JEE Main 2005 — Algebra

A real valued function $f(x)$ satisfies the functional equation $f(x-y)=f(x) f(y)-f(a-x)$ $f(a+y)$ where $a$ is a given constant and $f(0)=1, f(2 a-x)$ is equal to

If $z_1$ and $z_2$ are two non-zero complex numbers such that $\left|z_1+z_2\right|=\left|z_1\right|+\left|z_2\right|$ then $\arg z_1-\arg z_2$ is equal to

If $a^2+b^2+c^2=-2$ and $f(x)=\left|\begin{array}{ccc}1+a^2 x & \left(1+b^2\right) x & \left(1+c^2\right) x \\ \left(1+a^2\right) x & 1+b^2 x & \left(1+c^2\right) x \\ \left(1+a^2\right) x & \left(1+b^2\right) x & 1+c^2 x\end{array}\right|$ then $f(x)$ is $a$ polynomial of degree

If $\omega=\frac{z}{z-\frac{1}{3} i}$ and $|\omega|=1$, then $z$ lies on

If $a_1, a_2, a_3, \ldots, a_n, \ldots$ are in G.P., then the determinant $\Delta=\left|\begin{array}{lll}\log a_n & \log a_{n+1} & \log a_{n+2} \\ \log a_{n+3} & \log a_{n+4} & \log a_{n+5} \\ \log a_{n+6} & \log a_{n+7} & \log a_{n+8}\end{array}\right|$ is equal to

If both the roots of the quadratic equation $x^2-2 k x+k^2+k-5=0$ are less than 5 , then $\mathrm{k}$ lies in the interval

If in a triangle $\mathrm{ABC}$, the altitudes from the vertices $\mathrm{A}, \mathrm{B}, \mathrm{C}$ on opposite sides are in H.P., then $\sin A, \sin B, \sin C$ are in

If non-zero numbers $a, b, c$ are in H.P., then the straight line $\frac{x}{a}+\frac{y}{b}+\frac{1}{c}=0$ always passes through a fixed point. That point is

If roots of the equation $x^2-b x+c=0$ be two consectutive integers, then $b^2-4 c$ equals

If the coefficient of $x^7$ in $\left[a x^2+\left(\frac{1}{b x}\right)\right]^{11}$ equals the coefficient of $x^{-7}$ in $\left[a x^2-\left(\frac{1}{b x}\right)\right]^{11}$, then $a$ and $b$ satisfy the relation

If the coefficients of $r$ th, $(r+1)$ th and $(r+2)$ th terms in the binomial expansion of $(1+$ $y)^m$ are in A.P., then $m$ and $r$ satisfy the equation

If the cube roots of unity are $1, \omega, \omega^2$ then the roots of the equation $(x-1)^3+8=0$, are

If the letters of word SACHIN are arranged in all possible ways and these words are written out as in dictionary, then the word SACHIN appears at serial number

If $A^2-A+I=0$, then the inverse of $A$ is

If $x=\sum_{n=0}^{\infty} a^n, y=\sum_{n=0}^{\infty} b^n, z=\sum_{n=0}^{\infty} c^n$ where $a, b, c$ are in A.P. and $|a| < 1,|b| < 1,|c| < 1$, then $x, y, z$ are in

Let $f:(-1,1) \rightarrow B$, be a function defined by $f(x)=\tan ^{-1} \frac{2 x}{1-x^2}$, then $f$ is both one-one and onto when $B$ is the interval

Let $R=\{(3,3),(6,6),(9,9),(12,12),(6,12),(3,9),(3,12),(3,6)\}$ be a relation on the set $A=\{3,6,9,12\}$ . The relation is

The system of equations $$ \begin{aligned} & \alpha x+y+z=\alpha-1 \\ & x+\alpha y+z=\alpha-1 \\ & x+y+\alpha z=\alpha-1 \end{aligned} $$ has no solution, if $\alpha$ is

The value of $\alpha$ for which the sum of the squares of the roots of the equation $x^2-(a-2) x-a-1=0$ assume the least value is