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JEE Mathematics / Practice

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IIT-JEE 1982Question 376

$mn$ squares of equal size are arranged to form a rectangle of dimension $m$ by $n$, where $m$ and $n$ are natural numbers. Two squares will be called 'neighbours' if they have exactly one common side. A natural number is written in each square such that the number written in any square is the arithmetic mean of the numbers written in its neighbouring squares. Show that this is possible only if all the numbers used are equal.

Permutation and Combination

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IIT-JEE 1999Question 377

For complex numbers $z$ and $w$, prove that $|z|^{2}w-|w|^{2}z=z-w$ if and only if $z=w$ or $z\bar w=1$.

Complex Numbers

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IIT-JEE 1989Question 378

If $f$ and $g$ are continuous functions on $[0,a]$ satisfying $f(x)=f(a-x)$ and $g(x)+g(a-x)=2$, then show that $\displaystyle\int_0^{a}f(x)g(x)\,dx=\int_0^{a}f(x)\,dx$.

Definite Integration

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IIT-JEE 2000Question 379

Let $ABC$ and $PQR$ be any two triangles in the same plane. Assume that the perpendiculars from the points $A,B,C$ to the sides $QR,RP,PQ$ respectively are concurrent. Using vector methods or otherwise, prove that the perpendiculars from $P,Q,R$ to $BC,CA,AB$ respectively are also concurrent.

Vector Algebra

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IIT-JEE 1998Question 380

Prove, by vector methods or otherwise, that the point of intersection of the diagonals of a trapezium lies on the line passing through the mid-points of the parallel sides. (You may assume that the trapezium is not a parallelogram.)

Vector Algebra

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