Class 11 Mathematics Straight Lines MCQs Set 19

Question. If the point \( P(a^2, a) \) lies in the region corresponding to the acute angle between the lines \( 2y = x \) and \( 4y = x \), then
(a) \( a \in [2, 4] \)
(b) \( a \in (2, 4] \)
(c) \( a \in [2, 5) \)
(d) \( a \in (2, 4) \)
Answer: (d) \( a \in (2, 4) \)

Question. The set of values of ‘b’ for which the origin and the point (1,1) lie on the same side of the straight line \( a^2x + aby + 1 = 0 \), \( \forall a \in R, b > 0 \) are
(a) \( b \in (2, 4) \)
(b) \( b \in (0, 2) \)
(c) \( b \in [0, 2] \)
(d) \( b \in [0, 3] \)
Answer: (b) \( b \in (0, 2) \)

Question. The equations of sides of a triangle are \( 7x - 5y - 11 = 0 \), \( 8x + 3y + 31 = 0 \), \( x + 8y - 19 = 0 \). Then the point (0,0) lies
(a) inside of triangle
(b) outside of triangle
(c) on the triangle
(d) can’t say
Answer: (a) inside of triangle

Question. If the points of intersection of lines \( L_1 : y - m_1x - k = 0 \) and \( L_2 : y - m_2x - k = 0 \) \( (m_1 \neq m_2) \) lies inside the triangle formed by the lines \( 2x + 3y = 1 \), \( x + 2y = 3 \) and \( 5x - 6y - 1 = 0 \), then true set of values of \( k \) are
(a) \( \left( \frac{1}{3}, \frac{3}{2} \right) \)
(b) \( \left( \frac{1}{2}, 1 \right) \)
(c) \( \left( 0, \frac{3}{2} \right) \)
(d) \( \left( \frac{-3}{2}, 0 \right) \)
Answer: (a) \( \left( \frac{1}{3}, \frac{3}{2} \right) \)

Question. The range of value of \( \alpha \) such that \( (0, \alpha) \) lies on or inside the triangle formed by the lines \( y + 3x + 2 = 0 \), \( 3y - 2x - 5 = 0 \), \( 4y + x - 14 = 0 \) is
(a) \( 5 < \alpha \leq 7 \)
(b) \( \frac{1}{2} \leq \alpha \leq 1 \)
(c) \( \frac{5}{3} \leq \alpha \leq \frac{7}{2} \)
(d) \( \frac{1}{3} \leq \alpha \leq \frac{1}{2} \)
Answer: (c) \( \frac{5}{3} \leq \alpha \leq \frac{7}{2} \)

Question. The lines \( x + y = a \), and \( ax - y = 1 \) intersect each other in the first quadrant then the set of all possible values of a is the interval [ AIEEE-2011]
(a) \( (0, \infty) \)
(b) \( (1, \infty) \)
(c) \( (-1, \infty) \)
(d) \( [-1, 1] \)
Answer: (b) \( (1, \infty) \)

Question. Let a, b, c and d be non zero numbers. If the point of intersection of the lines \( 4ax + 2ay + c = 0 \) and \( 5bx + 2by + d = 0 \) lies in the fourth quadrant and is equidistance from the two axes then
(a) \( 2bc - 3ad = 0 \)
(b) \( 2bc + 3ad = 0 \)
(c) \( 3bc - 2ad = 0 \)
(d) \( 3bc + 2ad = 0 \)
Answer: (c) \( 3bc - 2ad = 0 \)

Question. If the lines \( x + ay + a = 0 \), \( bx + y + b = 0 \), \( cx + cy + 1 = 0 \) (a, b, c being distinct and \( \neq 1 \)) are concurrent then the value of \( \frac{a}{a-1} + \frac{b}{b-1} + \frac{c}{c-1} = \)
(a) -1
(b) 0
(c) 1
(d) 3
Answer: (c) 1

Question. If \( 4a^2 + 9b^2 - c^2 + 12ab = 0 \) then the family of straight lines \( ax + by + c = 0 \) is concurrent at
(a) (2, 3) or (-2, -3)
(b) (2, -3) or (-2, 6)
(c) (-2, -4) or (-2, 3)
(d) (2, 5) or (-1, -5)
Answer: (a) (2, 3) or (-2, -3)

Question. If \( a^2 - b^2 - c^2 - 2bc = 0 \), then the family of lines \( ax + by + c = 0 \) are concurrent at the points
(a) (1, -1)
(b) (-1, 1)
(c) (1, 0)
(d) (-1, -1)
Answer: (b) (-1, 1)

