JEE Advanced (Single Correct Type): Straight Lines & pair of Lines | Chapter-wise Tests for JEE Main & Advanced PDF Download

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Q.1. Two lines are said to be perpendicular if the product of their slope is equal to:
(a) -1
(b) 0
(c) 1
(d) ½

Correct Answer is option (a)

When two lines are perpendicular, then the product of their slope is equal to -1. If two lines are perpendicular with slope m₁ and m₂, then m₁.m₂ = -1.

Q.2. What is the distance of (5, 12) from the origin?
(a) 5 units
(b) 8 units
(c) 12 units
(d) 13 units

Correct Answer is option (d) 
Let the points be A(0, 0) and B(5, 12).
A (0, 0) = (x₁, y₁)
B(5, 12) = (x₂, y₂)
The distance between two points, AB = √[(x₂-x₁)² + (y₂-y₁)²]
AB = √[(5-0)² + (12-0)²]
AB = √(25+144)
AB = √(169)
AB = 13
Hence, the distance of (5, 12) from the origin is 13 units.

Q.3. Two lines are said to be parallel if the difference of their slope is
(a) -1
(b) 0
(c) 1
(d) None of these

Correct Answer is option (b)
We know that two lines are said to be parallel if their slope is equal. If m₁ and m₂ are the slopes of two parallel lines, then it is represented as m₁ = m₂.
Hence, the difference of their slope should be m₁-m₂ = 0.
So, option (b) 0 is the correct answer.

Q.4. The equation of a straight line that passes through the point (3, 4) and perpendicular to the line 3x+2y+5=0 is
(a) 2x-3y+6 = 0
(b) 2x+3y+6 = 0
(c) 2x-3y-6 = 0
(d) 2x+3y-6 = 0

Correct Answer is option (a) 

The equation of a straight line perpendicular to 3x+2y+5 = 0 is 2x-3y+λ =0 …(1)

This passes through the point (3, 4).

Now, substitute in equation (1), we get

2(2) – 3(4) +λ = 0

4-12+λ =0

-6+λ =0

λ=6

Substituting λ=6 in (1), we get 2x-3y+6 = 0, which is the required equation.

Hence, option (a) 2x-3y+6 =0 is the correct answer.

Q.5. The slope of a line ax+by+c =0 is
(a) a/b
(b) -a/b
(c) c/b
(d) -c/b

Correct Answer is option (b)

We know that the general equation of a line is ax+by+c = 0.

Rearranging the equation, we get

⟹ by = -ax -c

⟹y = (-a/b)x -(c/b) …(1)

This is of the form, y= mx+c …(2)

By comparing (1) and (2), we get

Slope, m = -a/b.

Q.6. The equation of a line that passes through the points (1, 5) and (2, 3) is:
(a) 2x + y – 7 = 0
(b) 2x – y – 7 = 0
(c) x + 2y – 7 = 0
(d) 2x + y + 7 = 0

Correct Answer is option (a)

We know that the equation of a line passes through two points (x₁, y₁) and (x₂ y₂) is

(y-y₁)/(x-x₁) = (y₂-y₁)/(x₂-x₁)

(x₁, y₁) = (1, 5)

(x₂, y₂) = (2, 3)

Now, substitute the values in the formula, we get

(y-5)/(x-1) = (3-5)/(2-1)

(y-5)/(x-1) = (-2)/(1)

y-5 = -2(x-1)

y-5 = -2x+2

2x+y-5-2 =0

2x+y-7 =0

Therefore, the equation of a line that passes through the points (1, 5) and (2, 3) is 2x+y-7=0.

Q.7. The distance between the lines 3x+4y=9 and 6x+8y=15 is
(a) 2/3
(b) 3/2
(c) 3/10
(d) 7/10

Correct Answer is option (c) 3/10

Given equations:

3x + 4y = 9 …(i)

6x + 8y =15 …(ii)

⟹3x + 4y = 15/2

The slope of line (i) is -3/4.

The slope of line (ii) is -3/4.

Since slopes are equal, the lines are parallel.

Hence, the distance between two parallel lines = |(c₁ – c₂)|/√(a² + b²)

= |(9 – (15/2))|/√(3² + 4²)

= 3/10

Therefore, the distance between the lines 3x+4y=9 and 6x+8y=15 is 3/10.

Q.8. The locus of a point, whose abscissa and ordinate are always equal is
(a) x-y = 0
(b) x+y =1
(c) x+y+1 = 0
(d) None of the above

Correct Answer is option (a)

Let the abscissa and ordinate of a point “P” be(x, y)

Given condition: Abscissa = Ordinate

(i.e) x =y

Hence, the locus of a point is x-y = 0.

Therefore, option (a) x-y =0 is the correct answer.

Q.9. If A(6, 4) and B(2, 12) are the two points, then the slope of a line perpendicular to line AB is
(a) -2
(b) 2
(c) ½
(d) -½

Correct Answer is option (c)

Given points: A(6, 4) = (x₁, y₁)

B(2, 12) = (x₂, y₂)

We know that the slope of a line passing through two points (x1, y₁) and (x₂, y₂) is (y₂-y₁)/(x₂-x₁).

m = (12-4)/(2-6) = 8/-4 = -2.

We know that the slope of two perpendicular lines m₁.m₂ = -1.

So, m₂ = -1/m₁

Hence, the slope of a line perpendicular to line AB is -1/m = -1/-2 = ½.

Q.10. What can be said regarding a line if its slope is negative?
(a) θ is an acute angle
(b) θ is an obtuse angle
(c) Either the line is x-axis or it is parallel to the x-axis.
(d) None of these

Correct Answer is option (b)

The line with a negative slope makes an obtuse angle with a positive x-axis when measured in the anti-clockwise direction.