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IIT-JEE 2000Question 346

For every positive integer $n$, prove that $$\sqrt{4n+1}<\sqrt{n}+\sqrt{n+1}<\sqrt{4n+2}.$$ Hence or otherwise, prove that $$\left[\sqrt{n}+\sqrt{n+1}\right]=\left[\sqrt{4n+1}\right],$$ where $[x]$ denotes the greatest integer not exceeding $x$.

Sequence and Series · Miscellaneous

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IIT-JEE 2002Question 347

Let a complex number $\alpha$, $\alpha\neq1$, be a root of the equation $z^{p+q}-z^{p}-z^{q}+1=0$, where $p,q$ are distinct primes. Show that either $1+\alpha+\alpha^{2}+\cdots+\alpha^{p-1}=0$ or $1+\alpha+\alpha^{2}+\cdots+\alpha^{q-1}=0$, but not both together.

Complex Numbers

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IIT-JEE 2000Question 348

Let $a,b,c$ be positive real numbers such that $b^{2}-4ac>0$ and let $\alpha_1=c$. Prove by induction that $$\alpha_{n+1}=\frac{a\alpha_n^{2}}{b^{2}-2a\left(\alpha_1+\alpha_2+\cdots+\alpha_n\right)}$$ is well-defined and $\alpha_{n+1}<\dfrac{\alpha_n}{2}$ for all $n=1,2,\ldots$. (Here, 'well-defined' means that the denominator in the expression for $\alpha_{n+1}$ is not zero.)

Sequence and Series · Logarithmic and Exponential Series, and Recursive Sequences

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IIT-JEE 2000Question 349

A coin has probability $p$ of showing head when tossed. It is tossed $n$ times. Let $p_n$ denote the probability that no two (or more) consecutive heads occur. Prove that $p_1=1$, $p_2=1-p^{2}$ and $$p_n=(1-p)\cdot p_{n-1}+p(1-p)\,p_{n-2}\ \text{ for all } n\ge3.$$ Prove by induction on $n$ that $p_n=A\alpha^{n}+B\beta^{n}$ for all $n\ge1$, where $\alpha$ and $\beta$ are the roots of the quadratic equation $x^{2}-(1-p)x-p(1-p)=0$ and $A=\dfrac{p^{2}+\beta-1}{\alpha\beta-\alpha^{2}}$, $B=\dfrac{p^{2}+\alpha-1}{\alpha\beta-\beta^{2}}$.

Probability

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IIT-JEE 2000Question 350

In any triangle $ABC$, prove that $$\cot\frac{A}{2}+\cot\frac{B}{2}+\cot\frac{C}{2}=\cot\frac{A}{2}\cot\frac{B}{2}\cot\frac{C}{2}.$$

Trigonometric Ratio & Identities

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