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JEE Mathematics / Practice

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IIT-JEE 2003Question 356

Prove that $$2^k\binom{n}{0}\binom{n}{k}-2^{k-1}\binom{n}{1}\binom{n-1}{k-1}+2^{k-2}\binom{n}{2}\binom{n-2}{k-2}-\cdots+(-1)^k\binom{n}{k}\binom{n-k}{0}=\binom{n}{k}$$

Binomial Theorem · Properties and Sums of Binomial Coefficients

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IIT-JEE 2002Question 357

A straight line $L$ through the origin meets the lines $x+y=1$ and $x+y=3$ at $P$ and $Q$ respectively. Through $P$ and $Q$ two straight lines $L_1$ and $L_2$ are drawn, parallel to $2x-y=5$ and $3x+y=5$ respectively. Lines $L_1$ and $L_2$ intersect at $R$. Show that the locus of $R$, as $L$ varies, is a straight line.

Points and Straight Lines

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IIT-JEE 2003Question 358

If $P(1)=0$ and $\dfrac{dP(x)}{dx}>P(x)$ for all $x\ge1$ then prove that $P(x)>0$ for all $x>1$.

Differentiation and its Application

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IIT-JEE 2003Question 359

If $\vec u,\vec v,\vec w$ are three non-coplanar unit vectors and $\alpha,\beta,\gamma$ are the angles between $\vec u$ and $\vec v$, $\vec v$ and $\vec w$, $\vec w$ and $\vec u$ respectively and $\vec x,\vec y,\vec z$ are unit vectors along the bisectors of the angles $\alpha,\beta,\gamma$ respectively. Prove that $$\left[\vec x\times\vec y\ \ \vec y\times\vec z\ \ \vec z\times\vec x\right]=\frac1{16}\left[\vec u\ \vec v\ \vec w\right]^2\sec^2\frac\alpha2\sec^2\frac\beta2\sec^2\frac\gamma2.$$

Vector Algebra

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IIT-JEE 2004Question 360

Using Rolle's theorem, prove that there is at least one root in $\left(45^{1/100},\,46\right)$ of the polynomial $P(x)=51x^{101}-2323x^{100}-45x+1035$.

Differentiation and its Application

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