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JEE Mathematics / Practice

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IIT-JEE 1999Question 341

Let $ABC$ be a triangle having $O$ and $I$ as its circumcenter and incentre respectively. If $R$ and $r$ are the circumradius and the inradius respectively, then prove that $(IO)^{2}=R^{2}-2Rr$. Further show that the triangle $BIO$ is a right-angled triangle if and only if $b$ is the arithmetic mean of $a$ and $c$.

Trigonometric Ratio & Identities

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IIT-JEE 2001Question 342

If $\Delta$ is the area of a triangle with side lengths $a,b,c$, then show that $\Delta\le\dfrac14\sqrt{(a+b+c)abc}$. Also show that the equality occurs in the above inequality if and only if $a=b=c$.

Trigonometric Ratio & Identities

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IIT-JEE 1999Question 343

Let $\vec u$ and $\vec v$ be unit vectors. If $\vec w$ is a vector such that $\vec w+(\vec w\times\vec u)=\vec v$, then prove that $|(\vec u\times\vec v)\cdot\vec w|\le1/2$ and that the equality holds if and only if $\vec u$ is perpendicular to $\vec v$.

Vector Algebra

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IIT-JEE 2000Question 344

Let $ABC$ be a triangle with incentre $I$ and inradius $r$. Let $D,E,F$ be the feet of the perpendiculars from $I$ to the sides $BC,CA$ and $AB$ respectively. If $r_1,r_2$ and $r_3$ are the radii of circles inscribed in the quadrilaterals $AFIE$, $BDIF$ and $CEID$ respectively, prove that $$\frac{r_1}{r-r_1}+\frac{r_2}{r-r_2}+\frac{r_3}{r-r_3}=\frac{r_1r_2r_3}{(r-r_1)(r-r_2)(r-r_3)}.$$

Trigonometric Ratio & Identities

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IIT-JEE 2000Question 345

If $\alpha,\beta$ are the roots of $ax^{2}+bx+c=0$, $(a\neq0)$ and $\alpha+\delta,\ \beta+\delta$ are the roots of $Ax^{2}+Bx+C=0$, $(A\neq0)$ for some constant $\delta$, then prove that $$\frac{b^{2}-4ac}{a^{2}}=\frac{B^{2}-4AC}{A^{2}}.$$

Quadratic Equation · Relation Between Roots and Coefficients

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