JEE Mains · Maths · STD 12 - 8. Application and integration
Let \( f(\alpha) \) denote the area of the region in the first quadrant bounded by \( x=0, x=1, y^{2}=x \) and \( y=|\alpha x-5|-|1-\alpha x|+\alpha x^{2}. \) Then \( (f(0)+f(1)) \) is equal to
- A 9
- B 14
- C 7
- D 12
Answer & Solution
Correct Answer
(C) 7
Step-by-step Solution
Detailed explanation
\(\text {at } \alpha=0 \Rightarrow f(0)\) \(x=0, x=1, y^2=x\) \(y=|0 \cdot x-5|-|1-0 \cdot x|+0 \cdot x^2\) \(y=4\) \(A_1=\int_0^1(4-\sqrt{x}) d x\) \(=4 x-\left.\frac{x^{\frac{3}{2}}}{\frac{3}{2}}\right|_0 ^1\) \(=4-\frac{2}{3}(1)=\frac{10}{3}\)…
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 \(\bar{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}\) and \(\vec{c}=\hat{j}-\hat{k}\) be three vectors such that \(\vec{a} \times \vec{b}=\vec{c}\) and \(\vec{a} \cdot \vec{b}=1\). If the length of projection vector of the vector \(\vec{b}\) on the vector \(\vec{a} \times \vec{c}\) is \(l\), then the value of \(3l^{2}\) is equal to \(.....\)JEE Mains 2021 Medium
- The area of the region (in sq. units), in the first quadrant bounded by the parabola \(y = 9x^2\) and the lines \(x = 0,y = 1\) and \(y = 4,\) isJEE Mains 2013 Hard
- Let \(A (2, 3, 5), B (- 1, 3, 2)\) and \(C (\lambda, 5, \mu)\) be the vertices of a \(\Delta ABC\). If the median through \(A\) is equally inclined to the coordinate axes, thenJEE Mains 2014 Hard
- The general solution of the differential equation, \(\sin \,2x\,\left( {\frac{{dy}}{{dx}} - \sqrt {\tan \,x} } \right) - y = 0,\) isJEE Mains 2014 Hard
- Let \( A=\{-2,-1,0,1,2,3,4\} \). Let R be a relation on A defined by xRy if and only if \( 2x+y \le 2 \). Let \(l\) be the number of elements in R. Let m and n be the minimum number of elements required to be added in R to make it reflexive and symmetric relations respectively. Then \( l+m+n \) is equal to:JEE Mains 2026 Medium
- Let \(S=\left\{\left(\begin{array}{cc}-1 & a \\ 0 & b\end{array}\right) ; a, b \in\{1,2,3, \ldots 100\}\right\}\) and let \(T_{n}=\left\{A \in S: A^{n(n+1)}=I\right\}\). Then the number of elements in \(\bigcap \limits_{n=1}^{100} T_{n}\) isJEE Mains 2022 Hard
More PYQs from JEE Mains
- Let \(y=x\) be the equation of a chord of the circle \(C_{1}\) (in the closed half-plane \(x\ge0)\) of diameter 10 passing through the origin. Let \(C_{2}\) be another circle described on the given chord as its diameter. If the equation of the chord of the circle \(C_{2}\), which passes through the point (2, 3) and is farthest from the center of \(C_{2}\), is \(x+ay+b=0,\) then \(a-b\) is equal to:JEE Mains 2026 Easy
- Let \(\mathrm{E}: \frac{x^2}{\mathrm{a}^2}+\frac{y^2}{\mathrm{~b}^2}=1, \mathrm{a}\gt\mathrm{b}\) and \(\mathrm{H}: \frac{x^2}{\mathrm{~A}^2}-\frac{y^2}{\mathrm{~B}^2}=1\). Let the distance between the foci of E and the foci of \(H\) be \(2 \sqrt{3}\). If \(a-A=2\), and the ratio of the eccentricities of \(E\) and \(H\) is \(\frac{1}{3}\), then the sum of the lengths of their latus rectums is equal to:JEE Mains 2025 Hard
- If for \(p \neq q \neq 0\), then function,\(f(x)=\frac{\sqrt[7]{p(729+x)}-3}{\sqrt[3]{729+q x}-9}\)is continuous at \(x=0\), thenJEE Mains 2022 Hard
- Two dice \(A\) and \(B\) are rolled, Let the numbers obtained on \(A\) and \(B\) be \(\alpha\) and \(\beta\) respectively. If the variance of \(\alpha-\beta\) is \(\frac{p}{q}\), where \(p\) and \(q\) are coprime, then the sum of the positive divisors of \(p\) is equal toJEE Mains 2023 Hard
- Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three non zero vectors such that \(\vec{b} \cdot \vec{c}=0\) and \(\vec{a} \times(\vec{b} \times \vec{c})=\frac{\vec{b}-\vec{c}}{2}\). If \(\vec{d}\) be a vector such that \(\vec{b} \cdot \vec{d}=\vec{a} \cdot \vec{b}\), then \((\vec{a} \times \vec{b}) \cdot(\vec{c} \times \vec{d})\) is equal toJEE Mains 2023 Medium
- The number of seven digits odd numbers, that can be formed using all the seven digits \(1, 2, 2, 2, 3, 3, 5\) is \(.......\)JEE Mains 2023 Hard