Questions and Answers of Edexcel AS And A Level Mathematics

Find all the solutions, in the interval 0 ≤ x ≤ 360°, to the equation 8 sin2 x + 6 cos x – 9 = 0 giving each solution to one decimal place.
a. Sketch for 0 ≤ x ≤ 360° the graph of y = sin (x + 60°)b. Write down the exact coordinates of the points where the graph meets the coordinate axes.c. Solve, for 0 ≤ x ≤ 360°, the
Solve the following equations for θ, in the interval 0 < θ < 360°:a. √3 sin θ = cos θ b. sin θ + cos θ = 0 c. 3 sin θ = 4 cos θd. 2 sin θ − 3 cos θ = 0 e. √2 sin
Given that θ is an acute angle, express in terms of sin θ:a. sin (−θ) b. sin (180° + θ) c. sin (360° − θ)d. sin (−(180° + θ)) e. sin (−180° + θ) f. sin
The line l1 has equation y = 3x − 6. The line l2 is perpendicular to l1 and passes through the point (6, 2).a. Find an equation for l2 in the form y = mx + c, where m and c are constants The
Given that angle B is obtuse and tan B = +√21/2, find the exact value of: a. sin B b. cos B
Solve for θ, in the interval −180° ≤ θ ≤ 180°, the following equations.Give your answers to 3 significant figures where they are not exact.a. 5sin2 θ = 4cos2 θ b. tan θ = cos θ
Solve for 0 ≤ x ≤ 180° the equations:a. sin(x + 20°) = 1/2b. cos 2x = −0.8, giving your answers to 1 decimal place.
Solve the following equations for θ, in the interval 0 < θ < 360°:a. 7 sin θ = 5 b. 2 cos θ = −√2 c. 3 cos θ = −2 d. 4 sin θ = −3e. 7 tan θ = 1 f. 8 tan θ =
Without attempting to solve them, state how many solutions the following equations have in the interval 0 ≤ θ ≤ 360°. Give a brief reason for your answer.a. 2 sin θ = 3 b. sin θ = − cos
a. Show that x2 + y2 − 6x + 2y − 10 = 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. Hence write down the centre and radius of the circle
Given that cos θ = 3/4 and that θ is reflex, find the exact value of: a. sin θ b. tan θ
Find all the values of θ, to 1 decimal place, in the interval 0 < θ < 360° for which tan2 θ = 9.
a. Show that tan x = p ± √q where p and q are numbers to be found.b. Hence solve the equation tan2 x – 2tan x – 4 = 0 in the interval 0 ≤ x ≤ 540°.tan2 x – 2tan x – 4 = 0
a. Factorise 4xy − y2 + 4x − y.b. Solve the equation 4 sin θ cos θ − cos2 θ + 4 sin θ − cos θ = 0, in the interval 0 ≤ θ ≤ 360°.
The line 3x + y = 14 intersects the circle (x − 2)2 + (y − 3)2 = 5 at the points A and B.a. Find the coordinates of A and B.b. Determine the length of the chord AB.
In each of the following, eliminate θ to give an equation relating x and y:a. x = sin θ, y = cos θ b. x = sin θ, y = 2 cos θc. x = sin θ, y = cos2 θ d. x = sin θ, y = tan θe. x =
a. Show that 4 sin2 x – 3 cos2 x = 2 can be written as 7 sin2 x = 5.b. Hence solve, for 0 < x < 360°, the equation 4 sin2 x – 3 cos2 x = 2. Give your answers to 1 decimal place.
a. Express 4 cos 3θ − sin (90° − 3θ) as a single trigonometric function.b. Hence solve 4 cos 3θ − sin (90° − 3θ) = 2 in the interval 0 ≤ θ ≤ 360°. Give your answers to 3
The diagram shows the triangle ABC with AB = 12 cm, BC = 8 cm and AC = 10 cm.a. Show that cos B = 9/16b. Hence find the exact value of sin B. B 8 cm 12 cm 10 cm
The line with equation y = 3x − 2 does not intersect the circle with centre (0, 0) and radius r. Find the range of possible values of r.
a. Show that the equation 2 sin2x + 5 cos2x = 1 can be written as 3 sin2x = 4.b. Use your result in part a to explain why the equation 2 sin2 x + 5 cos2 x = 1 has no solutions.
