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mathematics
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 the equation 2cos2 x = 4 − 5 sin x may be written as 2sin2 x − 5sin x + 2 = 0.b. Hence solve, for 0 ≤ x < 360°, the equation 2cos2 x = 4 − 5 sin x.
Find all of the solutions in the interval 0 ≤ x < 360° of 2 tan2 x − 4 = 5 tan x giving each solution, in degrees, to one decimal place.
OP(vector) = 4i − 3j, OQ(vector) = 3i + 2ja. Find PQ(vector)b. Find, in surd form: i |OP| ii |00| iii |PQ|
Find all of the solutions in the interval 0 ≤ x < 360° of 5 sin2 x = 6(1 − cos x) giving each solution, in degrees, to one decimal place.
Prove that cos2 x (tan2 x + 1) = 1 for all values of x where cos x and tan x are defined.
Find the speed of a particle moving with these velocities:a. (3i + 4j) m s−1 b. (24i − 7j) km h−1c. (5i + 2j) m s−1 d. (−7i + 4j) cm s−1
OAB is a triangle. OA(vector) = a and OB(vector) = b. The point M divides OA in the ratio 2 : 1.MN is parallel to OB.a. Express the vector ON(vector) in terms of a and b.b. Show that AN : NB = 1 : 2 M N B
Find the magnitude of each of these vectors.a. 3i + 4j b. 6i − 8j c. 5i + 12j d. 2i + 4je. 3i − 5j f. 4i + 7j g. −3i + 5j h. −4i − j
Given thatfind:a. 5a b. −1/2 c c. a + b + c d. 2a − b + ce. 2b + 2c − 3a f. 1/2 a + 1/2 b = (9), b a = = (13) and c = = (-₁)
Two forces F1 and F2 act on a particle.F1 = −3i + 7j 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.b. Calculate, to the nearest degree, the angle between the line of action of R and the vector j.
Find the distance moved by a particle which travels for:a. 5 hours at velocity (8i + 6j) km h−1b. 10 seconds at velocity (5i − j) m s−1c. 45 minutes at velocity (6i + 2j) km h−1d. 2 minutes at velocity (−4i − 7j) cm s−1.
OQ = 4i − 3j, PQ(vector) = 5i + 6ja. Find OP(vector)b. Find, in surd form: i |OP| ii |OQ| iii |PQ|
In the triangle PQR, PQ = 2a and QR = 2b. The midpoint of PR is M. Find, in terms of a and b: a PR b PM с QM ом
Given that a = 2i + 3j and b = 4i − j, find these vectors in terms of i and j.a. 4a b.1/2 a c. −b d. 2b + ae. 3a − 2b f. b − 3a g. 4b − a h. 2a − 3b
Find the exact value of the magnitude of:a = 2i + 3j, b = 3i − 4j and c = 5i − j. a. a + b b. 2a − c c. 3b − 2c
A small boat S, drifting in the sea, is modelled as a particle moving in a straight line at constant speed. When first sighted at 09:00, S is at a point with position vector (−2i − 4j) km relative to a fixed origin O, where i and j are unit vectors due east and due north respectively. At 09:40,
ABCD is a trapezium with AB parallel to DC and DC = 4AB. M divides DC such that DM : MC = 3 : 2, AB(vector) = a and BC(vector) = b.Find, in terms of a and b: 1 a AM b BD с MB d DA
OABC is a square. M is the midpoint of OA, and Q divides BC in the ratio 1 : 3. AC and MQ meet at P.a. If OA(vector) = a and OC(vector) = c, express OP(vector) in terms of a and c.b. Show that P divides AC in the ratio 2 : 3. A M B 12 C
In triangle ABC the position vectors of the vertices A, B and C areFind:a. |AB(vector) b. |AC(vector)| c. |BC(vector)|d. The size of ∠BAC, ∠ABC and ∠ACB to the nearest degree. (3), (3) and (7).
Find the speed and the distance travelled by a particle moving in a straight line with:a. velocity (−3i + 4j) m s−1 for 15 seconds b. velocity (2i + 5j) m s−1 for 3 secondsc. velocity (5i − 2j) km h−1 for 3 hours d. velocity (12i − 5j) km h−1 for 30 minutes.
For each of the following vectors, find the unit vector in the same direction.a. a = 4i + 3j b. b = 5i − 12j c. c = −7i + 24j d d = i − 3j
OABC is a parallelogram.The point P divides OB in the ratio 5:3. Find, in terms of a and b: ОА a and OC = b.
A football player kicks a ball from point A on a flat football field. The motion of the ball is modelled as that of a particle travelling with constant velocity (4i + 9j) m s−1.a. Find the speed of the ball.b. Find the distance of the ball from A after 6 seconds.c. Comment on the validity of this
The position vectors of 3 vertices of a parallelogram areFind the possible position vectors of the fourth vertex. (2), (3) and (8).
