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JEE Mathematics / Practice

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IIT-JEE 1999Question 396

Let $a,b,c,d$ be real numbers in G.P. If $u,v,w$ satisfy the system of equations $u+2v+3w=6$ $4u+5v+6w=12$ $6u+9v=4$ then show that the roots of the equation $\left(\dfrac1u+\dfrac1v+\dfrac1w\right)x^{2}+\big[(b-c)^{2}+(c-a)^{2}+(d-b)^{2}\big]x+u+v+w=0$ and $20x^{2}+10(a-d)^{2}x-9=0$ are reciprocals of each other.

Sequence and Series · Geometric Progression

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IIT-JEE 2003Question 397

Prove that there exists no complex number $z$ such that $|z|<\dfrac13$ and $\displaystyle\sum_{r=1}^{n}a_rz^r=1$ where $|a_r|<2$.

Complex Numbers

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IIT-JEE 2002Question 398

Let $V$ be the volume of the parallelopiped formed by the vectors $\vec a=a_1\hat i+a_2\hat j+a_3\hat k$, $\vec b=b_1\hat i+b_2\hat j+b_3\hat k$, $\vec c=c_1\hat i+c_2\hat j+c_3\hat k$, where $r=1,2,3$, are non-negative real numbers and $\displaystyle\sum_{r=1}^{3}(a_r+b_r+c_r)=3L$, show that $V\le L^{3}$.

Vector Algebra

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IIT-JEE 2001Question 399

Let $\vec A(t)=f_1(t)\hat i+f_2(t)\hat j$ and $\vec B(t)=g_1(t)\hat i+g_2(t)\hat j$, $t\in[0,1]$, where $f_1,f_2,g_1,g_2$ are continuous functions. If $\vec A(t)$ and $\vec B(t)$ are nonzero vectors for all $t$ and $\vec A(0)=2\hat i+3\hat j$, $\vec A(1)=6\hat i+2\hat j$, $\vec B(0)=3\hat i+2\hat j$ and $\vec B(1)=2\hat i+6\hat j$. Then show that $\vec A(t)$ and $\vec B(t)$ are parallel for some $t$.

Limits & Continuity

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IIT-JEE 2000Question 400

Suppose $p(x)=a_0+a_1x+a_2x^{2}+\cdots+a_nx^{n}$. If $|p(x)|\le\left|e^{x-1}-1\right|$ for all $x\ge0$, prove that $|a_1+2a_2+\cdots+na_n|\le1$.

Differentiation and its Application

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