Given that 2 sin 2θ = cos 2θ:a. Show that tan 2θ = 0.5.b. Hence find the values of θ, to one decimal place, in the interval 0 ≤ θ ≤ 360° for which 2 sin 2θ = cos 2θ.
The circle C has centre (1, 5) and passes through the point P(4, −2). Find:a. An equation for the circle C.b. An equation for the tangent to the circle at P.
The diagram shows triangle PQR with PR = 8 cm, QR = 6 cm and angle QPR = 30°.a. Show that sin Q = 2/3b. Given that Q is obtuse, find the exact value of cos Q P. 30° 8 cm 6 cm R
Find the values of x in the interval 0 ≤ x ≤ 270° which satisfy the equation cos 2x + 0.5 1 - cos 2x = 2 ||
Find all the values of θ in the interval 0 ≤ θ ≤ 360° for which:a. cos (θ + 75°) = 0.5,b. sin 2θ = 0.7, giving your answers to one decimal place.
The points A(2, 1), B(6, 5) and C(8, 3) lie on a circle.a. Show that ∠ABC = 90°.b. Deduce a geometrical property of the line segment AC.c. Hence find the equation of the circle.
Where a, b and c are constants. Work outthe values of a, b and c. 2x² + 20x + 42 224x + 4x2 - 4x³ = x + a bx(x + c)
Find, in degrees, the values of θ in the interval 0 ≤ θ ≤ 360° for which 2cos2 θ – cos θ – 1 = sin2 θ Give your answers to 1 decimal place, where appropriate.
Find the area of this trapezium in cm2. Give your answer in the form a + b√2, where a and b are integers to be found. -3+ √2 cm- 2√2 cm -(5 + 3√2) cm-
The diagram shows the graph of the quadratic function f(x). The graph meets the x-axis at (1, 0) and (3, 0) and the minimum point is (2, −1).a. Find the equation of the graph in the form y = ax2 +
The graph of y = f(x) where f(x) = 1/x is translated so that the asymptotes are at x = 4 and y = 0. Write down the equation for the transformed function in the form V 1 x + a
Given that p = 3 − 2√2 and q = 2 − √2,find the value ofGive your answer in the form m + n√2, where m and n are rational numbers to be found. p + q p-q
a. On the same axes sketch the curve y = x3 – 3x2 – 4x and the line y = 6x.b. Find the coordinates of the points of intersection.
The point P(4, –1) lies on the curve with equation y = f(x).a. State the coordinates that point P is transformed to on the curve with equation y = f(x – 2).b. State the coordinates that point P
f(x) = (x − 1)(x − 2)(x + 1).a. State the coordinates of the point at which the graph y = f(x) intersects the y-axis.b. The graph of y = af(x) intersects the y-axis at (0, −4). Find the value
The point Q(−2, 8) lies on the curve with equation y = f(x).State the coordinates that point Q is transformed to on the curve with equation y = f(1/2 x).
Write these lines in the form ax + by + c = 0. a y = 4x + 3 d y = x-6 g y = 2x - jy = -x + 1/2 by = 3x - 2 e_y=x+2 h y=-3x + ² k y = x + ²/ c y = -6x + 7 f_y=zx i_y=-6x - 3 1 y = x + 1/ ان |
The cost of electricity, E, in pounds and the number of kilowatt hours, h, are shown in the table.a. Draw a graph of the data.b. Explain how you know a linear model would be appropriate.c. Deduce an
a. On the same axes sketch the curve y = (x2 – 1)(x – 2) and the line y = 14x + 2.b. Find the coordinates of the points of intersection.
