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edexcel as and a level mathematics

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

a. Show that (2 + 5 √x)2 can be written as 4 + k√x + 25x, where k is a constant to be found.b. Hence find ∫(2 + 5 √x)2 dx.
Given that y = 3x5 − 4/√x, x > 0, find ∫y dx in its simplest form.
The shaded area under the graph of the function f(x) = 3x2 − 2x + 2,bounded by the curve, the x-axis and the lines x = 0 and x = k, is 8. Work out the value of k. y = 3x² - 2x +2 k
The curve C has equationa. Verify that C crosses the x-axis at the point (1, 0).b. Show that the point A(8, 4) also lies on C.c. The point B is (4, 0). Find the equation of the line through AB. The finite region R is bounded by C, AB and the positive x-axis.d. Find the area of R. y=x²_2 +1. 1 X3
The speed, v ms−1, of a train at time t seconds is given by v = 20 + 5t, 0 ≤ t ≤ 10.The distance, s metres, travelled by the train in 10 seconds is given byFind the value of s. 10 S= = "0 (20 + 5t)dt.
Given that y1/2 = 3x1/4 − 4x−1/4 (x > 0):a. Find dy/dxb. Find ∫y dx.
The height, in metres, of an arrow fired horizontally from the top of a castle is modelled by the function f(t), where f(0) = 35. Given that f′(t) = −9.8t,a. Find f(t).b. Determine the height of the arrow when t = 1.5.c. Write down the height of the castle according to this model.d. Estimate
The diagram shows the curve C with equation y = x(8 − x) and the line with equation y = 12 which meet at the points L and M.a. Determine the coordinates of the point M.b. Given that N is the foot of the perpendicular from M on to the x-axis, calculate the area of the shaded region which is
f(x) = (2 − x)10Given that x is small, and so terms in x3 and higher powers of x can be ignored:a. Find an approximation for f(x) in the form A + Bx + Cx2b. Find an approximation for ∫f(x)dx.
Rewrite using a power.a. log2 16 = 4 b. log5 25 = 2 c. log9 3 = 1/2d. log5 0.2 = −1 e. log10 100 000 = 5
Solve these equations, giving your answers to 3 significant figures.a. 4x = 23 b. 72x + 1 = 1000 c. 10x = 6x + 2
A student is asked to evaluate the integralThe student’s working is shown belowa. Identify two errors made by the student.b. Evaluate the definite integral, giving your answer correct to 3 significant figures. 1,² (+ + - + 7/3 + + √x + 2 dx
Each of the sketch graphs below is of the form y = Aebx + C, where A, b and C are constants. Find the values of A and C for each graph, and state whether b is positive or negative. a y X b ya 0 8 2 X
The data below follows a trend of the form y = abx, where a and b are constants.a. Copy and complete the table of values of x and log y, giving your answers to 2 decimal places.b. Plot a graph of log y against x and draw in a line of best fit.c. Use your graph to estimate the values of a and b to
The data below follows a trend of the form y = axn, where a and n are constants.a. Copy and complete the table of values of log x and log y, giving your answers to 2 decimal places.b. Plot a graph of log y against log x and draw in a line of best fit.c. Use your graph to estimate the values of a
Write in terms of loga x, loga y and loga z. a loga (x³y4z) d loga X yz, b loga 312 e loga vax c loga (a²x²)
The number of rabbits, R, in a population after m months is modelled by the formula R = 12e0.2ma. Use this model to estimate the number of rabbits afteri. 1 month ii. 1 yearb. Interpret the meaning of the constant 12 in this model.c. Show that after 6 months, the rabbit population is
Given the four points X(9, 6), Y(13, −2), Z(0, −15), and C(1, −3),a. Show thatb Using your answer to part a or otherwise, find the equation of the circle which passes through the points X, Y and Z. |CX| = |CY| = |CZ|.
