Class 11 Mathematics Straight Lines MCQs Set E

Question. A ray of light coming from the point (1, 2) is reflected at a point A on the x-axis and then passes through the point (5, 3). The co-ordinates of the point A is
(a) (13/5 , 0)
(b) (5/13 , 0)
(c) (–7, 0)
(d) None of these
Answer: a

Question. The lines a₁x + b₁y + c₁= 0 and a₂x + b₂y + c₂ = 0 are perpendicular to each other
(a) a₁b₁ – b₁a₂ = 0
(b) a₁²b₂ + b₁²a₂ = 0
(c) a₁b₁ + a₂b₂ = 0
(d) a₁b₂ + b₁b₂ = 0
Answer: d

Question. The line parallel to the x-axis and passing through the intersection of the lines ax + 2by + 3b = 0 and bx – 2ay – 3a = 0, where (a, b) ≠ (0, 0) is
(a) Above the x-axis at a distance of 3/2 from it
(b) Above the x-axis at a distance of 2/3 from it
(c) Below the x-axis at a distance of 3/2 from it
(d) Below the x-axis at a distance of 2/3 from it
Answer: c

Question. A line passes through P (1, 2) such that its intercept between the axes is bisected at P. The equation of the line is
(a) x + 2y = 5
(b) x – y + 1 = 0
(c) x + y – 3 = 0
(d) 2x + y – 4 = 0
Answer: d

Question. If two vertical poles 20 m and 80 m high stand apart on a hori ontal plane, then the height (in m) of the point of intersection of the lines oining the top of each pole to the foot of other is
(a) 16
(b) 18
(c) 50
(d) 15
Answer: a

Question. Suppose that the points (h, k), (1, 2) and (– 3, 4) lie on the line L₁. If a line L₂ passing through the points (h, k) and (4, 3) is perpendicular on L₁, then equals :
(a) 1/3
(b) 0
(c) 3
(d) – 1/7
Answer: a

Question. If the area of the triangle with vertices (x, 0), (1, 1) and (0, 2) is 4 square unit, then the value of x is :
(a) – 2
(b) – 4
(c) – 6
(d) 8
Answer: c

Question. Let the perpendiculars from any point on the line 7x + 56y = 0 upon 3x + 4y = 0 and 5x – 12y = 0 be p and p’, then
(a) 2p = p’
(b) p = 2p’
(c) p = p’
(d) None of these
Answer: c

Question. The lines p(p² +1) x – y + q = 0 and (p² + 1)²x + (p² + 1) y + 2q = 0 are perpendicular to a common line for
(a) exactly one value of p
(b) exactly two values of p
(c) more than two values of p
(d) no value of p
Answer: a

Question. Let θ₁ be the angle between two lines 2x + 3y + c₁ = 0 and – x + 5y + c₂ = 0 and θ₂ be the angle between two lines 2x + 3y + c₁ = 0 and – x + 5y + c₃ = 0, where c₁, c₂, c₃ are any real numbers :
Statement-1: If c₂ and c₃ are proportional, then θ₁ = θ₂.
Statement-2: θ₁ = θ₂ for all c₂ and c₃.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation of Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation of Statement-1.
(c) Statement-1 is false; Statement-2 is true.
(d) Statement-1 is true; Statement-2 is false.
Answer: a

Question. The point (x, y) lies on the line with slope m and through the fixed point (x₀, y₀) if and only if its coordinates satisfy the equation y – y₀ is equal to ……… .
(a) m(x – x₀)
(b) m(y – x₀)
(c) m(y – x)
(d) m(x – y₀)
Answer: a

Question. The perpendicular bisector of the line segment oining P (1, 4) and Q(k, 3) has y-intercept –4. Then a possible value of k is
(a) 1
(b) 2
(c) –2
(d) – 4
Answer: d

