IIT-JEE 1978Question 371
Show that the square of $\dfrac{\sqrt{26-15\sqrt3}}{5\sqrt2-\sqrt{38+5\sqrt3}}$ is a rational number.
Quadratic Equation · Miscellaneous
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JEE Mathematics / Practice
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IIT-JEE 1984Question 372
Given a function $f(x)$ such that (i) it is integrable over every interval on the real line and (ii) $f(t+x)=f(x)$, for every $x$ and a real $t$, then show that the integral $\displaystyle\int_a^{a+t}f(x)\,dx$ is independent of $a$.
Definite Integration
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IIT-JEE 1981Question 373
Show that $\displaystyle\lim_{n\to\infty}\left(\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{6n}\right)=\log6$.
Definite Integration
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IIT-JEE 1978Question 374
From a point $O$ inside a triangle $ABC$, perpendiculars $OD$, $OE$, $OF$ are drawn to the sides $BC$, $CA$, $AB$ respectively. Prove that the perpendiculars from $A$, $B$, $C$ to the sides $EF$, $FD$, $DE$ are concurrent.
Points and Straight Lines
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IIT-JEE 1984Question 375
If $\alpha$ be a repeated root of a quadratic equation $f(x)=0$ and $A(x)$, $B(x)$ and $C(x)$ be polynomials of degree $3$, $4$ and $5$ respectively, then show that $\begin{vmatrix}A(x)&B(x)&C(x)\\A(\alpha)&B(\alpha)&C(\alpha)\\A'(\alpha)&B'(\alpha)&C'(\alpha)\end{vmatrix}$ is divisible by $f(x)$, where prime denotes the derivatives.
Determinant
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