www.vineetloomba.com Class XI
P REPARED BY : E R . V INEET L OOMBA EC H . IIT R OORKEE ) (B.T ECH
ALP ADVANCED LEVEL PROBLEMS
TARGET TA RGET : JEE Main/Adv
Straight Lines
60 Best JEE Main and Advanced Level Problems (IIT-JEE). Prepared by IITians. Straight Lines-ADVANCED LEVEL PROBLEMS
Q.1
[STRAIGHT OBJECTIVE TYPE] A variab variable le rectan rectangle gle PQRS has its sides sides parallel parallel to to fixed fixed directions. directions. Q and and S lie respectivel respectively y on the lines lines x = a, a, x = P lies on the x axis. Then the locus of R is (A) a straight line (B) a circle (C) a parabola (D) pair of straight lines
a and
Q.2
A, B and and C are points points in the the xy xy plane such that A(1, 2) ; B (5, 6) and AC = 3BC. 3BC. Then Then (A) ABC is a unique triangle (B) There can be only two such triangles. (C) (C) No suc such tria triang ngle le is poss possib ible le (D) (D) The There can can be infi infin nite ite num number ber of suc such tria trian ngles. les.
Q.3
If A (1, p2) ; B (0, 1) and C (p, 0) are the coordinates of three points then the value of p for which the area of the triangle ABC is minimum, is (A)
1 3
(B) –
1 3
(C)
1 3
or –
1 3
(D) none
Q.4
Each Each membe memberr of the family family of parabo parabolas las y = ax 2 + 2x + 3 has a maximum or a minimum point depending upon the value of a. The equation to the locus of the maxima or minima for all possible values of 'a' is (A) a straight line with slope 1 and y intercept 3. (B) a straight line with slope 2 and y intercept 2. (C) a straight line with slope 1 and x intercept 3. (D) a circle
Q.5
m, n are integer with 0 < n < m. A is the point (m, n) n) on the cartesian plane. plane. B is the reflection reflection of A in the line y = x. C is the 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 ABCDE is (A) 2m(m + n) (B) m(m + 3n) (C) m(2m + 3n) (D) 2m(m + 3n)
Q.6
The area enclosed enclosed by the graphs of | x + y | = 2 and | x | = 1 is (A) 2 (B) 4 (C) 6
(D) 8
Q.7
If P = (1, (1, 0) ; Q = ( 1, 0) and R = (2, 0) are three given points, then the locus of the points S satisfying the relation, SQ2 2 + SR = 2 SP2 is : (A) a straight line parallel to xaxis (B) a circle passing through the origin (C) (C) a circ circle le with ith the the cent centre re at the the orig origin in (D) (D) a stra straig ight ht line line para parall llel el to y axis .
Q.8
Two poin points ts A(x1, y 1) and B(x2, y2) are chosen on the graph of f (x) = l n x with 0 < x 1 < x2. The points C and D trisect line segment AB with AC < CB. Through C a horizontal line is drawn to cut the curve at E( x 3, y3). If x1 = 1 and x2 = 1000 then the value of x 3 equals (A) 10
Q.9
Q.10
(B )
10
(C) (10)2/3
(D) (10)1/3
What is the y-intercept of the line that that is parallel to y = 3x, and which which bisects the the area of a rectangle with corners at (0, 0), (4, 0), (4, 2) and (0, 2)? (A) (0, – 7) (B) (0, – 6) (C) (0, – 5) (D) (0, – 4) Given A (1, 1) and AB AB is any line through it cutting the x-axis in B. If AC is perpendicular to AB and meets meets the y-axis in
Class (XI)
2
C, then the equation of locus of mid- point P of BC is (A) x + y = 1 (B ) x + y = 2 (C) x + y = 2xy Q.11
(D) 2x + 2y = 1
AB is the diameter of a semicircle k, C is an arbitrary point on the semicircle semicircle (other than than A or B) and S is the centre of the circle inscribed into triangle ABC, then measure of (A) angle ASB changes as C moves on k. (B) angle ASB is the same for all positions of C but it cannot be determined without knowing the radius. (C) angle angle ASB = 135° 135° for for all all C. C. (D) angle angle ASB = 150° 150° for for all all C.
