𝑥2
𝑑𝑦
1. (a) Show that 𝑑𝑥 (𝑥 − tan−1 𝑥) = 1+𝑥 2 .
2
√3
[2]
1
(i) Show that ∫0 𝑥 tan−1 𝑥 𝑑𝑥 = 3 𝜋 − 2 √3.
[5]
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1
2. The variables 𝑥 and 𝑦 satisfy the relation sin 𝑦 = tan 𝑥, where − 2 𝜋 < 𝑦 < 2 𝜋. Show
that
𝑑𝑦
1
=
𝑑𝑥 cos 𝑥 √cos 2𝑥
[5]
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3. (i) Differentiate sin2 𝜃 with respect to 𝜃.
[2]
−𝟐 𝐜𝐨𝐬 𝜽
−𝟐 𝐬𝐢𝐧 𝜽 𝐜𝐨𝐬 𝜽
−𝟐 𝐜𝐨𝐬𝐞𝐜 𝟐 𝜽 𝐜𝐨𝐭 𝜽 , 𝒐𝒓
𝒐𝒓
𝟑
𝐬𝐢𝐧 𝜽
𝐬𝐢𝐧𝟒 𝜽
(ii) The variables 𝑥 and 𝜃 satisfy the differential equation
𝑑𝑥
𝑥 tan 𝜃
+ cosec 2 𝜃 = 0,
𝑑𝜃
1
1
for 0 < 𝜃 < 2 𝜋 and 𝑥 > 0. It is given that 𝑥 = 4 when 𝜃 = 6 𝜋. Solve the
differential equation, obtaining an expression for 𝑥 in terms of 𝜃.
[6]
𝒙 = √𝐜𝐨𝐬𝐞𝐜 𝟐 𝜽 + 𝟏𝟐
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1
4. The diagram shows the curve 𝑦 = sin 3𝑥 cos 𝑥 for 0 ≤ 𝑥 ≤ 2 𝜋 and its minimum point
𝑀. The shaded region 𝑅 is bounded by the curve and the 𝑥-axis.
(i) By expanding sin(3𝑥 + 𝑥) and sin(3𝑥 − 𝑥) show that
1
sin 3𝑥 cos 𝑥 = 2 (sin 4𝑥 + sin 2𝑥)
[3]
(ii) Using the result of part (i) and showing all necessary working, find the exact area of
region 𝑅.
[4]
𝟗
𝟏𝟔
𝑑𝑦
(iii)Using the result of part (i), express 𝑑𝑥 in terms of cos 2𝑥 and hence find the 𝑥coordinate of 𝑀, giving your answer correct to 2 decimal places.
[5]
𝟏. 𝟐𝟗
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1
1
5. The curve 𝑦 = sin (𝑥 + 3 𝜋) cos 𝑥 has two stationary points in the interval 0 ≤ 𝑥 ≤ 𝜋.
(i)
(ii)
(iii)
6. (i)
(ii)
𝑑𝑦
Find 𝑑𝑥 .
[2]
𝟏
𝐜𝐨𝐬 (𝟐𝒙 + 𝝅)
𝟑
By considering the formula for cos(𝐴 + 𝐵), show that, at the stationary points on
1
the curve, cos (2𝑥 + 3 𝜋) = 0.
[2]
Hence find the exact 𝑥-coordinate of the stationary points.
[3]
𝟏
𝟕
𝝅 𝒐𝒓
𝝅
𝟏𝟐
𝟏𝟐
9709/33/M/J/19/#7
By differentiating
1
𝜋
2
1
𝜋
4
cos 𝑥
sin 𝑥
𝑑𝑦
, show that if 𝑦 = cot 𝑥 then 𝑑𝑥 = − 𝐜𝐨𝐬𝐞𝐜 𝟐 𝒙.
1
Show that ∫ 𝑥 cosec 2 𝑥 𝑑𝑥 = 4 (𝜋 + ln 4).
[2]
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7. The variable 𝑥 and 𝜃 satisfy the differential equation
1 𝑑𝑥
1
sin 𝜃
= (𝑥 + 2) cos 𝜃
2 𝑑𝜃
2
1
for 0 < 𝜃 < 𝜋. It is given that 𝑥 = 1 when 𝜃 = 3 𝜋. Solve the differential equation and
obtain an expression for 𝑥 in terms of cos 𝜃.
[8]
𝒙 = 𝟒 − 𝟔 𝒄𝒐𝒔 𝜽
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8. The diagram shows the graph of 𝑦 = 𝑒 cos 𝑥 sin3 𝑥 for 0 ≤ 𝑥 ≤ 𝜋, and its maximum point
𝑀. The shaded region 𝑅 is bounded by the curve and the 𝑥-axis.
