Column -1 represents equation of straight lines Column -2 represents position of points A -1-4 and B 4-3 w-r-t- equation of line in Column - 1 Column -3 represents the sum of length of perpendiculars from points A and B to the lines in Column-1 beginarray -1- hline Column 1 - Column 2 - Column 3NllhlineI L - 1- y - x -1-0 - i Both points lie on the same side of the line - p - frac 7 sqrt 55- unitsllthlineIII L3- y -2 x -2-0 - iii Both points lie on the line - R - frac 25 sqrt 3737 unitslllhlineend array Which of the following options is the only INCORRECT combination-A- III ii PB- II i PC- I iv QD- IV iii S

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Algebra of Derivatives

Column -1 rep...

Question

Column – 1 represents equation of straight lines
Column – 2 represents position of points A $\equiv$ (1, 4) and B(4, 3) w.r.t. equation of line in Column - 1
Column – 3 represents the sum of length of perpendiculars from points A and B to the lines in Column-1
$\begin{matrix} C o l u m n 1 & C o l u m n 2 & C o l u m n 3 (I) L_{1} : y - x + 1 = 0 & (i) B o t h p o i n t s l i e o n t h e s a m e s i d e o f t h e l i n e & (p) \frac{7 \sqrt{5}}{5} u n i t s (I I) L_{2} : 6 y + x - 11 = 0 & (i i) B o t h p o i n t s l i e o n t h e o p p o s i t e s i d e o f t h e l i n e & (Q) 2 \sqrt{2} u n i t s (I I I) L_{3} : y - 2 x + 2 = 0 & (i i i) B o t h p o i n t s l i e o n t h e l i n e & (R) \frac{25 \sqrt{37}}{37} u n i t s (I V) L_{4} : 3 y + x - 13 = 0 & (i v) O n e p o i n t (o u t o f A, B) l i e o n t h e l i n e & (S) 0 u n i t s \end{matrix}$
Which of the following options is the only INCORRECT combination?

(III) (ii) (P)

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(II) (i) (P)

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(IV) (iii) (S)

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(I) (iv) (Q)

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Solution

The correct option is B (II) (i) (P)
Using $S_{1} a n d S_{2}$ , we can say that
$L_{1}$ passes through one of the points
$L_{2}$ have both the points on the same side
$L_{3}$ have both the points on the opposite side
$L_{4}$ have both the points on the line
For perpendicular distance, use formula $∣ ∣ ∣ \frac{a x_{1} + b y_{1} + c}{\sqrt{a^{2} + b^{2}}} ∣ ∣ ∣$
$\begin{matrix} P e r p e n d i c u l a r & d i s t a n c e f r o m L i n e & P o i n t A (1, 4) & P o i n t B (4, 3) & S u m o f d i s t a n c e L_{1} : y - x + 1 = 0 & 2 \sqrt{2} & 0 & 2 \sqrt{2} L_{2} : 6 y + x - 11 = 0 & \frac{14}{\sqrt{37}} & \frac{11}{\sqrt{37}} & \frac{25 \sqrt{27}}{37} L_{3} : y - 2 x + 2 = 0 & \frac{4}{\sqrt{5}} & \frac{3}{\sqrt{5}} & \frac{7 \sqrt{5}}{5} L_{4} : 3 y + x - 13 = 0 & 0 & 0 & 0 \end{matrix}$
Now, Match the options

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Q. Column – 1 represents equation of straight lines
Column – 2 represents position of points A

\equiv

(1, 4) and B(4, 3) w.r.t. equation of line in Column - 1
Column – 3 represents the sum of length of perpendiculars from points A and B to the lines in Column-1

\begin{matrix} C o l u m n 1 & C o l u m n 2 & C o l u m n 3 (I) L_{1} : y - x + 1 = 0 & (i) B o t h p o i n t s l i e o n t h e s a m e s i d e o f t h e l i n e & (p) \frac{7 \sqrt{5}}{5} u n i t s (I I) L_{2} : 6 y + x - 11 = 0 & (i i) B o t h p o i n t s l i e o n t h e o p p o s i t e s i d e o f t h e l i n e & (Q) 2 \sqrt{2} u n i t s (I I I) L_{3} : y - 2 x + 2 = 0 & (i i i) B o t h p o i n t s l i e o n t h e l i n e & (R) \frac{25 \sqrt{37}}{37} u n i t s (I V) L_{4} : 3 y + x - 13 = 0 & (i v) O n e p o i n t (o u t o f A, B) l i e o n t h e l i n e & (S) 0 u n i t s \end{matrix}

Which of the following options is the only correct combination?

