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NDA 2010
Exam
NDA
Year
2010
Subject
Mathematics
Chapter
Determinants

NDA 2010 · Mathematics · Determinants

What is the value of | array cc 15^ ^ & 15^ ^ 45^ ^ & 45^ ^ array |...

What is the value of \left|cos⁡15∘sin⁡15∘cos⁡45∘sin⁡45∘\begin{array}{cc}\cos 15^{{}^{\circ}} & \sin 15^{{}^{\circ}} \\ \cos 45^{{}^{\circ}} & \sin 45^{{}^{\circ}}\end{array}\right| \times\left|cos⁡45∘cos⁡15∘sin⁡45∘sin⁡15∘\begin{array}{cc}\cos 45^{{}^{\circ}} & \cos 15^{{}^{\circ}} \\ \sin 45^{{}^{\circ}} & \sin 15^{{}^{\circ}}\end{array}\right| ?

Options

  1. A14\frac{1}{4}
  2. B32\frac{\sqrt{3}}{2}
  3. C−14-\frac{1}{4}
  4. D−34-\frac{3}{4}

Correct answer

C. −14-\frac{1}{4}

Step-by-step solution

∣cos⁡15∘sin⁡15∘cos⁡45∘sin⁡45∘∣×∣cos⁡45∘cos⁡15∘sin⁡45∘sin⁡15∘∣=[cos⁡15∘sin⁡45∘−sin⁡15∘cos⁡45∘]×[cos⁡45∘sin⁡15∘−sin⁡45∘cos⁡15∘]=sin⁡30∘×−sin⁡30∘=+12×−12=−14\begin{aligned} & \left|\begin{array}{cc} \cos 15^{{}^{\circ}} & \sin 15^{{}^{\circ}} \\ \cos 45^{{}^{\circ}} & \sin 45^{{}^{\circ}} \end{array}\right| \times\left|\begin{array}{cc} \cos 45^{{}^{\circ}} & \cos 15^{{}^{\circ}} \\ \sin 45^{{}^{\circ}} & \sin 15^{{}^{\circ}} \end{array}\right| \\ & =\left[\cos 15^{{}^{\circ}} \sin 45^{{}^{\circ}}-\sin 15^{{}^{\circ}} \cos 45^{{}^{\circ}}\right] \times \\ & {\left[\cos 45^{{}^{\circ}} \sin 15^{{}^{\circ}}-\sin 45^{{}^{\circ}} \cos 15^{{}^{\circ}}\right]} \\ & =\sin 30^{{}^{\circ}} \times-\sin 30^{{}^{\circ}} \\ & =+\frac{1}{2} \times-\frac{1}{2}=-\frac{1}{4} \end{aligned}
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