimum value for the perimeter. 200 100 TT ا < P = 2r + r ,r
A shape consists of a rectangular base with a semicircular top, as shown.a. Given that the perimeter of the shape is 40 cm, show that its area, A cm2, is given by the formulawhere r cm is the radius
A curve C has equation y = x3 − 2x2 − 4x − 1 and cuts the y-axis at a point P. The line L is a tangent to the curve at P, and cuts the curve at the point Q. Show that the distance PQ
A curve has equation y = 12x1/2 − x3/2.a. Show thatb. Find the coordinates of the point on the curve where the gradient is zero. dy 3 dx = 2x-¹(4- x).
The normals to the curve 2y = 3x3 − 7x2 + 4x, at the points O(0, 0) and A(1, 0), meet at the point N.a. Find the coordinates of N.b. Calculate the area of triangle OAN.
A curve has equation y = 8/x − x + 3x2, x > 0. Find the equations of the tangent and the normal to the curve at the point where x = 2.
A curve C has equation y = x3 − 5x2 + 5x + 2.a. Find dy/dx in terms of x.b. The points P and Q lie on C. The gradient of C at both P and Q is 2. The x-coordinate of P is 3.i. Find the x-coordinate
The curve with equation y = ax2 + bx + c passes through the point (1, 2). The gradient of the curve is zero at the point (2, 1). Find the values of a, b and c.
a. Expand (x3/2 − 1)( x−1/2 + 1).b. A curve has equation y = (x3/2 − 1)(x−1/2 + 1), x > 0. Find dy/dx.c. Use your answer to part b to calculate the gradient of the curve at the point where
f(x) = x2 − 2x − 8a. Sketch the graph of y = f(x).b. On the same set of axes, sketch the graph of y = f'(x).c. Explain why the x-coordinate of the turning point of y = f(x) is the same as the
f(x) = ax2, where a is a constant. Prove, from first principles, that f'(x) = 2ax.
A curve has equation y = x3 − 5x2 + 7x − 14. Determine, by calculation, the coordinates of the stationary points of the curve.
A cylindrical biscuit tin has a close-fitting lid which overlaps the tin by 1 cm, as shown. The radii of the tin and the lid are both x cm. The tin and the lid are made from a thin sheet of metal of
The diagram shows part of the curve with equation y = 3 + 5x + x2 − x3. The curve touches the x-axis at A and crosses the x-axis at C. The points A and B are stationary points on the curve.a. Show
The diagram shows an open tank for storing water, ABCDEF. The sides ABFE and CDEF are rectangles. The triangular ends ADE and BCF are isosceles, and ∠AED = ∠BFC = 90°. The ends ADE and BCF are
The motion of a damped spring is modelled using this graph.On a separate graph, sketch the gradient function for this model. Choose suitable labels and units for each axis, and indicate the
The total surface area, A cm2, of a cylinder with a fixed volume of 1000 cm3 is given by the formulawhere x cm is the radius. Show that when the rate of change of the area with respect to the radius
A wire is bent into the plane shape ABCDE as shown. Shape ABDE is a rectangle and BCD is a semicircle with diameter BD. The area of the region enclosed by the wire is R m2, AE = x metres, and AB = ED
Find the equation of the curve with the given derivative of y with respect to x that passes through the given point: D с e dy dx dy d.x dy dx = 3x² + 2x; = √x + = x²; = (x + 2)²; point (2,
Find f(x) when f ′(x) is given by the following expressions. In each case simplify your answer. 12x + x² + 5 d 10x4 + 8x-3 b 6x5 + 6x-7-x- e 2x-³+4x- с f 9x² + 4x−³+√x-² -
Find an expression for y when dy/dx is the following: a x5 g -21-6 m -3x- b 10x4 h x- n -5 c -x-2 i 5x-12/ o 6x d -4x-³ j 6x P 2x-0.4 e xś k 36x¹1 f 4x² 1-14x-8
The graph shows a sketch of part of the curve C with equation y = (x − 4)(2x + 3). The curve C crosses the x-axis at the points A and B.a. Write down the x-coordinates of A and B.The finite region
The volume, V cm3, of a tin of radius r cm is given by the formula V = π(40r − r2 − r3).Find the positive value of r for which dV/dr = 0, and find the value of V which corresponds to this value
Find the following integrals:a. ∫x3 dx b. ∫x7 dx c. ∫3x−4 dx d. ∫5x2 dx
Find the area between the curve with equation y = f(x), the x-axis and the lines x = a and x = b in each of the following cases:a. f(x) = −3x2 + 17x − 10; a = 1, b = 3b. f(x) = 2x3 + 7x2 − 4x;
Sketch the following and find the total area of the finite region or regions bounded by the curves and the x-axis:a. y = x(x + 2) b. y = (x + 1)(x − 4) c. y = (x + 3)x(x − 3)d. y = x2(x
Find:a. ∫(x + 1)(2x − 5)dx b. ∫(x1/3 + x−1/3) dx
Find the following integrals:a. ∫(x4 + 2x3)dx b. ∫(2x3 − x2 + 5x)dx c. ∫(5 x3/2 − 3x2)dx
The function f, defined for x ∈ R, x > 0, is such that:a. Find the value of f"(x) at x = 4.b. Prove that f is an increasing function. f'(x) = x² - 2 + 1 x²
The diagram shows part of the curve with equation y = f(x), where:The curve cuts the x-axis at the points A and C.The point B is the maximum point of the curve.a. Find f'(x).b. Use your answer to
A curve has equation y = x3 − 6x2 + 9x. Find the coordinates of its local maximum.
