JEE Mains · Maths · STD 12 - 6. Application of derivatives
The lengths of the sides of a triangle are \(10+ x ^{2}\), \(10+ x ^{2}\) and \(20-2 x ^{2}\). If for \(x = k\), the area of the triangle is maximum, then \(3 K ^{2}\) is equal to
- A \(5\)
- B \(10\)
- C \(8\)
- D \(12\)
Answer & Solution
Correct Answer
(B) \(10\)
Step-by-step Solution
Detailed explanation
\(a =20-2 x ^{2}, b =10+ x ^{2}, c =10+ x ^{2}\) \(=\frac{a+b+c}{2}\) \(=20\) \(\Delta=\sqrt{s(s-a)(s-b)(s-c)}\) \(=\sqrt{20\left(2 x^{2}\right)\left(10-x^{2}\right)\left(10-x^{2}\right)}\) \(=2 \sqrt{10} \sqrt{x^{2}\left(10-x^{2}\right)^{2}}\)…
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 \(\mathrm{P}\) be a point on the hyperbola \(\mathrm{H}: \frac{\mathrm{x}^2}{9}-\frac{\mathrm{y}^2}{4}=1\), in the first quadrant such that the area of triangle formed by \(\mathrm{P}\) and the two foci of \(\mathrm{H}\) is \(2 \sqrt{13}\). Then, the square of the distance of \(\mathrm{P}\) from the origin isJEE Mains 2024 Hard
- Let \(f:[0,3] \rightarrow\) A be defined by \(f(x)=2 x^3-15 x^2+36 x+7\) and \(g:[0, \infty) \rightarrow B\) be defined by \(\mathrm{g}(x)=\frac{x^{2025}}{x^{2025}+1}\). If both the functions are onto and \(\mathrm{S}=\{x \in \mathbf{Z}: x \in \mathrm{~A}\) or \(x \in \mathrm{~B}\}\), then \(\mathrm{n}(\mathrm{S})\) is equal to :JEE Mains 2025 Medium
- The values of \(\alpha\), for which \(\left|\begin{array}{ccc}1 & \frac{3}{2} & \alpha+\frac{3}{2} \\ 1 & \frac{1}{3} & \alpha+\frac{1}{3} \\ 2 \alpha+3 & 3 \alpha+1 & 0\end{array}\right|=0\), lie in the intervalJEE Mains 2024 Hard
- Let \(\alpha = 3+4+8+9+13+14+\ldots\) upto 40 terms. If \((\tan\beta)^{\frac{\alpha}{1020}}\) is a root of the equation \(x^2+x-2=0\), \(\beta \in \left(0, \dfrac{\pi}{2}\right)\), then \(\sin^2\beta + 3\cos^2\beta\) is equal to:JEE Mains 2026 Medium
- The lowest integer which is greater than \(\left(1+\frac{1}{10^{100}}\right)^{10^{100}}\) is \(.....\)JEE Mains 2021 Hard
- Let the tangent and normal at the point \((3 \sqrt{3}, 1)\) on the ellipse \(\frac{x^2}{36}+\frac{y^2}{4}=1\) meet the \(y\)-axis at the points \(A\) and \(B\) respectively. Let the circle \(C\) be drawn taking \(A B\) as a diameter and the line \(x =2 \sqrt{5}\) intersect \(C\) at the points \(P\) and \(Q\). If the tangents at the points \(P\) and \(Q\) on the circle intersect at the point \((\alpha, \beta)\), then \(\alpha^2-\beta^2\) is equal toJEE Mains 2023 Hard
More PYQs from JEE Mains
- Let a be an integer such that \(\lim \limits_{x \rightarrow 7} \frac{18-[1-x]}{[x-3 a]}\) exists, where \([ t ]\) is greatest integer \(\leq t\). Then a is equal toJEE Mains 2022 Hard
- Let \(A\) and \(B\) be two invertible matrices of order \(3 \times 3\). If det \((ABA^T) = 8\) and \(det\,(AB^{-1}) = 8\), then \(det\, (BA^{-1} B^T)\) is equal toJEE Mains 2019 Hard
- Let a line having direction ratios \(1,-4,2\) intersect the lines \(\frac{x-7}{3}=\frac{y-1}{-1}=\frac{z+2}{1}\) and \(\frac{x}{2}=\frac{y-7}{3}=\frac{z}{1}\) at the point \(A\) and \(B\). Then \(( AB )^{2}\) is equal toJEE Mains 2022 Hard
- The square of the distance of the point of intersection of the lines \(\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(a\hat{i} - \hat{j})\), \(a \neq 0\) and \(\vec{r} = (4\hat{i} - \hat{k}) + \mu(2\hat{i} + a\hat{k})\) from the origin is:JEE Mains 2026 Hard
- Let \(f\left( x \right) = 5 - \left| {x - 2} \right|\) and \(g\left( x \right) = \left| {x + 1} \right|,x \in R\). If \(f(x)\) attains maximum value at \(\alpha \) and \(g(x)\) attains minimum value at \(\beta \), then \(\mathop {\lim }\limits_{x \to \alpha \beta } \frac{{\left( {x - 1} \right)\left( {{x^2} - 5x + 6} \right)}}{{{x^2} - 6x + 8}}\) is equal toJEE Mains 2019 Hard
- The real valued function \(f(x)=\frac{\operatorname{cosec}^{-1} x}{\sqrt{x-[x]}, \text { where }}\) \([ x ]\) denotes the greatest integer less than or equal to \(x,\) is defined for all \(x\) belonging toJEE Mains 2021 Hard