JEE Mains · Maths · STD 11 - 3. trignometrical ratios,functions and identities
Let \({f_k}\left( x \right) = \frac{1}{k}\left( {{{\sin }^k}x + {{\cos }^k}x} \right)\) where \(x \in R\;\) and \(k \ge 1\). Then \({f_4}\left( x \right) - {f_6}\left( x \right) \) is equals
- A \(\frac{1}{4}\)
- B \(\frac{1}{{12}}\)
- C \(\frac{1}{6}\)
- D \(\frac{1}{3}\)
Answer & Solution
Correct Answer
(B) \(\frac{1}{{12}}\)
Step-by-step Solution
Detailed explanation
\({f_4}(x)\, - \,{f_6}(x)\, = \frac{1}{4}({\sin ^4} + {\cos ^4}x) - \frac{1}{6}({\sin ^6} + {\cos ^6}x)\) \( = \frac{1}{4}(1 - 2{\sin ^2}{\cos ^2}x) - \frac{1}{6}(1 - 3{\sin ^2}{\cos ^2}x)\) \( = \frac{1}{4} - \frac{1}{6} = \frac{{3 - 2}}{{12}} = \frac{1}{{12}}\)
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
- If \(\alpha\) is the positive root of the equation, \(p(x)=x^{2}-x-2=0,\) then \(\lim \limits_{x \rightarrow \alpha^{+}} \frac{\sqrt{1-\cos (p(x))}}{x+\alpha-4}\) is equal toJEE Mains 2020 Hard
- Let \(f\) be a differentiable function such that \(2(x+2)^2 f(x)-3(x+2)^2=10 \int_0^x(t+2) f(t) d t, x \geq 0\). Then \(f(2)\) is equal to ______.JEE Mains 2025 Hard
- The coefficient of \(x^{18}\) in the expansion of \(\left(x^4-\frac{1}{x^3}\right)^{15}\) is \(...........\).JEE Mains 2023 Hard
- If for \(x \geq 0, y=y(x)\) is the solution of the differential equation \((\mathrm{x}+1) \mathrm{d} \mathrm{y}=\left((\mathrm{x}+1)^{2}+\mathrm{y}-3\right) \mathrm{d} \mathrm{x}, \mathrm{y}(2)=0\) then \(y(3)\) is equal toJEE Mains 2020 Hard
- Let the solution \(y=y(x)\) of the differential equation \(\frac{d y}{d x}-y=1+4 \sin x\) satisfy \(y(\pi)=1\). Then \(y\left(\frac{\pi}{2}\right)+10\) is equal to ...........JEE Mains 2024 Medium
- Let a rectangle \(A B C D\) of sides \(2\) and \(4\) be inscribed in another rectangle \(P Q R S\) such that the vertices of the rectangle \(A B C D\) lie on the sides of the rectangle \(P Q R S\). Let \(a\) and \(b\) be the sides of the rectangle \(P Q R S\) when its area is maximum. Then \((a+b)^2\) is equal to :JEE Mains 2024 Hard
More PYQs from JEE Mains
- Let \(A\) be a \(3 \times 3\) matrix such that \(|\operatorname{adj}(\operatorname{adj}(\operatorname{adj} A ))|=12^4\). Then \(\left| A ^{-1} \operatorname{adj} A \right|\) is equal toJEE Mains 2023 Hard
- A value of \(\theta \in (0, \pi /3)\), for which \(\left| {\begin{array}{*{20}{c}}
{1 + {{\cos }^2}\,\theta }&{{{\sin }^2}\,\theta }&{4\,\cos \,6\theta } \\
{{{\cos }^2}\,\theta }&{1 + {{\sin }^2}\,\theta }&{4\,\cos \,6\theta } \\
{{{\cos }^2}\,\theta }&{{{\sin }^2}\,\theta }&{1 + 4\,\cos \,6\theta }
\end{array}} \right| = 0\), isJEE Mains 2019 Hard - If \(S=\frac{7}{5}+\frac{9}{5^{2}}+\frac{13}{5^{3}}+\frac{19}{5^{4}}+\ldots .\), then \(160 \mathrm{~S}\) is equal to....... .JEE Mains 2021 Hard
- The remainder when \(19^{200}+23^{200}\) is divided by \(49\) , is \(.........\).JEE Mains 2023 Hard
- If \(f(x)=\left|\begin{array}{ccc}x^3 & 2 x^2+1 & 1+3 x \\ 3 x^2+2 & 2 x & x^3+6 \\ x^3-x & 4 & x^2-2\end{array}\right|\) for all \(x \in \mathbb{R}\), then \(2 f(0)+f^{\prime}(0)\) is equal toJEE Mains 2024 Hard
- The number of solutions of \(sin \,3x\, = cos\, 2x\) , in the interval \(\left( {\frac{\pi }{2},\pi } \right)\) isJEE Mains 2018 Hard