Question. If \( t_1 \neq t_2 \neq t_3 \) and the lines \( t_1x + y = 2at_1 + at_1^3 \); \( t_2x + y = 2at_2 + at_2^3 \); \( t_3x + y = 2at_3 + at_3^3 \) are concurrent then \( t_1 + t_2 + t_3 \) is
(a) 0
(b) -1
(c) 1
(d) 2
Answer: (a) 0

Question. Consider the family of lines \( (x + y - 1) + \lambda(2x + 3y - 5) = 0 \) and \( (3x + 2y - 4) + \mu(x + 2y - 6) = 0 \), equation of a straight line that belongs to both the families is
(a) \( x - 2y - 8 = 0 \)
(b) \( x - 2y + 8 = 0 \)
(c) \( 2x + y - 8 = 0 \)
(d) \( 2x - y - 8 = 0 \)
Answer: (b) \( x - 2y + 8 = 0 \)

Question. If a, b and c are three consecutive odd integers then the variable line \( ax + by + c = 0 \) always passes through
(a) (2, 1)
(b) (1, 2)
(c) (-1, 2)
(d) (1, -2)
Answer: (d) (1, -2)

Question. One vertex of an equilateral triangle is (2,3) and the equation of one side is x-y+5=0 then the equations to the other sides are
(a) \( y - 3 = -(2 \pm \sqrt{3})(x - 2) \)
(b) \( y - 3 = (\sqrt{2} \pm 1)(x - 2) \)
(c) \( y - 3 = (\sqrt{3} \pm 1)(x - 2) \)
(d) \( y - 3 = (\sqrt{5} \pm 1)(x - 2) \)
Answer: (a) \( y - 3 = -(2 \pm \sqrt{3})(x - 2) \)

Question. Let P (2, -4) and Q (3, 1) be two given points. Let R (x, y) be a point such that \( (x - 2)(x - 3) + (y - 1)(y + 4) = 0 \). If area of \( \Delta PQR \) is \( \frac{13}{2} \), then the number of possible positions of R are
(a) 2
(b) 3
(c) 4
(d) 6
Answer: (a) 2

Question. If the base of an isosceles triangle is of length 2P and the length of the altitude dropped to the base is q, then the distance from the mid point of the base to the side of the triangle is
(a) \( \frac{pq}{\sqrt{p^2 + q^2}} \)
(b) \( \frac{2pq}{\sqrt{p^2 + q^2}} \)
(c) \( \frac{3pq}{\sqrt{p^2 + q^2}} \)
(d) \( \frac{4pq}{\sqrt{p^2 + q^2}} \)
Answer: (a) \( \frac{pq}{\sqrt{p^2 + q^2}} \)

Question. If \( m_1 \) and \( m_2 \) are the roots of the equation \( x^2 - ax - a - 1 = 0 \), then the area of the triangle formed by the three straight lines \( y = m_1x \), \( y = m_2x \) and \( y = a \) \( (a \neq -1) \) is
(a) \( \frac{-a^2(a+2)}{2(a+1)} \) if \( a > -1 \)
(b) \( \frac{+a^2(a+2)}{2(a+1)} \) if \( a < -1 \)
(c) \( \frac{-a^2(a+2)}{2(a+1)} \) if \( -2 < a < -1 \)
(d) \( \frac{a^2(a+2)}{2(a+1)} \) if \( a > -2 \)
Answer: (c) \( \frac{-a^2(a+2)}{2(a+1)} \) if \( -2 < a < -1 \)

Question. The equation of a straight line L is x+y=2, and \( L_1 \) is another straight line perpendicular to L and passes through the point (1/2, 0), then area of the triangle formed by the y-axis and the lines L, \( L_1 \) is
(a) 25/8
(b) 25/16
(c) 25/4
(d) 25/12
Answer: (b) 25/16

Question. In an isosceles triangle OAB, O is the origin and OA = OB = 6. The equation of the side AB is x-y+1=0. Then the area of the triangle is
(a) \( 2\sqrt{21} \)
(b) \( \sqrt{142} \)
(c) \( \frac{\sqrt{142}}{2} \)
(d) \( \frac{\sqrt{71}}{2} \)
Answer: (d) \( \frac{\sqrt{71}}{2} \)

