JEE Main 2003 — Algebra

A function $\mathrm{f}$ from the set of natural numbers to integers defined by $\mathrm{f}(\mathrm{n})=\left\{\begin{array}{l}\frac{\mathrm{n}-1}{2} \text {, when } n \text { isodd } \\ \frac{\mathrm{n}}{2} \text {, when } \mathrm{n} \text { is even }\end{array}\right.$ is

A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is

Domain of definition of the function $f(x)=\frac{3}{4-x^2}+\log _{10}\left(x^3-x\right)$, is

If $\mathrm{x}_1, \mathrm{x}_2, \mathrm{x}_3$ and $\mathrm{y}_1, \mathrm{y}_2, \mathrm{y}_3$ are both in G.P. with the same common ratio, then the points $\left(\mathrm{x}_1, \mathrm{y}_1\right),\left(\mathrm{x}_2, \mathrm{y}_2\right)$ and $\left(\mathrm{x}_3\right.$, $\mathrm{y}_3$ )

If $\mathrm{z}$ and $\omega$ are two non-zero complex numbers such that $|z \omega|=1$ and $\operatorname{Arg}(z)-\operatorname{Arg}(\omega)=\frac{\pi}{2}$, then $\bar{z} \omega$ is equal to

If $A=\left[\begin{array}{ll}a & b \\ b & a\end{array}\right]$ and $A_2=\left[\begin{array}{ll}\alpha & \beta \\ \beta & \alpha\end{array}\right]$, then

If $1, \omega, \omega^2$ are the cube roots of unity, then $\Delta=\left|\begin{array}{ccc}1 & \omega^n & \omega^{2 n} \\ \omega^n & \omega^{2 n} & 1 \\ \omega^{2 n} & 1 & \omega^n\end{array}\right|$ is equal to

If ${ }^n C_r$ denotes the number of combination of $n$ things taken $r$ at a time, then the expression ${ }^n C_{r+1}+{ }^n C_{r-1}+2 x^n C_r$ equals

If $\mathrm{x}$ is positive, the first negative term in the expansion of $(1+x)^{27 / 5}$ is

If $f: R \rightarrow R$ satisfies $f(x+y)=f(x)+f(y)$, for all $x, y \in R$ and $f(1)=7$, then $\sum_{r=1}^n f(r)$ is

If the sum of the roots of the quadratic equation $a x^2+b x+c=0$ is equal to the sum of the squares of their reciprocals, then $\frac{\mathrm{a}}{\mathrm{c}}, \frac{\mathrm{b}}{\mathrm{a}}$ and $\frac{\mathrm{c}}{\mathrm{b}}$ are in

If the system of linear equations $x+2 a y+a z=0$ $x+3 b y+b z=0$ $x+4 c y+c z=0$ has a non-zero solution, then $\mathrm{a}, \mathrm{b}, \mathrm{c}$

If $\left(\frac{1+i}{1-i}\right)^x=1$ then

Let $Z_1$ and $Z_2$ be two roots of the equation $x^2+a Z+b=0$ being complex. Further, assume that the origin, $Z_1$ and $\mathrm{Z}_2$ form an equilateral triangle. Then

Let $R_1$ and $R_2$ respectively be the maximum ranges up and down an inclined plane and $R$ be the maximum range on the horizontal plane. Then $\mathrm{R}_1, \mathrm{R}, \mathrm{R}_2$ are in

Let $f(x)$ be a polynomial function of second degree. If $f(1)=f(-1)$ and $a, b, c$ are in A.P, then $f^{\prime}(a), f^{\prime}(c)$ are in

The function $f(x)=\log \left(x+\sqrt{x^2+1}\right)$, is

The number of integral terms in the expansion of $(\sqrt{3}+8 \sqrt{5})^{256}$ is

The number of real solutions of the equation $x^2-3|x|+2=0$ is

The number of ways in which 6 men and 5 women can dine at a found table if no two women are to sit together is given by

The value of ' $a$ ' for which one root of the quadratic equation $\left(a^2-5 a+3\right) x^3+(3 a-1) x+2=0$ is twice as large as the other is