MCQ
Choose the correct answer: Area lying between the curves $y^2=4 x$ and $y=2 x$ is:
- A$\frac23$
- ✓$\frac13$
- C$\frac14$
- D$\frac34.$
✓
Answer
Correct option: B.
$\frac13$
Equation of curve (parabola) is $y^2=4 x \ldots$ (i)
$\Rightarrow\text{y}=2\sqrt{\text{x}}=2\text{x}^{\frac12}...(\text{ii})$ Equation of another curve (line) is y = 2x ...(iii) Solving eq. (i) and (iii), we get x = 0 or x = 1 and y = 0 or y = 2 Therefore, Points of intersections of circle (i) and line (ii) are O(0, 0) and A(1, 2). Now Area OBAM = Area bounded by parabola (i) and x-axis $=\Bigg|\int\limits^1_0\text{y dx}\Bigg|=\Bigg|\int\limits^1_02\text{x}^{\frac12}\text{dx}\Bigg|=$ $2\frac{\Big(\text{x}^{\frac32}\Big)^1_0}{\frac32}$ $=\frac43(1-0)=\frac43\dots(\text{iv})$ Also, Area $\Delta\text{ OAM}=$ Area bounded by parabola (iii) and x-axis $=\Bigg|\int\limits^1_0\text{y dx}\Bigg|=\Bigg|\int\limits^1_02\text{x dx}\Bigg|=2\Big(\frac{\text{x}^2}{2}\Big)^1_0$ = (1 - 0) = 1 ...(v) Now Required shaded area OBA = Area OBAM - Area of $\Delta\text{ OAM}$ $=\frac43-1=\frac{4-3}{3}=\frac13\text{ sq. units}$ Therefore, option (B) is correct.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
If the position vector of one end of the line segment $AB$ be $2i + 3j - k$ and the position vector of its middle point be $3\,(i + j + k),$ then the position vector of the other end is
→If $y = {{2{{(x - \sin x)}^{3/2}}} \over {\sqrt x }}$, then ${{dy} \over {dx}} = $
→The degree of differential equation $[1+(\frac{\text{dy}}{\text{dx}})^2]^{\frac{1}{2}}=\frac{\text{d}^2\text{y}}{\text{dx}^2}$ is:
→If ${a^{ - 1}} + {b^{ - 1}} + {c^{ - 1}} = 0$ such that $\left| {\,\begin{array}{*{20}{c}}{1 + a}&1&1\\1&{1 + b}&1\\1&1&{1 + c}\end{array}\,} \right| = \lambda $, then the value of $\lambda $is
→Consider the binary operation $\times$ on $Q$ defind by $a \times b = a + 12b + ab$ for $a, b \in Q.$ Find $2\times\frac{1}{3}$
→If $f(x)=\int_{0}^{x}(5+|1-t|) d t, \quad x>2$
→$\quad \quad \quad \quad \quad 5 x+1,\quad \quad \quad \quad \quad x \leq 2$, then
Differential coefficient of $\sec(\tan^{-1}\text{x})$ is:
→Let $A=\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right]$. If the sum of the diagonal elements of $\mathrm{A}^{13}$ is $3^{\mathrm{n}}$, then $\mathrm{n}$ is equal to ..........
→$\int(3\text{x}^2-1)\text{dx} :$
→If $A$ and $B$ are matrices of order $3 \times 2$ and $C$ is of order $2 \times 3$, then which of the following matrices is not defined :
→