The area bounded by $y=x+1$,$y=\cos x$ and the $X$-axis is
- A$1 \text{ sq unit}$
- B$\frac{3}{2} \text{ sq unit}$
- C$\frac{1}{4} \text{ sq unit}$
- D$\frac{1}{8} \text{ sq unit}$
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If the area of the region bounded by $y=\cos x$,$y=\sin x$,$x=\frac{\pi}{4}$ and $x=\pi$ is bisected by the line $x=a$,then $\sin \left(a+\frac{\pi}{4}\right)=$
The area bounded by the parabola $y^2=x$ and the line $x+y=2$ in the first quadrant is
Column-$I$ Column-$II$ $(A)$ In a triangle $\triangle XYZ$,let $a, b$ and $c$ be the lengths of the sides opposite to the angles $X, Y$ and $Z$,respectively. If $2(a^2-b^2)=c^2$ and $\lambda=\frac{\sin(X-Y)}{\sin Z}$,then possible values of $n$ for which $\cos(n\pi\lambda)=0$ is (are) $(P)$ $1$ $(B)$ In a triangle $\triangle XYZ$,let $a, b$ and $c$ be the lengths of the sides opposite to the angles $X, Y$ and $Z$,respectively. If $1+\cos 2X-2\cos 2Y=2\sin X\sin Y$,then possible value$(s)$ of $\frac{a}{b}$ is (are) $(Q)$ $2$ $(C)$ In $\mathbb{R}^2$,let $\sqrt{3}\hat{i}+\hat{j}$,$\hat{i}+\sqrt{3}\hat{j}$ and $\beta\hat{i}+(1-\beta)\hat{j}$ be the position vectors of $X, Y$ and $Z$ with respect to the origin $O$,respectively. If the distance of $Z$ from the bisector of the acute angle of $\overline{OX}$ with $\overline{OY}$ is $\frac{3}{\sqrt{2}}$,then possible value$(s)$ of $|\beta|$ is (are) $(R)$ $3$ $(D)$ Suppose that $F(\alpha)$ denotes the area of the region bounded by $x=0, x=2, y^2=4x$ and $y=|\alpha x-1|+|\alpha x-2|+\alpha x$,where $\alpha \in \{0, 1\}$. Then the value$(s)$ of $F(\alpha)+\frac{8}{3}\sqrt{2}$,when $\alpha=0$ and $\alpha=1$,is (are) $(S)$ $5$ $(T)$ $6$
| Column-$I$ | Column-$II$ |
| $(A)$ In a triangle $\triangle XYZ$,let $a, b$ and $c$ be the lengths of the sides opposite to the angles $X, Y$ and $Z$,respectively. If $2(a^2-b^2)=c^2$ and $\lambda=\frac{\sin(X-Y)}{\sin Z}$,then possible values of $n$ for which $\cos(n\pi\lambda)=0$ is (are) | $(P)$ $1$ |
| $(B)$ In a triangle $\triangle XYZ$,let $a, b$ and $c$ be the lengths of the sides opposite to the angles $X, Y$ and $Z$,respectively. If $1+\cos 2X-2\cos 2Y=2\sin X\sin Y$,then possible value$(s)$ of $\frac{a}{b}$ is (are) | $(Q)$ $2$ |
| $(C)$ In $\mathbb{R}^2$,let $\sqrt{3}\hat{i}+\hat{j}$,$\hat{i}+\sqrt{3}\hat{j}$ and $\beta\hat{i}+(1-\beta)\hat{j}$ be the position vectors of $X, Y$ and $Z$ with respect to the origin $O$,respectively. If the distance of $Z$ from the bisector of the acute angle of $\overline{OX}$ with $\overline{OY}$ is $\frac{3}{\sqrt{2}}$,then possible value$(s)$ of $|\beta|$ is (are) | $(R)$ $3$ |
| $(D)$ Suppose that $F(\alpha)$ denotes the area of the region bounded by $x=0, x=2, y^2=4x$ and $y=|\alpha x-1|+|\alpha x-2|+\alpha x$,where $\alpha \in \{0, 1\}$. Then the value$(s)$ of $F(\alpha)+\frac{8}{3}\sqrt{2}$,when $\alpha=0$ and $\alpha=1$,is (are) | $(S)$ $5$ |
| $(T)$ $6$ |
DifficultIIT 2015
View SolutionThe area (in sq. units) of the region $\{(x, y) \in R^{2}: x^{2} \leq y \leq 3-2x\}$ is
DifficultJEE MAIN 2020
View SolutionThe area (in sq. units) bounded by the curve $y=2x-x^2$ and the line $y=-x$ is
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