2005 年

$A(2,5)$ and $B(6,-3)$ are two points. If $P$ is a point lying on the straight line $x=y$ such that $AP=PB$, then the coordinates of $P$ are

In the figure, $ABCD$ is a parallelogram. The coordinates of $B$ are

If the equation of the straight line $L$ is $x-2y+3=0$, then the equation of the straight line passing through the point $(2,-1)$ and perpendicular to $L$ is

If the mean of five numbers $15$, $x+4$, $x+1$, $2x-7$ and $x-3$ is $6$, then the mode of the five numbers is

Bag $X$ contains $1$ white ball and $3$ red balls while bag $Y$ contains $3$ yellow balls and $6$ red balls. A ball is randomly drawn from bag $X$ and put into bag $Y$. If a ball is now randomly drawn from bag $Y$, then the probability that the ball drawn is red is

If a fair die is thrown three times, then the probability that the three numbers thrown are all different is

If $n$ is a positive integer, then $\dfrac{1}{1+2\sqrt{n}}-\dfrac{1}{1-2\sqrt{n}}=$

The H.C.F. of $x^2(x+1)(x+2)$ and $x(x+1)^3$ is

If $a$ and $b$ are positive integers, then $\log(a^b b^a)=$

Let $k$ be a positive integer. When $x^{2k+1}+kx+k$ is divided by $x+1$, the remainder is