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cambridge igcse and o level accounting coursebook

Cambridge IGCSE And O Level Additional Mathematics Coursebook 3rd Edition Sue Pemberton - Solutions

a Given that find the value of a and the value ofb. (x ^ 2 + 4/x) ^ 3 - (x ^ 2 - 4/x) ^ 3 = a * x ^ 3 + b/(x ^ 3)b Hence, without using a calculator, find the exact value of (2 + 4/(sqrt(2))) ^ 3 - (2 - 4/(sqrt(2))) ^ 3
The coefficient of x in the expansion of (3 + ax) ^ 5 12 times the coefficient of x in the expansion of (1 + (ax)/2) ^ 4 Find the value of a.
a Find the first three terms, in ascending powers of y, in the expansion of (2 + y) ^ 5 b By replacing y with 3x4x², find the coefficient of x in the expansion of (2 + 3x - 4x ^ 2) ^ 5
Find the term independent of x in the expansion of (x ^ 2 + 1/(2x)) ^ 3
Find the coefficient of x in the expansion or
a Expand (2-x).b Find the coefficient of x in the expansion of (1 + 3x ^ 2) * (2 - x ^ 2) ^ 4
The diagram shows an isosceles triangle PQR inscribed in a circle, centre O, radius rem.PRQR and angle ORP = 0 radians.Triangle PQR has an area of A cm².a Show that A = r ^ 2 * sin 2theta + r ^ 2 * sin 2theta * cos 2theta b Find the value of 6 for which A has a stationary value and determine the
The diagram shows a semi-circle with diameter EF of length 12cm.Angle GEF radians and the shaded region has an area of Ac * m ^ 2 a Show that A = 36theta + 18sin 2theta b Given that @ is increasing at a rate of 0.05 radians per second, find the rate of change of A when theta = pi/6 radians. E 12 cm
A curve has equation y = x ^ 2 * e ^ x The curve has a minimum point at P and a maximum point at Q.a. Find the coordinates of P and the coordinates of Q.b. The tangent to the curve at the point A(1,e) meets the y-axis at the point B. The normal to the curve at the point A(1,e) meets the y-axis at
A curve has equation y = x * ln(x) The curve crosses the x-axis at the point A and has a minimum point at B. Find the coordinates of A and the coordinates of B.
A curve has equation y = A * e ^ (2x) + B * e ^ (-2x) The gradient of the tangent at the point (0, 10) is-12.a Find the value of A and the value of B.b Find the coordinates of the turning point on the curve and determine its nature.
Find the coordinates of the stationary points on these curves and determine their nature. a y=4sinx + 3cos.x for 0
Find the coordinates of the stationary points on these curves and determine their nature.
A curve has equation y = (ln(x ^ 2 - 2))/(x ^ 2 - 2)Find the approximate change in y as x increases from sqrt(3) to sqrt(3) + p where p is small.
Variables x and y are connected by the equation y = 3 + 2x - 5e ^ (- x) Find the approximate change in y as x increases from In 2 to In 2 + p where p is small.
Variables x and y are connected by the equation y = (ln(x))/(x ^ 2 + 3)Find the approximate change in y as x increases from 1 to 1 + p where p is small.
Variables x and y are connected by the equation y = 3 + ln(2x - 5) Find the approximate change in y as x increases from 4 to 4 + p where p is small.
Variables x and y are connected by the equation y = sin 2x T Find the approximate increase in y as x increases from top, where p is small.
A curve has equation y = xe". The tangent to the curve at the point P (1,e) meets the y-axis at the point A. The normal to the curve at P meets the x-axis at the point B. Find the area of triangle OAB, where O is the origin.
A curve has equation y = 5 - e ^ (2x) The curve crosses the x-axis at A and the y-axis at B.a Find the coordinates of A and B.b The normal to the curve at B meets the x-axis at the point C. Find the coordinates of C.
3 A curve has equation y = e ^ (1/2 * x) + 1 The curve crosses the y-axis at P.The normal to the curve at P meets the x-axis at Q.Find the coordinates of Q.
