JEE Mains · Maths · STD 12 - 6. Application of derivatives
If the normal to the curve \(y(x)=\int_{0}^{x}\left(2 t^{2}-15 t+10\right) d t\) at a point \((a, b)\) is parallel to the line \(x+3 y=-5, a>1,\) then the value of \(|a +6 b|\) is equal to..........
- A \(324\)
- B \(406\)
- C \(512\)
- D \(376\)
Answer & Solution
Correct Answer
(B) \(406\)
Step-by-step Solution
Detailed explanation
\(y(x)=\int_{0}^{x}\left(2 t^{2}-15 t+10\right) d t\) \(\left.y^{\prime}(x)\right]_{x=a}=\left[2 x^{2}-15 x+10\right]_{a}=2 a^{2}-15 a+10\) Slope of normal \(=-\frac{1}{3}\) \(\Rightarrow \quad 2 a^{2}-15 a+10=3 \Rightarrow a=7\) \(a=\frac{1}{2}(\) rejected \()\)…
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 a random variable \(X\) follows the Binomial distribution \(B (33, p )\) such that \(3 P ( X =0)= P ( X =1)\), then the value of \(\frac{ P ( X =15)}{ P ( X =18)}-\frac{ P ( X =16)}{ P ( X =17)}\) is equal toJEE Mains 2022 Hard
- Let the length of the focal chord \(P Q\) of the parabola \(y^2=12 x\) be \(15\) units. If the distance of \(P Q\) from the origin is \(\mathrm{p}\), then \(10 \mathrm{p}^2\) is equal to ...........JEE Mains 2024 Hard
- Let \(R _{1}\) and \(R _{2}\) be two relations defined as follows : \(R _{1}=\left\{( a , b ) \in R ^{2}: a ^{2}+ b ^{2} \in Q \right\}\) and \(R _{2}=\left\{( a , b ) \in R ^{2}: a ^{2}+ b ^{2} \notin Q \right\}\) where \(Q\) is the set of all rational numbers. ThenJEE Mains 2020 Hard
- The area bounded by the curve \(4 y^{2}=x^{2}(4-x)(x-2)\) is equal to ...... .JEE Mains 2021 Hard
- Let \(A=\{0,1,2,3,4,5,6,7\} .\) Then the number of bijective functions \(f: A \rightarrow A\)such that \(f(1)+f(2)=3-f(3)\) is equal to \(.....\)JEE Mains 2021 Hard
- Let \(\overrightarrow{ a }=\hat{ i }+2 \hat{ j }-\hat{ k }, \overrightarrow{ b }=\hat{ i }-\hat{ j }\) and \(\overrightarrow{ c }=\hat{ i }-\hat{ j }-\hat{ k }\) be three given vectors. If \(\overrightarrow{ r }\) is a vector such that \(\overrightarrow{ r } \times \overrightarrow{ a }=\overrightarrow{ c } \times \overrightarrow{ a }\) and \(\overrightarrow{ r } \cdot \overrightarrow{ b }=0,\) then \(\overrightarrow{ r } \cdot \overrightarrow{ a } \quad\) is equal to ...........JEE Mains 2021 Medium
More PYQs from JEE Mains
- Let the function \(f :[0,2] \rightarrow R\) be defined as \(f(x)=\left\{\begin{array}{cc}e^{\min \left[x^2, x-[x]\right\}}, & x \in[0,1) \\e^{\left[x-\log _e x\right]}, & x \in[1,2]\end{array}\right.\) where [t] denotes the greatest integer less than or equal to \(t\). Then the value of the integral \(\int \limits_0^2 x f(x) d x\) isJEE Mains 2023 Hard
- Let \(A\,(4, -4)\) and \(B\,(9,6)\) be points on the parabola \(y^2 = 4x\) . Let \(C\) be chosen on the arc \(AOB\) of the parabola, where \(O\) is the origin, such that the area of \(\Delta ACB\) is maximum. Then, the area (in sq. units) of \(\Delta ACB,\) isJEE Mains 2019 Hard
- Let a random variable \(X\) have a binomial distribution with mean \(8\) and variance \(4\). If \(P\left( {X \le 2} \right) = \frac{k}{{{2^{16}}}}\), then \(k\) is equal toJEE Mains 2019 Hard
- If the area of the region \(\left\{(\mathrm{x}, \mathrm{y}): \frac{\mathrm{a}}{\mathrm{x}^2} \leq \mathrm{y} \leq \frac{1}{\mathrm{x}}, 1 \leq \mathrm{x} \leq 2,0<\mathrm{a}<1\right\}\) is \(\left(\log _e 2\right)-\frac{1}{7}\) then the value of \(7 a-3\) is equal to :JEE Mains 2024 Hard
- Let \(A=\{1,2,3, \ldots, 10\}\) and \(B=\left\{\frac{m}{n}: m, n \in A, m \lt n\right.\) and \(\left.\operatorname{gcd}(m, n)=1\right\}\). Then \(n(B)\) is equal to :JEE Mains 2025 Medium
- Let, \(\alpha, \beta\) be the distinct roots of the equation \(\mathrm{x}^2-\left(\mathrm{t}^2-5 \mathrm{t}+6\right) \mathrm{x}+1=0, \mathrm{t} \in \mathrm{R}\) and \(\mathrm{a}_{\mathrm{n}}=\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}\). Then the minimum value of \(\frac{\mathrm{a}_{2023}+\mathrm{a}_{2025}}{\mathrm{a}_{2024}}\) isJEE Mains 2024 Hard