Column – 1 represents equation of straight lines
Column – 2 represents position of points A $\equiv$ (1, 4) and B(4, 3) w.r.t. equation of line in Column - 1
Column – 3 represents the sum of length of perpendiculars from points A and B to the lines in Column-1
$\begin{matrix} C o l u m n 1 & C o l u m n 2 & C o l u m n 3 (I) L_{1} : y - x + 1 = 0 & (i) B o t h p o i n t s l i e o n t h e s a m e s i d e o f t h e l i n e & (p) \frac{7 \sqrt{5}}{5} u n i t s (I I) L_{2} : 6 y + x - 11 = 0 & (i i) B o t h p o i n t s l i e o n t h e o p p o s i t e s i d e o f t h e l i n e & (Q) 2 \sqrt{2} u n i t s (I I I) L_{3} : y - 2 x + 2 = 0 & (i i i) B o t h p o i n t s l i e o n t h e l i n e & (R) \frac{25 \sqrt{37}}{37} u n i t s (I V) L_{4} : 3 y + x - 13 = 0 & (i v) O n e p o i n t (o u t o f A, B) l i e o n t h e l i n e & (S) 0 u n i t s \end{matrix}$
Which of the following options is the only CORRECT combination?

Column – 1 : represent different words
Column – 2 : represent number ways of selecting five letters from the word in column – 1
Column – 3 : represent total number of possible words with or without meaning, using all the alphabets of word in
column - 1 such that all the vowels are together.
$\begin{matrix} C o l u m n 1 & C o l u m n 2 & C o l u m n 3 (I) I N D E P E N D E N T & (i) 41 & (p) {24.}^{8} C_{6} .^{6} C_{3} .^{3} C_{2} .^{3} C_{2} (I I) I N S T I T U T E & (i i) 60 & (Q) {24.}^{8} C_{5} .^{5} C_{3} (I I I) C U R R I C U L U M & (i i i) 72 & (R) {12.}^{7} C_{3} .^{4} C_{3} .^{3} C_{2} (I V) M A T H E M A T I C S & (i v) 179 & (S) {24.}^{6} C_{3} .^{3} C_{2} \end{matrix}$
Which of the following options is the only INCORRECT combination?

$\begin{matrix} C o l u m n I & C o l u m n 2 & C o l u m n 3 (I) & f (x) = t a n (2 t a n^{- 1} \sqrt{\frac{\sqrt{1 + \sqrt{x}} - 1}{\sqrt{1 + \sqrt{x}} + 1}}) & (i) & \int f (x) d x = \frac{4}{3} x^{\frac{3}{4}} + C & (P) & \int_{0}^{1} f (x) d x = \frac{4}{3} (I I) & f (x) = c o t (2 t a n^{- 1} \sqrt{\frac{\sqrt{1 + \sqrt{x}} - \sqrt[4]{x}}{\sqrt{1 + \sqrt{x}} + \sqrt[4]{x}}}) & (i i) & \int f (x) d x = \frac{4}{5} x^{\frac{5}{4}} + C & (Q) & \int_{0}^{1} f (x) d x = \frac{4}{5} (I I I) & f (x) ⎛ ⎜ ⎝ \frac{1 - t a n (\frac{1}{2} s i n^{- 1} (\frac{1 - \sqrt{x}}{1 + \sqrt{x}}))}{1 + t a n (\frac{1}{2} s i n^{- 1} (\frac{1 - \sqrt{x}}{1 + \sqrt{x}}))} ⎞ ⎟ ⎠ & (i i i) & \int f (x) d x = \frac{2}{3} x^{\frac{3}{4}} + C & (R) & \int_{0}^{1} f (x) d x = \frac{2}{3} (I V) & f (x) = \sqrt{x} t a n (2 t a n^{- 1} (\frac{\sqrt{\sqrt{1 + \sqrt{x} + 1}} - \sqrt{\sqrt{1 + \sqrt{x} - 1}}}{\sqrt{\sqrt{1 + \sqrt{x} + 1}} + \sqrt{\sqrt{1 + \sqrt{x} - 1}}})) & (i v) & \int f (x) d x = \frac{2}{5} x^{\frac{5}{4}} + C & (S) & \int_{0}^{1} f (x) d x = \frac{2}{5} \end{matrix}$

Which of the following options is only correct combination?

Q. Type of terms and corresponding number is written in Column 2 and Column 3 respectively of the Binomial in Column 1.

\begin{matrix} C o l u m n I & C o l u m n I I & C o l u m n 3 (I) & {(5^{\frac{1}{6}} + 7^{\frac{1}{9}})}^{1824} & (i) & T o t a l n u m b e r o f r a t i o n a l t e r m s & (P) & 4 (I I) & {(5^{\frac{1}{6}} + 2^{\frac{1}{8}})}^{100} & (i i) & T o t a l n u m b e r o f i r r a t i o n a l t e r m s & (Q) & 102 (I I I) & {(3^{\frac{1}{4}} + 4^{\frac{1}{3}})}^{99} & (i i i) & \frac{1}{2} [\frac{T o t a l n u m b e r o f t e r m s}{N u m b e r o f r a t i o n a l t e r m s}], & (R) & 224 w h e r e [.] r e p r e s e n t s g r e a t e s t i n t e g e r f u n c t i o n . (I V) & {(7^{\frac{1}{3}} + 11^{\frac{1}{9}})}^{2007} & (i v) & T o t a l n u m b e r o f t e r m s & (S) & I s d i v i s i b l e b y 13 \end{matrix}

Select the correct combination

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