A curve has equation y = x * sin 2x for 0
A curve has equation y = 3sin(2x + pi/2)Find the equation of the normal to the curve at the point on the curve where x = pi/4
A curve has equation y = A * sin x + B * cos 2x The curve has a gradient of 5sqrt(3) when x = pi/6 and has a gradient of 6 + 2sqrt(2) when x = pi/4 Find the value of A and the value of B.
Find the gradient of the tangent to a y=2xcos 3x when x = b 2 cosx TT y= when x= 3 tan.x
Differentiate with respect to x. de(sinx + cosx) e*(cosx+sinx) here cosx a b C tan x e e sin x f c'cosx g In(sin x) j xIn(cos x) sin 3x k e2x-1 xsin x
Differentiate with respect to x. a xsin x d xtan 22 e cos 3x b 2 sin 2x cos 3x xtan x 5 X f COSX tan.x sin.x g h 2+cos x 1 3x j k sin 2x sin 2x sin.x 3x-1 sinx + cos x sin.x-cos.x
Differentiate with respect to x. a sinx b 5cos(3x) sinx 2cosx d (3-cos.x) e 2sin (2x+) f 3 cosx+2tan(2x- tan 2 (2x-7)
Differentiate with respect to x. a 2+ sin x b 2 sin x + 3 cos x C 2cosx tanx d 3 sin 2x e g tan (3x+2) h 4 tan 5x sin(2x+ f 2 cos 3x - sin 2x i 2 cos(3x-7)
Find dy/dx for each of the following. a e=4x-1 be=5x-2x ce=(x+3)(x-4)
Find dy/dx for each of the following. a y = log3x by=log2x2 y = log(5x-1)
Use the laws of logarithms to help differentiate these expressions with respect to x. a In3x+1 b In C (2x-5) In [x(x-5)] g In d In (2x+1) 2x + 3 (x-5)(x+1)] [x(x+1) In f In x+2 2 [(x+1)(2x-3)] h In i In (x+3)(x-1) x(x-1)
Differentiate with respect to x. a xlnx b 2xInx c (x-1)Inx d 5xln.x2 e xIn (Inx) In 2x f x 4 In (2x+1) In (x-1) g h Inx x 2x+3
Differentiate with respect to x. a In 5x b In 12x d 2+ In(1-x) e In (3x+1) c In(2x+3) f Inx+2 g In (2-5x) h 2x + In (4) 3 i 5-In (2-3x) j In (Inx) k In(x+1) In(x+ Inx)
Differentiate with respect to x. Inx a 5x In 2x b 3x
Differentiate with respect to x. a In 8x b In (5x-7) c In(2x+5) d Inx-10
A curve has equation y = xe".a Find, in terms ofe, the coordinates of the stationary point on this curve and determine its nature.b Find, in terms ofe, the equation of the normal to the curve at the point P(1, e).c The normal at P meets the x-axis at A and the y-axis at B. Find, in terms ofe, the
A curve has equation y = 5e ^ (2x) - 4x - 3 The tangent to the curve at the point (0, 2) meets the x-axis at the point A.Find the coordinates of A.
Find the equation of the tangent to 5 y= at x=0 +3 by ve+1 at x = In 5 cy=x(1+e) at x = 1.
Differentiate with respect to x. 10 4 b xx C 3.xe ex+1 dxe xe-5 g h i e +1
Differentiate with respect to x. e e i e7x b e3x C 3e5x 6e d 2e4* f e3x+1 ge h 5x-3ex 2+ 3x j 2(3-e2x) k e*+e* 2 1 5(x + e*)
At time t = 0 boat P leaves the origin and travels with velocity (3i+4j) kmh. Also at timer t = 0 boat Q leaves the point with position vector (- 10i + 17i) * km and travels with velocity (5i+2j) kmh.a Write down the position vectors of boats A and B after 2 hours.b Find the distance between boats
At 1200 hours, boats A and B have position vectors (- 10i + 40j) km and (70i + 10i) km and are moving with velocities (20i+10j) kmh and (-10i+30j) kmh¹ respectively.a Find the position vectors of A and B at 1500 hours.b Find the distance between A and B at 1500 hours.
