Class 11 Mathematics Straight Lines MCQs Set 21

COMPREHENSION QUESTIONS

PASSAGE : I

Given point A (6, 30) and point B (24, 6) equation of line AB is \( 4x + 3y = 114 \). Point P (0, \( \lambda \)) is a point on y - axis such that \( 0 < \lambda < 38 \) and point Q (0, k) is a point on y axis such that \( k > 38 \).

Question. For all positions of point P, angle APB is maximum when point P is
(a) (0, 12)
(b) (0, 15)
(c) (0, 18)
(d) (0, 21)
Answer: (c) (0, 18)

Question. The maximum value of angle APB is
(a) \( \frac{\pi}{3} \)
(b) \( \frac{\pi}{2} \)
(c) \( \frac{2\pi}{3} \)
(d) \( \frac{3\pi}{4} \)
Answer: (b) \( \frac{\pi}{2} \)

Question. For all position of point Q, angle AQB is maximum when point Q is:
(a) (0, 54)
(b) (0, 58)
(c) (0, 60)
(d) (0, 1)
Answer: (b) (0, 58)

PASSAGE : II

Consider a variable line ‘L’ which passes through the point of intersection P of the lines \( 3x + 4y - 12 = 0 \) and \( x + 2y - 5 = 0 \) meeting the coordinate axes at point A and B.

Question. Locus of the middle point of the segment AB has the equation
(a) \( 3x + 4y = 4xy \)
(b) \( 3x + 4y = 3xy \)
(c) \( 4x + 3y = 4xy \)
(d) \( 4x + 3y = 3xy \)
Answer: (a) \( 3x + 4y = 4xy \)

Question. Locus of the feet of the perpendicular from the origin on the variable line L has the equation
(a) \( 2(x^2 + y^2) - 3x - 4y = 0 \)
(b) \( 2(x^2 + y^2) - 4x - 3y = 0 \)
(c) \( x^2 + y^2 - 3x - y = 0 \)
(d) \( x^2 + y^2 - x - 2y = 0 \)
Answer: (b) \( 2(x^2 + y^2) - 4x - 3y = 0 \)

Question. Locus of the centroid of the variable triangle OAB has the equation (where O is origin)
(a) \( 3x + 4y + 6xy = 0 \)
(b) \( 4x + 3y - 6xy = 0 \)
(c) \( 3x + 4y - 6xy = 0 \)
(d) \( 4x + 3y + 6xy = 0 \)
Answer: (c) \( 3x + 4y - 6xy = 0 \)

PASSAGE : III

Let a and b be the lengths of the legs of a right triangle with following properties
(i) All 3 sides of the triangle are integers
(ii) the perimeter of the triangle is numerically equal to area of the triangle, it is given that a < b

Question. The number of ordered pairs (a, b) will be
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (b) 2

Question. Maximum possible perimeter of the triangle is
(a) 27
(b) 28
(c) 29
(d) 30
Answer: (d) 30

Question. Minimum possible area of the triangle is
(a) 24
(b) 25
(c) 26
(d) 27
Answer: (a) 24

PASSAGE : IV

Consider three 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 \( H(x, y) \) are the coordinates of the orthocentre of triangle ABC whose area be denoted by \( \Delta \).

Question. If D, E, F are the middle points of 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 is
(a) 3
(b) 12
(c) 13
(d) None of the options
Answer: (b) 12

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

PASSAGE : V

An equilateral triangle ABC has its centroid at the origin and one side ‘AC’ is \( x + y = 1 \)

Question. If \( A(x_1, y_1), B(x_2, y_2), C(x_3, y_3) \) be the vertices of the equilateral triangle. Then \( x_1x_2 + x_2x_3 + x_3x_1 = \)
(a) -1
(b) -3/2
(c) 1
(d) None of the options
Answer: (b) -3/2

Question. A point P is taken on the minor arc AC of the circumcircle of triangle ABC. Then maximum value of (PA + PB + PC) is equal to
(a) \( 4\sqrt{2} \)
(b) \( \sqrt{2} \)
(c) \( 3\sqrt{2} \)
(d) \( 9/\sqrt{2} \)
Answer: (a) \( 4\sqrt{2} \)

Question. Another point ‘\( P_1 \)’ is taken in the plane of ABCP. (P being PA+PB+PC is maximum). If \( P_1A + P_1B + P_1C + P_1P = \sqrt{6} + \sqrt{8} \), then locus of \( P_1 \) is
(a) square
(b) circle
(c) a point
(d) None of the options
Answer: (d) None of the options

PASSAGE : VI

Let the lines represented by the equation \( x^2y^2 - x^2 - y^2 + 1 = 0 \) form a square ABCD. A point ‘P’ is taken on the same plane of square such that atleast two sides of all the triangles PAB, PBC, PCD and PDA are equal.

