IIT JEE MAINS & ADVANCED MATHEMATICS BY OM SIR

MASTERSHEET: Straight Line

EXERCISE # 1 Q.1

The cartesian coordinates of the points whose

Q.7

P and Q are points on the line joining A(– 2, 5) and B(3, 1) such that AP = PQ = QB then the mid point of PQ is (A) (1/2, 3) (B) (– 1/2, 4) (C) (2, 3) (D) (1, 4)

Q.8

The line segment joining the points (1, 2) and (–2, 1) is divided by the line 3x + 4y = 7 in the ratio (A) 3 : 4 (B) 4 : 3 (C) 9 : 4 (D) 4 : 9

Q.9

The line segment joining the points (–3, –4) and (1, –2) is divided by y-axis in the ratio (A) 1 : 3 (B) 2 : 3 (C) 3 : 1 (D) 3 : 2

Q.10

If m1 and m2 are roots of the equation

  polar coordinates are  – 5,–  equal to 4 

Q.2

Q.3

Q.4

 5 5   (A)  ,  2 2

 5 –5  (B)  ,  2 2

 –5 5   (C)  ,  2 2

 –5 –5  (D)  ,  2 2

The polar form of the equation x2 + y2 = ax is (A) r = a sin  (B) r = a cos  (C) r = – a sin  (D) None of these The abscissa of two points A, B are the roots of the equation x2 + 2ax – b2 = 0 and their ordinate are the roots of x2 + 2px – q2 = 0 then the distance AB in terms of a, b, p, q is (A)

a 2  b2  p2  q 2

(B)

a 2  b2  2p2q 2

(C) 2

a 2  b2  p2  q 2

1 (D) 2

a b p q 2

2

2

x2 + ( 3 + 2) x + ( 3 – 1) = 0 then the area of the triangle formed by the lines y = m1x, y = m2x and y = c is 2

The distance between point (2, 15º) & (1, 75º) is (A) 1

(B) 3

(C) 2

3

(D)

3

 33 – 11  2 c (A)    4  

 33  11  2 c (B)    4  

 33  11  2 c (C)    2  

 33 – 11  2 c (D)    2  

Q.5

The coordinates of base BC of an isosceles triangle ABC are given by B (1, 3) and C (–2, 7) which of the following points can be the possible coordinates of the vertex A. (A) (–7, 1/8) (B) (1, 6) (C) (–1/2, 5) (D) (–5/6, 6)

Q.11

The points (0, 1), (–2, 3), (6, 7) and (8, 3) are(A) Collinear (B) Vertices of parallelogram which is not a rectangle (C) Vertices of rectangle which is not a square (D) None of these

Q.6

If vertices of a quadrilateral are A(0, 0), B(3, 4), C(7, 7) and D (4, 3), then quadrilateral ABCD is (A) parallelogram (B) rectangle (C) square (D) rhombus

Q.12

The area of the pentagon whose vertices are (4, 1), (3, 6), (–5, 1), (–3, –3) and (–3, 0) is(A) 30 unit2 (B) 60 unit2 (C) 120 unit2 (D) none of these

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IIT JEE MAINS & ADVANCED MATHEMATICS BY OM SIR Q.13

Q.14

If the vertices of a triangle be (0, 0), (6, 0) and (6, 8), then its incentre will be(A) (2, 1) (B) (1, 2) (C) (4, 2) (D) (2, 4) The coordinates of the middle points of the sides of a triangle are (4, 2), (3, 3) and (2, 2) then the coordinates of its centroid are 7 (A)  3,   3 (C) (4, 3)

Q.15

Q.16

keeping the axis parallel the new coordinates of the point (5, –3) will be-

Q.21

(A) (–3, –2)

(B) (3, 2)

(C) (–7, 8)

(D) None of these

At what point the origin be shifted, if the coordinates of point (4, 5) becomes (–3, 9) (A) (7, –4)

(B) (–7, 4)

(D) None of these

(C) (–7, –4)

(D) None of these

The orthocentre of the triangle formed by the lines 4x – 7y +10 = 0, x + y = 5 and 7x + 4y = 15, is(A) (1, 2) (B) (1, –2) (C) (–1, –2) (D) (–1, 2)

Q.22

negative direction without shifting the origin . The new coordinates of the point are-

Q.23

The locus of the mid point of the portion intercept between the axes by the line x cos + y sin = P where P is a constant is1 1 4 (A) x2 + y2 = 4P2 (B) 2  2  2 x y P 1 1 2 (D) 2  2  2 x y P

Reflecting the point (2, –1) about Y axis coordinate axis are rotated at 45º angle in

The locus of the points of intersection of the lines x cos + y sin = a and x sin – y cos = b

