C3 Exam Workshop 2 - Elite Tuition€¦ · C3 Exam Workshop 2 Workbook 1. (a) Express 7 cos x − 24 sin x in the form R cos (x + α) where R > 0 and 0 < α < 2 π. Give

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C3 Exam Workshop 2 Workbook

1. (a) Express 7 cos x − 24 sin x in the form R cos (x + α) where R > 0 and 0 < α < 2π .

Give the value of α to 3 decimal places. (3)

(b) Hence write down the minimum value of 7 cos x – 24 sin x. (1)

(c) Solve, for 0 ≤ x < 2π, the equation

7 cos x − 24 sin x = 10, giving your answers to 2 decimal places.

(5)


2. (a) Express 1.5 sin 2x + 2 cos 2x in the form R sin (2x + α), where R > 0 and

0 < α < π21 , giving your values of R and α to 3 decimal places where appropriate. (4)

(b) Express 3 sin x cos x + 4 cos2 x in the form a cos 2x + b sin 2x + c, where a, b and c are constants to be found.

(2)

(c) Hence, using your answer to part (a), deduce the maximum value of 3 sin x cos x + 4 cos2 x.

(2)


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3. (a) Prove that

θθ

2sin2cos1− ≡ tan θ , θ ≠

2πn , n ∈ ℤ.

(3) (b) Solve, giving exact answers in terms of π, 2(1 – cos 2θ ) = tan θ , 0 < θ < π .

(6)


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4. (a) Express sin x + √3 cos x in the form R sin (x + α), where R > 0 and 0 < α < 90°. (4)

(b) Show that the equation sec x + √3 cosec x = 4 can be written in the form

sin x + √3 cos x = 2 sin 2x. (3)

(c) Deduce from parts (a) and (b) that sec x + √3 cosec x = 4 can be written in the

form

sin 2x – sin (x + 60°) = 0. (1)

(d) Hence, using the identity sin X – sin Y = 2 cos 2

sin2

YXYX −+ , or otherwise,

find the values of x in the interval 0 ≤ x ≤ 180°, for which sec x + √3 cosec x = 4. (5)


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5. (i) (a) Express (12 cos θ – 5 sin θ) in the form R cos (θ + α), where R > 0 and 0 < α < 90°.

(4)

(b) Hence solve the equation 12 cos θ – 5 sin θ = 4, for 0 < θ < 90°, giving your answer to 1 decimal place.

(3)

(ii) Solve 8 cot θ – 3 tan θ = 2, for 0 < θ < 90°, giving your answer to 1 decimal place.

(5)


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6. (i) Given that sin x = 53 , use an appropriate double angle formula to find the exact

value of sec 2x. (4)

(ii) Prove that

cot 2x + cosec 2x ≡ cot x, ⎟⎠⎞⎜

⎝⎛ ∈≠ Z,

2nnx π .

(4)


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7. (a) Prove that

θθθ 2cos

tan1tan1

2

2

≡+− .

(4)

(b) Hence, or otherwise, prove

tan2 8π = 3 – 2√2.

(5)


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8. (a) Prove that for all values of x

2 tan x − sin2x = 2sin2 x tan x .

(5)

(b) Hence, or otherwise, find the values of x in the interval 0 ≤ x ≤ 360º, for which

2 tan x − sin2x = sin2 x

giving your answers to an appropriate degree of accuracy.

(6)


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9. (a) Using the half-angle formulae, or otherwise, prove that for all values of x

1+ cos x1− cos x

≡ cot2 x2

.

(5)

(b) Hence, or otherwise, find the values of x in the interval 0 ≤ x ≤ 2π for which

1+ cos x1− cos x

= 6cosec x2−10

(7)


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10. (a) Prove that there are no real values of θ for which cos2θ + cosθ + 2 = 0 .

(4)

(b) Find the values of x in the interval 0 ≤ x ≤ 360º, for which

3sin x − 2cos2 x = 0 (5)

(c) Hence, find the values of y in the interval 0 ≤ y ≤ 180º, for which 3sec2y − 2cot 2y = 0

(4)


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11. (a) Prove that for all values of x

cos2 x − sin2 2x ≡ cos2 x 4 cos2 x − 3( ) . (5)

(b) Hence, or otherwise, find the values of x in the interval 0 ≤ x ≤ 2π for which

cos2 x − sin2 2x = 0 giving your answer in terms of π.

(6)


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12. (a) Show that for all values of x, where x is measured in degrees,

cos x + 60°( ) − 3sin x − 60°( ) ≡ 2cos x − 3sin x . (5)

(b) Hence, find the values of x in the interval -180º ≤ x ≤ 180º, for which

cos x + 60º( ) − 3sin x − 60º( ) = 0 (4)


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13. (a) Using the identity cos (A + B) ≡ cos A cos B – sin A sin B, prove that

cos 2A ≡ 1 – 2 sin2 A. (2)

(b) Show that

2 sin 2θ – 3 cos 2θ – 3 sin θ + 3 ≡ sin θ (4 cos θ + 6 sin θ – 3). (4)

(c) Express 4 cos θ + 6 sin θ in the form R sin (θ + α ), where R > 0 and 0 < α < π21 .

(4)

(d) Hence, for 0 ≤ θ < π, solve

2 sin 2θ = 3(cos 2θ + sin θ – 1),

giving your answers in radians to 3 significant figures, where appropriate. (5)


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14. (a) Use the double angle formulae and the identity

cos(A + B) ≡ cosA cosB − sinA sinB to obtain an expression for cos 3x in terms of powers of cos x only.

(4)

(b) (i) Prove that

xx

sin1cos+

+ xx

cossin1+ ≡ 2 sec x, x ≠ (2n + 1)

2π .

(4)

(ii) Hence find, for 0 < x < 2π, all the solutions of

xx

sin1cos+

+ xx

cossin1+ = 4.

(3)


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15.

a) Given that sin θ +α( ) = 2.5sinθ , show that tanθ =sinα

2.5 − cosα.

(3)

b) Hence, solve the equation sin θ + 45°( ) = 2.5sinθ , given 0° ≤θ ≤ 360° . (4)

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C3 Exam Workshop 2 - Elite Tuition€¦ · C3 Exam Workshop 2 Workbook 1. (a) Express 7 cos x − 24 sin x in the form R cos (x + α) where R > 0 and 0 < α < 2 π. Give