At 1500 hours, a submarine departs from point A and travels a distance of 120 km to a point B.The position vector, r km, of the submarine relative to an origin O, 1 hours after a Write down the position vector of the point A.b Write down the velocity vector of the submarine.C Find the position
At 1200 hours, a boat sails from a point P. The position vector, rkm, of the boat relative to an origin O, 1 hours after 1200 is given by r = binomial(10,6) + t * binomial(5,12)a Write down the position vector of the point P.b Write down the velocity vector of the boat.C Find the speed of the
At 1200 hours, a tanker sails from a point P with position vector (5i + 12j) km relative to an origin O. The tanker sails south-east with a speed of 12sqrt(2) * km * h ^ - 1 a Find the velocity vector of the tanker.b Find the position vector of the tanker at i 1400 hours ii 1245 hours.C Find the
At 1200 hours, a ship leaves a point Q with position vector (10i + 38i) km relative to an origin O. The ship travels with velocity (6i8j) kmh¹.a Find the speed of the ship.b Find the position vector of the ship at 3 pm.c Find the position vector of the ship / hours after leaving Q.d Find the time
A particle starts at a point P with position vector (- 80i + 60j) m relative to an origin O. The particle travels with velocity (12i -16j) ms.a Find the speed of the particle.b Find the position vector of the particle after i I second ii 2 seconds iii 3 seconds.c Find the position vector of the
4 a A car travels north-east with a speed of 18sqrt(2) * km * h ^ - 1 Find the velocity vector of the car.b A boat sails on a bearing of 030° with a speed of 20 kmh.Find the velocity vector of the boat.c A plane flies on a bearing of 240° with a speed of 100ms. Find the velocity vector of the
A helicopter flies from a point P with position vector (50i + 100j) km to a point Q. The helicopter flies with a constant velocity of (30i-40j) kmh¹ and takes 2.5 hours to complete the journey. Find the position vector of the point Q.
A car travels from a point A with position vector (60i40j) km to a point B with position vector (- 50i + 18j) km. The car travels with constant velocity and takes 5 hours to complete the journey. Find the velocity vector.
a Displacement = (21i + 54j) * m time taken = 6 seconds. Find the velocity.b Velocity = (5i6j)ms, time taken = 6 seconds. Find the displacement.C Velocity = (-4i+4j) kmh, displacement = (-50i+50j) km. Find the time taken.
The vector OA has a magnitude of 25 units and is parallel to the vector -31 +4j.The vector OB has a magnitude of 26 units and is parallel to the vector 12i + 5j. Find 1 OA b OB AB d |AB|
a O is the origin, P is the point (1, 5) and Find 00 vec PQ = [[3], [5]] .b O is the origin, E is the point (-3, 4) and Find the position vector of F. C O is the origin, M is the point (4,-2) and Find the position vector of N. vec EF = binomial(- 2,7) vec NM = [[3], [- 5]]
Find AB, in the form ai + bj, for each of the following.a A(4, 7) and B(3, 4)b (0, 6) and B(2, -4) b c A (3, 3) and B (6, -2)d A (7,0) and B(-5, 3)e (-4, -2) and B(-3, 5) e f A(5,-6) and B(-1, -7).
Relative to an origin O, the position vector of A is 4i2j and the position vector of B is ¿i + 2j. The unit vector in the direction of AB is 0.31 +0.4j. Find the value of λ.
Relative to an origin O, the position vector of Pis-2i4j and the position vector of Q is 8i + 20j.a Find PQ.b Find [PQ].C Find the unit vector in the direction of PQ.d Find the position vector of M, the midpoint of PQ.
4 Relative to an origin O, the position vector of A is-7i7j and the position vector of B is 9i + 5j.The point Clies on AB such that AC = 3CB.a Find AB.b Find the unit vector in the direction of AB.C Find the position vector of C.
Find and such that a + b =c. c = - 13i + 18j a = 5i - 6j b = - i + 2j
p = 7i - 2j q = i + j. Find and such that lambda*p + q = 36i - 13j
p = 9i + 12j q = 3i - 3j and r = 7i + j Find a p+q b p+q+r.