4 (C) x2 + y2 = 2 P

On shifting the origin to the point (2, –5) and

(B) (3, 3)

(where  is a variable) is – (A) x2 + y2 = a2 + b2 (B) x2 – y2 = a2 + b2 (C) x2 + y2 = a2 – b2 (D) None of these Q.17

Q.20

 –1 – 3  (A)  ,  2 2

–3  (B)  ,– 2   2 

 1 3   (C)  ,  2 2

(D) None of these

If the axes are rotated through an angle of 30º in the clockwise direction, the point (4, –2

3)

in the new system was formerly (A) (2,

3 )

(C) ( 3 , 2) Q.24

(B) ( 3 , –5) (D) (2, 3)

The equation of a line which makes an angle of tan–1 (3) with the x-axis anticlockwise & cuts

Q.18

Q.19

The locus of a point which moves so that the algebraic sum of the perpendiculars let fall from it on two given straight lines is constant, is(A) a circle (B) a straight line (C) a pair of lines (D) none of these If P = (1, 0) and Q = (–1, 0) and R = (2, 0) are three given points, the locus of the point S satisfying the relation SQ2 + SR2 = 2SP2 is(A) a straight line parallel to the x-axis (B) a circle passing through the origin (C) a circle with the centre at the origin (D) a straight line parallel to the y-axis

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off an intercept of 4 units on negative direction of y-axis is -

Q.25

(A) y = 3x + 4

(B) y = 3x – 4

(C) x = 3y + 4

(D) None of these

A line passes through (x1, y1). This point bisects the segment of the line between the axes. Its equation isx y 1   x1 y1 2

(A)

x y  2 x1 y1

(B)

(C)

x y  1 x1 y1

(D) None

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IIT JEE MAINS & ADVANCED MATHEMATICS BY OM SIR Q.26

The equation of the straight line on which the length of the perpendicular from the origin is 2

Q.32

and the perpendicular makes an angle  with

Q.27

x-axis such that sin =

1 is 3

(A) 2 2 x – y = 6

(B) 2 2 x + y = 6

(C) 3 2 x + y = 6

(D) 2 2 x – y = 5

Q.33

(A)

132 12 3  5 132

(C)

3 –5

(B)

The equation of the line through the point of intersection of the lines 2x + 3y – 7 = 0 and 3x + 2y – 8 = 0 which cuts equal intercepts on the axes is (A) x + y = 3 (B) 2x + 2y = 7 (C) x + y = 1 (D) 3x + 3y = 8

Q.35

The point of intersection of the lines

12 3 – 5

(D) None of these

opposite to it if length of square is 2 2 (A) x – y = ± 4 (C) x + y = 4 Q.29

x y   1 does not lies on the line b a (A) x – y = 0 (B) (x + y) (a + b) = 2ab (C) (x + my) (a + b) = ( + m) ab.

The equation of a line perpendicular to the line

(D) (x – my) (a – b) = (1 – m) ab

x y  = 1 and passing through the point where a b

Q.30

Q.31

(A)

x y a   0 a b b

(B)

x y a   b a b

(C)

x y  0 b a

(D)

x y b   b a a

The equation of the perpendicular bisector of the line segment joining points (1, 5) and (–3, 2) is (A) 4x + 3y – 29 = 0 (B) 4x + 3y – 13 = 0 (C) 8x + 6y – 13 = 0 (D) 8x + 6y + 13 = 0 The image of the point (2, 1) with respect to the line mirror be (5, 2). Then the equation of the mirror is (A) 3x + y – 12 = 0 (B) 3x – y + 12 = 0 (C) 3x + y + 12 = 0 (D) 3x – y – 12 = 0

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x y  1 a b

and

(B) x + y = ± 4 (D) x – y = 4

it meets x-axis is -

The coordinates of a point on x + y + 3 = 0,

Q.34

132

One side of square is x – y = 0 find side

1 3 (D)  ,  5 5

whose distance from x + 2y + 2 = 0 is 5 is equal to (A) (9, 6) (B) (–9, 6) (C) (–9, –6) (D) None of these

 with x-axis and meets the 6

line 12x + 5y + 10 = 0 at Q. Then the length of PQ is -

Q.28

 1 3 (C)   ,   5 5

If the straight line through the point P (3, 4) makes an angle

Perpendicular bisector of segment PQ is 3x + 4y – 2 = 0. If P is (1, 1) then point Q is 1 3  1 3 (A)  ,  (B)   ,  5 5    5 5

Q.36

For what value of , the three lines 2x – 5y + 3 = 0, 5x – 9y +  = 0 & x – 2y + 1 = 0, are concurrent (A) 4 (B) 5 (C) 3 (D) 2

