Section A — MCQ (Single Correct)
Question 1
Find the equation of the straight line passing through the point $(2, 3)$ that is perpendicular to the line $3x - 4y + 7 = 0$.
A
$4x + 3y - 17 = 0$
B
$4x - 3y + 1 = 0$
C
$3x + 4y - 18 = 0$
D
$4x + 3y + 17 = 0$
Question 2
The acute angle between the straight lines $x - \sqrt{3}y + 5 = 0$ and $\sqrt{3}x - y + 1 = 0$ is:
A
$30^\circ$
B
$45^\circ$
C
$60^\circ$
D
$90^\circ$
Question 3
Calculate the perpendicular distance from the coordinate point $P(-1, 4)$ to the line path given by $5x + 12y + 7 = 0$.
A
$2$
B
$4$
C
$3$
D
$5$
Question 4
Find the perpendicular distance separating the two parallel lines $8x + 15y - 34 = 0$ and $8x + 15y + 17 = 0$.
A
$1$
B
$2$
C
$3$
D
$4$
Question 5
Find the coordinates of the foot of the perpendicular dropped from the point $(1, 3)$ onto the line $2x - y + 4 = 0$.
A
$(-1, 2)$
B
$(1, 6)$
C
$(0, 4)$
D
$(-0.2, 3.6)$
Question 6
Find the mirror reflection image of the coordinate point $P(3, 4)$ across the line mirror $x + y - 2 = 0$.
A
$(-1, 0)$
B
$(0, -1)$
C
$(-2, -1)$
D
$(1, 2)$
Question 7
If the three lines $x + 2y - 3 = 0$, $3x - y + 1 = 0$, and $kx - y + 5 = 0$ are concurrent, find the value of $k$.
A
$1$
B
$2$
C
$3$
D
$4$
Question 8
The angle between the pair of straight lines represented by the homogeneous quadratic equation $x^2 - 3xy + 2y^2 = 0$ is:
A
$\tan^{-1}(1/3)$
B
$\tan^{-1}(3)$
C
$45^\circ$
D
$90^\circ$
Question 9
Find the joint equation of the angle bisectors for the origin line pair $x^2 - 4xy + y^2 = 0$.
A
$x^2 - y^2 = 0$
B
$xy = 0$
C
$x^2 + y^2 = 0$
D
$x^2 - 2xy - y^2 = 0$
Question 10
A line passes through $(1, 2)$ and makes an angle of $45^\circ$ with the positive x-axis. Find the coordinates of a point on this line located at a distance of $\sqrt{2}$ units away from the starting point in the positive direction.
A
$(2, 3)$
B
$(3, 4)$
C
$(0, 1)$
D
$(2, 2)$
Section B — Integer Type
Question 11 — Integer answer
If the pair of straight lines represented by the equation $kx^2 + 4xy + y^2 = 0$ are coincident, find the value of the integer parameter $k$.
Question 12 — Integer answer
Find the total number of independent arbitrary constants that must be eliminated to form the differential equation representing a general family of straight lines ($y = mx + c$).
Question 13 — Integer answer
If the perpendicular distance from the origin $(0, 0)$ to the line $3x + 4y + c = 0$ is exactly $5$ units, find the positive value of the constant parameter $c$.
Section C — Assertion & Reasoning
Question 14 — Assertion / Reason
Assertion (A): The straight lines $2x - 3y + 5 = 0$ and $4x - 6y + 11 = 0$ are parallel to each other.Reason (R): Two straight lines are parallel if and only if their slopes are identically equal ($m_1 = m_2$).
A
Both A and R are true and R is the correct explanation of A
B
Both A and R are true but R is NOT the correct explanation of A
C
A is true but R is false
D
A is false but R is true
Solution: Both A and R are true, and R is the correct explanation.
Question 15 — Assertion / Reason
Assertion (A): The joint homogeneous equation $x^2 + 2xy + y^2 = 0$ represents two distinct, mutually perpendicular lines passing through the origin.Reason (R): A pair of lines $ax^2 + 2hxy + by^2 = 0$ are perpendicular if and only if the sum of their quadratic coefficients satisfies $a + b = 0$.
A
Both A and R are true and R is the correct explanation of A
B
Both A and R are true but R is NOT the correct explanation of A
C
A is true but R is false
D
A is false but R is true
Solution: A is false (the equation factors to $(x+y)^2 = 0$, representing coincident lines, not perpendicular lines), but R is true.