For more such free Assignments visit https://vineetloomba.com Q.12
Given
x
y
= 1 and ax + by = 1 are two variable lines, 'a' and 'b' being the parameters connected connected by the relation a 2 + b2 =
a b
ab. The locus of the point of intersection has the equation (A) x2 + y2 + xy 1 = 0 (B) x2 + y2 – xy + 1 = 0 (C) x2 + y2 + xy + 1 = 0 (D) x2 + y2 – xy – 1 = 0
y + = 0, where 2 + 2 = 2, are concurrent then (C) = – 1, = ± 1 (D) = ± 1, = 1
Q.13
If the lines x + y + 1 = 0 ; 4x + 3y + 4 = 0 and x + (A) = 1, = – 1 (B) = 1, = ± 1
Q.14
Let (x1, y1) ; (x2, y2) and (x3, y3) are the vertices of a triangle ABC respectively. D is a point on BC such that BC = 3BD. The equation of the line through A and D, is
x (A) x1 x2
y y1 y2
1 x 1 + 2 x1 1 x3
y y1 y3
1 1 =0 1
x (B ) 3 x 1 x2
y y1 y2
1 x 1 + x1 1 x3
y y1 y3
1 1 =0 1
x (C) x1 x2
y y1 y2
1 x 1 + 3 x1 1 x3
y 1 y1 1 = 0 y3 1
x (D) 2 x1 x2
y y1 y2
1 x 1 + x1 1 x3
y 1 y1 1 = 0 y3 1
Q.15 Q.15
If the straig straight ht lines lines , ax + amy + 1 = 0 , b x + (m + 1) b y + 1 = 0 and cx + (m + 2)cy + 1 = 0, m 0 are concurrent then a, b, c are in : (A) A.P. only for m = 1 (B) A.P. for all m (C) G.P. for all m (D) H.P. for all m.
Q.16 Q.16
If in tria triang ngle le ABC ABC , A (1, 10) , circumcentre of side opposite to A is : (A) (1, 11/3)
Q.17
(B) (1, 5)
A is a point on either either of two lines y +
31 , 23
and orthocentre
,4 11 3 3
(C) (1, 3)
3 x = 2 at a distance of
then the co-ordinates of mid-point
(D) (1, 6)
4 3
units from their their point of intersection. The co-ordinates
of the foot of perpendicular from A on the bisector of the angle between them are (A)
2 3
, 2
(B) (0, 0)
(C)
2 , 2 3
(D) (0, 4)
Q.18 Q.18
Point Point 'P' lies lies on the line line l { { (x, y) | 3x + 5y = 15}. If 'P' is also equidistant from the coordinate axes, then P can be located in which of the four quadrants. (A) I only (B) II only (C) I or II only (D) IV only
Q.19
An equilatera equilaterall triang triangle le has has each each of of its sides of length length 6 cm . If (x 1, y 1) ; (x2, y 2) and (x3, y 3) are its vertices then the value of the determinant,
x1 x2 x3
y1 y2 y3
1 1 1
2
is equal equal to :
Prepared By: Er. Vineet Loomba (IIT Roorkee)
Jupiter (XI)
(A)
4 (B) 12
2 3
(C) 24
(D) 144
Q.32
Through Through a point point A on the x-axis x-axis a straight straight line line is drawn drawn parallel parallel to to y-axis y-axis so as as to meet meet the pair of straight straight lines lines ax 2 + 2hxy + by2 = 0 in B and C. If AB = BC then then 2 (A) h = 4ab (B) 8h2 = 9ab (C) 9h2 = 8ab (D) 4h2 = ab
Q.33
The equatio equation n of the pair pair of bisectors bisectors of the angles angles betwee between n two two straight straight lines lines is, 12x2 7xy 12y2 = 0 . If the equation equation of one line line is 2y x = 0 then the the equation equation of the other line line is : (A) 41x 38y = 0 (B) 11x + 2y = 0 (C) 38x + 41y = 0 (D) 11x – 2y = 0
Q.34
