Class 11 Mathematics Straight Lines MCQs Set 12

Question. Let \( P = (-1, 0) \), \( Q = (0, 0) \) and \( R = (3, 3\sqrt{3}) \) be three points. Then the equation of the bisector of the angle \( PQR \) is
(a) \( \frac{\sqrt{3}}{2}x + y = 0 \)
(b) \( x + \sqrt{3}y = 0 \)
(c) \( \sqrt{3}x + y = 0 \)
(d) \( x + \frac{\sqrt{3}}{2}y = 0 \)
Answer: (c) \( \sqrt{3}x + y = 0 \)

Question. A straight line through the origin O meets the parallel lines \( 4x + 2y = 9 \) and \( 2x + y + 6 = 0 \) at points P and Q respectively. Then the point O divides the segment PQ in the ratio
(a) 1 : 2
(b) 3 : 4
(c) 2 : 1
(d) 4 : 3
Answer: (b) 3 : 4

Question. The number of integral points (integral point means both the coordinates should be integers) exactly in the interior of the triangle with vertices (0, 0), (0, 21) and (21, 0) is
(a) 133
(b) 190
(c) 233
(d) 105
Answer: (b) 190

Question. \( m, n \) are integers with \( 0 < n < m \). A is the point \( (m, n) \) on the cartesian plane. B is the reflection of A in the line \( y = x \). C is the reflection of B in the \( y \)-axis, D is the reflection of C in the \( x \)-axis and E is the reflection of D in the \( y \)-axis. The area of the pentagon ABCDE is
(a) \( 2m(m + n) \)
(b) \( m(m + 3n) \)
(c) \( m(2m + 3n) \)
(d) \( 2m(m + 3n) \)
Answer: (b) \( m(m + 3n) \)

Question. Let \( O(0, 0), P(3, 4), Q(6, 0) \) be the vertices of the triangle OPQ. The point R inside the triangle OPQ is such that the triangles OPR, PQR, OQR are of equal area. The coordinates of R are
(a) \( \left( \frac{4}{3}, 3 \right) \)
(b) \( \left( 3, \frac{2}{3} \right) \)
(c) \( \left( 3, \frac{4}{3} \right) \)
(d) \( \left( \frac{4}{3}, \frac{2}{3} \right) \)
Answer: (c) \( \left( 3, \frac{4}{3} \right) \)

Question. A straight line L through the point (3, -2) is inclined at an angle \( 60^\circ \) to the line \( \sqrt{3}x + y = 1 \), if L also intersects the x-axis, then the equation of L is
(a) \( y + \sqrt{3}x + 2 - 3\sqrt{3} = 0 \)
(b) \( y - \sqrt{3}x + 2 + 3\sqrt{3} = 0 \)
(c) \( \sqrt{3}y - x + 3 + 2\sqrt{3} = 0 \)
(d) \( \sqrt{3}y + x - 3 + 2\sqrt{3} = 0 \)
Answer: (b) \( y - \sqrt{3}x + 2 + 3\sqrt{3} = 0 \)

Multiple Answer Questions

Question. One of the bisectors of the angle between the lines \( a(x - 1)^2 + 24(x - 1)(y - 2) + b(y - 2)^2 = 0 \) is \( x + 2y - 5 = 0 \) then the other bisector
(a) \( 2x - y = 0 \)
(b) \( 2x - y - 1 = 0 \)
(c) Passes through origin
(d) Passes through (1, 1)
Answer: (a) \( 2x - y = 0 \)
(c) Passes through origin

Question. For all values of \( \theta \), the lines represented by the equation \( (2\cos\theta + 3\sin\theta)x + (3\cos\theta - 5\sin\theta)y - (5\cos\theta - 2\sin\theta) = 0 \)
(a) Pass through a fixed point
(b) Pass through the point (1, 1)
(c) Pass through a fixed point whose reflection in the line \( x + y = \sqrt{2} \) is \( (\sqrt{2} - 1, \sqrt{2} - 1) \)
(d) Pass through the origin
Answer: (a) Pass through a fixed point
(b) Pass through the point (1, 1)
(c) Pass through a fixed point whose reflection in the line \( x + y = \sqrt{2} \) is \( (\sqrt{2} - 1, \sqrt{2} - 1) \)

