enEnglishguગુજરાતી
JEE Mains · Maths · STD 12 - 6. Application of derivatives
Let \(P ( x )\) be a real polynomial of degree \(3\) which vanishes at \(x =-3 .\) Let \(P ( x )\) have local minima at \(x=1,\) local maxima at \(x=-1\) and \(\int_{-1}^{1} P ( x ) d x =18,\) then the sum of all the coefficients of the polynomial \(P ( x )\) is equal to ....... .
- A \(16\)
- B \(8\)
- C \(4\)
- D \(12\)
Answer & Solution
Correct Answer
(B) \(8\)
Step-by-step Solution
Detailed explanation
Let \(p^{\prime}(x)=a(x-1)(x+1)=a\left(x^{2}-1\right)\) \(p ( x )= a \int\left( x ^{2}-1\right) d x + c\) \(=a\left(\frac{x^{3}}{3}-x\right)+c\) Now \(p(-3)=0\) \(\Rightarrow a \left(-\frac{27}{3}+3\right)+ c =0\) \(\Rightarrow-6 a+c=0\) Now…
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 the system of linear equations \(x+y+3 z=0\) \(x+3 y+k^{2} z=0\) \(3 x+y+3 z=0\) has a non-zero solution \((x, y, z)\) for some \(k \in R ,\) then \(x +\left(\frac{ y }{ z }\right)\) is equal toJEE Mains 2020 Medium
- Let \(\quad S=\left\{z \in C-\{i, 2 i\}: \frac{z^2+8 i z-15}{z^2-3 i z-2} \in R \right\}\). \(\alpha-\frac{13}{11} i \in S , \alpha \in R -\{0\}\), then \(242 \alpha^2\) is equal toJEE Mains 2023 Hard
- The number of values of \(a \in N\) such that the variance of \(3,7,12 a, 43-a\) is a natural number is (Mean \(=13\))JEE Mains 2022 Medium
- The number of points, at which the function \(f ( x )\) \(=|2 x+1|-3|x+2|+\left|x^{2}+x-2\right|, x \in R\) is not differentiable, is ............JEE Mains 2021 Medium
- Let the area of a \(\triangle P Q R\) with vertices \(P(5,4), Q(-2,4)\) and \(R(a, b)\) be 35 square units. If its orthocenter and centroid are \(O\left(2, \frac{14}{5}\right)\) and \(C(c, d)\) respectively, then \(c+2 d\) is equal toJEE Mains 2025 Easy
- For \(t \gt -1\), let \(\alpha_t\) and \(\beta_t\) be the roots of the equation
\(((t+2)^{\frac{1}{7}}-1) x^2+((t+2)^{\frac{1}{6}}-1) x~+\) \(((t+2)^{\frac{1}{21}}\) \(-~1)=0\)
If \(\lim _{t \rightarrow-1^{+}} \alpha_t=a\) and \(\lim _{t \rightarrow-1^{+}} \beta_t=b\), then \(72(a+b)^2\) is equal to ________.JEE Mains 2025 Medium
More PYQs from JEE Mains
- Let \(x =\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]\) and \(A =\left[\begin{array}{ccc}-1 & 2 & 3 \\ 0 & 1 & 6 \\ 0 & 0 & -1\end{array}\right]\). For \(k \in N\), if \(X ^{\prime} A ^{ k } X =33\), then \(k\) is equal to.JEE Mains 2022 Hard
- If the shortest distance between the line joining the points \((1, 2, 3)\) and \((2,3,4)\), and the line \(\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-2}{0}\) is \(\alpha\), then \(28 \alpha^2\) is equal to \(........\).JEE Mains 2023 Hard
- If the midpoint of a chord of the ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\) is \((\sqrt{2}, 4 / 3)\), and the length of the chord is \(\frac{2 \sqrt{\alpha}}{3}\), then \(\alpha\) is :JEE Mains 2025 Medium
- The sum, of the coefficients of the first \(50\) terms in the binomial expansion of \((1-x)^{100}\), is equal toJEE Mains 2023 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
- If \(\mathop {\lim }\limits_{n \to \infty } \frac{{{1^a} + {2^a} + ....... + {n^a}}}{{{{\left( {n + 1} \right)}^{a - 1}}\left[ {\left( {na + 2} \right) + ......\left( {na + n} \right)} \right]}} = \frac{1}{{60}}\) for some positive real number \(a\), then \(a\) is equal toJEE Mains 2017 Hard