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JEE Mathematics / Practice

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Showing 351-355 of 408 questions.

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IIT-JEE 2000Question 351

Let $ABC$ be an equilateral triangle inscribed in the circle $x^{2}+y^{2}=a^{2}$. Suppose perpendiculars from $A,B,C$ to the major axis of the ellipse $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$, $(a>b)$ meet the ellipse respectively at $P,Q,R$ so that $P,Q,R$ lie on the same side of the major axis as $A,B,C$ respectively. Prove that the normals to the ellipse drawn at the points $P,Q$ and $R$ are concurrent.

Ellipse

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IIT-JEE 2002Question 352

Use mathematical induction to show that $(25)^{n+1}-24n+5735$ is divisible by $(24)^{2}$ for all $n=1,2,\ldots$

Mathematical Reasoning

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IIT-JEE 2002Question 353

Prove that $\cos\tan^{-1}\sin\cot^{-1}x=\sqrt{\dfrac{x^{2}+1}{x^{2}+2}}$.

Inverse Trigonometric Functions

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IIT-JEE 2003Question 354

If $z_1$ and $z_2$ are two complex numbers such that $|z_1|<1<|z_2|$ then prove that $\left|\dfrac{1-z_1\overline{z_2}}{z_1-z_2}\right|<1$.

Complex Numbers

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IIT-JEE 2002Question 355

Let $a,b$ be positive real numbers. If $a,A_1,A_2,b$ are in arithmetic progression, $a,G_1,G_2,b$ are in geometric progression and $a,H_1,H_2,b$ are in harmonic progression, show that $$\frac{G_1G_2}{H_1H_2}=\frac{A_1+A_2}{H_1+H_2}=\frac{(2a+b)(a+2b)}{9ab}.$$

Sequence and Series · Means and Inequalities (AM, GM, HM)

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