Consider Consider a quadrati quadraticc equatio equation n in Z with with parameters parameters x and and y as 2 2 Z – xZ + (x ( x – y) = 0 The parameters x and y are the co-ordinates of a variable point P w.r.t. an orthonormal co-ordinate system in a plane. If the quadratic equation has equal roots then the locus of P is (A) a circle (B) a line pair through the origin of co-ordinates with slope 1/2 and 2/3 (C) a line pair through the origin of co-ordinates with slope 3/2 and 2 (D) a line pair through the origin of co-ordinates with slope 3/2 and 1/2
Q.35 Q.35
The image image of the pair pair of lines lines represe represente nted d by by ax 2 + 2h xy + by 2 = 0 by the the line line mirro mirrorr y = 0 is 2 2 2 2 (A) (A) ax 2h xy by = 0 (B) bx 2h xy + ay = 0 2 2 (C) (C) bx + 2h xy + ay = 0 (D) ax2 2h xy + by2 = 0
Q.36
A r ea ea o f t he h e t ri ri a n g le le f or or me me d b y t he h e l in in e x + y = 2 2 x – y + 4y – 4 = 0 is (A) 1/2 (B ) 1 (C) 3/2
3 a n d t he h e a ng ng l e b is i s e ct ct or or s o f t he he l in in e p ai air (D) 2
Q.37 Q.37
The dista distanc ncee of the the point point P(x1, y1) from each of of the two two straight straight lines lines through through the origin origin is d. d. The equation equation of the two two straight straight lines is (A) (xy1 – yx1)2 = d 2(x2 + y 2) (B) d2(xy1 – yx1)2 = x 2 + y 2 (C) d2(xy1 + yx1)2 = x 2 + y 2 (D) (xy1 + yx1)2 = d 2(x2 + y 2)
Q.38
Let PQR be be a right angled angled isoscel isosceles es triangle, triangle, right right angled angled at P (2, 1). If the the equation equation of the line line QR is 2x 2x + y = 3, then then the equation representing the pair of lines PQ and PR is (A) 3x2 3y2 + 8x 8xy + 20 20x + 10 10y + 25 = 0 (B) 3x 2 3y2 + 8xy 8xy 20x 10y + 25 = 0 2 2 2 2 (C) 3x 3y + 8xy + 10 10x + 15 15y + 20 = 0 (D) 3x 3y 8xy 10x 15y 20 = 0
Q.39
The greatest greatest slope along the graph represented represented by the equation equation 4x 2 – y 2 + 2y – 1 = 0, is (A) – 3 (B ) – 2 (C) 2 (D) 3
Q.40
If the straight straight lines lines joining joining the the origin origin and and the points of intersec intersection tion of the curve curve 2 2 5x + 12xy 6y + 4x 2y + 3 = 0 and and x + ky ky 1 = 0 are equally equally inclined inclined to the co-ordin co-ordinate ate axes then then the value value of k : (A) is equal to 1 (B) is equal to 1 (C) is equal to 2 (D) does not exist in the set of real numbers .
Q.41
Vertices ertices of a parallel parallelogramABCD ogramABCD are A(3, 1), 1), B(13, 6), 6), C(13, 21) 21) and D(3, D(3, 16). If a line passing passing through through the the origin origin divides divides the parallelogram into two congruent parts then the slope of the line is (A)
11
(B)
12
11
(C)
8
25
(D)
8
13 8
[REASONING TYPE] (A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1. (B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1. (C) Statement-1 is true, statement-2 is false. (D) Statement-1 is false, statement-2 is true. Q.42 Q.42
Cons Consid ider er the the line lines, s, L1: Stateme Statement-1: nt-1: because
x
3
y 4
1;
L2 =
x 4
y 3
1;
L3 :
x 3
y 4
2 and L4 :
The quadrilatera quadrilaterall formed formed by these four lines is a rhombus. rhombus.
Prepared By: Er. Vineet Loomba (IIT Roorkee)
x 4
y 3
2
Jupiter (XI)
Statement-2: Statement-2: Q.43 .43
5 If diagonals of a quadrilateral formed by any four lines lines are unequal and intersect at right angle then it is a rhombus.
