Algebra PYQ — Page 80

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

The coefficient of $x^n$ in expansion of $(1+x)(1-x)^n$ is

Let $R=\{(1,3),(4,2),(2,4),(2,3),(3,1)\}$ be a relation on the set $A=\{1,2,3,4\}$. The relation $R$ is

If $z=x-i y$ and $z^{\frac{1}{3}}=p+i q$, then $\frac{\left(\frac{x}{p}+\frac{y}{q}\right)}{\left(p^2+q^2\right)}$ is equal to

Let $A=\left(\begin{array}{ccc}1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1\end{array}\right)(10) B=\left(\begin{array}{ccc}4 & 2 & 2 \\ -5 & 0 & \alpha \\ 1 & -2 & 3\end{array}\right)$. If $B$

Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation

Let $A=\left(\begin{array}{ccc}0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0\end{array}\right)$ The only correct statement about the matrix $A$ is

If $f: R \rightarrow S$, defined by $f(x)=\sin x-\sqrt{3} \cos x+1$, is onto, then the interval of $S$ is

The graph of the function $y=f(x)$ is symmetrical about the line $x=2$, then

The coefficient of the middle term in the binomial expansion in powers of $x$ of $(1+\alpha x)^4$ and of $(1-\alpha x)^6$ is the same if $\alpha$ equals

The range of the function $f(x)={ }^{7-x} P_{x-3}$ is

If $u=\sqrt{a^2 \cos ^2 \theta+b^2 \sin ^2 \theta}+\sqrt{a^2 \sin ^2 \theta+b^2 \cos ^2 \theta}$, then the difference between the maximum and minimum values of $u^2$ is given by

If one root of the equation $x^2+p x+12=0$ is 4 , while the equation $x^2+p x+q=0$ has equal roots, then the value of ' $q$ ' is

If $(1-p)$ is a root of quadratic equation $x^2+p x+(1-p)=0$, then its roots are

If $\left|z^2-1\right|=|z|^2+1$, then $z$ lies on

The domain of the function $f(x)=\frac{\sin ^{-1}(x-3)}{\sqrt{9-x^2}}$ is

Let $T_r$ be the rth term of an A.P. whose first term is a and common difference is $d$. If for some positive integers $m, n, m \neq n, T_m=\frac{1}{n}$ and $T_n=\frac{1}{m}$, then $a-d$ equals

The sum of the first $n$ terms of the series $1^2+2 \cdot 2^2+3^2+2 \cdot 4^2+5^2+2 \cdot 6^2+\ldots$ is $\frac{n(n+1)^2}{2}$ when $n$ is even. When $n$ is odd the sum is

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

How many ways are there to arrange the letters in the word GARDEN with the vowels in alphabetical order?

Let $z$, $w$ be complex numbers such that $\bar{z}+i \bar{w}=0$ and $\arg z w=\pi$. Then arg $z$ equals

The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty 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

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