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?

(I) (ii) (P)

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(II) (i) (Q)

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(I) (iv) (Q)

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(IV) (i) (S)

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Solution

The correct option is C (I) (iv) (Q)
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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Similar questions

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?

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 INCORRECT combination?

Q. Column - 1: Trigonometric expression
Column - 2: Dependency on '

α

' and '

β

'
Column - 3: Equivalent value of Trigonometric expression 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) & {cos}^{2} (α + β) - {sin}^{2} (α - β) - cos (2 α) cos (2 β) + 1 & (i) & Independent of α only & (P) & 1 (I I) & {cos}^{2} α + {cos}^{2} (α + β) - 2 cos (α) cos (β) cos (α + β) & (i i) & Independent of β only & (Q) & {sin}^{2} β (I I I) & \frac{sin (α - β) sin (α + β)}{1 - {tan}^{2} α {cot}^{2} β} & (i i i) & Independent of α and β & (R) & - {cos}^{2} α {sin}^{2} β (I V) & 2 {sin}^{2} β + 4 cos (α + β) sin α sin β + cos 2 (α + β) & (S) & cos (2 α) \end{matrix}

Which of the following options is the only correct combination ?

From the given graph (in red colour) match the mentioned points (column I)
with their slope (column II)
Column I Column II
(p) X = 2 (i) 2
(q) X = 6 (ii) 6
(r) X = 10 (iii) 10
(iv) 0
(v) $\frac{1}{2}$
(vi) $\frac{- 1}{2}$

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

Which of the following options is only correct combination?

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