JEE Main 2007 — Algebra

If $p$ and $q$ are positive real numbers such that $p^2+q^2=1$, then the maximum value of $(p+q)$ is

If $D=\left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & 1+x & 1 \\ 1 & 1 & 1+y\end{array}\right|$ for $x \neq 0, y \neq 0$ then $D$ is

If the difference between the roots of the equation $x^2+a x+1=0$ is less than $\sqrt{5}$, then the set of possible values of $a$ is

If $|z+4| \leq 3$, then the maximum value of $|z+1|$ is

In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of this progression equals

In the binomial expansion of $(a-b)^n, n \geq 5$, the sum of $5^{\text {th }}$ and $6^{\text {th }}$ terms is zero, then $\frac{a}{b}$ equals

Let $A=\left[\begin{array}{ccc}5 & 5 \alpha & \alpha \\ 0 & \alpha & 5 \alpha \\ 0 & 0 & 5\end{array}\right]$. If $\left|A^2\right|=25$, then $|\alpha|$ equals

The largest interval lying in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ for which the function $\left[f(x)=4^{-x^2}+\cos ^{-1}\left(\frac{x}{2}-1\right)+\log (\cos x)\right]$ is defined, is

The set $S=\{1,2,3, \ldots, 12)$ is to be partitioned into three sets $A, B, C$ of equal size. Thus, $A \cup B \cup C=S, A \cap B=B \cap C=A \cap C=\phi$. The number of ways to partition $S$ is