Questions and Answers of Pearson Edexcel A Level Mathematics

Using known trigonometric identities, prove the following: a sec 0 cosec 0 = 2 cosec 20 e sin (x + y) sin(x - y) = cos² y cos²x - n (4+x) - tan (-x) = 2 tan 2x d 1 + 2 cos 20 + cos 40 = 4 cos²0
The vectors a and b are defined byShow that the vector 3a + 2b is parallel to 6i + 4j + 10k. a = 3 -31 2 | and b = -2 -1 4
The vectors a and b are defined byGiven that a + 2b = 5i + 4j, find the values of p, q and r. a = 1 2 and b = 9 -4/ ·().
Given that a = ti + 2j + 3k, and that |a| = 7, find the possible values of t.
Given that a = 5√3 or – √3 ti + 2tj + tk, and that |a| = 3 √10, find the possible values of t.
The points A, B and C have coordinates (2, 1, 4), (3, −2, 4) and (−1, 2, 2).a. Find, in terms of i, j and k:i. The position vectors of A, B and Cii. ACb. Find the exact value of:i. |AC|ii. |OC|
A is the point (2, 3, −2), B is the point (0, −2, 1) and C is the point (4, −2, −5). When A is reflected in the line BC it is mapped to the point D. a. Work out the coordinates of the
OA is the vector 4i − j − 2k and OB is the vector −2i + 3j + k.Find:a. The vector ABb. The distance between A and Bc. The unit vector in the direction of AB .
A particle of mass 2 kg is acted upon by three forces:F1 = (bi + 2j + k) NF2 = (3i − bj + 2k) NF3 = (−2i + 2j + (4 − b)k) N Given that the particle accelerates at 3.5 m s−2, work out the
a. Solve the equation f(x) = 0.b. Find ∫f(x) dx.c. Evaluate ∫14 f(x) dx, giving your answer in the form p + q ln r, where p, q and r are rational numbers. f(x) = 1-8 4 /... 0
a. Find the value of the constants A, B and C.b. Hence find the exact value ofdx writing your answer in the form 2 + k ln m, giving the values of the rational constants k and m. 32x² + 4 (4x + 1)(4x
a. Given that y = cosec x, complete the table with values of y corresponding togive your answers to 5 decimal places.b. Use the trapezium rule, with all the values of y in the completed table, to
a. Find ∫ x2 ln 2x dx.b. Hence show that the exact value of ∫31 x2 ln 2x dx is 9 ln 6 − 215/72.
Given that y = 1 at x = π/2, solve the differential equation dy/dx = xy sin x.
By using an appropriate trigonometric substitution, find 1 1 + x² dx.
The points A(2, 7, 3) and B(4, 3, 5) are joined to form the line segment AB. The point M is the midpoint of AB. Find the distance from M to the point C(5, 8, 7).
The coordinates of P and Q are (2, 3, a) and (a − 2, 6, 7). Given that the distance from P to Q is √14, find the possible values of a.
Given thatfind in column vector form:a. p + qb. q − rc. p + q + rd. 3p − re. p − 2q + r 5 2 7 P-(1)-4-(1) and (2) p= q= r = -4 -3/
The points D, E and F have position vectorsrespectively.a. Find the vectors DE ⟶, ⟶EF and ⟶FD .b. Find |DE|, |EF| and |FD| giving your answers in exact form.c. Describe triangle DEF. 2 (1). (1)
AB is the vector −3i + tj + 5k, where t > 0. Given that |AB | = 5 √2, show that AB is parallel to 6i − 8j − 5/2tk.
P is the point (5, 6, −2), Q is the point (2, −2, 1) and R is the point (2, −3, 6). a. Find the vectors PQ, PR and QR.b. Hence, or otherwise, find the area of triangle PQR.
The position vector of the point A is 2i − 7j + 3k and AB = 5i + 4j − k. Find the position vector of the point B.
The diagram shows a tetrahedron OABC. a, b and c are the position vectors of A, B and C respectively. P, Q, R, S, T and U are the midpoints of OC, AB, OA, BC, OB and AC respectively. Prove that the
Find the unit vector in the direction of each of the following vectors.a.b.c. P = 3 -4 -2/
P is the point (3, 0, 7) and Q is the point (−1, 3, −5).Finda. The vector PQb. The distance between P and Qc. The unit vector in the direction of PQ .