Question. An equilateral triangle is constructed between two parallel lines \( \sqrt{3}x + y - 6 = 0 \) and \( \sqrt{3}x + y + 9 = 0 \) with base on one and vertex on the other. Then the area of triangle is
(a) \( \frac{200}{\sqrt{3}} \)
(b) \( \frac{225}{4\sqrt{3}} \)
(c) \( \frac{225}{\sqrt{3}} \)
(d) \( \frac{200}{4\sqrt{3}} \)
Answer: (b) \( \frac{225}{4\sqrt{3}} \)

Question. Area of triangle formed by the lines \( 2x + y - 3 = 0 \), \( x + 4y - 5 = 0 \) and \( 3x + 5y - 1 = 0 \) is
(a) 15/2
(b) 49/2
(c) 27/56
(d) 7/2
Answer: (d) 7/2

Question. If \( f(x + y) = f(x) f(y) \) for all x and y if \( f(1) = 2 \), then area enclosed by \( 3|x| + 2|y| \leq 8 \) is
(a) \( f(5) \) sq.units
(b) \( f(6) \) sq.units
(c) \( 1/3 f(6) \) sq.units
(d) \( f(4) \) sq.units
Answer: (c) \( 1/3 f(6) \) sq.units

Question. Four sides of a quadrilateral are given by the equation \( xy(x - 2)(y - 3) = 0 \), then the equation of the line parallel to \( x - 4y = 0 \) that divides the quadrilateral into two equal parts is
(a) \( x - 4y + 5 = 0 \)
(b) \( x - 4y - 5 = 0 \)
(c) \( x - 4y + 1 = 0 \)
(d) \( x - 4y - 1 = 0 \)
Answer: (a) \( x - 4y + 5 = 0 \)

Question. \( L_1 \) and \( L_2 \) are two intersecting lines and the angle between the image of \( L_1 \) w.r.t \( L_2 \) and that of \( L_2 \) w.r.t. \( L_1 \) is \( 45^\circ \). Then the angle between \( L_1 \) and \( L_2 \) is
(a) \( 20^\circ \)
(b) \( 15^\circ \)
(c) \( 45^\circ \)
(d) \( 60^\circ \)
Answer: (b) \( 15^\circ \)

Question. \( L_1 \) and \( L_2 \) are two intersecting lines. If the image of \( L_1 \) w.r.t. \( L_2 \) and that of \( L_2 \) w.r.t. \( L_1 \) coincide, then the angle between \( L_1 \) and \( L_2 \) is
(a) \( 35^\circ \)
(b) \( 60^\circ \)
(c) \( 90^\circ \)
(d) \( 45^\circ \)
Answer: (b) \( 60^\circ \)

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

Question. The combined equation of straight lines that can be obtained by reflecting the lines \( y = |x - 2| \) in the y-axis is
(a) \( y^2 + x^2 + 4x + 4 = 0 \)
(b) \( y^2 + x^2 - 4x + 4 = 0 \)
(c) \( y^2 - x^2 + 4x - 4 = 0 \)
(d) \( y^2 - x^2 - 4x - 4 = 0 \)
Answer: (d) \( y^2 - x^2 - 4x - 4 = 0 \)

Question. In \( \Delta ABC \), B=(0, 0), AB=2, \( \angle ABC = \frac{\pi}{3} \) and the middle point of BC has the co-ordinates (2, 0). Then the centroid of triangle is
(a) \( \left( \frac{5}{3}, \frac{1}{3} \right) \)
(b) \( \left( \frac{5}{3}, \frac{1}{\sqrt{3}} \right) \)
(c) \( \left( \frac{5}{\sqrt{3}}, \frac{1}{3} \right) \)
(d) \( \left( \frac{5}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right) \)
Answer: (b) \( \left( \frac{5}{3}, \frac{1}{\sqrt{3}} \right) \)

Question. In triangle ABC, co-ordinates of A are \( (-1, 3) \) and equation of medians and altitude through point B are \( 2x + y = 8 \) and \( 2x + 3y = 8 \) respectively, then
(a) coordinates of C are (4,0)
(b) coordinates of C are (3,9)
(c) coordinates of C are (3,3)
(d) coordinates of centroid are (2,2)
Answer: (b) coordinates of C are (3,9)

Question. The sides of a triangle are x+y=1, 7y = x and \( \sqrt{3}y + x = 0 \). Then the following is an interior point of the triangle
(a) Circumcentre
(b) Centroid
(c) Orthocentre
(d) Cannot say
Answer: (b) Centroid

