JEE Main 2004 — Algebra

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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