For each of these statements, decide whether it is true or false, justifying your answer or offering a counter-example.a. The graph of y = ax passes through (0, 1) for all positive real numbers a.b. The function f(x) = ax is always an increasing function for a > 0.c. The graph of y = ax, where a
Solve these equations, giving your answers in exact form.a. e2x − 8ex + 12 = 0 b. e4x − 3e2x = −2c. (ln x)2 + 2 ln x − 15 = 0 d. ex − 5 + 4e−x = 0e. 3e2x + 5 = 16ex f. (ln x)2 = 4(ln x + 3)
Solve the following equations, giving your answers to 3 significant figures where appropriate.a. 3x + 1 = 2000b. log5 (x − 3) = −1
Without using a calculator, find the value ofa. log2 8 b. log5 25 c. log10 10 000 000 d. log12 12e. log3 729 f. log10 √10 g. log4 (0.25) h. log0.25 16i. loga (a10) j. log2/3 (9/4)
The number of people infected with a disease is modelled by the formula N = 300 − 100e−0.5t where N is the number of people infected with the disease and t is the time in years after detection.a. How many people were first diagnosed with the disease?b. What is the long term prediction of how
Sketch the graphs of:a. y = ex + 1 b. y = 4e−2xc. y = 2ex − 3d. y = 4 − ex e. y = 6 + 10e1/2 x f. y = 100e−x + 10
Given that p = logq 16, express in terms of p,a. logq 2b. logq (8q)
Sketch the graph of y = 1x.
Find the equation of the line parallel to 2x − 3y + 4 = 0 that passes through the point (5, 6). Give your answer in the form y = ax + b where a and b are rational numbers.
Solve these equations, giving your answers in exact form.a. ln x = 2 b. ln (4x) = 1 c. ln (2x + 3) = 4d. 2 ln (6x − 2) = 5 e. ln (18 − x) = 1/2 f. ln (x2 − 7x + 11) = 0
Write as a single logarithm, then simplify your answer.a. log2 40 − log2 5 b. log6 4 + log6 9 c. 2log12 3 + 4log12 2d. log8 25 + log8 10 − 3log8 5 e. 2log10 2 − (log10 5 + log10 8)
Solve, giving your answers to 3 significant figures.a. 22x − 6(2x) + 5 = 0 b. 32x − 15(3x) + 44 = 0c. 52x − 6(5x) − 7 = 0 d. 32x + 3x + 1 − 10 = 0e. 72x + 12 = 7x + 1 f. 22x + 3(2x) − 4 = 0g. 32x + 1 − 26(3x) − 9 = 0 h. 4(32x + 1) + 17(3x) − 7 = 0
Given that |5i − kj| = |2ki + 2j|, find the exact value of the positive constant k.
The population of a country is modelled using the formula P = 20 + 10 et/50 where P is the population in thousands and t is the time in years after the year 2000.a. State the population in the year 2000.b. Use the model to predict the population in the year 2030.c. Sketch the graph of P against t
The graph shows a sketch of part of the curve C with equation y = x(x − 3)(x + 2).The curve crosses the x-axis at the origin O and the points A and B.a. Write down the x-coordinates of the points A and B.The finite region shown shaded is bounded by the curve C and the x-axis.b. Use integration to
a. Draw an accurate graph of y = ex for −4 ≤ x ≤ 4.b. By drawing appropriate tangent lines, estimate the gradient at x = 1 and x = 3.c. Compare your answers to the actual values of e and e3.
The finite region S, which is shown shaded, is bounded by the x-axis and the curve with equation y = 3 − 5x − 2x2.The curve meets the x-axis at points A and B.a. Find the coordinates of point A and point B.b. Find the area of the region S. S O y=3-5x-2x² B
a. Express log a (p2q) in terms of log a p and log a q.b. Given that loga (pq) = 5 and loga (p2q) = 9, find the values of loga p and loga q.
a. Show that f′(x) = 8x−2 − 12 + Ax2 + Bx4, where A and B are constants to be found.b. Find f″(x). Given that the point (−2, 9) lies on the curve with equation y = f(x),c. Find f(x). f'(x) = (2-x²)³ x2 , X = 0
a. Draw an accurate graph of y = (0.6)x, for −4 ≤ x ≤ 4.b. Use your graph to solve the equation (0.6)x = 2.