Question. If a, b, c ∈ R and 1 is a root of equation ax² + bx + c = 0, then the curve y = 4ax² + 3bx + 2c, a ≠ 0 intersect x-axis at
(a) two distinct points whose coordinates are always rational numbers
(b) no point
(c) exactly two distinct points
(d) exactly one point
Answer: d

Question. A square of side a lies above the x-axis and has one vertex at the origin. The side passing through the origin makes an angle α(o < α < π/4) with the positive direction of x-axis. The equation of its diagonal not passing through the origin is
(a) y(cos α + sin α) + x(cos α – sin α) = α
(b) y(cos α – sin α) – x(sin α – cos α) = α
(c) y(cos α + sin α) + x(sin α – cos α) = α
(d) y(cos α + sin α) + x(sin α + cos α) = α
Answer: a

Question. If the line, 2x – y + 3 = 0 is at a distance 1/√5 and 2/√5 from the lines 4x – 2y + α = 0 and 6x – 3y + β = 0, respectively, then the sum of all possible value of α and β is ______.
Answer: 30

STATEMENT TYPE QUESTIONS 

Question. Slope of the lines passing through the points
I. (3, – 2) and (– 1, 4) is −3/2
II. (3, – 2) and (7, – 2) is 0.
III. (3, – 2) and (3, 4) is 1.
Choose the correct option.
(a) Only I and III are true
(b) Only I and II are true
(c) Only II and III are true
(d) None of these
Answer: b

Question. The distances of the point (1, 2, 3) from the coordinate axes are A, B and C respectively. Now consider the following equations:
I. A² = B² + C²
II. B² = 2C²
III. 2A²C² = 13 B²
Which of these hold(s) true?
(a) Only I
(b) I and III
(c) I and II
(d) II and III
Answer: d

Question. Equation of a line is 3x – 4y + 10 = 0
I. Slope of the given line is 3/4.
II. x-intercept of the given line is −10/3.
III. y-intercept of the given line is 5/2.
Choose the correct option.
(a) Only I and II are true
(b) Only II and III are true
(c) Only I and III are true
(d) All I, II and III are true
Answer: d

Question. Consider the following statements.
I. If (a, b), (c, d) and (a – c, b – d) are collinear, then bc – ad = 0
II. If the points A (1, 2), B (2, 4) and C (3, a) are collinear, then the length BC = 5 unit.
Choose the correct option.
(a) Only I is true
(b) Only II is true
(c) Both are true
(d) Both are false
Answer: a

Question. Consider the following statements.
The three given points A, B, C are collinear i.e., lie on the same straight line, if
I. area of ΔABC is zero.
II. slope of AB = Slope of BC.
III. any one of the three points lie on the straight line joining the other two points.
Choose the correct option
(a) Only I is true
(b) Only II is true
(c) Only III is true
(d) All are true
Answer: d

Question. Consider the following statements.
I. Let A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) be the vertices of a triangle then centroid is
(x₁ + x₂ + x₃/3, y₁ + y₂ + y₃/3)
II. If the point P(x , y) divides the line joining the points A(x₁, y₁) and B(x₂, y₂) in the ratio m : n (internally), then
x = mx₂ + nx₁/m + n , y = my₂ + ny₁/m + n
Choose the correct option.
(a) Only I is true
(b) Only II is true
(c) Both are true
(d) Both are false
Answer: c

Question. Consider the following statements about straight lines :
I. Slope of horizontal line is zero and slope of vertical line is undefined.
II. Two lines are parallel if and only if their slopes are equal.
III. Two lines are perpendicular if and only if product of their slope is – 1.
Which of the above statements are true ?
(a) Only I
(b) Only II
(c) Only III
(d) All the above
Answer: d

Question. Consider the straight lines
L₁ : x – y = 1
L₂ : x + y = 1
L₃ : 2x + 2y = 5
L₄ : 2x – 2y = 7
The correct statement is
(a) L₁ || L₄ , L₂ || L₃ , L₁ intersect L₄.
(a) L₁ ⊥ L₂ , L₁ || L₃ , L₁ intersect L₂.
(a) L₁ ⊥ L₂ , L₂ || L₃ , L₁ intersect L₄.
(a) L₁ ⊥ L₂ , L₁ ⊥ L₃ , L₂ intersect L₄.
Answer: d