The point P(4, 3) lies on a curve y = f(x).a. State the coordinates of the point to which P is transformed on the curve with equation:i. y = f(3x) ii. 1/2 y = f(x) iii. y = f(x −
a. Sketch the graph of y = (x – 2)(x – 3)2.b. The graph of y = (ax – 2)(ax – 3)2 passes through the point (1, 0).Find two possible values for a.
a. Factorise the expression x2 − 10x +16.b. Hence, or otherwise, solve the equation 82y − 10(8y) + 16 = 0.
The line joining (3, −5) to (6, a) has a gradient 4. Work out the value of a.
Consider the points P(11, −8), Q(4, −3) and R(7, 5). Show that the line segment joining the points P and Q is not congruent to the line joining the points Q and R.
Find the equation of the line l which passes through the points A(7, 2) and B(9, −8). Give your answer in the form ax + by + c = 0.
The line L1 has gradient 1/7 and passes through the point A(2, 2). The line L2 has gradient −1 and passes through the point B(4, 8). The lines L1 and L2 intersect at the point C.a. Find an equation
The line y = 1/2x + 6 meets the x-axis at the point C. Find the equation of the line with gradient 2/3 that passes through the point C. Write your answer in the form ax + by + c = 0, where a, b and c
The coordinates of a quadrilateral ABCD are A(−6, 2), B(4, 8), C(6, 1) and D(−9, −8). Show that the quadrilateral is a trapezium.
A line is perpendicular to the line 3x + 8y − 11 = 0 and passes through the point (0, −8). Find an equation of the line.
A racing car accelerates from rest to 90 m/s in 10 seconds. The table shows the total distance travelled by the racing car in each of the first 10 seconds.a. Draw a graph of the data.b. Explain how
The line joining (5, b) to (8, 3) has gradient −3. Work out the value of b.
The distance between the points (−1, 13) and (x, 9) is √65. Find two possible values of x.
The line y = 6x − 18 meets the x-axis at the point P. Work out the coordinates of P.
The vertices of the triangle ABC have coordinates A(3, 5), B(−2, 0) and C(4, −1). Find the equations of the sides of the triangle.
a. Find an equation of the line l which passes through the points A(1, 0) and B(5, 6). The line m with equation 2x + 3y = 15 meets l at the point C.b. Determine the coordinates of C.
The line y = 1/4x + 2 meets the y-axis at the point B. The point C has coordinates (−5, 3). Find the gradient of the line joining the points B and C.
A line is parallel to the line y = 5x + 8 and its y-intercept is (0, 3).Write down the equation of the line.
Find an equation of the line that passes through the point (6, −2) and is perpendicular to the line y = 3x + 5.
The line joining (c, 4) to (7, 6) has gradient 3/4. Work out the value of c.
The distance between the points (2, y) and (5, 7) is 3√10. Find two possible values of y.
The line 3x + 2y = 0 meets the x-axis at the point R. Work out the coordinates of R.
A website designer charges a flat fee and then a daily rate in order to design new websites for companies. Company A’s new website takes 6 days and they are charged £7100. Company B’s new
The straight line l passes through (a, 4) and (3a, 3). An equation of l is x + 6y + c = 0. Find the value of a and the value of c.
The line L passes through the points A(1, 3) and B(−19, −19). Find an equation of L in the form ax + by + c = 0. where a, b and c are integers.
The line that passes through the points (2, −5) and (−7, 4) meets the x-axis at the point P. Work out the coordinates of the point P.
A line is parallel to the line y = −2/5x + 1 and its y-intercept is (0, −4). Work out the equation of the line. Write your answer in the form ax + by + c = 0, where a, b and c are integers.
Find an equation of the line that passes through the point (−2, 5) and is perpendicular to the line y = 3x + 6.
The line joining (−1, 2d) to (1, 4) has gradient −1/4. Work out the value of d.
a. Show that the straight line l1 with equation y = 2x + 4 is parallel to the straight line l2 with equation 6x − 3y − 9 = 0.b. Find the equation of the straight line l 3 that is perpendicular to
The line 5x − 4y + 20 = 0 meets the y-axis at the point A and the x-axis at the point B. Work out the coordinates of A and B.
