Algebra PYQ — Page 81

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 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

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

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 $\mathrm{a}, \mathrm{b}, \mathrm{c}$

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

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

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 $1, \omega, \omega^2$ 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

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

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 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

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

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

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

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 $\mathrm{z}$ 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

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

Difference between the corresponding roots of $x^2+a x+b=0$ and $x^2+b x+a=0$ is same and $a \neq b$, then

Five digit number divisible by 3 is formed using 0, 1, 2, 3, 4, 6 and 7 without repetition. Total number of such numbers are

If $1, \log _9\left(3^{1-x}+2\right), \log _3\left(4.3^x-1\right)$ are in A.P. then $x$ equals

The domain of $\sin ^{-1}\left[\log _3(x / 3)\right]$ is

$r$ and $n$ are positive integers $r>1, n>2$ and coefficient of $(r+2)^{\text {th }}$ term and $3 r^{\text {th }}$ term in the expansion of $(1+x)^{2 n}$ are equal, then $n$ equals