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JEE Advanced
Exam
JEE Advanced
Subject
Mathematics
Chapter
Trigonometry

JEE Advanced · Mathematics · Trigonometry

Which of the following statement(s) is(are) correct?

Which of the following statement(s) is(are) correct?

Options

  1. AIf A=130∘A=130^{{}^{{}^{\circ}}} and S=sin⁡A+cos⁡AS=\sin A+\cos A, then S>0S\gt0.
  2. BIf cosec⁡230∘+k(cos⁡36∘−sin⁡18∘)=5\operatorname{cosec}^2 30^{{}^{{}^{\circ}}}+k\left(\cos 36^{{}^{{}^{\circ}}}-\sin 18^{{}^{{}^{\circ}}}\right)=5, then kk equals 2 .
  3. CThe minimum value of 27tan⁡2θ+3cot⁡2θ27 \tan ^2 \theta+3 \cot ^2 \theta is 18 .
  4. DIf a,ba, b are constant real numbers such that a2<ba^2 \lt b, then x2+2ax+bx^2+2 a x+b is always positive ∀x∈R\forall x \in R.

Correct answer

D. If a,ba, b are constant real numbers such that a2<ba^2 \lt b, then x2+2ax+bx^2+2 a x+b is always positive ∀x∈R\forall x \in R.

Step-by-step solution

(A) Clearly, S=sin⁡130∘+cos⁡130∘=(sin⁡130∘−sin⁡40∘)=2cos⁡85∘sin⁡45∘>0S=\sin 130^{\circ}+\cos 130^{\circ}=\left(\sin 130^{\circ}-\sin 40^{\circ}\right)=2 \cos 85^{\circ} \sin 45^{\circ}\gt0. (B) Given, cosec⁡230∘+k(cos⁡36∘−sin⁡18∘)=5\operatorname{cosec}^2 30^{\circ}+\mathrm{k}\left(\cos 36^{\circ}-\sin 18^{\circ}\right)=5⇒4+k[(5+14)−(5−14)]=5⇒4+k2=5⇒k2=1⇒k=2.\begin{aligned} & \Rightarrow 4+k\left[\left(\frac{\sqrt{5}+1}{4}\right)-\left(\frac{\sqrt{5}-1}{4}\right)\right]=5 \\ & \Rightarrow 4+\frac{k}{2}=5 \Rightarrow \frac{k}{2}=1 \Rightarrow k=2 .\end{aligned}$(C)Let(C) Let\mathrm{E}=27 \tan ^2 \theta+3 \cot ^2 \theta=(3 \sqrt{3} \tan \theta-\sqrt{3} \cot \theta)^2+18 ∴Emin⁡=18.]\left.\therefore \quad \mathrm{E}_{\min }=18.\right] (D) Let P(x)=x2+2ax+bP(x)=x^2+2 a x+b. If P(x)>0∀x∈RP(x)\gt0 \forall x \in R then $\mathrm{D} \lt 0\Rightarrow 4 a^2-4 b \lt 0 \Rightarrow a^2 \lt b
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