Question

Matching type questions:

$\begin{array}{cc}Column-I& Column-II\\ \left(P\right)Numberofpointsonthehyperbolaxy=c^{2}fromwhich& \left(1\right)1\\ twotangentsdrawntotheellipse\frac{x^2}{a^2}+\frac{y^2}{b^2}=1& \\ (whereb<a<c)areperpendiculartoeachotheris& \\ \left(Q\right)Theconstanttermofthequadraticexpression& \left(2\right)0\\ {\sum }_{k=2}^n(x-\frac{1}{k-1})(x-\frac{1}{k}),asn\to \infty & \\ \left(R\right)f\left(x\right)=\left[tanx\right[cotx\left]\right],xϵ[\frac{\pi }{12},\frac{\pi }{2}]\left(where\right[.]represents& \left(3\right)3\\ thegreatestintegerfunction)thenthenumberofpoints,& \\ wheref\left(x\right)isnotcontinuousis& \\ \left(S\right)Equationofplanethrough(2,-1,0),(3,-4,5)and& \left(4\right)4\\ paralleltoalinewhosedirectioncosineproportionalto2,& \\ 3,4is9x-2y-3z=5kthenkis& \end{array}$

• A. P -2, Q-1. R-3, S-4
• B. P -2, Q-4. R-1, S-3
• C. P -4, Q-1. R-3, S-2
• D. P -4, Q-3. R-1, S-2

Answer 1

The correct option is B

P -2, Q-4. R-1, S-3

Required number of points will be equal to number of real points of intersection of the given hyperbola and the director circle of the given ellipse.
As director circle is $x^2+y^2=a^2+b^2<2c^2$ which does not intersect the hyperbola xy = c^2.
which does not intersect the hyperbola $xy=c^2.$
(B) constant term =${\sum }_{k=2}^n\frac{1}{k(k-1)}={\sum }_{k=2}^n(\frac{1}{k-1}-\frac{1}{k})=-1-\frac{1}{n}$
As $n\to \infty$
constant term $\to 1$
$\left[cotx\right]=cotxifcotxϵI$
$<cotxifcotx\text{/}ϵI.\Rightarrow \left[tanx\right[cotx\left]\right]=1ifcotxϵI=0ifcotx\text{/}ϵI$
so points of discontinuity are those where $cotxϵI$
As $\frac{\pi }{12}\le x<\frac{\pi }{2}$
$0<cotx<2+\sqrt{3}$
$\therefore$ number of points of discontinuity = 3.
(D) equation of plane through (2, -1, 0) is
a(x - 2) + b(y + 1) + c(z - 0) = 0 \\
As it passes through (3, -4, 5)
then a - 3b + 5c = 0 \\
Also 2a + 3b + c = 0
$\therefore \frac{a}{-27}=\frac{b}{6}=\frac{c}{9}$
equation of plane
$9(x-2)-2(y+1)-3z=0$
or, $9x-2y-3z=20$

$\therefore k=4.$