Q.37

The equation of the bisector of the acute angle between the lines 3x – 4y + 7 = 0 and 12x + 5y – 2 = 0 is (A) 21x + 77y – 101 = 0 (B) 11x – 3y + 9 = 0 (C) 11x – 3y – 9 = 0 (D) none of these On the portion of the straight line x + y – 7 = 0 which is intercepted between the axes a square is constructed on the side of the line away from the origin. Then the equation to the diagonals are (A) –x + y = 7 ; x – y = 7 (B) x = –7 ; y = 7 (C) x = 7 ; y = – 7 (D) x = 7 ; y = 7

Q.38

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IIT JEE MAINS & ADVANCED MATHEMATICS BY OM SIR Q.39

The equation of the line through the point of

Q.45

Only in case of acute angled triangle centroid divides line joining circumcentre and orthocentre in the ratio 2 : 1

Q.46

If L1 = 0 and L2 = 0 are parallel lines, then family of lines will be L1 + L2 = 0

Q.47

For a triangle there exists a unique point whose distance from all three sides is same and it is called incentre of triangle

Q.48

If a and b are real numbers between 0 and 1, such that the points (a, 1) (1, b) and (0, 0) form an equilateral triangle then 2(a + b) – ab is equal to ..............

Q.49

The no. of points (p, q) such that p, q  {1, 2, 3, 4} and the equation px2 + qx + 1 = 0 has real roots is............

Q.50

If , ,  are the real roots of the equation x3 – 3px2 + 3qx – 1 = 0 then centroid of the triangle with vertices (, 1/), (, 1/) and (, 1/) is at the point ............

Q.51

The integral values of  for which origin lies in the bisector of acute angle between lines (2 + 3) x + 4y + 3 = 0 and x + y + 1 = 0 is ..............

Q.52

In a triangle if vertex A is (2, 3) and angle bisector through B is x + 2y = 3 and median through C is x – 2y = – 1, then co-ordinate of vertex B is............... The distance between lines whose combined

intersection of the lines x – y + 4 = 0 and y – 2x – 5 = 0 and passing through the point (3, 2) is -

Q.40

(A) x – 4y + 5 = 0

(B) x + 4y – 11 = 0

(C) 2x – y – 4 = 0

(D) none of these

The equation of the line passing through the intersection of x –

3y +

3 – 1 = 0 and

x + y – 2 = 0 and making an angle of 15º with the first line is

Q.41

(A) x – y = 0

(B) x – y + 1 = 0

(C) y = 1

(D) 3 x –y +1 – 3 =0

If a + b + c = 0 then the straight line 2ax + 3by + 4c = 0 passes through the fixed point-

Q.42

(A) (2, 4/3)

(B) (2,2)

(C) (4/3, 4/2)

(D) none

Find the separate equations of the straight lines whose joint equations is ab (x2 – y2) + (a2 – b2) xy = 0 (A) bx + ay = 0 and ax – by = 0 (B) bx – ay = 0 and ax + by = 0 (C) bx – ay = 0 and ax – by = 0 (D) None of these

Q.43

Q.53

equation are x2 + 2 2 xy + 2y2 + 4x + 4 2 y + 1 = 0 is

The equation ax2 + by2 + c (x + y) = 0 represents a pair of straight lines if -

Q.44

(A) c = 0

(B) a + b = 0

(C) Both (A) & (B)

(D) none of these

Q.54

The slopes of two lines represented by x2 (tan2 + cos2) – 2xy tan + y2 sin2 = 0 are m1 and m2, then |m1 – m2| is equal to......

If lines px2 – qxy – y2 = 0 make angle '' and '' with x-axis then value of tan(+ ) is (A)

–q 1 p

(B)

(C)

p 1 q

(D) –

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q 1 p p 1 q

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IIT JEE MAINS & ADVANCED MATHEMATICS BY OM SIR

EXERCISE # 2 Part-A Q.1

Only single correct answer type questions

Q.7

AB = 2, ABC = /3 and the middle point of BC has the coordinates (2, 0). The centroid of the triangle is –

If one vertex of an equilateral triangle of side 'a' lies at the origin and the other lies on the line x – 3 y = 0, the co-ordinates of the third vertex are

Q.2

(A) (0, a)

(B)( 3 a/2, –a/2)

(C) (0, –a)

(D) All of these

Let P = (1, 1) and Q = (3, 2). The point R on the x-axis such that PR + RQ is the minimum is 5 3



(A)  , 0  

(C) (3, 0) Q.3

Q.4

Q.5

1 3

(C)

36 7

(D)

31 7

The vertex O of an isosceles triangle OAB lies at the origin and the equation of the base AB is x – y + 1 = 0. If OA = OB = 6, the area of the triangle OAB (A) 71 / 2 sq. units

