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JEE Mathematics / Practice

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IIT-JEE 1995Question 386

Let $I_m=\displaystyle\int_0^{\pi}\dfrac{1-\cos mx}{1-\cos x}\,dx$. Use mathematical induction to prove that $I_m=m\pi$, $m=0,1,2,\ldots$

Definite Integration

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IIT-JEE 1997Question 387

If $A$, $B$ and $C$ are vectors such that $|B|=|C|$. Prove that $[(A+B)\times(A+C)]\times(B\times C)\cdot(B+C)=0$.

Vector Algebra

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IIT-JEE 1997Question 388

Let $C$ be any circle with centre $\left(0,\sqrt2\right)$. Prove that at the most two rational points can lie on $C$. (A rational point is a point both of whose coordinates are rational numbers.)

Circle

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IIT-JEE 1997Question 389

A tangent to the ellipse $x^{2}+4y^{2}=4$ meets the ellipse $x^{2}+2y^{2}=6$ at $P$ and $Q$. Prove that the tangents at $P$ and $Q$ of the ellipse $x^{2}+2y^{2}=6$ are at right angles.

Ellipse

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IIT-JEE 1996Question 390

Let $A_n$ be the area bounded by the curve $y=(\tan x)^{n}$ and the lines $x=0$, $y=0$, and $x=\dfrac{\pi}{4}$. Prove that for $n>2$, $A_n+A_{n-2}=\dfrac{1}{n-1}$ and deduce $\dfrac{1}{2n+2}<A_n<\dfrac{1}{2n-2}$.

Application of Integrals

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