Algebra PYQ — Page 79

Let $a_1, a_2, a_3, \ldots$ be terms of an A.P. If $\frac{a_1+a_2+\cdots a_p}{a_1+a_2+\cdots+a_q}=\frac{p^2}{q^2}, p \neq q$, then $\frac{a_6}{a_{21}}$ equals

If $z^2+z+1=0$, where $z$ is a complex number, then the value of $$ \left(z+\frac{1}{z}\right)^2+\left(z^2+\frac{1}{z^2}\right)^2+\left(z^3+\frac{1}{z^3}\right)^2+\cdots+\left(z^6+\frac{1}{z^6}\right)^2 $$

If the roots of the quadratic equation $x^2+p x+q=0$ are $\tan 30^{\circ}$ and $\tan 15^{\circ}$, respectively then the value of $2+q-p$ is

At an election, a voter may vote for any number of candidates, not greater than the number to be elected. There are 10 candidates and 4 are of be elected. If a voter votes for at least one candidate, then the number of ways in which he can vote is

If the expansion in powers of $x$ of the function $\frac{1}{(1-a x)(1-b x)}$ is $a_0+a_1 x+a_2 x^2+a_3 x^3+\ldots$, then $a_n$ is

For natural numbers $m, n$ if $(1-y)^m(1+y)^n=1+a_1 y+a_2 y^2+\ldots$, and $a_1=a_2=10$ then $(\mathrm{m}, \mathrm{n})$ is

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

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

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

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

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

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 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 $z_1$ 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 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 $a_1, a_2, a_3, \ldots, a_n, \ldots$ 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

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

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