(B) 142 / 2 sq. units

(C) 2 71 sq. units

(D) 142 sq. units

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The sides of a triangle are x + y = 1, 7y = x 3 y + x = 0 Then the following is an

Q.9

If coordinates of orthocentre and centroid of a triangle are (4, –1) and (2, 1), then coordinates of a point which is equidistant from the vertices of the triangle is (A) (2, 2) (B) (3, 2) (C) (2, 3) (D) (1, 2)

Q.10

One vertex of the equilateral triangle with circumcentre at (1, 1) and one side as x + y = 3 is(A) (2, 2) (B) (0, 0) (C) (–2, –2) (D) None

Q. 11

A point moves such that its distance from the point (4, 0) is half that of its distance from the line x = 16. The locus of this point is (A) 3x2 + 4y2 = 192 (B) 4x2 + 3y2 = 192 (C) x2 + y2 = 192 (D) None of these

Q. 12

Let A = (1, 0) and B = (2, 1). The line AB turn

Let A = (1, 2), B = (3, 4) and let C = (x, y) be a point such that (x–1) (x–3) + (y – 2) (y – 4) = 0.

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(D) none of these

interior point of the triangle (A) Circumcentre (B) Centroid (C) Orthocentre (D) None of these

If the line segment joining (2, 3) and (–1, 2) is divided internally in the ratio 3 : 4 by the line x + 2y = k then k is 5 7

 4  3 1 (C)  ,  3  3



The four points whose coordinates are (2,1), (1,4), (4,5), (5,2) form : (A) a rectangle which is not a square (B) a trapezium which is not a parallelogram (C) a square (D) a rhombus which is not a square

(B)

5 1   (B)  , 3 3

and

(D) none

41 7

1 3  (A)  , 2 2   



If ar (ABC) = 1 then maximum number of positions of C in the x-y plane is (A) 2 (B) 4 (C) 8 (D) none Q.6

Q.8

(B)  , 0 

(A)

In the ABC, the coordinates of B are (0, 0)

about A through an angle /6 in the clockwise sense, and the new position of B is B. Then B has the coordinates  3   3  3 1  (A)  , 2   2 1  3 1  3   (C)  , 2   2

 3   3  3 1  (B)  , 2   2

(D) none of these

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IIT JEE MAINS & ADVANCED MATHEMATICS BY OM SIR Q. 13

Q. 14

Q.15

The point (4, 1) undergoes the following three transformation successively : (i) Reflection about the line y = x (ii) Transformation through a distance 2 units along the positive direction of x-axis (iii) Rotation through angle /4 about the origin in the anticlockwise direction. The final position of the point is given by the coordinates :  7 1   (A)  ,  2 2

(B) (2,7 2 )

 1 7   (C)   , 2 2 

(D) ( 2 ,7 2 )

The image of the point A (1, 2) by the mirror y = x is the point B and the image of B by the line mirror y = 0 is the point (, ). Then(A) = 1,  = –2 (B) = 0,  = 0 (C)  = 2,  = –1 (D) none of these

Q.18

If the vertices of a quadrilateral is given by (x2 – 4)2 + (y2 – 9) 2 = 0 then area of quadrilatural is(A) 36 (B) 24 (C) 16 (D) 81

Q.19

The equation of the line through the point (–5, 4) such that its segment intercepted by the lines x + 2y + 1 = 0 and x + 2y – 1 = 0 is of length

is 5 (A) 2x – y = 4 (C) 2x – y = 0 Q.20

The distance of the line x + y – 8 = 0 from (4, 1) measured along the direction whose slope is –2 is (A) 3

Q.16

5 (B) 6 5

(C) 2 5

Q.21

In what direction a line be drawn through the point (1, 2) so that its point of intersection with the line x + y = 4 is at a distance given point (A) 75º (B) 60º

Q.17

(D) None

(C) 45º

6 from the 3

(D) 30º

2

Q.22

P is a point on either of the two lines

(B) 2x – y = – 14 (D) none

If the point (cos, sin) does not fall in that angle between the lines y = | x – 1 | in which the origin lies then  belongs to   3  (A)  ,  . 2 2 

   (B)   ,   2 2

(C) (0, )

  (D) 0,   2

The straight line y = x – 2 rotates about a point where it cuts x-axis and becomes perpendicular on the straight line ax + by + c = 0 then its equation is (A) ax + by + 2a = 0 (B) ay – bx + 2b = 0 (C) ax + by + 2b = 0 (D) none of these cos   sin  If A   1,  1 and B(1, 1),– 2  3 

y – 3 |x| = 2 at a distance of 5 units from their point of intersection. The coordinates of the foot of the perpendicular from P on the bisector of the angle between them are -

are two points on the same side of the line 3x – 2y + 1 = 0, then  belongs to the interval-