Question. Let L be the line y = 2x, in the two dimensional plane.
Statement 1: The image of the point (0, 1) in L is the point (4/5, 3/5).
Statement 2: The points (0, 1) and (4/5, 3/5) lie on opposite sides of the line L and are at equal distance from it.
(a) Statement 1 is true, Statement 2 is false.
(b) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.
(c) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.
(d) Statement 1 is false, Statement 2 is true.
Answer: c

Question. The angle between the lines xy = 0 is equal to:       
(a) 45°
(b) 60°
(c) 90°
(d) 180°

Answer: C

Question. If the line 2x + 3ay − 1 = 0 and 3x + 4y + 1 = 0 are mutually perpendicular, then the value of a will be:         
(a) 1/2
(b) 2
(c) −1/2
(d) None of these

Answer: C

Question. If the line passing through (4, 3) and (2, k) is perpendicular to y = 2x + 3 , then k = ?       
(a) –1
(b) 1
(c) – 4
(d) 4

Answer: D

Question. Let P(−1,0), Q(0,0) and R(3,3 3) be three points. Then the equation of the bisector of the angle PQR is:         
(a) √3/2 x+y = 0
(b) x+√3y = 0
(c) √3x+y = 0
(d) x+√3/2 y = 0

Answer: C

Question. Equation of angle bisectors between x and y -axes are:         
(a) y = ±x
(b) y = ±2x
(c) y = ± 1√2 x
(d) y = ±3x

Answer: A

Question. The bisector of the acute angle formed between the lines 4x − 3y + 7 = 0 and 3x − 4 y +14 = 0 has the equation:         
(a) x + y + 3 = 0
(b) x − y − 3 = 0
(c) x − y + 3 = 0
(d) 3x + y − 7 = 0

Answer: C

Question. If the sum of the slopes of the lines given by x² −2cxy−7y² =0 is four times their product. Then c has the value:     
(a) – 2
(b) – 1
(c) 2
(d) 1

Answer: C

Question. The straight lines 4ax + 3by + c = 0 where a + b + c = 0 , will be concurrent, if point is :         
(a) (4, 3)
(b) (1/4, 1/3)
(c) (1/2, 1/3)
(d) None of these

Answer: B

Question. If the equation 12x² −10xy + 2y² +11x−5y+K=0 represent two straight lines, then the value of K is:       
(a) 1
(b) 2
(c) 0
(d) 3

Answer: B

Question. The angle between the lines x² − xy − 6y² − 7x +31y −18 = 0 is:       
(a) 45°
(b) 60°
(c) 90°
(d) 30°

Answer: B

Question. The new equation of curve 12x² + 7xy −12y −17x − 31y −7 = 0 after removing the first degree terms:       
(a) 12x² − 7XY −12y² = 0
(b) 12x² + 7XY +12y² = 0
(c) 12x² + 7XY −12y² = 0
(d) None of these

Answer: C

Question. Distance between the pair of lines represented by the equation 2x² − 6xy + 9y² + 3x − 9y − 4 = 0 ?V           
(a) 15/√10
(b) 1/2
(c)√(5/2)
(d) 1/√10

Answer: C

Question. The angle between the pair of straight lines represented by 2x² − 7xy + 3y² = 0 is:         
(a) 60º
(b) 45º
(c) tan–1 (7/6)
(d) 30º

Answer: B

Question. The equation of the bisectors of the angles between the lines represented by x² + 2xy cotθ + y² = 0 is:   
(a) x² − y² = 0
(b) x² − y² = xy
(c) (x² − y² )cotθ = 2xy
(d) None of these