Two variables A and x satisfy the formula A = 6x4.a. Show that log A = log 6 + 4 log x.b. The straight line graph of log A against log x is plotted. Write down the gradient and the value of the intercept on the vertical axis.
a. Given that 4 = 64n, find the value of n.b. Write √50 in the form k√2 where k is an integer to be determined.
Given that A is constant andshow that there are two possible values for A and find these values. 3 L² (2²/7 - 4) dx = X A x = 4²
Solve these equations, giving your answers in exact form.a. ex = 6 b. e2x = 11 c. e−x + 3 = 20d. 3e4x = 1 e. e2x + 6 = 3 f. e5 − x = 19
Solve, giving your answers to 3 significant figures.a. 2x = 75 b. 3x = 10 c. 5x = 2 d. 42x = 100e. 9x + 5 = 50 f. 72x − 1 = 23 g. 113x − 2 = 65 h. 23 − 2x = 88
Write as a single logarithm.a. log2 7 + log2 3 b. log2 36 − log2 4 c. 3 log5 2 + log5 10d. 2 log6 8 − 4 log6 3 e. log10 5 + log10 6 − log10 (1/4)
Rewrite using a logarithm.a. 44 = 256 b. 3−2 = 1/9 c. 106 = 1000000d. 111 = 11 e. (0.2)3 = 0.008
The vector 9i + qj is parallel to the vector 2i − j. Find the value of the constant q.
The diagram shows the line y = x − 1 meeting the curve with equation y = (x − 1)(x − 5) at A and C.The curve meets the x-axis at A and B.a. Write down the coordinates of A and B and find the coordinates of C.b. Find the area of the shaded region bounded by the line, the curve and the x-axis.
The value of a car is modelled by the formula V = 20 000 e−t/12 where V is the value in £s and t is its age in years from new.a. State its value when new.b. Find its value (to the nearest £) after 4 years.c. Sketch the graph of V against t.
Use a calculator to find the value of ex to 4 decimal places whena. x = 1 b. x = 4 c. x = −10 d. x = 0.2
The diagram shows part of the curve with equation y = p + 10x − x2, where p is a constant, and part of the line l with equation y = qx + 25, where q is a constant. The line l cuts the curve at the points A and B. The x-coordinates of A and B are 4 and 8 respectively. The line through A parallel
a. Draw an accurate graph of y = (1.7)x, for −4 ≤ x ≤ 4.b. Use your graph to solve the equation (1.7)x = 4.
Sketch each of the following graphs, labelling all intersections and asymptotes.a. y = 2−x b. y = 5ex − 1 c. y = ln x
Two variables, S and x satisfy the formula S = 4 × 7x.a. Show that log S = log 4 + x log 7.b. The straight line graph of log S against x is plotted. Write down the gradient and the value of the intercept on the vertical axis.
Given that f(x) = 9/x2 − 8√x + 4x − 5, x > 0, find ∫f(x)dx.
A rectangular box has sides measuring x cm, x + 3 cm and 2x cm.a. Write down an expression for the volume of the box. Given that the volume of the box is 980 cm3,b. Show that x3 + 3x2 − 490 = 0.c. Show that x = 7 is a solution to this equation.d. Prove that the equation has no other real
Two forces, F1 and F2, act on a particle.F1 = 2i − 5j newtonsF2 = i + j newtonsThe resultant force R acting on the particle is given by R = F1 + F2.a. Calculate the magnitude of R in newtons. A third force, F3 begins to act on the particle, where F3 = kj newtons and k is a positive constant. The
Differentiate the following with respect to x.a. e6x b. e−1/3 x c. 7e2xd. 5e0.4x e. e3x + 2ex f. ex(ex + 1)
The population, P, of a colony of endangered Caledonian owlet-nightjars can be modelled by the equation P = abt where a and b are constants and t is the time, in months, since the population was first recorded.The line l shown in figure 2 shows the relationship between t and log10P for the
Solve 3xe4x − 1 = 5, giving your answer in the form a + lnb c + Ind
Zipf’s law is an empirical law which relates how frequently a word is used, f, to its ranking in a list of the most common words of a language, R. The law follows the form f = ARb, where A and b are constants to be found.The table below contains data on four words.a. Copy and complete this table
a. Without using a calculator, justify why the value of log2 50 must be between 5 and 6.b. Use a calculator to find the exact value of log2 50 to 4 significant figures.