 45 3     or  0, 4  5 3  depending (A)  0,    2  2    on which the point P is takes.  45 3   (B)  0,   2  

(C) 

 45 3   (C)  0,  2   5 5 3  (D)  , 2 2   

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3      (A)  ,    ,  (B) [– , ] 4  4  

Q.23

(D) none of these

Given four lines whose equations are x + 2y – 3 = 0, 2x + 3y – 4 = 0, 3x + 4y – 7 = 0 and 4x + 5y – 6 = 0 then (A) they are all concurrent (B) they are sides of a quadrilateral (C) They are sides of trapezium (D) none of these

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IIT JEE MAINS & ADVANCED MATHEMATICS BY OM SIR Q.24

If a, b, c are in A.P., then ax + by + c = 0 represents (A) a single line (B) a family of concurrent lines (C) a family of parallel lines (D) none of these

Q.31

The four sides of a quadrilateral are given by the equation xy (x – 2) (y – 3) = 0. The equation of the line parallel to x – 4y = 0 that divides the quadrilateral in two equal areas is (A) x – 4y + 5 = 0 (B) x – 4y – 5 = 0 (C) 4y = x + 1 (D) 4y + 1 = x

Q.25

If the lines represented by x2 – 2pxy – y2 = 0 are rotated about the origin through an angle  one in clockwise direction and other in anticlockwise direction, then the equation of the bisectors of the angle between the lines in the new position is – (A) px2 + 2xy – py2 = 0 (B) px2 + 2xy + py2 = 0 (C) x2 – 2pxy + y2 = 0 (D) None of these

Q.32

The number of lines passing through (2, 3) each having distance equal to 5 units from the point (7, 8) is : (A) two (B) zero (C) one (D) infinite

Part-B

One or more than one correct answer type questions

Q.33

The line 3x + 2y = 24 meets the y-axis at A and the x-axis at B. C is a point on the perpendicular bisector of AB such that the area of the triangle ABC is 91 sq. units. The coordinates of C are (A) (29/2, –1) (B) (29/2, 13) (C) (–13/2, –3/2) (D) (–13/2, 13)

Q.34

Three vertices of a quadrilateral in order are (6, 1), (7, 2) and (–1, 0). If the area of the quadrilateral is 4 unit2 then the locus of the fourth vertex has the equation (A) x – 7y = 1 (B) x –7y + 15 = 0 (C) (x –7y)2 + 14(x – 7y) –15 = 0 (D) none of these

Q.26

If the equation 12x2 +7xy – py2 – 18x + qy + 6 = 0 represent a pair of perpendicular straight line then (A) p = 12, q = 1 (B) p = 1, q = 12 (C) p = –1, q = 12 (D) p = 1, q = –12

Q.27

One of the bisectors of the angle between the lines a (x – 1)2 + 2h (x – 1) (y – 2) + b (y – 2)2 = 0 is x + 2y – 5 = 0 The other bisector is (A) 2x – y = 0 (B) 2x + y = 0 (C) 2x + y – 4 = 0 (D) x – 2y + 3 = 0

Q.28

If the two pairs of lines x2 – 2mxy – y2 = 0 and x2 – 2nxy – y2 = 0 are such that one of them represents the bisectors of the angles between the other, then (A) mn + 1 = 0 (B) mn – 1 = 0 (C) 1/m + 1/n = 0 (D) 1/m – 1/n = 0

Q. 35

The number of values of  for which bisectors of the angle between the lines ax2 + 2hxy + by2 +  (x2 + y2) = 0 are the same is (A) two (B) one (C) zero (D) infinite

If P is a point which is at a distance of 4 units and 3 units from x-axis and y-axis respectively then co-ordinate of P may be (A) (3, 4) (B) (–3, 4) (C) (3, –4) (D) (–3, –4)

Q. 36

Let x(y – 3) = 5 where x, y  Integers then

Q.29

value of x + y is equal to(A) 6

Q.30

If the slope of one line is double the slope of another line and the combined equation of the x 2 2xy y 2 pair of lines is    0 then ab : h2 a h b is equal to (A) 9 : 8 (B) 3 : 2 (C) 8 : 3 (D) none of these

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Q. 37

(B) 9

(C) –6

(D) –3

If P is a point on the line joining points A (2, 3) & B (4, 5) such that AP = ordinates of P are(A) (3, 4) (C) (1, 2)

2 then co-

(B) (2, 4) (D) (3, 2)