Find p = 8i - 6j q = - 2i + 3j and r = 10i a 2q b 2p+q 2P-3r d r-p-q
Find the unit vector in the direction of each of these vectors.a 6i +8j b Si + 12j C -4i-3j d 8i-15j e 3i+3j
4 The vector PQ has a magnitude of 39 units and is parallel to the vector 121-5j. Find PQ.
The vector vec AB has a magnitude of 20 units and is parallel to the vector 4i + 3j. Find vec AB
Find the magnitude of each of these vectors. a -21 b 4i + 3j C 5i-12j d -8i-6j e 7i+24j f 15i-8j g-4i+4j h 5i-10j
Write each vector in the form ai + bj. 699 3 f i
The radius, rem, of a circle is increasing at the rate of 5cm * s ^ - 1 Find, in terms of ㅠ, the rate at which the area of the circle is increasing when r = 3.
The volume, V, and surface area, S, of a sphere of radius rare given by V = 4/3 * pi * r ^ 3 and and S = 4pi * r ^ 2 respectively. The volume of a sphere increases at a rate of 200c * m ^ 3 per second. At the the radius of the sphere is 10cm, find i the rate of increase of the radius of the sphere,
The diagram shows a right circular cone of base radius rem and height hem cut from a solid sphere of radius 10cm. The volume of the cone is V_{c} * m ^ 3 a Express r in terms of h.b Show that V = 1/3 * pi * h ^ 2 * (20 - h)c Find the value for h for which there is a stationary value of V.d
The diagram shows a hollow cone with base radius 12cm and height 24 cm. A solid cylinder stands on the base of the cone and the upper edge touches the inside of the cone. The cylinder has base radius rem, height hem and volume V cm ^ 3 .a Express h in terms of r.b Show that V = 2pi * r ^ 2 * (12 -
The diagram shows a solid cylinder of radius rem and height 2hcm cut from a solid sphere of radius 5cm. The volume of the cylinder is V cm ^ 3 a Express in terms of h.b Show that V = 2pi*h(25 - h ^ 2)c Find the value for h for which there is a stationary value of V.d Determine the nature of this
A piece of wire, of length 50cm, is cut into two pieces.One piece is bent to make a square of side x cm and the other is bent to make a circle of radius rcm. The total area enclosed by the two shapes is Ac * m ^ 2
a Express r in terms of x.b Show that C Find the stationary value of A and the value of x for which this occurs.Give your answers correct to 3 sf. (+4)x-100x + 625 A=
The diagram shows a solid formed by joining a hemisphere of radius rm to a cylinder of radius rem and height hem. The surface area of the solid is 288pi*c * m ^ 2 and the volume is Vcm³.a Express h in terms of r.b Show that V = 144pi*r - 5/6 * pi * r ^ 3.c Find the exact value of such that is a
A solid cylinder has radius rem and height hcm.The volume of this cylinder is 250pi*c * m ^ 3 and the surface area is Ac * m ^ 2 a Express h in terms of r. b Show that A=2r+- dA Find and dA dr dr 500T d Find the value for r for which there is a stationary value of A. e Determine the magnitude and
ABCD is a rectangle with base length 2p units, and area A units².The points A and B lie on the x-axis and the points Cand D lie on the curve y = 4 - x ^ 2 a Express BC in terms of p.b Show that A = 2p(4 - p ^ 2)c Find the value of p for which A has a stationary value.d Find this stationary value
The diagram shows a window made from a rectangle with base 2rm and height /m and a semicircle of radius rm. The perimeter of the window is 6m and the surface area is A * m ^ 2 rm hm a Express h in terms of r. 2r m 6r-2r2-r. b Show that A = 6r-2r2 dA Find and dr d e d dr Find the value for r for
A piece of wire, of length 60cm, is bent to form a sector of a circle with radius rem and sector angle & radians. The total area enclosed by the shape is Ac * m ^ 2 a Express in terms of r. b C Show that A = 30r-r. dA Find and dA dr dr d Find the value for r for which there is a stationary value of
A cuboid has a total surface area of 400c * m ^ 2 and a volume of Vc * m ^ 3 The dimensions of the cuboid are 4x cm by x cm by hem.a Express h in terms of V and x.b Show that V = 160x - 16/5 * x ^ 3 c Find the value of x when V is a maximum.