Give iven a ABC whose vertices are A(x 1, y 1) ; B(x2, y 2) ; C(x3, y 3). Let there exists a point P(a, b) such that 6a = 2x 1 + x2 + 3x3 ; 6b = 2y1 + y2 + 3y3
Q.44
Statement-1:
Area of triangle PBC must be less than the area of ABC
because Statement-2:
P lies inside the triangle ABC
Let points A, B, C are represented represented by (a cos
i, a sin i) i = 1, 2, 3 and
cos (1 – 2) + cos (2 – 3) + cos (3 – 1) = – Statement-1 Statement-1 : because Statement-2:
3 2
.
Orthocentre of ABC is at origin
ABC is equilateral triangle.
Q.45
Given the lines y + 2x = 3 and y + 2x = 5 cut the axes at A, B and and C, D respectively. respectively. Stateme Statement-1 nt-1 : ABDC forms quadrilater quadrilateral al and and point point (2, 3) 3) lies lies inside inside the quadrilate quadrilateral ral because Stateme Statement-2 nt-2 : Point lies on same same side of the lines. lines.
Q.46
Consider a triangle whose vertices are A(– 2, 1), B(1, 3) and C(3x, 2x – 3) where where x is a real number. number. Statement-1 Statement-1 : The area of the triangle ABC is independent of x because Statement-2 Statement-2 : The vertex C of the triangle ABC always moves on a line parallel to the base AB.
Q.47
Statement-1: Statement-1: Centroid of the triangle whose whose vertices are A(–1, 11); 11); B(– 9, – 8) and and C(15, – 2) lies on the internal angle angle bisector of the vertex A. because Stateme Statement-2: nt-2: Triangle Triangle ABC is isosceles isosceles with B and C as base angles. angles.
Q.48
Consider Consider the the line line L: L: = 3x + y + 4 = 0 and the points points A(–5, 6) and and B(3, 2) There is exactly one point on the line L which is equidistant from the point A and B. Statement-1: because Statement-2: The point A and B are on the different sides of the line.
Q.49
Consider Consider the following following statement statementss Stat Statem emen entt-1: 1: because State Stateme mentnt-2: 2:
The equa equati tion on x 2 + 2y2 –
two real lines lines on the cartesian plane. 2 3 x – 4y + 5 = 0 represents two
A genera generall equati equation on of degree degree two two 2 2 ax + 2hxy + by + 2gx + 2fy + c = 0 denotes a line pair if abc + 2fgh – af 2 – bg2 – ch2 = 0
For more such free Assignments visit https://vineetloomba.com [COMPREHENSION TYPE] Paragraph for Question Nos. 50 to 52 Consider a family of lines (4a + 3)x – (a + 1)y – (2a + 1) = 0 where a R Q.50
The locus of the foot of the perpendicular from the origin on each member member of this family, family, is (A) (2x – 1)2 + 4(y + 1)2 = 5 (B) (2x – 1)2 + (y + 1)2 = 5 (C) (2x + 1)2 + 4(y – 1)2 = 5 (D) (2x – 1)2 + 4(y – 1)2 = 5
Q.51
A membe memberr of this this famil family y with with positiv positivee gradien gradientt making making an angle angle of (A) 7x – y – 5 = 0
Q.52
(B) 4x – 3y + 2 = 0
4 with the line 3x – 4y = 2, is
(C) x + 7y = 15
(D) 5x – 3y – 4 = 0
Minimum area of the the triangle which a member member of this family with negative negative gradient gradient can make with with the positive semi semi axes, is (A) 8 (B) 6 (C) 4 (D) 2 Paragraph for Question Nos. 53 to 55
Prepared By: Er. Vineet Loomba (IIT Roorkee)
Jupiter (XI)
7
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STRAIGHT LINE Q.1
A
Q.2
D
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A,B
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Q.60 Q.60
(A) Q; (B) R; (C) S; (D) P
Prepared By: Er. Vineet Loomba (IIT Roorkee)