Question. A line through \( A(-5, -4) \) with slope \( \tan\theta \) meets the lines \( x + 3y + 2 = 0 \), \( 2x + y + 4 = 0 \), \( x - y - 5 = 0 \) at B, C, D respectively, such that \( \left( \frac{15}{AB} \right)^2 + \left( \frac{10}{AC} \right)^2 = \left( \frac{6}{AD} \right)^2 \) then
(a) \( \frac{15}{AB} = \cos\theta + 3\sin\theta \)
(b) \( \frac{10}{AC} = 2\cos\theta + \sin\theta \)
(c) \( \frac{6}{AD} = \cos\theta - \sin\theta \)
(d) Slope of the line is \( -\frac{2}{3} \)
Answer: (a) \( \frac{15}{AB} = \cos\theta + 3\sin\theta \)
(b) \( \frac{10}{AC} = 2\cos\theta + \sin\theta \)
(c) \( \frac{6}{AD} = \cos\theta - \sin\theta \)
(d) Slope of the line is \( -\frac{2}{3} \)

Question. The lines \( L_1 \) and \( L_2 \) denoted by \( 3x^2 + 10xy + 8y^2 + 14x + 22y + 15 = 0 \) intersects at the point P and having slopes \( m_1 \) and \( m_2 \) respectively. The acute angle between them is \( \theta \). Which of the following relations hold good.
(a) \( m_1 + m_2 = -\frac{5}{4} \)
(b) \( m_1 m_2 = \frac{3}{8} \)
(c) \( \theta = \sin^{-1} \left( \frac{2}{5\sqrt{5}} \right) \)
(d) Sum of abscissa and ordinate of point P is -1
Answer: (b) \( m_1 m_2 = \frac{3}{8} \)
(c) \( \theta = \sin^{-1} \left( \frac{2}{5\sqrt{5}} \right) \)
(d) Sum of abscissa and ordinate of point P is -1

Question. Two sides of a rhombus ABCD are parallel to lines \( y = x + 2 \) and \( y = 7x + 3 \). If the diagonals of the rhombus intersect at point (1, 2) and the vertex A is on the y-axis is, then the possible coordinates of A are
(a) \( \left( 0, \frac{5}{2} \right) \)
(b) (0, 0)
(c) (0, 5)
(d) (0, 3)
Answer: (a) \( \left( 0, \frac{5}{2} \right) \)
(b) (0, 0)

Question. Two lines from the family of lines \( (1 + 2\lambda)x - (1 + \lambda)y + 1 = 0 \) and the line \( x + y = 5 \) form an equilateral triangle. the equation of the two lines can be
(a) \( y - 2 = (2 + \sqrt{3})(x - 1) \)
(b) \( y - 2 = (2 - \sqrt{3})(x - 1) \)
(c) \( y - 2 = (3 - \sqrt{2})(x - 1) \)
(d) \( y - 1 = (3 - \sqrt{2})(x - 1) \)
Answer: (a) \( y - 2 = (2 + \sqrt{3})(x - 1) \)
(b) \( y - 2 = (2 - \sqrt{3})(x - 1) \)

Question. Let \( x_1 \) and \( y_1 \) be the roots of \( x^2 + 8x - 2009 = 0 \); \( x_2 \) and \( y_2 \) be the roots of \( 3x^2 + 24x - 2010 = 0 \) and \( x_3 \) and \( y_3 \) be the roots of \( 9x^2 + 72x - 2011 = 0 \). The points \( A(x_1, y_1) \), \( B(x_2, y_2) \) and \( C(x_3, y_3) \)
(a) can not lie on a circle
(b) forma triangle of area 2 sq. units
(c) form a right angled triangle
(d) are collinear
Answer: (a) can not lie on a circle
(d) are collinear

Question. The sides of a triangle are the straight lines \( x + y = 1 \), \( 7y = x \) and \( \sqrt{3}y + x = 0 \). Then which of the following is an interior point of the triangle
(a) Circumcentre
(b) Centroid
(c) Incentre
(d) orthocentre
Answer: (b) Centroid
(c) Incentre

Question. A ray travelling along the line \( 3x - 4y = 5 \) after being reflected from a line \( l \) travel along the line \( 5x + 12y = 13 \). Then the equation of the line \( l \) is
(a) \( x + 8y = 0 \)
(b) \( x = 8y \)
(c) \( 32x + 4y = 65 \)
(d) \( 32x - 4y + 65 = 0 \)
Answer: (b) \( x = 8y \)
(c) \( 32x + 4y = 65 \)

Question. All the point lying inside the triangle formed by the points (1, 3), (5, 6) and (-1, 2) satisfy
(a) \( 3x + 2y \ge 0 \)
(b) \( 2x + y + 1 \ge 0 \)
(c) \( -2x + 11 \ge 0 \)
(d) \( 2x + 3y - 12 \ge 0 \)
Answer: (a) \( 3x + 2y \ge 0 \)
(b) \( 2x + y + 1 \ge 0 \)
(c) \( -2x + 11 \ge 0 \)

