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JEE Main 202623 JanuaryEvening Shift
Exam
JEE Main
Year
2026
Date
23 January
Shift
Evening Shift
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

JEE Main 2026 · 23 January · Evening Shift · Mathematics · Trigonometric Ratios & Identities

The least value of ( ^ 2 θ-6 θ θ+3 ^ 2 θ+2) is

The least value of (cos⁡2θ−6sin⁡θcos⁡θ+3sin⁡2θ+2)\left(\cos ^{2} \theta-6 \sin \theta \cos \theta+3 \sin ^{2} \theta+2\right) is

Options

  1. A−1-1
  2. B1
  3. C4−104-\sqrt{10}
  4. D4+104+\sqrt{10}

Correct answer

C. 4−104-\sqrt{10}

Step-by-step solution

cos⁡2θ+3sin⁡2θ−3sin⁡2θ+2\cos^2\theta + 3\sin^2\theta - 3\sin 2\theta + 2.
Using double angle formulas:
=1+cos⁡2θ2+3(1−cos⁡2θ)2−3sin⁡2θ+2= \frac{1+\cos 2\theta}{2} + \frac{3(1-\cos 2\theta)}{2} - 3\sin 2\theta + 2
=4−cos⁡2θ−3sin⁡2θ= 4 - \cos 2\theta - 3\sin 2\theta.
Minimum of −cos⁡2θ−3sin⁡2θ=−1+9=−10-\cos 2\theta - 3\sin 2\theta = -\sqrt{1+9} = -\sqrt{10}.
Least value =4−10= 4 - \sqrt{10}.
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