Answer: A

Question. The point of intersection of the lines represented by the equation 2x² + 3y² + 7xy + 8x +14y + 8 = 0 is:         
(a) (0,2)
(b) (1,2)
(c) (–2, 0)
(d) (–2,1)

Answer: C

Question. A triangle with vertices (4, 0); (–1, –1); (3, 5) is:         
(a) Isosceles and right angled
(b) Isosceles but not right angled
(c) Right angled but not isosceles
(d) Neither right angled nor isosceles

Answer: A

Question. The co-ordinate of the point dividing internally the line joining the points (4,–2) and (8,6) in the ratio 7: 5 will be:         
(a) (16, 18)
(b) (18, 16)
(c) (19/3 , 8/3)
(d) (8/3 , 19/3)

Answer: C

Question. If the pair of straight lines xy − x − y +1 = 0 and line ax + 2 y − 3 = 0 are concurrent, then a =?         
(a) – 1
(b) 0
(c) 3
(d) 1

Answer: D

Question. The lines joining the origin to the point of intersection of the circle x2 + y2 = 3 and the line x + y = 2 are: 
(a) y − (3 + 2√2)x = 0
(b) x − (3 + 2√2) y = 0
(c) x − (3 − 2√2) y = 0
(d) y − (3 − 2√2)x = 0

Answer: ALL

Question. The pair of straight lines joining the origin to the points of intersection of the line y = 2√2x + c and the circle x² + y² = 2 are at right angles, if:         
(a) 2 c − 4 = 0
(b) 2 c − 8 = 0
(c) 2 c − 9 = 0
(d) 2 c −10 = 0

Answer: C

Question. The coordinates of the foot of perpendicular drawn from (2, 4) to the line x + y =1 is:         
(a) (1/3 , 3/2)
(b) (−1/2 , 3/2)
(c) (4/3 , 1/2)
(d) (3/4 , −1/2)

Answer: B

Question. The area enclosed within the curve | x | + | y |=1 is:       
(a) 2
(b) 1
(c) 3
(d) 2

Answer: D

Question. The area of the triangle formed by the lines 4x² −9xy−9y² = 0 and x = 2 is:           
(a) 2
(b) 3
(c) 10/3
(d) 20/3

Answer: C

Question. The gradient of the line joining the points on the curve y² = x + 2x , whose abscissae are 1 and 3, is:         
(a) 6
(b) 5
(c) 4
(d) 3

Answer: A

Question. Equation to the straight line cutting off an intercept 2 from the negative direction of the axis of y and inclined at 30° to the positive direction of x, is:           
(a) y + x − 3 = 0
(b) y − x + 2 = 0
(c) y − √3x − 2 = 0
(d) √3y − x + 2√3 = 0

Answer: D

Question. The vertex of an equilateral triangle is (2, –1) and the equation of its base is x + 2y =1.The length of its sides is:
(a) 4/√15
(b) 2/√15
(c) 4/3√3
(d) None of these

Answer: B

Question. The reflection of the point (4,–13) in the line 5x+y+6=0 is:
(a) (–1, –14)
(b) (3, 4)
(c) (1, 2)
(d) (–4, 13)

Answer: A

Question. If the lines 4x+3y =1, y = x+5 and 5y + bx = 3 are concurrent, then b equals:
(a) 1
(b) 3
(c) 6
(d) 0

Answer: C

Question. If the lines ax+by+c=0, bx+cy+a=0 and cx + ay + b = 0 be concurrent, then:
(a) a³ + b³ + c³ + 3abc = 0
(b) a³ + b³ + c³ – abc = 0
(c)  a³ + b³ + c³ – 3abc = 0
(d) None of these

Answer: C

QuestionLet L1 be a straight line passing through the origin and L2 be the straight line x + y = 1. If the intercepts made by the circlex2 + y2 – x + 3y = 0 on L1 and L2 are equal, then
which of the following equation can represent L1?         
(a) x + y = 0
(b) x –y = 0
(c) x +7y = 0
(d) x –7y = 0

