JEE Mains · Maths · STD 12 - 7.2 definite integral
If \(x \phi(x)=\int_{5}^{x}\left(3 t^{2}-2 \phi^{\prime}(t)\right) d t, x\,>\,-2\), and \(\phi(0)=4\) then \(\phi(2)\) is .... .
- A \(4\)
- B \(6\)
- C \(8\)
- D \(10\)
Answer & Solution
Correct Answer
(A) \(4\)
Step-by-step Solution
Detailed explanation
\(\mathrm{x} \phi(\mathrm{x})=\int_{5}^{\mathrm{x}} 3 \mathrm{t}^{2}-2 \phi^{\prime}(\mathrm{t}) \mathrm{dt}\) \(\mathrm{x} \phi(\mathrm{x})=\mathrm{x}^{3}-125-2[\phi(\mathrm{x})-\phi(5)]\) \(\mathrm{x} \phi(\mathrm{x})=\mathrm{x}^{3}-125-2 \phi(\mathrm{x})-2 \phi(5)\)…
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 function \(f(x)=\frac{\sin 3 x+\alpha \sin x-\beta \cos 3 x}{x^3}\), \(x \in R\), is continuous at \(x=0\), then \(f(0)\) is equal to :JEE Mains 2024 Medium
- If \(\overrightarrow x = 3\hat i - 6\hat j - \hat k\) , \(\overrightarrow y = \hat i + 4\hat j - 3\hat k\) and \(\,\,\overrightarrow z = 3\hat i - 4\hat j - 12\hat k\) , then the magnitude of the projection of \(\overrightarrow x \times \overrightarrow y \) on \(\overrightarrow z\) isJEE Mains 2014 Medium
- Let \(f :R \to R\) be a function defined as \(f\left( x \right) = \left\{ \begin{array}{l}
5,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if\,\,\,\,\,\,\,x \le 1\,\,\,\,\,\,\,\\
a + bx,\,\,\,\,if\,\,\,\,\,\,1 < x < 3\\
b + 5x,\,\,\,\,if\,\,\,\,\,\,3 \le x < 5\\
30,\,\,\,\,\,\,\,\,\,\,if\,\,\,\,\,\,\,x \ge 5
\end{array} \right.\,\,\,\,\) Then \(f\) isJEE Mains 2019 Hard - Let \(S_n\) denote the sum of the first \(n\) terms of an arithmetic progression. If \(\mathrm{S}_{10}=390\) and the ratio of the tenth and the fifth terms is \(15: 7\), then \(S_{15}-S_5\) is equal to :JEE Mains 2024 Hard
- If the system of equations \(x-2 y+3 z=9\) \(2 x+y+z=b\) \(x-7 y+a z=24\) has infinitely many solutions, then \(a - b\) is equal toJEE Mains 2020 Medium
- Let \(A =\{ x \in R :| x +1|<2\}\) and \(B=\{x \in R:|x-1| \geq 2\}\). Then which one of the following statements is NOT true ?JEE Mains 2022 Medium
More PYQs from JEE Mains
- When a missile is fired from a ship, the probability that it is intercepted is \(\frac{1}{3}\) and the probability that the missile hits the target, given that it is not intercepted, is \(\frac{3}{4}\). If three missiles are fired independently from the ship, then the probability that all three hit the target, isJEE Mains 2021 Easy
- Let \(A\) be a \(3 \times 3\) invertible matrix. If \(|adj (24 A ) \mid=\) \(\operatorname{adj}(3 \operatorname{adj}(2 A )) \mid\), then \(\mid A ^{2}|=\dots\dots\dots\) is equal toJEE Mains 2022 Hard
- If the distances of the point (1, 2, a) from the line \(\frac{x-1}{1}=\frac{y}{2}=\frac{z-1}{1}\) along the lines
\(L_{1}:\frac{x-1}{3}=\frac{y-2}{4}=\frac{z-a}{b}\) and
\(L_{2}:\frac{x-1}{1}=\frac{y-2}{4}=\frac{z-a}{c}\) are equal,
then \(a+b+c\) is equal toJEE Mains 2026 Easy - Let \(f(x)=\left|\begin{array}{ccc}a & -1 & 0 \\ a x & a & -1 \\ a x^{2} & a x & a\end{array}\right|, a \in R\). Then the sum of which the squares of all the values of a for \(2 f^{\prime}(10)-f^{\prime}(5)+100=0\) isJEE Mains 2022 Hard
- If the domain of the function \(\log _5\left(18 x-x^2-77\right)\) is \((\alpha, \beta)\) and the domain of the function \(\log _{(x-1)}\left(\frac{2 x^2+3 x-2}{x^2-3 x-4}\right)\) is \((\gamma, \delta)\), then \(\alpha^2+\beta^2+\gamma^2\) is equal to :JEE Mains 2025 Easy
- The number of real solutions of \(x^{7}+5 x^{3}+3 x+1=\) \(0\) is equal to............JEE Mains 2022 Medium