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JEE Mathematics / Practice

Question Bank

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IIT-JEE 1997Question 336

Let $0<A_i<\pi$ for $i=1,2,\ldots,n$. Use mathematical induction to prove that $$\sin A_1+\sin A_2+\cdots+\sin A_n\le n\sin\left(\dfrac{A_1+A_2+\cdots+A_n}{n}\right)$$ where $n\ge1$ is a natural number. {You may use the fact that $p\sin x+(1-p)\sin y\le\sin[px+(1-p)y]$, where $0\le p\le1$ and $0\le x,y\le\pi$.}

Trigonometric Ratio & Identities

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IIT-JEE 1999Question 337

Let $n$ be any positive integer. Prove that $$\sum_{k=0}^{m}\dfrac{\dbinom{2n-k}{k}}{\dbinom{2n-k}{n}}\cdot\dfrac{(2n-4k+1)}{(2n-2k+1)}\,2^{n-2k}=\dfrac{\dbinom{n}{m}}{\dbinom{2n-2m}{n-m}}\,2^{n-2m}$$ for each non-negative integer $m\le n$. $\left(\text{Here }\dbinom{p}{q}={}^{p}C_q\right)$.

Binomial Theorem · Properties and Sums of Binomial Coefficients

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IIT-JEE 1998Question 338

Let $p$ be a prime and $m$ a positive integer. By mathematical induction on $m$, or otherwise, prove that whenever $r$ is an integer such that $p$ does not divide $r$, $p$ divides ${}^{mp}C_r$. [Hint: You may use the fact that $(1+x)^{(m+1)p}=(1+x)^{p}(1+x)^{mp}$]

Permutation and Combination

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IIT-JEE 1998Question 339

The angle between a pair of tangents drawn from a point $P$ to the parabola $y^{2}=4ax$ is $45^\circ$. Show that the locus of the point $P$ is a hyperbola.

Parabola

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IIT-JEE 2000Question 340

The fourth power of the common difference of an arithmetic progression with integer entries is added to the product of any four consecutive terms of it. Prove that the resulting sum is the square of an integer.

Sequence and Series · Arithmetic Progression

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