Given that a and b are positive constants, and that a > b, solve the simultaneous equationsa + b = 13log6 a + log6 b = 2
Nigel has bought a tractor for £20 000. He wants to model the depreciation of the value of his tractor, £T, in t years. His friend suggests two models:Model 1: T = 20 000e−0.24tModel 2: T = 19 000e−0.255t + 1000a. Use both models to predict the value of the tractor after one year. Compare
The point P with x-coordinate −1 lies on the curve y = f(x). Find the equation of the normal to the curve at P, giving your answer in the form ax + by + c = 0 where a, b and c are positive integers. f(x) = x³5x² - 2 + -1 Xx²
Kleiber’s law is an empirical law in biology which connects the mass of an animal, m, to its resting metabolic rate, R. The law follows the form R = amb, where a and b are constants. The table below contains data on five animals.a. Copy and complete this table giving values of log R and log m to
The vectors a, b and c are given aswhere x is an integer. Given that a + b is parallel to b − c, find the value of x. a = = (23), b = (-15) and c = -13) 2 "
Solve the equation log2 (x + 10) − log2 (x − 5) = 4.
The graph of y = kax passes through the points (1, 6) and (4, 48). Find the values of the constants k and a.
Solve these equations, giving your answers in exact form.a. ln (8x − 3) = 2 b. e5(x − 8) = 3 c. e10x − 8e5x + 7 = 0d. (ln x − 1)2 = 4
Solve the following equations, giving your answers to four decimal places.a. 5x = 2x + 1 b. 3x + 5 = 6x c. 7x + 1 = 3x + 2
a. Given that log3 (x + 1) = 1 + 2 log3 (x − 1), show that 3x2 − 7x + 2 = 0.b. Hence, or otherwise, solve log3 (x + 1) = 1 + 2 log3 (x − 1).
Use your calculator to evaluate these logarithms to three decimal places.a. log9 230 b. log5 33 c. log10 1020 d. loge 3
On Earth, the atmospheric pressure, p, in bars can be modelled approximately by the formula p = e−0.13h where h is the height above sea level in kilometres.a. Use this model to estimate the pressure at the top of Mount Rainier, which has an altitude of 4.394 km.b. Demonstrate that dp/dh = kp
Rearrange f(x) = e3x + 2 into the form f(x) = Aebx, where A and b are constants whose values are to be found. Hence, or otherwise, sketch the graph of y = f(x).
a. Using the substitution u = 2x, show that the equation 4x − 2x + 1 − 15 = 0 can be written in the form u2 − 2u − 15 = 0.b. Hence solve the equation 4x − 2x + 1 − 15 = 0, giving your answer to 2 decimal places.
The function f(x) is defined as f(x) = 3x, x ∈ ℝ. On the same axes, sketch the graphs of:a. y = f(x) b. y = 2f(x) c. y = f(x) − 4 d. y = f(1/2 x)Write down the coordinates of the point where each graph crosses the y-axis, and give the equations of any asymptotes.
Find all the solutions in the interval 0 ≤ x ≤ 180° of 2sin2(2x) − cos(2x) − 1 = 0 giving each solution in degrees.
Find the exact solutions to the equation ex + 12e−x = 7.
a. Sketch the graph of y = 4x, stating the coordinates of any points where the graph crosses the axes.b. Solve the equation 42x − 10(4x) + 16 = 0.
