JEE Mains · Maths · STD 11 - Trigonometrical equations
Let,\(S=\left\{\theta \in[0,2 \pi]: 8^{2 \sin ^{2} \theta}+8^{2 \cos ^{2} \theta}=16\right\}\). Then \(n ( S )+\sum_{\theta \in S}\left(\sec \left(\frac{\pi}{4}+2 \theta\right) \operatorname{cosec}\left(\frac{\pi}{4}+2 \theta\right)\right)\) is equal to.
- A \(0\)
- B \(-2\)
- C \(-4\)
- D \(12\)
Answer & Solution
Correct Answer
(C) \(-4\)
Step-by-step Solution
Detailed explanation
\(8^{2 \sin ^{2} \theta}+8^{2-2 \sin ^{2} \theta}=16\) \(y+\frac{64}{y}=16\) \(\Rightarrow y =8\) \(\Rightarrow \sin ^{2} \theta=1 / 2\) \(n ( S )+\sum_{\theta \in S} \frac{1}{\cos (\pi / 4+2 \theta) \sin (\pi / 4+2 \theta)}\) \(=4+(-2) \times 4=-4\)
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- Let \(\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\mathrm{k}, \overrightarrow{\mathrm{b}}=3(\hat{\mathrm{i}}-\hat{\mathrm{j}}+\mathrm{k})\). Let \(\overrightarrow{\mathrm{c}}\) be the vector such that \(\vec{a} \times \vec{c}=\vec{b}\) and \(\vec{a} \cdot \vec{c}=3\). Then \(\overrightarrow{\mathrm{a}} \cdot((\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{b}})-\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{c}})\) is equal to :JEE Mains 2024 Hard
- If the coefficients of \(x^4, x^5\) and \(x^6\) in the expansion of \((1+x)^n\) are in the arithmetic progression, then the maximum value of \(n\) is :JEE Mains 2024 Hard
- The product \(2^{\frac{1}{4}} \cdot 4^{\frac{1}{16}} \cdot 8^{\frac{1}{48}} \cdot 16^{\frac{1}{128}} \cdot \ldots .\) to \(\infty\) is equal toJEE Mains 2020 Hard
- If \(\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}-x+1}-a x\right)=b\), then the ordered pair \((a, b)\) is:JEE Mains 2021 Hard
- Let \(\mathrm{I}(x)=\int \frac{d x}{(x-11)^{\frac{11}{13}}(x+15)^{\frac{15}{13}}}\). If \(\mathrm{I}(37)-\mathrm{I}(24)=\frac{1}{4}\left(\frac{1}{\mathrm{~b}^{\frac{1}{13}}}-\frac{1}{\mathrm{c}^{\frac{1}{13}}}\right), \mathrm{b}, \mathrm{c} \in \mathrm{N}\), then \(3(\mathrm{~b}+\mathrm{c})\) is equal toJEE Mains 2025 Hard
- Let the line L pass through \((1,1,1)\) and intersect the lines \(\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}\) and \(\frac{x-3}{1}=\frac{y-4}{2}=\frac{z}{1}\) . Then, which of the following points lies on the line L ?JEE Mains 2025 Medium
More PYQs from JEE Mains
- Let \(n \ge 2\) be a natural number and \(0 < \theta < \frac{\pi }{2}\). Then \(\int {\frac{{{{\left( {{{\sin }^n}\,\theta - \sin \,\theta } \right)}^{\frac{1}{n}}}\,\cos \,\theta }}{{{{\sin }^{n + 1}}\,\theta }}} d\theta \) is equal toJEE Mains 2019 Hard
- A spherical iron ball of \(10 \;\mathrm{cm}\) radius is coated with a layer of ice of uniform thickness the melts at a rate of \(50\; \mathrm{cm}^{3} / \mathrm{min}\). When the thickness of ice is \(5 \;\mathrm{cm},\) then the rate (in \(\mathrm{cm} / \mathrm{min.}\) ) at which of the thickness of ice decreases, isJEE Mains 2020 Medium
- An ellipse \(E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) passes through the vertices of the hyperbola \(H: \frac{x^{2}}{49}-\frac{y^{2}}{64}=-1\). Let the major and minor axes of the ellipse \(E\) coincide with the transverse and conjugate axes of the hyperbola \(H\). Let the product of the eccentricities of \(E\) and \(H\) be \(\frac{1}{2}\). If \(l\) is the length of the latus rectum of the ellipse \(E\), then the value of \(113\,l\) is equal to \(....\)JEE Mains 2022 Hard
- Let f: R→R be a twice differentiable function such that the quadratic equation \( f(x)m^{2}-2f^{\prime}(x)m+f^{\prime\prime}(x)=0 \) in m, has two equal roots for every \( x\in R \). If \( f(0)=1, f^{\prime}(0)=2 \) and \( (\alpha, \beta) \) is the largest interval in which the function \( f(\log_{e}x-x) \) is increasing, then \( \alpha+\beta \) is equal to:JEE Mains 2026 Hard
- Let \(\mathrm{g}(\mathrm{x})\) be a linear function and \(f(x)=\left\{\begin{array}{cl}g(x) & , x \leq 0 \\ \left(\frac{1+x}{2+x}\right)^{\frac{1}{x}} & , x>0\end{array}\right.\), is continuous at \(x=0\). If \(f^{\prime}(1)=f(-1)\), then the value of \(g(3)\) isJEE Mains 2024 Hard
- Let \(\vec{a}=-5 \hat{i}+\hat{j}-3 \hat{k}, \vec{b}=\hat{i}+2 \hat{j}-4 \hat{k}\) and \(\vec{c}=(((\vec{a} \times \vec{b}) \times \hat{i}) \times \hat{i}) \times \hat{i}\). Then \(\vec{c} \cdot(-\hat{i}+\hat{j}+\hat{k})\) is equal toJEE Mains 2024 Hard