The graph of function $f$ contains the point $P (1, 2)$ and $Q (s, r)$ . The equation of the secant line through $P$ and $Q$ is $y = (\frac{s^{2} + 2 s - 3}{s - 1}) x - 1 - s$ . The value of $f^{'} (1)$ is

2

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3

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4

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non existent

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Solution

The correct option is D $4$
Given equation of secant line is
$y = (\frac{s^{2} + 2 s - 3}{s - 1}) x - 1 - s$
Slope of this line is $(\frac{s^{2} + 2 s - 3}{s - 1})$
So, $f^{'} (1) = {lim}_{x \to 1} (\frac{s^{2} + 2 s - 3}{s - 1}) (\frac{0}{0}) = {lim}_{x \to 1} \frac{2 s + 2}{1} = 4$

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Which of the following is only correct combination?

Consider the lines
$L_{1} : \frac{x - 1}{2} = \frac{y}{- 1} = \frac{z + 3}{1}, L_{2} : \frac{x - 4}{1} = \frac{y + 3}{1} = \frac{z + 3}{2},$ and the planes $P_{1} : 7 x + y + 2 z = 3, P_{2} : 3 x + 5 y - 6 z = 4.$ Let $a x + b y + c z = d$ the equation of the plane passing through the point of intersection of lines $L_{1}$ and $L_{2}$ and perpendicular to planes $P_{1}$ and $P_{2}$ .
Match List I with List II and select the correct answer using the code given below the lists.

$\begin{matrix} L i s t I & L i s t I I P & a = & 1 & 13 Q & b = & 2 & - 3 R & c = & 3 & 1 S & d = & 4 & - 2 \end{matrix}$

Matching type questions:

$\begin{matrix} C o l u m n - I & C o l u m n - I I (P) N u m b e r o f p o i n t s o n t h e h y p e r b o l a x y = c^{2} f r o m w h i c h & (1) 1 t w o t a n g e n t s d r a w n t o t h e e l l i p s e \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (w h e r e b < a < c) a r e p e r p e n d i c u l a r t o e a c h o t h e r i s (Q) T h e c o n s t a n t t e r m o f t h e q u a d r a t i c e x p r e s s i o n & (2) 0 \sum_{k = 2}^{n} (x - \frac{1}{k - 1}) (x - \frac{1}{k}), a s n \to \infty (R) f (x) = [t a n x [c o t x]], x ϵ [\frac{π}{12}, \frac{π}{2}] (w h e r e [.] r e p r e s e n t s & (3) 3 t h e g r e a t e s t i n t e g e r f u n c t i o n) t h e n t h e n u m b e r o f p o i n t s, w h e r e f (x) i s n o t c o n t i n u o u s i s (S) E q u a t i o n o f p l a n e t h r o u g h (2, - 1, 0), (3, - 4, 5) a n d & (4) 4 p a r a l l e l t o a l i n e w h o s e d i r e c t i o n c o s i n e p r o p o r t i o n a l t o 2, 3, 4 i s 9 x - 2 y - 3 z = 5 k t h e n k i s \end{matrix}$