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Question

Evaluate the following determinants:

(i) x-7x5x+1

(ii) cos θ-sin θsin θcos θ

(iii) cos 15°sin 15°sin 75°cos 75°

(iv) a+ibc+id-c+ida-ib

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Solution

(i)
= x(5x + 1) + 7x = 5x2 + x + 7x=5x2 + 8x

(ii)
= cos2θ - - sin2θ = cos2θ + sin2θ = 1

(iii)
=cos15°cos75°- sin15°sin75°=cos15°cos75°-sin(90°- 75°)sin(90° - 15°) sin90° - θ = cosθ=cos15°cos75° - cos75°cos15°=cos15°cos75° - cos15°cos75°= 0

(iv)
=a2 - i2b2 - i2d2 - c2=a2 - i2b2 - i2d2 + c2=a2 + c2 - i2b2 + d2 i2 =-1=a2 + c2 + b2 + d2

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