f(x) = 3x4 − 8x3 − 6x2 + 24x + 20a. Find the coordinates of the stationary points of f(x), and determine the nature of each of them.b. Sketch the graph of y = f(x).
The diagram shows the part of the curve with equation y = 5 − 1/2 x2 for which y > 0. The point P(x, y) lies on the curve and O is the origin.a. Show that OP2 = 1/4 x4 − 4x2 + 25. Taking f(x)
The probability of throwing exactly 10 heads when a fair coin is tossed 20 times is given byCalculate the probability and describe the likelihood of this occurring. (20) 0.520. 10
a. By considering the graphs of the functions, or otherwise, verify that:i. cos θ = cos (−θ)ii. sin θ = −sin (−θ)iii. sin (θ − 90°) = −cos θ.b. Use the results in a ii and iii to
In each triangle below, find the size of x and the area of the triangle. a 2.4 cm 3 cm 1.2 cm X b X 6 cm 80° 5cm C 3 cm 40° X 5cm
The circle with equation (x − k)2 + y2 = 41 passes through the point (3, 4). Find the two possible values of k.
Prove that 2x3 + x2 − 43x – 60 ≡ (x + 4)(x – 5)(2x + 3).
If x is so small that terms of x3 and higher can be ignored, and (2 − x)(3 + x)4 ≈ a + bx + cx2 find the values of the constants a, b and c.
A teacher asks two students to solve the equation 2 cos x = 3 sin x for −180° a. Identify the mistake made by Student A.b. Identify the mistake made by Student B and explain the effect it has on
Given that:(2 − x)13 ≡ A + Bx + Cx2 + …Find the values of the integers A, B and C.
Expand (2 + y)3. Hence or otherwise, write down the expansion of (2 + x − x2)3 in ascending powers of x.
Work out the 5th number on the 12th row from Pascal’s triangle.
Find the first 3 terms, in ascending powers of x, of the binomial expansion of (2 − x)6 and simplify each term.
A teacher asks one of his students to solve the equation 2 sin 3x = 1 for –360° ≤ x ≤ 360°. The attempt is shown below:a. Identify two mistakes made by the student.b. Solve the equation. sin
The diagram shows triangle ABC, with AB = 5 cm, BC = (2x − 3) cm, CA = (x + 1) cm and ∠ABC = 60°.a. Show that x satisfies the equation x2 − 8x + 16 = 0.b. Find the value of x.c. Calculate the
The diagram shows the triangle ABC with AB = 11 cm, BC = 6 cm and AC = 7 cm.a. Find the exact value of cos B, giving your answer in simplest form.b. Hence find the exact value of sin B. B 6cm 11 cm 7
The diagram shows triangle PQR with PR = 6 cm, QR = 5 cm and angle QPR = 45°.a. Show that sin Q = 3√2/5b. Given that Q is obtuse, find the exact value of cos Q. P 45° 6 cm 5 cm R
a. Show that (2x − 1) is a factor of 2x3 − 7x2 − 17x + 10.b. Factorise 2x3 − 7x2 − 17x + 10 completely.c. Hence, or otherwise, sketch the graph of y = 2x3 − 7x2 − 17x + 10, labelling
f(x) = 3x3 + x2 − 38x + c Given that f(3) = 0,a. find the value of c,b. factorise f(x) completely,
a. Sketch the graphs of y = 3 sin x and y = 2 cos x on the same set of axes (0 ≤ x ≤ 360°).b. Write down how many solutions there are in the given range for the equation 3 sin x = 2 cos x.c.
g(x) = x3 − 13x + 12a. Use the factor theorem to show that (x − 3) is a factor of g(x).b. Factorise g(x) completely.