The volume of the solid cuboid is 576c * m ^ 3 and the surface area is Ac * m ^ 2 a Express y in terms of x.b Show that A = 4x ^ 2 + 1728/x Find the maximum value of A and state the dimensions of the cuboid for which this occurs. 2x cm y cm xcm
The diagram shows a rectangular garden with a fence on three of its sides and a wall on its fourth side. The total length of the fence is 100m and the area enclosed is A * m ^ 2 a Show that A = 1/2 * x(100 - x)b Find the the garden enclosed and the value of x for which this occurs. xm
The sum of two numbers x and y is 8.a Express y in terms of x.bi Given that P = xy write down an expression for P in terms of x.ii Find the maximum value of P.C Given that S = x ^ 2 + y ^ 2 write down an expression for S, in terms of x.ii Find the minimum value of S.
The curve y = x ^ 2 + a/x + b has a stationary point at (1, -1).a Find the value of a and the value of b.b Determine the nature of the stationary point (1, -1).
The curve y = x ^ 3 + ax + b has a stationary point at (1, 3).a Find the value of a and the value of b.b Determine the nature of the stationary point (1, 3).c Find the coordinates of the other stationary point on the curve and determine the nature of this stationary point.
The curve y = 2x ^ 3 - 3x ^ 2 + ax + b has a stationary point at the point (3, -77).a Find the value of a and the value of b.b Find the coordinates of the second stationary point on the curve.C Determine the nature of the two stationary points.d Find the coordinates of the point on the curve where
The curve has a stationary point at (-1,-12). y = ax + b/(x ^ 2)a Find the value of a and the value of b.b Determine the nature of the stationary point (-1,-12).
The curve y = 2x ^ 3 + a * x ^ 2 - 12x + 7 has a maximum point at x = - 2 .Find the value of a.
The equation of a curve is y = (2x + 5)/(x + 1)Find and hence explain why the curve has no turning points. d/dx (y)
Find the coordinates of the stationary points on each of the following curves and determine the nature of each of the stationary points. a y=x+ d y 2x x+9 b y=x-2 B y=+ x y= x+1 f y= yux-5x+3 x+1
Find the coordinates of the stationary points on each of the following curves and determine the nature of each of the stationary points. a y=x-12x+8 c y=x-12x+2 e y= x(3-4x-x b y=(5+x)(1-x) d y=x+x-16x-16 f y=(x-1)(x-6x+2)
Given that y = 8√x, show that dy 4x dr dy +4x- d.x
Given that y = x2 - 2x + 5, d'y dy +(x-1) 2y. dx show that 4- dx
A curve has equation y = 2x ^ 3 + 3x ^ 2 - 36x + 5 Find the range of values of x for which both and (d ^ 2 * y)/(d * x ^ 2) are both positive d/dx (y)
A curve has equation y = 2x ^ 3 - 15x ^ 2 + 24x + 6.Copy and complete the table to show whether negative (-) or zero (0) for the given values of x. d/dx (y) and (d ^ 2 * y)/(d * x ^ 2) are positive (+), x dy dr dy dr 0 2 3 5
A curve has equation y = 4x ^ 3 + 3x ^ 2 - 6x - 1 Show that when Show d/dx (y) = 0 when x = - 1 and when x = 0.5 a b Find the value of (d ^ 2 * y)/(d * x ^ 2) when x = - 1 and when x = 0.5
Given that f(x) = x ^ 3 - 7x ^ 2 + 2x + 1 find a f(1)b f' * (1)c f^ prime prime (1)
Find for each 2 Find (d ^ 2 * y)/(d * x ^ 2) for each of the following functions. a y=x(x-4) d y=x+2 x-1 b y= " 4x-1 22 x-5 x+1 y= x-3 f y= 2x+5 3.x-1
Find (d ^ 2 * y)/(d * x ^ 2) for each of the following functions. y = 5x-7x+3 by=2x+3x-1 3. y=4- x d y = (4x+1) e y=2x+1 4 f y= x+3
Water is poured into the hemispherical bowl at a rate of 4pi*c * m ^ 3 * s ^ - 1 After seconds, the volume of water in the bowl, Vc * m ^ 3 is given by where hcm is the height of the water in the bowl. V = 8pi * h ^ 2 - 1/3 * pi * h ^ 3 a Find the rate of change of h when h = 2.b Find the rate of

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