Yukti Vault

JEE Mathematics / Practice

Question Bank

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IIT-JEE 1998Question 391

$C_1$ and $C_2$ are two concentric circles, the radius of $C_2$ being twice that of $C_1$. From a point $P$ on $C_2$, tangents $PA$ and $PB$ are drawn to $C_1$. Prove that the centroid of the triangle $PAB$ lies on $C_1$.

Circle

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IIT-JEE 1997Question 392

Let $u(x)$ and $v(x)$ satisfy the differential equation $\dfrac{du}{dx}+p(x)u=f(x)$ and $\dfrac{dv}{dx}+p(x)v=g(x)$, where $p(x)$, $f(x)$ and $g(x)$ are continuous functions. If $u(x_1)>v(x_1)$ for some $x_1$ and $f(x)>g(x)$ for all $x>x_1$, prove that any point $(x,y)$, where $x>x_1$, does not satisfy the equations $y=u(x)$ and $y=v(x)$.

Differential Equation

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IIT-JEE 1994Question 393

Show that $\displaystyle\int_{0}^{n\pi+v}|\sin x|\,dx=2n+1-\cos v$ where $n$ is a positive integer and $0\le v<\pi$.

Definite Integration

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IIT-JEE 1994Question 394

Consider the following statements connecting a triangle $ABC$: (i) The sides $a,b,c$ and area $\Delta$ are rational. (ii) $a,\tan\dfrac{B}{2},\tan\dfrac{C}{2}$ are rational. (iii) $a,\sin A,\sin B,\sin C$ are rational. Prove that $(i)\Rightarrow(ii)\Rightarrow(iii)\Rightarrow(i)$.

Trigonometric Ratio & Identities

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IIT-JEE 1987Question 395

If $A,B,C,D$ are any four points in space, prove that $\left|\overrightarrow{AB}\times\overrightarrow{CD}+\overrightarrow{BC}\times\overrightarrow{AD}+\overrightarrow{CA}\times\overrightarrow{BD}\right|=4\,(\text{area of triangle }ABC)$.

Vector Algebra

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