The points A, B and C have position vectors respectively.a. Find the vectors AB ,AC and BC.b. Find |AB| |AC| and |BC| giving your answers in exact form.c. Describe triangle ABC. 8 -7 4 " 8 12 -3 and
In this question i and j are the unit vectors due east and due north respectively, and k is the unit vector acting vertically upwards. A BASE jumper descending with a parachute is modeled as a
A is the point (3, 4, 8), B is the point (1, −2, 5) and C is the point (7, −5, 7).a. Find the vectors AB, AC and BC .b. Hence find the lengths of the sides of triangle ABC.c. Given that angle ABC
The diagram shows the triangle PQR. Given that PQ = 3i − j + 2k and QR = −2i + 4j + 3k, show that ∠PQR = 78.5° to 1 d.p. P Q R
A scalene triangle has the coordinates (2, 0, 0), (5, 0, 0) and (4, 2, 3). Work out the area of the triangle.
The diagram shows the quadrilateral ABCD. Given thatandfind the area of the quadrilateral. 6 AB= -2 \11/
Integrate the following:a. cot2 xb. cos2 xc. sin 2x cos 2xd. (1 + sin x)2e. tan2 3xf. (cot x − cosec x)2g. (sin x + cos x)2h. sin2 x cos2 x F 1 sin² x cos²x j (cos2x - 1)²
Find the following intervals. a fx sin x dx b fxe³dx S -dx X sin² x d fx sec sec x tan x dx e с fx sec² x dx
Find the following intervals. a - sin x cos²x dx d cos²x d.x sin² x g f(cos.x - sin. x)² dx 1 + cos x sin² x (1 + cos x)² sin² x h [(cos.x - sec. x)² dx b e dx dx с S dx cos 2x cos²x f
Use the substitutions given to find the exact values of: a с d fx√x + √x + 4 dx; u = x + 4 sin x√3 cos x + 1 dx; u = cos x secx sec x tan x√/sec x + 2 dx; u = sec x b_f ²x(2 + x)³ dx; u =
Find the following intervals. a [3 In x dx a fa f(lnx)² dx b fx lnx dx e с f(x² + 1) In xdx In x x3 -dx
The diagram shows the curve with equation y = (x − 2) ln x + 1, x > 0.a. Complete the table with the values of y corresponding to x = 2 and x = 2.5.Given thatb. Use the trapezium rulei. With
Find the following intervals. a fx²e-dx bfx² cos xdx с [12x²(3 + 2x) dx d [2.x² sin 2x dx e [2x² sec²x tan x dx
a. By expanding sin (3x + 2x) and sin (3x − 2x) using the double-angle formulae, or otherwise, show that sin 5x + sin x ≡ 2 sin 3x cos 2x.b. Hence find ∫sin 3x cos 2xdx
a. Use integration by parts to find ∫ x cos 4x dx.b. Use your answer to part a to find ∫ x2 sin 4x dx.
a. Show that f(x) = cos 2x + 6.b. Hence, find the exact value of ∫0π/4 f(x) dx.f(x) = 5 sin2 x + 7 cos2 x
Given thata. Find the value of x and the value of y when dy/dx = 0.b. Show that the value of y which you found is a minimum. The finite region R is bounded by the curve with equation y = x3/2 + 48/x,
Using a suitable trigonometric substitution for x, find xP zx - I/\zxz ६०
Find the particular solution to the differential equation dy/dx = cos2 y + cos 2x cos2y , with boundary condition y = π/4 at x = π/4. Give your answer in the form tan y = f(x).
Givenfind the value of k. ㅠ 3k “(1 -ㅠ sin kx) dx = (7 – 6/2), 피 4k
a. Find ∫ sec2 3x dx.b. Using integration by parts, or otherwise, find ∫ x sec2 3x dx.c. Hence show thatfinding the exact values of the constants p and q. x sec² 3x dx = pm - q ln 3 18
a. Show thatb. Hence find ∫cos4 x dx. 1 3 costx = cos 4x + cos2x + 8 8
A. Given that y = e√2x + 1 complete the table of values of y corresponding to x = 0.5, 1 and 1.5.B. Use the trapezium rule, with all the values of y in the completed table, to obtain an
a. Find ∫√8 - x dxb. Using integration by parts, or otherwise, show thatc. Hence find f(x - 2)√8 - x dx = -(8 - x)²(x + 2) + c
Using the substitution t2 = x + 1, where x > −1, X 17 √x + 1 a find f d.x. X b Hence evaluate S√x+1 d.x.
