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JEE Mathematics / Practice

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IIT-JEE 1997Question 381

Let $a+b=4$, where $a<2$, and let $g(x)$ be a differentiable function. If $\dfrac{dg}{dx}>0$ for all $x$, prove that $\displaystyle\int_0^{a}g(x)\,dx+\int_0^{b}g(x)\,dx$ increases as $(b-a)$ increases.

Definite Integration

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IIT-JEE 1990Question 382

Show that $\displaystyle\int_{0}^{\pi/2}f(\sin2x)\sin x\,dx=\sqrt2\int_{0}^{\pi/4}f(\cos2x)\cos x\,dx$

Definite Integration

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IIT-JEE 1991Question 383

If $f$ is a continuous function with $\displaystyle\int_0^{x}f(t)\,dt\to\infty$ as $|x|\to\infty$, then show that every line $y=mx$ intersects the curve $y^{2}+\displaystyle\int_0^{x}f(t)\,dt=2$. [figure: curve through $A(0,\sqrt2)$ and $B(0,-\sqrt2)$ cutting the $x$-axis at $(x_p,0)$]

Definite Integration

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IIT-JEE 1990Question 384

Prove that for any positive integer $k$, $\dfrac{\sin2kx}{\sin x}=2\left[\cos x+\cos3x+\cdots+\cos(2k-1)x\right]$ Hence prove that $\displaystyle\int_{0}^{\pi/2}\sin2kx\,\cot x\,dx=\dfrac{\pi}{2}$

Definite Integration

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IIT-JEE 2000Question 385

For any positive integers $m,n$ (with $n\ge m$), let $\dbinom{n}{m}={}^{n}C_m$. Prove that $$\dbinom{n}{m}+\dbinom{n-1}{m}+\dbinom{n-2}{m}+\cdots+\dbinom{m}{m}=\dbinom{n+1}{m+1}.$$ Hence or otherwise, prove that $$\dbinom{n}{m}+2\dbinom{n-1}{m}+3\dbinom{n-2}{m}+\cdots+(n-m+1)\dbinom{m}{m}=\dbinom{n+2}{m+2}.$$

Binomial Theorem · Properties and Sums of Binomial Coefficients

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