Two statements are given, one labelled Assertion (A) and ...

CBSE Class 12 Math 2026 All Sets Solved Paper

Question : 20 of 20

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Two statements are given, one labelled Assertion (A) and other labelled Reason (R).
Assertion (A): Lines given by

x=p y+q,

z=r y+s

and

x=p^{'} y+q^{'}, z=r^{'} y+s^{'}

are perpendicular to each other when

p p^{'}+r r^{'}=1

.
Reason (R): Two lines

{r}↖{→}={a}↖{→}_{1}+λ {b}↖{→}_{1}

and

{r}↖{→}={a}↖{→}_{2}+µ {b}↖{→}_{2}

are perpendicular to each other if

{b}↖{→}_{1} ⋅ {b}↖{→}_{2}=0

.
Select the correct answer from the options (A), (B), (C) and (D) as given below.

Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A).
Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
Assertion (A) is true, but Reason (R) is false.
Assertion (A) is false, but Reason (R) is true.

Solution:

x=p y+q, z=r y+s

here, we put y = t

x=p t+q, z=r t+s

(x, y, z)=(p t+q, t, r t+s)

Direction vector:

{d}↖{→}_{1}=(p, 1, r)

Line 2:

x=p^{'} y+q^{'}, z=r^{'} y+s^{'}

Here, we put

y=t^{'}

x=p^{'} t^{'}+q^{'}, z=r^{'} t^{'}+s^{'}

(x, y, z)= (p^{'} t^{'}+q^{'}, t^{'}, r^{'} t^{'}+s^{'})

Direction vector:

{d}↖{→}_{2}= (p^{'}, 1, r^{'})

Two lines are perpendicular when dot product of their direction vectors is zero.

{d_{1}↖{→}} ⋅ {d_{2}↖{→}}=0

p p^{'}+1+r r^{'}=0

p p^{'}+r r^{'}=-1

, but in assertion it is given that

p p^{'}+r r^{'}=1

. So assertion is false.
Two lines

{r}↖{→}={a_{1}↖{→}}+λ {b_{1}↖{→}}

and

{r}↖{→}={a_{2}↖{→}}+µ {b_{2}↖{→}}

are perpendicular when the dot product of their direction vector is zero.
Here, direction vectors are

{b_{1}↖{→}}

and

{b_{2}↖{→}}

respectively.
So

{b_{1}↖{→}} ⋅ {b_{2}↖{→}}=0

Reason is true.

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