Find the particular solution to the differential equation dy/dx = xe−y , with boundary condition y = ln 2 at x = 4. Give your answer in the form y = f(x).
Use the substitution u = cos x to show 47 ³sin³x cos²x dx = = 480 F|M
a. Given thatfind the values of the constants A, B and C.b. Hence find ∫ f(x) dx.c. Hence show that f(x) A B с xx-1(x - 1)²²
a. Findb. Given that y = 16 at x = 1, solve the differential equationgiving your answer in the form y = f(x). 3x + 4 -dx, x > 0.
The diagram shows the graph ofa. Show thatb. Hence find the area of the shaded region R.c. Find the coordinates of A, the turning point on the graph. Зп y = (1 + sin 2x)2, 0
a. Given thatfind the values of the constants A, B and C.b. Given that x = 2 at t = 1, solve the differential equation.You do not need to simplify your final answer. B 1=4+BC A + + x-1 x+1 F x² -
The diagram shows part of the curve y = e3x + 1 and the line y = 8. The curve and the line intersect at the point (h, 8).a. Find h, giving your answer in terms of natural logarithms.The region R is
a. Find ∫ x2e −xdx.b. Use your answer to part a to find the solution to the differential equationgiven that y = 0 when x = 0. Express your answer in the form y = f(x). dy d.x ==x²e³y-x₂
a. Using the substitution u = 1 + 2x, or otherwise, findb. Given that y = π/4 when x = 0, solve the differential equation 4x (1 + 2x)² dx, x ± − 1/
a. Obtain the general solution to the differential equationb. Given also that y = 1 at x = 1, show thatis a particular solution to the differential equation. The curve C has equationc. Write down the
a. Show that the general solution to the differential equation dy/dx = −x/y can be written in the form x2 + y2 = c.b. On the same axes, sketch three different particular solutions to this
a. Find the general solution to the differential equationb. On the same axes, sketch three different particular solutions to this differential equation.c. Write down the particular solution that
a. Find ∫ x sin 2x dx.b. Given that y = 0 at x = π/4, solve the differential equation dy/dx = x sin 2x cos2y.
a. Find ∫xe−x dx.b. Given that y = π4 at x = 0, solve the differential equation dy ex. d.x X sin 2y
a. Find the general solution of dy/dx = 2x - 4.b. On the same axes, sketch three different particular solutions to this differential equation.
a. Express 8x - 18/(3x − 8)(x − 2) in partial fractions.b Given that x ≥ 3, find the general solution to the differential equationc. Hence find the particular solution to this differential
The curve with equation y = e2x − e−x, 0 ≤ x ≤ 1, is shown in the diagram. The finite region enclosed by the curve, the x-axis and the line x = 1 is shaded. The table below shows the
The rate, in cm3 s−1, at which oil is leaking from an engine sump at any time t seconds is proportional to the volume of oil, V cm3, in the sump at that instant. At time t = 0, V = A.a. By forming
a. Show that the general solution to the differential equation dy/dx = x/k - y can be written in the form x2 + ( y − k)2 = c.b. Describe the family of curves that satisfy this differential equation
The diagram shows a sketch of the curve y = f(x), whereThe region R, shown in the diagram, is bounded by the curve, the x-axis and the lines with equations x = 1 and x = 4. The table below shows the
An oil spill is modelled as a circular disc with radius r km and area A km2. The rate of increase of the area of the oil spill, in km2/day at time t days after it occurs is modelled as:a. Show
a. Find ∫ x (1 + 2x2)5 dx.b. Given that y = π/8 at x = 0, solve the differential equation dy d.x = x(1 + 2x²)5 cos² 2y
The vectors a and b are defined bya. Findi. a − bii −a + 3bb. State with a reason whether each of these vectors is parallel to 6i − 10j + 18k a = 1 2 and b = -4, 4 دل در -3 5
Givenfind the value of k. es 1 e² kx dx = 1 4'
a. Use integration by parts to find ∫ x sin 8x dx.b. Use your answer to part a to find ∫ x2 cos 8x dx.