Question. Two straight lines \( u = 0 \) and \( v = 0 \) passes the origin and the angle between them is \( \tan^{-1} \left( \frac{7}{9} \right) \). If the ratio of slopes of \( v = 0 \) and \( u = 0 \) is \( \frac{9}{2} \), then their equations are
(a) \( y = 3x \) and \( 3y = 2x \)
(b) \( 2y = 3x \) and \( 3y = x \)
(c) \( y + 3x = 0 \) and \( 3y + 2x = 0 \)
(d) \( 2y + 3x = 0 \) and \( 3y + x = 0 \)
Answer: (a) \( y = 3x \) and \( 3y = 2x \)
(b) \( 2y = 3x \) and \( 3y = x \)
(c) \( y + 3x = 0 \) and \( 3y + 2x = 0 \)
(d) \( 2y + 3x = 0 \) and \( 3y + x = 0 \)

Question. A ray of light travelling along the line \( x + y = 1 \) is incident on the x-axis and after refraction it enters the other side of the x-axis by turning \( \pi/6 \) away from the x-axis. The equation of the line along which the refracted ray travels is
(a) \( x + (2 - \sqrt{3})y = 1 \)
(b) \( (2 - \sqrt{3})x + y = 1 \)
(c) \( y + (2 + \sqrt{3})x = 2 + \sqrt{3} \)
(d) \( y + (2 - \sqrt{3})x = 2 - \sqrt{3} \)
Answer: (a) \( x + (2 - \sqrt{3})y = 1 \)
(c) \( y + (2 + \sqrt{3})x = 2 + \sqrt{3} \)

Comprehension Questions

PASSAGE : I
Consider 3 non-collinear points \( A(9, 3), B(7, -1), \) and \( C(1, -1) \). Let \( P(a, b) \) be the centre and \( R \) is the radius of circle 'S' passing through points A, B, C. Also \( H(\bar{x}, \bar{y}) \) are the coordinates of the orthocentre of triangle ABC whose area be denoted by \( \Delta \).

Question. If D, E and F are the middle points BC, CA and AB respectively then the area of the triangle DEF is :
(a) 12
(b) 6
(c) 4
(d) 3
Answer: (d) 3

Question. The value of \( a + b + R \) equals
(a) 3
(b) 12
(c) 13
(d) None of the options
Answer: (b) 12

Question. The ordered pair \( (\bar{x}, \bar{y}) \) is :
(a) (9, 6)
(b) (9, -6)
(c) (9, -5)
(d) (9, 5)
Answer: (c) (9, -5)

PASSAGE : II
The equation of an altitude of an equilateral triangle is \( \sqrt{3}x + y = 2\sqrt{3} \) and one of its vertices is \( (3, \sqrt{3}) \) then

Question. The possible number of triangle(s) is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (b) 2

Question. Which of the following can not be the vertex of the triangle
(a) (0, 0)
(b) \( (0, 2\sqrt{3}) \)
(c) \( (3, -\sqrt{3}) \)
(d) \( (2\sqrt{3}, 0) \)
Answer: (d) \( (2\sqrt{3}, 0) \)

Question. Which of the following can be the possible orthocentre of the triangle
(a) \( (1, \sqrt{3}) \)
(b) \( (0, \sqrt{3}) \)
(c) (0, 2)
(d) (0, 3)
Answer: (a) \( (1, \sqrt{3}) \)

PASSAGE : III
Let \( A(0, 0), B(5, 0), C(5, 3) \) and \( D(0, 3) \) be the vertices of rectangle ABCD. If P is a variable point lying inside the rectangle ABCD and \( d(P, L) \) denotes perpendicular distance of point P from line L

Question. If \( d(P, AB) \le \min\{d(P, BC), d(P, CD), d(P, AD)\} \), then area of the region in which P lies is (in sq.units)
(a) 17/4
(b) 19/4
(c) 21/4
(d) 23/4
Answer: (c) 21/4

Question. If \( d(P, AB) \ge \max[d(P, BC), d(P, CD), d(P, AD)] \), then area of the region in which P lies is (in sq.units)
(a) 1
(b) 1/2
(c) 3/4
(d) 1/4
Answer: (d) 1/4

Question. If \( \left( d(P, AB) - \frac{3}{2} \right)^2 + (d(P, AD))^2 \ge 1 \), then area of region in which P lies is (in sq.units)
(a) \( 15 - 2\pi \)
(b) \( 10 - \frac{\pi}{2} \)
(c) \( 15 - \pi \)
(d) \( 15 - \frac{\pi}{2} \)
Answer: (d) \( 15 - \frac{\pi}{2} \)

PASSAGE : IV
The line \( 6x + 8y = 48 \) intersects the coordinate axes at A and B respectively. A line L bisects the area and perimeterof triangle OAB where ‘O’ is origin