The straight line l1 passes through the points A and B with coordinats (2, 2) and (6, 0) respectively.a. Find an equation of l1. The straight line l2 passes through the point C with coordinate (−9,
The average August temperature in Exeter is 20°C or 68°F. The average January temperature in the same place is 9°C or 48.2°F.a. Write an equation linking Fahrenheit F and Celsius C in the form F
The straight line l passes through (7a, 5) and (3a, 3). An equation of l is x + by − 12 = 0. Find the value of a and the value of b.
The line that passes through the points (−3, −5) and (4, 9) meets the y-axis at the point G. Work out the coordinates of the point G.
The line that passes through the pointsmeets the y-axis at the point J. Work out the coordinates of the point J. (3, 2) and (-11, 4)
A line is parallel to the line 3x + 6y + 11 = 0 and its intercept on the y-axis is (0, 7). Write down the equation of the line.
The scatter graph shows the height h and foot length f of 8 students. A line of best fit is drawn on the scatter graph.a. Explain why the data can be approximated to a linear model.b. Use points A
Find an equation of the line that passes through the point (3, 4) and is perpendicular to the line 4x − 6y + 7 = 0.
The line joining (−3, −2) to (2e, 5) has gradient 2. Work out the value of e.
A point P lies on the line with equation y = 4 − 3x. The point P is a distance √34 from the origin. Find the two possible positions of point P.
A line l passes through the points with coordinates (0, 5) and (6, 7).a. Find the gradient of the line.b. Find an equation of the line in the form ax + by + c = 0.
In 2004, in a city, there were 17,500 homes with internet connections. A service provider predicts that each year an additional 750 homes will get internet connections.a. Write a linear model for the
The straight line l passes through A(1, 3 √3) and B(2 + √3, 3 + 4√3). Show that l meets the x-axis at the point C(−2, 0).
A line is parallel to the line 2x − 3y − 1 = 0 and it passes through the point (0, 0). Write down the equation of the line.
Find an equation of the line that passes through the point (5, −5) and is perpendicular to the line y = 2/3x + 5. Write your answer in the form ax + by + c = 0, where a, b and c are integers.
The line joining (7, 2) to ( f, 3f) has gradient 4. Work out the value of f.
The vertices of a triangle are A(2, 7), B(5, −6) and C(8, −6).a. Show that the triangle is a scalene triangle.b. Find the area of the triangle ABC.
A line l cuts the x-axis at (5, 0) and the y-axis at (0, 2).a. Find the gradient of the line.b. Find an equation of the line in the form ax + by + c = 0.
The points A and B have coordinates (−4, 6) and (2, 8) respectively. A line p is drawn through B perpendicular to AB to meet the y-axis at the point C.a. Find an equation of the line p.b. Determine
The lines y = x and y = 2x − 5 intersect at the point A. Find the equation of the line with gradient 2/5 that passes through the point A.
Find an equation of the line that passes through the point (−2, 7) and is parallel to the line y = 4x + 1. Write your answer in the form ax + by + c = 0.
Find an equation of the line that passes through the point (−2, −3) and is perpendicular to the line y = − 4/7x + 5. Write your answer in the form ax + by + c = 0, where a, b and c are integers.
The line joining (3, −4) to (−g, 2g) has gradient −3. Work out the value of g.
The straight line l1 has equation y = 7x − 3. The straight line l2 has equation 4x + 3y − 41 = 0. The lines intersect at the point A.a. Work out the coordinates of A. The straight line l2
Show that the line with equation ax + by + c = 0 has gradient −a/b and cuts the y-axis at −c/b.
The price P of a good and the quantity Q of a good are linked. The demand for a new pair of trainers can be modelled using the equation P = −3/4Q + 35. The supply of the trainers can be
The line l has equation 2x − y − 1 = 0. The line m passes through the point A(0, 4) and is perpendicular to the line l.a. Find an equation of m.b. Show that the lines l and m intersect at the
The lines y = 4x − 10 and y = x − 1 intersect at the point T. Find the equation of the line with gradient −2/3 that passes through the point T. Write your answer in the form ax + by + c = 0,

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