Solve the following equations:a. log2 3 + log2 x = 2 b. log6 12 − log6 x = 3c. 2log5 x = 1 + log5 6 d. 2log9 (x + 1) = 2log9 (2x − 3) + 1
Without using a calculator, find the value of x for whicha. log5 x = 4b. logx 81 = 2 c. log7 x = 1d. log2 (x − 1) = 3 e. log3 (4x + 1) = 4 f. logx (2x) = 2
In the triangle ABC, AB(vector) = 9i + 2j and AC(vector) = 7i − 6j.a. Find BC(vector) .b. Prove that the triangle ABC is isosceles.c. Show that cos ∠ABC = 1/√5
A scientist is modelling the number of people, N, who have fallen sick with a virus after t days.From looking at this graph, the scientist suggests that the number of sick people can be modelled by the equation N = abt, where a and b are constants to be found.The graph passes through the points (0,
The table below shows the population of Mozambique between 1960 and 2010.This data can be modelled using an exponential function of the form P = abt, where t is the time in years since 1960 and a and b are constants.a. Copy and complete the table below.b. Show that P = abt can be rearranged into
Prove that 1 + cos4x − sin4x ≡ 2 cos2x.
Officials are testing athletes for doping at a sporting event. They model the concentration of a particular drug in an athlete’s bloodstream using the equation D = 6 e−t /10 where D is the concentration of the drug in mg/l and t is the time in hours since the athlete took the drug.a. Interpret
Find the values of:i. log2 2 ii. log3 3 iii. log17 17b. Explain why loga a has the same value for all positive values of a (a ≠ 1).
A helicopter takes off from its starting position O and travels 100 km on a bearing of 060°. It then travels 30 km due east before landing at point A. Given that the position vector of A relative to O is (mi + nj) km, find the exact values of m and n.
Find the gradient of the curve with equation y = e3x at the point wherea. x = 2 b. x = 0 c. x = −0.5
Differentiate each of the following expressions with respect to x.a. e−x b. e11x c. 6e5x
The graph of y = pqx passes through the points (−3, 150) and (2, 0.048).a. By drawing a sketch or otherwise, explain why 0 < q < 1.b. Find the values of the constants p and q.
Solve the following equations, giving exact solutions.a. ln (2x − 5) = 8 b. e4x = 5 c. 24 − e−2x = 10d. ln x + ln (x − 3) = 0 e. ex + e−x = 2 f. ln 2 + ln x = 4
The function f is defined as f(x) = e0.2x, x ∈ ℝ. Show that the tangent to the curve at the point (5, e) goes through the origin.
a. Find the values of:i. log2 1 ii. log3 1 iii. log17 1b. Explain why loga 1 has the same value for all positive values of a (a ≠ 1).
At the very end of a race, Boat A has a position vector of (−65i + 180j) m and Boat B has a position vector of (100i + 120j) m. The finish line has a position vector of 10i km.a. Show that Boat B is closer to the finish line than Boat A. Boat A is travelling at a constant velocity of (2.5i −
A rock is dropped off a cliff. The height in metres of the rock above the ground after t seconds is given by the function f(t). Given that f(0) = 70 and f′(t) = −9.8t, find the height of the rock above the ground after 3 seconds.
The sketch shows part of the curve with equation y = x2(x + 4). The finite region R1 is bounded by the curve and the negative x-axis. The finite region R2 is bounded by the curve, the positive x-axis and AB, where A(2, 24) and B(b, 0).The area of R1 = the area of R2.a. Find the area of R1.b. Find
Given that f(x) = 6/x2 + 4√x − 3x + 2, x > 0, find ∫f(x)dx.
A cyclist is travelling along a straight road. The distance in metres of the cyclist from a fixed point after t seconds is modelled by the function f(t), where f′(t) = 5 + 2t and f(0) = 0.a. Find an expression for f(t).b. Calculate the time taken for the cyclist to travel 100 m.
A triangular lawn ABC is shown in figure 3:Given that AB = 7.5 m, BC = 10.6 m and AC = 12.7 m,a. Find angle BAC. Grass seed costs £1.25 per square metre.b. Find the cost of seeding the whole lawn. A B Figure 3 Diagram not to scale C

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