OABC is a parallelogram. P divides AC in the ratio 3 : 2.Find in i, j format and column vector format: ОА OA = 2i + 4j, OČ = 7i. =
OPQ is a triangle.a. Show that OS(vector) = 2a + b.b. Point T is added to the diagram such that OT(vector) = −b. Prove that points T, P and S lie on a straight line. 2PR RQ and 3OR = OS OP = a and 00 = b. 18 =
A particle P is accelerating at a constant speed. When t = 0, P has velocity u = (2i + 3j) m s−1 and at time t = 5 s, P has velocity v = (16i − 5j) m s−1.The acceleration vector of the particle is given by the formula: a = (v − u)/tFind the acceleration of P in terms of i and j.
ABCD is a trapezium with AB parallel to DC and DC = 3AB. M divides DC such that DM : MC = 2 : 1.Find, in terms of a and b: AB= a and BC = b.
In triangle ABC,Find BC. AB = 4i + 3j and AC = 5i + 2j.
Given that a = 2i + 5j and b = 3i − j, find:a. λ if a + λb is parallel to the vector i b. μ if μa + b is parallel to the vector j
Find the angle that each of these vectors makes with the positive x-axis.a. 3i + 4j b. 6i − 8j c. 5i + 12j d. 2i + 4j
Given thatfind:a. a + b + c b. a − 2b + c c. 2a + 2b − 3c = (2), b = (12) and c = ( a = -5) -3)
OABCDE is a regular hexagon. The points A and B have position vectors a and b respectively, where O is the origin. Find, in terms of a and b, the position vectors ofa. C b. D c. E.
Write these vectors in i, j and column vector form. 450 15 bу 20⁰ X с у 25° 20 d yA AH
Given that c = 3i + 4j and d = i − 2j, find:a. λ if c + λd is parallel to i + j b. μ if μc + d is parallel to i + 3jc. s if c − sd is parallel to 2i + j d. t if d − tc is parallel to −2i + 3j
A particle P of mass m = 0.3 kg moves under the action of a single constant force F newtons. The acceleration of P is a = (5i + 7j) m s−2.a. Find the angle between the acceleration and i. Force, mass and acceleration are related by the formula F = ma.b. Find the magnitude of F.
The diagram shows a sketch of a field in the shape of a triangle ABC. Given AB(vector) = 30i + 40j metres and AC(vector) = 40i − 60j metres,a. Find BC(vector)b. Find the size of ∠BAC, in degrees, to one decimal placec. Find the area of the field in square metres. A B C
Find the angle that each of these vectors makes with j.a. 3i − 5j b. 4i + 7j c. −3i + 5j d. −4i − j
The vectors 5a + kb and 8a + 2b are parallel. Find the value of k.
In triangle ABC,P is the midpoint of AB and Q is the midpoint of AC.a. Write in terms of a and b:b. Show that PQ is parallel to BC. AB a and AC = b. =
Given that b − 2a = c, find the values of j and k. a = = ( ₁ )₁ b = (10), c = (²) C
Given that the point A has position vector 4i − 5j and the point B has position vector 6i + 3j,a. Find the vector AB(vector).b. Find |AB(vector)| giving your answer as a simplified surd.
In triangle ABC, AB(vector) = 3i + 5j and AC(vector) = 6i + 3j, find:a. BC(Vector)b. ∠BACc. The area of the triangle. A В
Two forces, F1 and F2, are given by the vectors F1 = (3i − 4j) N and F2 = ( pi + qj) N. The resultant force, R = F1 + F2 acts in a direction which is parallel to the vector (2i − j).a. Find the angle between R and the vector i.b. Show that p + 2q = 5.c. Given that p = 1, find the magnitude
The point A lies on the circle with equation x2 + y2 = 9. Given that OA(vector) = 2ki + kj, find the exact value of k.
Given that a + 2b = c, find the values of p and q. a = = ( ²₁ ), b = ( 2 ), c = (4)
Draw a sketch for each vector and work out the exact value of its magnitude and the angle it makes with the positive x-axis to one decimal place.a. 3i + 4j b. 2i − j c. −5i + 2j
In triangle PQR, PQ(vector) = 4i + j, PR(vector) = 6i − 8j.a. Find the size of ∠QPR, in degrees, to one decimal place.b. Find the area of triangle PQR. P R
In triangle ABC, AB(vector) = 4i + 3j, AC(vector) = 6i − 4j.a. Find the angle between AB(vector) and i.b. Find the angle between AC(vector) and i.c. Hence find the size of ∠BAC, in degrees, to one decimal place. A B C
State with a reason whether each of these vectors is parallel to the vector a − 3b:a. 2a − 6b b. 4a − 12b c. a + 3b d. 3b − a e. 9b − 3a f. 1/2a − 2/3b
Given that |2i − kj| = 2√10, find the exact value of k.