Question. The number of possible position of the point ‘P’
(a) 1
(b) 5
(c) 17
(d) 9
Answer: (d) 9

Question. For all the possible position of point ‘P’. The number of the given triangles have all the three sides equal
(a) 0
(b) 4
(c) 8
(d) 5
Answer: (c) 8

Question. One of the possible position of the ‘P’ such that atleast one of the given triangle is equilateral is given by
(a) (0, 0)
(b) \( (\sqrt{3} + 1, 0) \)
(c) \( (0, \sqrt{3} - 2) \)
(d) \( (\sqrt{3} + 1, \sqrt{3} - 1) \)
Answer: (b) \( (\sqrt{3} + 1, 0) \)

PASSAGE : VII

Consider \( \Delta ABC \) with incentre \( I(1,0) \). Equations of the straight lines AI, BI, CI are \( x=1 \), \( y+1=x \) and \( x+3y=1 \) respectively and \( \cot \frac{A}{2} = 2 \)

Question. Equation to the locus of centroid of \( \Delta ABC \) is
(a) \( x = 8y + 1 \)
(b) \( x = 2y + 1 \)
(c) \( 2x - 3y = 1 \)
(d) \( 3x = 5y \)
Answer: (b) \( x = 2y + 1 \)

Question. Slope of side BC is
(a) 1/2
(b) -1/3
(c) 1/4
(d) 2/3
Answer: (a) 1/2

Question. If point A lies above the x-axis and area of \( \Delta ABC \) is 30 sq, units, then the in radius of \( \Delta ABC \) is
(a) \( \sqrt{3} \)
(b) \( \sqrt{5} \)
(c) 2
(d) \( 3\sqrt{2} \)
Answer: (b) \( \sqrt{5} \)

MATRIX MATCHING QUESTIONS

Question. Match the following
Column I
A) Lines \( x - 2y - 6 = 0 \), \( 3x + y - 4 = 0 \) and \( \lambda x + 4y + \lambda^2 = 0 \) are concurrent. The value of \( \lambda \) is
B) The points \( (\lambda + 1, 1), (2\lambda + 1, 3) \) and \( (2\lambda + 2, 2\lambda) \) are collinear, then the value of \( \lambda \) is
C) If line \( x + y - 1 - \lambda = 0 \), passing through the intersections of \( x - y + 1 = 0 \) and \( 3x + y - 5 = 0 \) is perpendicular to one of them then the value of \( \lambda \) is
D) If the line \( y - x - 1 + \lambda = 0 \) is equally inclined to axes and equidistant from the points (1, -2) and (3, 4) then \( \lambda \) is
Column II
p) 2
q) -4
r) -1/2
s) 1
Answer: A-p,q; B-p,r; C-p; D-p

Question. Reflection of the line \( x + y + 1 = 0 \) in the line.
Column I
A) \( 2x + y + 1 = 0 \)
B) \( x - 2y + 1 = 0 \)
C) \( x + 2y - 1 = 0 \)
D) \( 2x + y - 1 = 0 \)
Column II
p) \( x + 7y - 11 = 0 \)
q) \( 7x + y + 1 = 0 \)
r) \( 7x + y - 11 = 0 \)
s) \( 7x + y + 7 = 0 \)
Answer: A-q; B-s; C-p; D-r