Answer: A,C

Question. Two roads are represented by the equations y − x = 6 and x + y = 8.An inspection bunglow has to be so constructed that it is at a distance of 100 from each of the roads.
Possible location of the bunglow is given by:         
(a) (100√3 +1,7)
(b) (1−100√2,7)
(c) (1,7 +100√2)
(d) (1,7 −100√2)

Answer: ALL

Question. The point (3, 2) is reflected in the y-axis and then moved a distance 5 units towards the negative side of y-axis. The co-ordinate of the point thus obtained are:       
(a) (3, –3)
(b) (–3, 3)
(c) (3, 3)
(d) (–3, –3)

Answer: D

Question. Let A (2,–3) and B(–2,1) be vertices of triangle AB(c) If the centroid of this triangle moves on the line 2x + 3y = 1 , then the locus of the vertex C is the line:         
(a) 3x − 2y = 3
(b) 2x − 3y = 7
(c) 3x + 2y = 5
(d) 2x + 3y = 9

Answer: D

Question. The centroid of a triangle is (2,7)and two of its vertices are (4, 8) and (–2, 6) the third vertex is:       
(a) (0, 0)
(b) (4, 7)
(c) (7, 4)
(d) (7, 7)

Answer: B

Question. If the vertices of a triangle be (2, 1); (5, 2) and (3,4) then its circumcentre is:         
(a) (13/2 , 9/2)
(b) (13/4 , 9/4)
(c) (9/4 , 13/4)
(d) None of these

Answer: B

Question. Equation of the straight line, inclined at 30° to the axis of x such that the length of its (each of their) line segment between the coordinate axes is 10 units, is: (are)         
(a) x − √3y − 5√3 = 0
(b) x − y/√3 − 5√3 = 0
(c) x − √3y + 5√3 = 0
(d) x − y/√3 + 5√3 = 0

Answer: A,C

Question. Let L be the line 2x + y = 2. If the axes are rotated by 45°, then the intercept made by the line L on the length of new axes are respectively:           
(a) √2 and 1
(b) 1 and √2
(c) 2√2 and 2√2/3
(d) 2√2/3 and 2√2

Answer: C,D

Question. The reflection of the point (4,–13) in the line 5x+y+6=0 is:           
(a) (–1, –14)
(b) (3, 4)
(c) (1, 2)
(d) (–4, 13)

Answer: A

Question. The image of a point A(3,8) in the line x + 3y − 7 = 0 , is:         
(a) (–1, –4)
(b) (–3, –8)
(c) (1, –4)
(d) (3, 8)

Answer: A

Assertion and Reason
Note: Read the Assertion (A) and Reason (R) carefully to mark the correct option out of the options given below:
(a) If both assertion and reason are true and the reason is the correct explanation of the assertion.
(b) If both assertion and reason are true but reason is not the correct explanation of the assertion.
(c) If assertion is true but reason is false.
(d) If the assertion and reason both are false.
e. If assertion is false but reason is true.

Question. Consider the lines L1:p2x + py –1=0; L2: q2x +qy +6=0;L1 passes through the point (3, 2) and L2 passes through the point (2, 7):         
Assertion: If the product of the slopes of L1 and L2 is 2 then they intersect at the point (–4, –5)
Reason: L1and L2 are neither parallel nor perpendicular

Answer: C

Question. Assertion: 4x² + 12xy + 9y² = 0 represents a pair of perpendicular lines through the origin.   
Reason: ax² + 2hxy + by² = 0 represents a pair of coincident lines if h2 = a(b)

Answer: D

Question. Assertion: x²y² – x² – y² + 1 = 0 represents the sides of a square of area 4 square units.         
Reason: 3x² + λxy – 3y² = 0 represents a pair of perpendicular lines for all values of λ.

