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Yukti Vault

JEE Mathematics / Practice

Question Bank

Showing 366-370 of 408 questions.

IIT-JEE 1982Question 366
$A_1,A_2,\ldots,A_n$ are the vertices of a regular plane polygon with $n$ sides and $O$ is its centre. Show that $$\sum_{i=1}^{n-1}\left(\overrightarrow{OA_i}\times\overrightarrow{OA_{i+1}}\right)=(1-n)\left(\overrightarrow{OA_2}\times\overrightarrow{OA_1}\right)$$
Vector Algebra

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IIT-JEE 1982Question 367
Show that $\displaystyle\int_{0}^{\pi}x\,f(\sin x)\,dx=\frac{\pi}{2}\int_{0}^{\pi}f(\sin x)\,dx$.
Definite Integration

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IIT-JEE 1982Question 368
If $f(x)$ and $g(x)$ are differentiable functions for $0\le x\le1$ such that $f(0)=2$, $g(0)=0$, $f(1)=6$, $g(1)=2$, then show that there exists $c$ satisfying $0<c<1$ and $f'(c)=2g'(c)$.
Differentiation and its Application

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JEE Advanced 2009 Paper 1Question 369
Match the conics in Column I with the statements/expressions in Column II. $$\begin{array}{ll}\textbf{Column I}&\textbf{Column II}\\ (A)\ \text{Circle}&(p)\ \text{The locus of the point }(h,k)\text{ for which the line }hx+ky=1\text{ touches the circle }x^2+y^2=4\\ (B)\ \text{Parabola}&(q)\ \text{Points }z\text{ in the complex plane satisfying }|z+2|-|z-2|=\pm3\\ (C)\ \text{Ellipse}&(r)\ \text{Points of the conic have parametric representation }x=\sqrt3\left(\frac{1-t^2}{1+t^2}\right),\ y=\frac{2t}{1+t^2}\\ (D)\ \text{Hyperbola}&(s)\ \text{The eccentricity of the conic lies in the interval }1\le x<\infty\\ &(t)\ \text{Points }z\text{ in the complex plane satisfying }\mathrm{Re}(z+1)^2=|z|^2+1\end{array}$$
Circle

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IIT-JEE 1982Question 370
For any real $t$, $x=\dfrac{e^{t}+e^{-t}}{2}$, $y=\dfrac{e^{t}-e^{-t}}{2}$ is a point on the hyperbola $x^{2}-y^{2}=1$. Show that the area bounded by this hyperbola and the lines joining its centre to the points corresponding to $t_1$ and $-t_1$ is $t_1$.
Application of Integrals

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