The diagram shows triangle ABC, with AB = √5 cm, ∠ABC = 45° and ∠BCA = 30°. Find the exact length of AC. C 30° A √5 cm 45° B
a. It is claimed that the following inequality is true for all real numbers a and b. Use a counter-example to show that the claim is false: a2 + b2 < (a + b)2b. Specify conditions on a and b that
a. Show that the equation 3 sin2 x – cos2 x = 2 can be written as 4 sin2 x = 3.b. Hence solve the equation 3 sin2 x – cos2 x = 2 in the interval –180° ≤ x ≤ 180°, giving your answers to 1
The graph shows the curve y = sin (x + 45°), −360° ≤ x ≤ 360°.a. Write down the coordinates of each point where the curve crosses the x-axis.b. Write down the coordinates of the point where
a. Use proof by exhaustion to prove that for all prime numbers p, 3 < p < 20, p2 is one greater than a multiple of 24.b. Find a counterexample that disproves the statement ‘All numbers which
Find all the solutions to the equation 3 cos2 x + 1 = 4 sin x in the interval –360° ≤ x ≤ 360°, giving your answers to 1 decimal place.
a. Show that x2 + y2 − 10x − 8y + 32 = 0 can be written in the form (x − a)2 + (y − b)2 = r2, where a, b and r are numbers to be found.b. Circle C has equation x2 + y2 − 10x − 8y + 32 = 0
a. Expand (1 − 2x)10 in ascending powers of x up to and including the term in x3.b. Use your answer to part a to evaluate (0.98)10 correct to 3 decimal places.
If x is so small that terms of x3 and higher can be ignored, (2 − x)(1 + 2x)5 ≈ a + bx + cx2. Find the values of the constants a, b and c.
The coefficient of x in the binomial expansion of (2 − 4x)q, where q is a positive integer, is −32q. Find the value of q.
ACGI is a square, B is the midpoint of AC, F is the midpoint of CG, H is the midpoint of GI, D is the midpoint of AI.Find, in terms of b and d: AB = b and AD = d.
Ship B is 8 km, on a bearing of 030°, from ship A. Ship C is 12 km, on a bearing of 140°, from ship B.a. Calculate the distance of ship C from ship A.b. Calculate the bearing of ship C from ship A.
The triangle ABC has vertices A(−2, 4), B(6, 10) and C(16, 10).a. Prove that ABC is an isosceles triangle.b. Calculate the size of ∠ABC .
The circle C has centre (5, 2) and radius 5. The points X (1, −1), Y (10, 2) and Z (8, k) lie on the circle, where k is a positive integer.a. Write down the equation of the circle.b. Calculate the
In the diagram, WX(vector) = a, WY(vector) = b and WZ(vector) = c. It is given that XY(vector) = YZ(vector). Prove that a + c = 2b. X a b W Y Z
OACB is a parallelogram. M, Q, N and P are the midpoints of OA, AC, BC and OB respectively. Vectors p and m are equal torespectively. Express in terms of p and m. OP and OM
a. On the same set of axes, in the interval 0 ≤ x ≤ 360°, sketch the graphs of y = tan (x − 90°) and y = sin x. Label clearly any points at which the graphs cross the coordinate axes.b. Hence
The diagram shows the vectors a, b, c and d. Draw a diagram to illustrate these vectors:a. a + c b. −bc. c − d d. b + c + de. a − 2b f. 2c + 3dg. a + b + c + d ay с
These vectors are drawn on a grid of unit squares.Express the vectors v1, v2, v3, v4, v5 and v6 in:(i) i, j notation (ii) Column vector form 5, V3 V6
A pyramid has four triangular faces and a square base. All the edges of the pyramid are the same length, s cm. Show that the total surface area of the pyramid is (√3 + 1)s2 cm2.
a. Given that sin θ = cos θ, find the value of tan θ.b. Find the values of θ in the interval 0 ≤ θ < 360° for which sin θ = cos θ.
Find all the values of x in the interval 0 ≤ x < 360° for which 3 tan2x = 1.
Find all the values of θ in the interval 0 ≤ θ < 360° for which 2 sin (θ − 30°) = √3.
OAB is a triangle. P, Q and R are the midpoints of OA, AB and OB respectively. OP and OR are equal to p and r respectively.a. Find i. OB(vector) ii. PQ(vector)b. Hence prove that triangle PAQ is

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