A boat has a position vector of (2i + j) km and a buoy has a position vector of (6i − 4j) km, relative to a fixed origin O.a. Find the distance of the boat from the buoy.b. Find the bearing of the boat from the buoy.The boat travels with constant velocity (8i − 10j) km/h.c. Verify that the boat
The resultant of the vectors a = 4i − 3j and b = 2 pi − pj is parallel to the vector c = 2i − 3j. Find:a. The value of p.b. The resultant of vectors a and b.
OAB is a triangle. OA(vector) = a and OB(vector) = b. The point M divides OA in the ratio 3 : 2. MN is parallel to OB.a. Express the vector ON(vector) in terms of a and b.b. Find vector MN(vector).c. Show that AN : NB = 2 : 3. M N B
For each graph given, sketch the graph of the corresponding gradient function on a separate set of axes. Show the coordinates of any points where the curve cuts or meets the x-axis, and give the equations of any asymptotes. a (6, 15) (-9, 12) M -11 8 X d 6 + y = 3 -2 b y = 10 Z X (4,3) y = 4 +
Vector a = pi + qj has magnitude 10 and makes an angle θ with the positive x-axis where sin θ = 3/5. Find the possible values of p and q.
The diagram shows the curve with equation y = x2 − 2x.a. Copy and complete this table showing estimates for the gradient of the curve.b. Write a hypothesis about the gradient of the curve at the point where x = p.c. Test your hypothesis by estimating the gradient of the graph at the point (1.5,
For each of the following vectors, findi. A unit vector in the same direction ii. The angle the vector makes with ia. a = 8i + 15j b. b = 24i − 7j c. c = −9i + 40j d. d = 3i − 2j
The resultant of the vectors a = 3i − 2j and b = pi − 2pj is parallel to the vector c = 2i − 3j. Find:a. The value of p.b. The resultant of vectors a and b.
The vector a = pi + qj, where p and q are positive constants, is such that |a| = 15.Given that a makes an angle of 55° with i, find the values of p and q.
The vectors 2a + kb and 5a + 3b are parallel. Find the value of k.
Given that |3i − kj| = 3√5, find the value of k.
Find f'(x) given that f(x) equals: a x7 g x-3 m x³ x x6 b x8 h x-4 n x² x x³ i 1 x2 0 x x X² d x³ 1 j Р $X X e x4 k b x2 f √√x 1 r x6 9-X x3
The diagram shows the curve with equationThe point A has coordinates (0.6, 0.8).The points B, C and D lie on the curve with x-coordinates 0.7, 0.8 and 0.9 respectively.a. Verify that point A lies on the curve.b. Use a ruler to estimate the gradient of the curve at point A.c. Find the gradient of
The displacement of a particle in metres at time t seconds is modelled by the functionThe acceleration of the particle in m s−2 is the second derivative of this function. Find an expression for the acceleration of the particle at time t seconds. f(t) = 1²+2 √T
Two forces, F1 and F2, are given by the vectors F1 = (4i − 5j) N and F2 = (pi + qj) N.The resultant force, R = F1 + F2 acts in a direction which is parallel to the vector (3i − j)a. Find the angle between R and the vector i.b. Show that p + 3q = 11.c. Given that p = 2, find the magnitude of R.
A particle P is accelerating at a constant speed. When t = 0, P has velocity u = (3i + 4j) m s−1 and at time t = 2 s, P has velocity v = (15i − 3j) m s−1.The acceleration vector of the particle is given by the formula: a = (v − u)/t.Find the magnitude of the acceleration of P.
Find the values of x for which f(x) is an increasing function, given that f(x) equals:a. 3x2 + 8x + 2 b. 4x − 3x2 c. 5 − 8x − 2x2 d. 2x3 − 15x2 + 36xe. 3 + 3x − 3x2 + x3 f. 5x3 + 12x g. x4 + 2x2 h. x4 − 8x3
Find dy/dx and d2y/dx2 when y equals:a. 12x2 + 3x + 8 b. 15x + 6 + 3/xc. 9 √x − 3/x2 d. (5x + 4)(3x − 2)
Find the least value of the following functions:a. f(x) = x2 − 12x + 8 b. f(x) = x2 − 8x − 1 c. f(x) = 5x2 + 2x
Find dθ/dt where θ = t2 − 3t.
f(x) = x2a. Show thatb. Hence deduce that f'(x) = 2x. f'(x) = lim (2x + h). h→0
Prove, from first principles, that the derivative of 10x2 is 20x.