Answer: B

Question. Assertion: Consider the point A(0,1) and B(2,0) and P be point on the line 4x+ 3y + 9= 0, then the coordinates of P such that |PA –PB| is maximum is (–24/5 , 17/5)         
Reason: If A and B are tow fixed point and P is any point in a plane, then |PA –PB| ≤ A(b)

Answer: A

Question. Assertion: If the circumcentre of a triangle lies at the origin and centroid is the mid point of the line joining the points (2,3) and (4,7), then its orthocenter line 5x – 3y = 0.     
Reason: Circumcentre, centroid and orthocenter of a triangle lie on the same line.

Answer: A

Question. Let A(2,–3) and B(–2,1) be the vertices of a triangle ABC:       
Assertion: If the centroid of the triangle moves on the line x + y =5, the vertex moves on the line x + y = 17.
Reason: If the centroid of the triangle moves the line x – y + 1 = 0 (x ≠ 0), the triangle is either isosceles or equilateral.

Answer: B

Question. Assertion: x2 y − 3xy − 2x2 + 6x − 4y + 8 = 0 represents three straight lines two of which are parallel and the third is perpendicular to the other two
Reason: xy − 2x + y − 2 = 0 represents a pair of straight lines one of which is common to the pair of straight lines xy + 2x – y = 0

Answer: C

Question. Assertion: If the perpendicular bisector of the line segment joining P(1,4) and Q(k,3) has y-intercept equal to –4, then k2 – 16 = 0       
Reason: Centroid of an isosceles triangle ABC lies on the perpendicular bisector of the base BC where B = (c)

Answer: B

Question. Assertion: If x +ky = 1and x = a are the equations of the hypotenuse and a side of a right angled isosceles triangle then k = ± (a)           
Reason: Each side of a right angled isosceles triangle makes an angle π/4 with the hypotenuse.

Answer: D

Comprehension Based
Paragraph –I
Let a,r,s,t be non-zero real numbers. Let P(at2, 2at), Q,R(ar2, 2ar), and S(as2,2as) be distinct point on the parabola y² = 4 ax.
Suppose that PQ is the focal chord and line QR and PK are parallel, where K is the point (2a, 0).

Question. The value of r is:         
(a) −1/t
(b) t² +1/t
(c) 1/t
(d) t² −1/t

Answer: D

Question. If st = 1, then the tangent at P and the normal at S to the parabola meet at a point whose ordinate is:           
(a) (t² +1/t)2t³
(b) a(t² +1)2t³
(c) a(t² +1)²/t³
(d) a(t² +2)²/t³

Answer: B

Paragraph –II
Let P(x, y) be the Cartesian coordinates with respect to axes OX and OY, then (r,θ) be its polar coordinates with respect to pole O and initial line OX i.e., OP = r (radius vector) and
∠XOP = θ (vectorial angle). Now let p be the length of perpendicular form O upon straight line (through A, B) ie., OM = p and ∠XOM = α We, have OM = OP cos (θ – α) or p = r cos
(θ – α) which is the required equation to the given line.

Question. Cartesian form of the curve √r = √asin (θ / 2), ∀ a > 0 is:       
(a) 4(x² – y²)(x² + y² + ax) = a²y²
(b) 4(x² – y²)(x² + y² – ax) = a²y²
(c) 4(x² + y²)(x² + y² + ax) = a²y²
(d) 4(x² + y²)(x² + y² – ax) = a²y²

Answer: C

Question. Polar form of the curve x³ + 3x² y – 3xy² – y³ = 5kxy is:       
(a) 2r (cos 3θ + sin 3θ) = 5k sin θ cos θ
(b) r (cos 3θ + sin 3θ) = 5k sin θ cos θ
(c) 2r (cos 3θ – sin 3θ) = 5k sin θ cos θ
(d) r (cos 3θ – sin 3θ) = 5k sin θ cos θ

Answer: B

Question. Locus of the point P(r,θ ), if 2/r = −1/2 + 4cosθ is an:           
(a) circle
(b) parabola
(c) ellipse
(d) hyperbola

Answer: D