(i)
(ii)
9. (i)
Find the 𝑥-coordinate of 𝑀. Show all necessary working and give your answer
correct to 2 decimal places.
[5]
𝒙 = 𝟏. 𝟐𝟔
By first using the substitution 𝑢 = cos 𝑥, find the exact of the area of 𝑅.
[7]
𝟒
𝒆
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Using the expansions of cos(3𝑥 + 𝑥) and cos(3𝑥 − 𝑥), show that
1
(cos 4𝑥 + cos 2𝑥) ≡ cos 3𝑥 cos 𝑥
2
[3]
(ii)
1
𝜋
6
1
− 𝜋
6
Hence show that ∫
3
cos 3𝑥 cos 𝑥 𝑑𝑥 = 8 √3.
[3]
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2
10. The variable 𝑥 and 𝜃 satisfy the differential equation
𝑑𝑥
𝑥 sin2 𝜃
= 2 tan 𝜃 + 1,
𝑑𝜃
1
1
for 0 ≤ 𝜃 ≤ 2 𝜋 and 𝑥 > 0. It is given that 𝑥 = 1 when 𝜃 = 4 𝜋.
(i)
(ii)
𝑑
Show that 𝑑𝜃 (tan2 𝜃) =
2 tan 𝜃
cos2 𝜃
.
[1]
1
Solve the differential equation and calculate the value of 𝑥 when 𝜃 = 3 𝜋, giving
your answer correct to 3 significant figures.
[7]
𝒙 = 𝟐. 𝟓𝟒
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11. A curve has equation 𝑦 =
𝑒 3𝑥
1
2
tan 𝑥
. Find the 𝑥-coordinates of the stationary points of the
curve in the interval 0 < 𝜃 < 𝜋. Give your answer correct to 3 decimal places
[6]
𝟎. 𝟑𝟒𝟎 𝒐𝒓 𝟐. 𝟖𝟎𝟐
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3
4
1
4
𝑥
12. Let that 𝐼 = ∫ √1−𝑥 𝑑𝑥.
1
𝜋
3
1
𝜋
6
2
(i)
Using the substitution that 𝑥 = cos 𝜃, show that that 𝐼 = ∫ 2 cos 2 𝜃 𝑑𝜃.
(ii)
Hence find the exact value of 𝐼.
(iv)
Find 𝑑𝑥 .
[4]
[4]
𝟏
𝝅
𝟔
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13. The curve 𝑦 = sin (𝑥 + 3 𝜋) cos 𝑥 has two stationary points in the interval 0 ≤ 𝑥 ≤ 𝜋.
(v)
(vi)
14. (i)
(iii)
𝑑𝑦
[2]
𝟏
𝐜𝐨𝐬 (𝟐𝒙 + 𝝅)
𝟑
By considering the formula for cos(𝐴 + 𝐵), show that, at the stationary points on
1
the curve, cos (2𝑥 + 3 𝜋) = 0.
[2]
Hence find the exact 𝑥-coordinate of the stationary points.
[3]
𝟏
𝟕
𝝅 𝒐𝒓
𝝅
𝟏𝟐
𝟏𝟐
9709/33/M/J/19/#7
By differentiating
1
𝜋
2
1
𝜋
4
cos 𝑥
𝑑𝑦
, show that if 𝑦 = cot 𝑥 then 𝑑𝑥 = − 𝐜𝐨𝐬𝐞𝐜 𝟐 𝒙.
sin 𝑥
1
Show that ∫ 𝑥 cosec 2 𝑥 𝑑𝑥 = 4 (𝜋 + ln 4).
[2]
[6]
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15. The variable 𝑥 and 𝜃 satisfy the differential equation
1 𝑑𝑥
1
sin 𝜃
= (𝑥 + 2) cos 𝜃
2 𝑑𝜃
2
1
for 0 < 𝜃 < 𝜋. It is given that 𝑥 = 1 when 𝜃 = 3 𝜋. Solve the differential equation and
obtain an expression for 𝑥 in terms of cos 𝜃.
[8]
𝒙 = 𝟒 − 𝟔 𝒄𝒐𝒔 𝜽
9709/32/O/N/19/#6
3
16. The diagram shows the graph of 𝑦 = 𝑒 cos 𝑥 sin3 𝑥 for 0 ≤ 𝑥 ≤ 𝜋, and its maximum point
𝑀. The shaded region 𝑅 is bounded by the curve and the 𝑥-axis.
(iii) Find the 𝑥-coordinate of 𝑀. Show all necessary working and give your answer
correct to 2 decimal places.
[5]
𝒙 = 𝟏. 𝟐𝟔
(iv)
By first using the substitution 𝑢 = cos 𝑥, find the exact of the area of 𝑅.
[7]
𝟒
𝒆
9709/33/O/N/19/#10
4