Question. If the equations of the sides of a triangle are \( 2x + y = 2 \), \( y = x \), \( \sqrt{3}y + x = 0 \) then which of the following is an exterior point of triangle.
(a) orthocentre
(b) incentre
(c) centroid
(d) Cannot say
Answer: (a) orthocentre

Question. One vertex of the equilateral triangle with centroid at the origin and one side as x+y-2=0 is
(a) \( (-2, -2) \)
(b) (2,2)
(c) (2, -2)
(d) (-2,2)
Answer: (a) \( (-2, -2) \)

Question. A ray of light is sent along the line x–2y–3=0. On reaching the line 3x–2y–5=0, the ray is reflected from it. The equation of the line containing the the reflected ray.
(a) \( 29x - 2y + 31 = 0 \)
(b) \( 29x + 2y - 31 = 0 \)
(c) \( 29x - 2y - 31 = 0 \)
(d) \( 29x + 2y + 31 = 0 \)
Answer: (c) \( 29x - 2y - 31 = 0 \)

Question. A light ray coming along the line \( 3x + 4y = 5 \) gets reflected from the line \( ax + by = 1 \) and goes along the line \( 5x - 2y = 10 \). Then,
(a) \( a = \frac{64}{115}, b = \frac{112}{15} \)
(b) \( a = \frac{14}{15}, b = \frac{-8}{115} \)
(c) \( a = \frac{64}{115}, b = \frac{-8}{115} \)
(d) \( a = \frac{64}{15}, b = \frac{14}{15} \)
Answer: (c) \( a = \frac{64}{115}, b = \frac{-8}{115} \)

Question. If \( x_1, y_1 \) are roots of \( x^2 + 8x - 20 = 0 \), \( x_2, y_2 \) are the roots of \( 4x^2 + 32x - 57 = 0 \) and \( x_3, y_3 \) are the roots of \( 9x^2 + 72x - 112 = 0 \), then the points \( (x_1, y_1), (x_2, y_2) \) and \( (x_3, y_3) \) where \( x_i < y_i \) for i =1,2,3
(a) are collinear
(b) form an equilateral triangle
(c) form a right angled isosceles triangle
(d) are concyclic
Answer: (a) are collinear

Question. Triangle is formed by the coordinates (0, 0), (0, 21) and (21, 0). The number of integral coordinates strictly inside triangle (integral coordinates has both x and y as integers) :
(a) 190
(b) 105
(c) 231
(d) 205
Answer: (a) 190

Question. Origin is the centre of the square with one of its vertices at (3,4) then the other vertices are
(a) (-3, 4), (-3, -4), (3, -4)
(b) (-4, 3), (-3, -4), (4, -3)
(c) (-4, 3), (-4, -3), (3, -4)
(d) (3, 4), (-4, -3), (4, -3)
Answer: (b) (-4, 3), (-3, -4), (4, -3)

Question. One side of a rectangle lies along the line 4x+7y+5=0. Two vertices are (-3,1), (1,1) then the remaining vertices are
(a) \( \left( \frac{1}{65}, \frac{-47}{65} \right), \left( \frac{-131}{65}, \frac{177}{65} \right) \)
(b) \( \left( \frac{-1}{65}, \frac{47}{65} \right), \left( \frac{-131}{65}, \frac{177}{65} \right) \)
(c) \( \left( \frac{1}{65}, \frac{-47}{65} \right), \left( \frac{131}{65}, \frac{-177}{65} \right) \)
(d) (1, -47), (131, 47)
Answer: (a) \( \left( \frac{1}{65}, \frac{-47}{65} \right), \left( \frac{-131}{65}, \frac{177}{65} \right) \)

Question. All points lying inside the triangle formed by the points (1,3), (5,0), (-1,2) satisfy
(a) \( 2x + y - 13 = 0 \)
(b) \( 3x + 2y \geq 0 \)
(c) \( 3x - 4y - 12 \leq 0 \)
(d) \( 4x + y = 0 \)
Answer: (b) \( 3x + 2y \geq 0 \)

Question. If one vertex of an equilateral triangle of side a lies at the origin and the other lies on the line \( x - \sqrt{3}y = 0 \), the coordinates of the third vertex are
(a) (0, –a)
(b) (a, 0)
(c) \( \left( \frac{a\sqrt{3}}{2}, \frac{a}{2} \right) \)
(d) \( \left( \frac{a\sqrt{3}}{2}, -\frac{a}{2} \right) \)
Answer: (d) \( \left( \frac{a\sqrt{3}}{2}, -\frac{a}{2} \right) \)