Find the particular solution to the differential equation (1 + x2) dy/dx = x − xy2, with boundary condition y = 2 at x = 0. Give your answer in the form y = f(x) .
Givenfind the value of b. J3 (2x - 6)² dx = 36.
Givenfind the value of b. £₂ (3e (3ex + 6e-2x) dx = 0,
Use the substitution u2 = ex − 2 to show thatIn n d, where a, b, c and d are integers to be found. In4 e4x In3 ex a - 2 = = 7. - 2 dx. b + C
Find the particular solution to the differential equation (1 − x2) dy/dx = xy + y , with boundary condition y = 6 at x = 0.5. Give your answer in the form y = f(x).
a. By writingb. Show that cot x = COS X sin x find fcot.x dx.
a. Express f(x) in partial fractions.b. Hence find the exact value ofwriting your answer in the form a + ln b, where a and b are rational numbers to be found. 46 + 3x - x² 2 x3 + 2x² dx,
Given that a is a positive constant andfind the exact value of a. Ina Simex + e-x dx = Inl 48 = 7,
Givenfind the smallest possible value of θ. 9 cos x sin³x dx = 4, where > 0_
Using the substitution u2 = 4x + 1, or otherwise, find the exact value of (20 6 8x √4x + 1 d.x.
Given thatfind the exact value of θ. S4 4 sin 2x cos4 2x dx = where 0 < 0 < T
a. Given thatfind the values of the constants A, B and C. b. Hence find the exact value of writing your answer in the form a + b ln c, where a, b and c are rational numbers to be found.
Given that a is positive constantfind the exact value of a. 2a 3x - 1 X S (7) dx = 6 + In
Given that x = 0 when y = 0, find the particular solution to the differential equationgiving your answer in the form y= g(x). dy (2y + 2yx) dx = 1-y².
Givenfind the value of b. 2 1² (²3/3 - 12/12) dx = 2/1, 9 X-3 45
By choosing a suitable substitution, find the exact values of: a S₂x√2 + x dx b 1 1+√√x-1 ܛ S₂²₁ dx с sin 20 1 + cos 0 5.²³. do
Given thatFind the value of k. Skx²ex³ dx = 3(e8 − 1)_ -
a. Express f(x) into partial fractions.b. Hence find the exact value ofwriting your answer in the form a + ln b, where a and b are constants to be found. 17 – 5x (3 + 2x)(2x)² dx,
a. Given that f(x) =A/2x + 1 + B/1 − 2x , find the value of the constants A and B.b. Hence find ∫ f(x) dx , writing your answer as a single logarithm.c. Find ∫12 f(x) dx, giving your answer in
Evaluate the following. Give your answers as exact values. a f2e* dx T -5 -5 sin x dx b fo 561 +3x dx df_sec x(sec x + tanx) dx
By choosing a suitable substitution, find: a fx(3 + 2x)³ dx X blitz S √1 + x =dx с √√x² + 4 dx X
a. Find the general solution to the differential equation x2dy/dx = y + xy, giving your answer in the form y = g(x).b. Find the particular solution to the differential equation that satisfies the
Find particular solutions to the following differential equations using the given boundary conditions. a с dy d.x dy dx = sin x cos²x; y = 0, x = = ㅠ ㅠ = 2 cos²y cos²x; y = ₁ x =
Find the following intervals. a f(x + 1)(x² + 2x + 3)4 dx [sin³ 3.x cos 3x dx e2x Sex + dx e2x + 3 g f(2x + 1)√√x² + x + 5 dx e i ſ sin x cos x cos 2x + 3 dx b fcosec² 2.x cot 2x dx d fcos x
By choosing a suitable method, evaluate the following definite integrals. Write your answers as exact values. L.x(x² + 3)5 dx 2 e f*(16x²-3) dx С 4 е e S₁ 16.x² + 8x - 3² -d.x b f³x sec² x

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