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IIT JEE MAINS & ADVANCED MATHEMATICS BY OM SIR Q.38

Let A (2, ), B (3, 5), C (4, 5) are the vertices

Part-C Column Matching type questions

of ABC whose area is 10(units) , then value 2

of  is/are -

Q.39

(A) 20

(B) 25

(C) –20

(D) –15

Q.44

Observe the following columns. Points given in the column -I are collinear then Column-I Column -II (A) (a, b + c), (b, c + a), (P) if ad = bc (c, a + b) 1 1 (B) (a, b), (c, d),(a + c, b + d) (Q) if + =1 a b (C) (a, 0), (0, b), (1, 1) (R) if a = 1/2 , –1 (D) (a, 2 –2a), (–a + 1,, 2a) (S) always (– 4– a,6 – a)

Q.45

Let P(x, y) be any point on the locus then observe the following column Column-I Column -II (A) The sum of the squares (P) x2 + y2 = 25 of distance from the coordinate axis is 25 (B) distances to the coordinate (Q) 4x2 – 9y2 = 0 axes are in the ratio 2 : 3 respectively (C) The square of whose (R) x2 + y2 = 4y distance from origin is 4 time its y-coordinate (D) Distance from P to (4, 0) (S) x2 – 3y2 – 8x is double the distance + 16 = 0 form P to the x-axis (T) 9x2 + 4y2 = 0

Q.46

The values of a, Column-I Column-II (A) If (0, a) lies on or (P) (–4, –3] inside the triangle formed by the lines y + 3x + 2 = 0, 3y – 2x – 5 = 0, 4y + x – 14 = 0 (B) If (2a – 5, a2) is (Q)(–3,0)(1/3,1) on the same side of the x + y = 3 as that of origin (C) If (a, 2) lies between (R) (5/3, 7/2) the lines x – y = 1 and 2(x – y) + 5 = 0 (D) Point (a2, a + 1) lies (S) (5/2, 3) between the angles of the lines 3x – y + 1 = 0 and x + 2y – 5 = 0 which contains origin if

If area of OPB = area of OPA when O is origin, A  (6, 0), B  (0, 4) and P lies on line x + y = 1 then possible co-ordinate of P is/are-

Q.40

3 2 (A)  ,  5 5

(B) (3, –2)

(C) (2, –1)

1 1 (D)  ,  2 2

The combined equation of two sides of an equilateral triangle is x2 – 3y2 – 2x + 1 = 0. If the length of a side of the triangle is 4 then the equation of the third side is -

Q.41

(A) x = 2 3 + 1

(B) y = 2 3 + 1

(C) x + 2 3 = 1

(D) x = 2 3

Two pairs of straight lines have the equations y2 + xy – 12x2 = 0 and ax2 + 2hxy + by2 = 0. One line will be common among them if-

Q.42

(A) a = – 3 (2h + 3b)

(B) a = 8 (h – 2b)

(C) a = 2 (b + h)

(D) a = – 3 (b + h)

The diagonals of a square are along the pair of lines whose equation is 2x2 – 3xy – 2y2 = 0. If (2, 1) is a vertex of the square then another vertex consecutive to it can be -

Q.43

(A) (1, –2)

(B) (1, 4)

(C) (–1, 2)

(D) (–1, –4)

The pairs of straight lines ax2 + 2hxy – ay2 = 0 and hx2 – 2axy – hy2 = 0 are such that(A) one pair bisects the angles between the other pair (B) the lines of one pair are equally inclined to the lines of the other pair (C) the lines of one pair are perpendicular to the lines of the other pair (D) none of these

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IIT JEE MAINS & ADVANCED MATHEMATICS BY OM SIR Q.47

The equation of the line through the intersection of the line 2x – 3y = 0 and 4x – 5y = 2 and Column-I Column-II (A) Through the point (P) 2x – y = 4 (2, 1) (B)  to line x+ 2y + 1 = 0 (Q) x + y– 5 = 0, x–y–1=0 (C) || to line (R) x – y – 1 = 0 3x – 4y + 5 = 0 (D) Equally inclined to (S) 3x – 4y – 1 = 0 axes

Q.48

Find the value of  if the family of straight lines (2x + 3y + 4) +  (6x – y + 12) = 0 is Column-I Column-II (A) || to y-axis (P)  = –3/4 (B)  to 7x + y – 4 = 0 (Q)  = – 1/3 (C) Passes through (1, 2) (R)  = –17/41 (D) || to x-axis (S)  = 3

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IIT JEE MAINS & ADVANCED MATHEMATICS BY OM SIR

EXERCISE # 3 JEE Main PYQ 1.

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 equidistant from the two axes then : [JEE Main-2014] (1) 2bc – 3ad = 0 (2) 2bc + 3ad = 0 (3) 3bc – 2ad = 0 (4) 3bc + 2ad = 0

2.

Let PS be the median of the triangle with vertices P(2, 2), Q(6, –1) and R(7, 3). The equation of The line passing through (1, –1) and parallel to PS is – [JEE Main-2014] (1) 4x –7y – 11 = 0 (2) 2x + 9y + 7 = 0 (3) 4x + 7y + 3 = 0 (4) 2x – 9y –11 = 0

3.