For the function f(x) = x2, use the definition of the derivative to show that:a. f'(2) = 4 b. f'(−3) = −6 c. f'(0) = 0 d. f'(50) = 100
Find dy/dx given that y equals: a 3.x2 f 10x1 b6x9 09 4x6 2x-3 C h 글라이 X 8.15 d 20x 2 √x i - e6x2 j 5x4 × 10x 2.x²
Find dy/dx when y equals:a. 2x2 − 6x + 3 b. 1/2x2 + 12x c. 4x2 − 6d. 8x2 + 7x + 12 e. 5 + 4x − 5x2
Find the coordinates of the points where the gradient is zero on the curves with the given equations. Establish whether these points are local maximum points, local minimum points or points of inflection in each case. a y = 4x² + 6x d y = x(x² - 4x - 3) g y=x-3√x by=9+x-x² 1 e y=x+ h y = x²(x
Differentiate:a. x4 + x−1 b. 2x5 + 3x−2 c. 6x3/2   + 2x−1/2 + 4
Find the equation of the normal to the curve:a. y = x2 − 5x at the point (6, 6)b. y = x2 − 8/√x at the point (4, 12)
Find the values of x for which f(x) is a decreasing function, given that f(x) equals:a. x2 − 9x b. 5x − x2 c. 4 − 2x − x2 d. 2x3 − 3x2 − 12xe. 1 − 27x + x3 f. x + 25/x g. x1/2 + 9x−1/2 h. x2(x + 3)
Find the greatest value of the following functions:a. f(x) = 10 − 5x2 b. f(x) = 3 + 2x − x2c. f(x) = (6 + x)(1 − x)
The point A with coordinates (−2, −8) lies on the curve with equation y = x3. At point A the curve has gradient g.a. Show thatb. Deduce the value of g. g= lim (12-6h + h²). h→0
f(x) = (x + 1)(x − 4)2a. Sketch the graph of y = f(x).b. On a separate set of axes, sketch the graph of y = f´(x).c. Show that f´(x) = (x − 4)(3x − 2).d. Use the derivative to determine the exact coordinates of the points where the gradient function cuts the coordinate axes.
Find dA/dr where A = 2πr.
The point A with coordinates (1, 4) lies on the curve with equation y = x3 + 3x.The point B also lies on the curve and has x-coordinate (1 + δx).a. Show that the gradient of the line segment AB is given by (δx)2 + 3δx + 6.b. Deduce the gradient of the curve at point A.
Find the gradient of the curve with equation:a. y = 3x2 at the point (2, 12) b. y = x2 + 4x at the point (1, 5)c. y = 2x2 − x − 1 at the point (2, 5) d. y = 1/2x2 + 3/2x at the point (1, 2)e. y = 3 − x2 at the point (1, 2) f. y = 4 − 2x2 at the point (−1, 2)
Find the gradient of the curve with equation y = f(x) at the point A where:a. f(x) = x3 − 3x + 2 and A is at (−1, 4) b. f(x) = 3x2 + 2x−1 and A is at (2, 13)
Find the coordinates of the point where the tangent to the curve y = x2 + 1 at the point (2, 5) meets the normal to the same curve at the point (1, 2).
Show that the function f(x) = 4 − x (2x2 + 3) is decreasing for all x ∈ R.
Given that y = (2x − 3)3, find the value of x when d2y/dx2 = 0.
Given that r = 12/t, find the value of dr/dt when t = 3.
A curve is given by the equation y = 3x2 + 3 + 1/x2 , where x > 0. At the points A, B and C on the curve, x = 1, 2 and 3 respectively. Find the gradient of the curve at A, B and C.
F is the point with coordinates (3, 9) on the curve with equation y = x2.a Find the gradients of the chords joining the point F to the points with coordinates:i. (4, 16) ii. (3.5, 12.25) iii. (3.1, 9.61)iv. (3.01, 9.0601) v. (3 + h, (3 + h)2)b. What do you deduce about the gradient
Find the gradient of the curve with equation y = 3√x at the point where:a. x = 4 b. x = 9c. x = 1/4 d. x = 9/16
Find the y-coordinate and the value of the gradient at the point P with x-coordinate 1 on the curve with equation y = 3 + 2x − x2.
Find the point or points on the curve with equation y = f(x), where the gradient is zero:a. f(x) = x2 − 5x b. f(x) = x3 − 9x2 + 24x − 20c. f(x) = x3/2 − 6x + 1 d. f(x) = x−1 + 4x
Find the equations of the normals to the curve y = x + x3 at the points (0, 0) and (1, 2), and find the coordinates of the point where these normals meet.
a. Given that the function f(x) = x2 + px is increasing on the interval [−1, 1], find one possible value for p.b. State with justification whether this is the only possible value for p.
f(x) = px3 − 3px2 + x2 − 4When x = 2, f"(x) = −1. Find the value of p.
Sketch the curves with equations given in question 3 parts a, b, c and d, labelling any stationary points with their coordinates.Data from Question 3Find the coordinates of the points where the gradient is zero on the curves with the given equations. Establish whether these points are local

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