Question. The number of possible such lines is
(a) 1
(b) 2
(c) 3
(d) more than 3
Answer: (a) 1

Question. The slope of the line L can be
(a) \( \frac{10 + 5\sqrt{6}}{10} \)
(b) \( \frac{10 - 5\sqrt{6}}{10} \)
(c) \( \frac{8 + 3\sqrt{6}}{10} \)
(d) \( \frac{8 - 3\sqrt{6}}{10} \)
Answer: (b) \( \frac{10 - 5\sqrt{6}}{10} \)

Question. The line L does not intersect the side____of the triangle OAB
(a) AB
(b) OB
(c) OA
(d) can intersect all sides
Answer: (c) OA

Matrix- Matching Questions

Question. A line cuts x - axis at A and y - axis at B such that \( AB = \ell \). Match the following
Column I
A) Circumcentre of \( \Delta OAB \)
B) Orthocentre of \( \Delta OAB \)
C) Incentre of the \( \Delta OAB \)
D) Centroid of the \( \Delta OAB \)
Column II (Locus of the point lies on)
p) \( x^2 + y^2 = \frac{\ell^2}{9} \)
q) \( x^2 + y^2 = \frac{\ell^2}{4} \)
r) \( x^2 + y^2 = 0 \)
s) \( y = x \)
Answer: A-q; B-r; C-s; D-p

Question. Vertex A of the triangle ABC is at origin. The equation of medians through B and C are \( 15x - 4y - 240 = 0 \) and \( 15x - 52y + 240 = 0 \) respectively
Column I
A) The coordinate of incentre of triangle ABC are
B) The coordinates of centroid of triangle ABC are
C) The coordinates of excentre opposite to vertex C of triangle ABC are
D) The coordinates of orthocentre of triangle ABC are
Column II
p) \( \left( \frac{56}{3}, 10 \right) \)
q) (21, 12)
r) (12, 21)
s) (-4, 7)
t) (0, 63)
Answer: A-q; B-p; C-s; D-t

Question. Match Column I with Column II
Column I
A) The number of integral values ‘a’ for which point \( (a, a^2) \) lies completely inside the triangle \( x = 0, y = 0, 2y + x = 3 \).
B) The number of values of a of the form \( \frac{k}{3} \) where \( k \in I \) so that point \( (a, a^2) \) lies between the lines \( x + y = 2 \) and \( 4x + 4y - 3 = 0 \)
C) The reflection of point \( (t - 1, 2t + 2) \) in a line is \( (2t + 1, t) \) then the slope of the line is
D) In a triangle ABC, the bisector of angles B and C lies along the lines \( y = x \) and \( y = 0 \). If A is (1, 2) then \( \sqrt{10}d(A, BC) \) equals (where \( d(A, BC) \) denotes the perpendicular distance of A from BC)
Column II
p) 0
q) 1
r) 2
s) 4
Answer: A-p; B-r; C-q; D-s

Question. Match Column I with Column II
Column I
A) If \( a, b, c \) are in A.P, then lines \( ax + by + c = 0 \) are concurrent
B) A point on the line \( x + y = 4 \) which lies at a unit distance from the line \( 4x + 3y = 10 \) is
C) Orthocentre of triangle made by lines \( x + y = 1, x - y + 3 = 0, 2x + y = 7 \) is
D) Two vertices of a triangle are (5, -1) and (-2, 3). If orthocentre is the origin then coordinate of the third vertex are
Column II
p) (-4, -7)
q) (-7, 11)
r) (1, -2)
s) (-1, 2)
t) (3, 1)
Answer: A-r; B-q, t; C-s; D-p

Question. Match Column I with Column II
Column I
A) If the lines \( x + 2ay + a = 0, x + 3by + b = 0, x + 4cy + c = 0 \), where \( a, b, c \in \mathbb{R} \) are concurrent, then \( a, b, c \) are in
B) The point with coordinates (2a, 3a), (3b, 2b), (c, c) where \( a, b, c \in \mathbb{R} \) are collinear, then \( a, b, c \) are in
C) If lines \( ax + 2y + 1 = 0, bx + 3y + 1 = 0 \) and \( cx + 4y + 1 = 0 \) where \( a, b, c \in \mathbb{R} \) passes through the same point, then \( a, b, c \) are in
D) Let \( a, b, c \) be distinct non-negative real numbers. If the lines \( ax + ay + c = 0, x + 1 = 0, cx + cy + b = 0 \) pass through the same point then \( a, b, c \) are in
Column II
p) A.P.
q) G.P.
r) H.P.
s) Neither A.P. nor G.P. nor H.P
Answer: A-r; B-s; C-p; D-q