Two sides of a rhombus are along the lines, x –y + 1 = 0 and 7x –y – 5 = 0. If its diagonals intersect at (–1, –2), then which one of the following is a vertex of this rhombus? [JEE- Main-2016]  10 7  (1)  – ,–  3  3

4.

(2) (–3, –9)

(3) (–3, –8)

1 8 `(4)  , –  3 3

A straight the through a fixed point (2, 3) intersects the coordinate axes at distinct points P and Q. If O is the origin and the rectangle OPRQ is completed, then the locus of R is : [JEE Main 2018]

(1) 2x + 3y = xy

(2) 3x + 2y = xy

(3) 3x + 2y = 6xy

(4) 3x + 2y = 6

5.

A point on the straight line, 3x + 5y = 15 which is equidistant from the coordinate axes will lie only in – [JEE Main 2019] st nd th st nd (1) 1 , 2 and 4 quadrants (2) 1 and 2 quadrants (3) 4th quadrants (4) 1st quadrants

6.

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 k (4, 3) is perpendicular to L1, then equals h [JEE Main 2019] 1 1 (1) – (2) (3) 0 (4) 3 7 3

7.

If the two lines x + (a – 1) y + 1 = 0 and 2x + a2y = 1 (a  R – {0, 1}) are perpendicular, then the distance of their point of intersection from the origin is : [JEE Main 2019] 2 2 2 (1) (2) 2 (3) (4) 5 5 5 5

8.

Slope of a line passing through P(2, 3) and intersecting the line, x + y = 7 at a distance of 4 units from P, is [JEE Main 2019] 5 –1 7 –1 1– 7 1– 5 (1) (2) (3) (4) 5 1 7 1 1 7 1 5

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IIT JEE MAINS & ADVANCED MATHEMATICS BY OM SIR 9.

Let the equations of two sides of a triangle be 3x – 2y + 6 = 0 and 4x + 5y – 20 = 0. If the orthocentre of this triangle is at (1, 1), then the equation of its third side is : [JEE Main 2019]

10.

(1) 26x – 122y – 1675 = 0

(2) 26x + 61y + 1675 = 0

(3) 122y – 26x – 1675 = 0

(4) 122y + 26x + 1675 = 0

The region represented by |x – y|  2 and |x + y|  2 is bounded by a : [JEE Main 2019]

11.

(1) rhombus of area 8 2 sq. units

(2) square of side length 2 2 units

(3) square of side 16 sq. units

(4) rhombus of side length 2 units

Lines are drawn parallel to the line 4x – 3y + 2 = 0, at a distance

3 from the origin. Then which one of the 5

following points lies on any of these lines? [JEE Main 2019] 1 1 (1)  , –   4 3

12.

 1 2 (2)  – ,   4 3

1 2 (3)  ,  4 3

2  1 (4)  – , –  3  4

A point P moves on the line 2x – 3y + 4 = 0. If Q(1, 4) and R(3, –2) are fixed points, then the locus of the centroid of PQR is a line [JEE Main 2019] 2 (1) parallel to y-axis (2) with slope 3 (3) parallel to x-axis

(4) with slope

3 2

13.

If the line 3x + 4y – 24 = 0 intersects the x-axis at the point A and the y-axis at the point B, then the incentre of the triangle OAB, where O is the origin, is – [JEE Main 2019] (1) (3, 4) (2) (2, 2) (3) (4, 4) (4) (4, 3)

14.

Two sides of a parallelogram are along the lines, x + y = 3 & x – y + 3 = 0. If its diagonals intersect at (2, 4), then one of its vertex is [JEE Main 2019] (1) (2, 1) (2) (2, 6) (3) (3, 5) (4) (3, 6)

15.

If in a parallelogram ABDC, the coordinates of A, B and C are respectively (1, 2), (3, 4) and (2, 5), then the equation of the diagonal AD is [JEE Main 2019] (1) 5x + 3y – 11 = 0 (2) 5x – 3y + 1 = 0 (3) 3x – 5y + 7 = 0 (4) 3x + 5y – 13 = 0

16.

If the straight line, 2x – 3y + 17 = 0 is perpendicular to the line passing through the points (7, 17) and (15, ), then b equals : [JEE Main 2019] 35 35 (1) (2) – 5 (3) – (4) 5 3 3

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IIT JEE MAINS & ADVANCED MATHEMATICS BY OM SIR 17.

If a straight line passing through the point P(–3, 4) is such that its intercepted portion between the coordinate axes is bisected at P, then its equation is : [JEE Main 2019] (1) x – y + 7 = 0 (2) 4x – 3y + 24 = 0 (3) 4x + 3y = 0 (4) 3x – 4y + 25 = 0

18.