Question. Vertex A of the \( \Delta ABC \) 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 coordinates of incentre of \( \Delta ABC \) are
b) The coordinates of centroid of \( \Delta ABC \) are
c) The coordinates of excenter opposite to vertex A are
d) The coordinates of orthocentre of \( \Delta ABC \) 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. Consider the line given by \( L_1 = x + 3y - 5 = 0 \), \( L_2 = 3x - ky - 1 = 0 \), match the statements / expressions in Column- I with the statements / expressions in Column-II. [IIT-JEE 2008]
Column I
a) \( L_1, L_2, L_3 \) are concurrent, if
b) One of \( L_1, L_2, L_3 \) is parallel to at least one of the other two, if
c) \( L_1, L_2, L_3 \) form a triangle, if
d) \( L_1, L_2, L_3 \) do not form a triangle, if
Column II
p) k = -9
q) k = -6/5
r) k = 5/6
s) k = 5
Answer: A-s; B-p,q; C-r; D-p,q,s

Question. In triangle ABC the equation of side BC is \( 2x - y = 3 \) and its circumcentre and orthocentre are at (2, 4) and (1, 2) respectively then match the items of column I with items of column II
Column I
(a) Tan B Tan C
(b) The length of BC
(c) Sin A
(d) Area of the triangle is
Column II
p) \( \frac{4\sqrt{13}}{5} \)
q) \( \frac{18\sqrt{13}}{5} \)
r) 3
s) \( \frac{2\sqrt{13}}{\sqrt{61}} \)
Answer: A-r, B-p, C-s, D-q

Question. The lines \( L_1 : y - x = 0 \) and \( L_2 : 2x + y = 0 \) intersect the line \( L_3 : y + 2 = 0 \) at P and Q respectively. The bisector of the acute angle between \( L_1 \) and \( L_2 \) intersect \( L_3 \) at R
Statement - 1 : The ratio PR : RQ equals \( 2\sqrt{2} : \sqrt{5} \).
Statement - 2 : In any triangle, bisector of an angle divides the triangle into two similar triangles. [AIEEE - 2011]
(a) Statement – 1 is true, Statement – 2 is true; Statement – 2 is not a correct explanation for Statement – 1
(b) Statement – 1 is true, Statement– 2 is false.
(c) Statement – 1 is false, Statement– 2 is true.
(d) Statement – 1 is true, Statement – 2 is true; Statement – 2 is a correct explanation for Statement – 1
Answer: (b) Statement – 1 is true, Statement– 2 is false.

Question. Observe the following list with respect to the line \( ax+by+c=0 \)
List I
A) Perpendicular distance from (0,0)
B) X-intercept of the line
C) Y-intercept of the line
D) Circumcentre of triangle OAB where A,B are X and Y intercepts
List II
1) \( -c/b \)
2) \( (-\frac{c}{a}, -\frac{c}{b}) \)
3) \( \frac{|c|}{\sqrt{a^2+b^2}} \)
4) \( [ -\frac{c}{2a}, -\frac{c}{2b} ] \)
5) \( -c/a \)
(a) A-3, B-5, C-1, D-2
(b) A-3, B-5, C-1, D-4
(c) A-3, B-4, C-1, D-5
(d) A-1, B-2, C-3, D-4
Answer: (b) A-3, B-5, C-1, D-4

Question. observe the following
column I
A) the area bounded by the curve \( \max\{|x|, |y|\} = 1 \) is
B) if the point (a,a) lies between the lines \( |x + y| = 3 \) then number of values of \( [a] \) is (where [.] denotes the greatest integer function)
C) Number of integral values of b for which the origin and the point(1,1) lie on the same side of the st.line \( a^2x + aby + 1 = 0 \) for all \( a \in R \sim \{0\} \)
column II
P) 3
Q) 2
R) 4
(a) A \(\rightarrow\) R, B \(\rightarrow\) Q, C \(\rightarrow\) P
(b) A \(\rightarrow\) R, B \(\rightarrow\) P, C \(\rightarrow\) Q
(c) A \(\rightarrow\) P, B \(\rightarrow\) Q, C \(\rightarrow\) R
(d) A \(\rightarrow\) P, B \(\rightarrow\) R, C \(\rightarrow\) Q
Answer: (a) A \(\rightarrow\) R, B \(\rightarrow\) Q, C \(\rightarrow\) P

Question. Equation of line passing through (1,3), perpendicular to \( 2x-3y+4 = 0 \) is \( ax+by+c = 0 \) (a>0) then ascending order of a, b, c is
(a) a, c, b
(b) c, b, a
(c) c, a, b
(d) a, b, c
Answer: (b) c, b, a