A straight line L at a distance of 4 units from the origin makes positive intercepts on the coordinate axes and the perpendicular from the origin to this line makes an angle of 60º with the line x + y = 0. Then an equation of the line L is : [JEE Main 2019] (1) x + 3y = 8 (2) 3x + y = 8 (3) ( 3 + 1)x + ( 3 –1)y = 8 2

(4) ( 3 – 1)x + ( 3 + 1)y = 8 2

19.

Consider the set of all lines px + qy + r = 0 such that 3p + 2q + 4r = 0. Which one of the following statements is true? [JEE Main 2019] 3 1 (1) The lines are not concurrent (2) the lines are concurrent at the point  ,  4 2 (3) The lines are all parallel (4) Each line passes through the origin

20.

The length of the perpendicular from the origin, on the normal to the curve, x2 + 2xy – 3y2 = 0 at the point (2, 2) is : [IIT JEE MAINS 2020]

(1) 2 21.

(2) 4 2

(3) 2

(4) 2 2

The locus of the mid-points of the perpendiculars drawn from points on the line, x = 2y to the line x = y is : [IIT JEE MAINS 2020]

(1) 7x – 5y = 0 22.

(2) 5x – 7y = 0

(3) 2x – 3y = 0

(4) 3x – 2y = 0

Let two points be A(1, –1) and B(0, 2). If a point P(x, y) be such that the area of PAB = 5 sq. units and it lies on the line, 3x + y – 4 = 0, then a value of  is : [IIT JEE MAINS 2020]

(1) 1 23.

(2) 4

(3) –3

(4) 3

Let C be the centroid of the triangle with vertices (3, –1), (1, 3) and (2, 4). Let P be the point of intersection of the lines x + 3y – 1 = 0 and 3x – y + 1 = 0. Then the line passing through the points C and P also passes through the point : [IIT JEE MAINS 2020]

(1) (–9, –7) 24.

(2) (9, 7)

(3) (7, 6)

If the line, 2x – y + 3 = 0 is at a distance

1 5

and

2 5

(4) (–9, –6)

from the lines 4x – 2y +  = 0 and 6x – 3y +  = 0,

respectively, then the sum of all possible values of  and  is ____. [IIT JEE MAINS 2020] 25.

A triangle ABC lying in the first quadrant has two vertices as A(1, 2) and B(3, 1). If BAC = 90º, and ar(ABC) = 5 5 sq. units, then the abscissa of the vertex C is : [IIT JEE MAINS 2020] (A) 2 +

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5

(B) 1 +

5

(C) 1 + 2 5

(D) 2 5 – 1

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IIT JEE MAINS & ADVANCED MATHEMATICS BY OM SIR 26.

The set of all possible values of  in the interval (0, ) for which the points (1, 2) and (sin , cos) lie on the same side of the line x + y = 1 is : [IIT JEE MAINS 2020]

 

(A)  0, 27.

  4

 3 3  5 5

(B)  – , 

(B) – 2

15

 

(D)  0,

  2

3 5

3 5

(D)  , – 

(C) (–3, 3)

(D) – 4

14

(C)

Let L denote the line in the xy-plane with x and y intercepts as 3 and 1 respectively. Then the image of the point (– 1, –4) in this line is : [IIT JEE MAINS 2020]

 8 29   5 5 

(A)  , 30.

  3  ,  4 4 

(C) 

If the perpendicular bisector of the line segment joining the points P(1, 4) and Q(k, 3) has y-intercept equal to – 4, then a value of k is – [IIT JEE MAINS 2020] (A)

29.

3   4 

If a ABC has vertices A(–1, 7), B(–7, 1) and C(5, –5), then its orthocentre has coordinates : [IIT JEE MAINS 2020] (A) (3, –3)

28.

 

(B)  0,

 29 11  ,   5 5

(B) 

 11 28  ,  5 5 

(C) 

 29 8  ,   5 5

(D) 

A ray of light coming from the point (2, 2 3 ) is incident at an angle 30º on the line x = 1 at the point A. The ray gets reflected on the line x = 1 and meets x-axis at the point B. Then, the line AB passes through the point : [IIT JEE MAINS 2020]



(A)  3, –



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1   3

(B) (3, – 3 )

  

(C)  4, –

3  2 

(D) (4, – 3 )

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IIT JEE MAINS & ADVANCED MATHEMATICS BY OM SIR JEE Advanced PYQ 1.

Lines L1 : y – x = 0 and L2 : 2x + y = 0 intersect the line L3 : y + 2 = 0 at P and Q respectively, The bisector of the acute angle between L1 and L2 intersects L3 at R. [IIT-JEE-2007, AIEEE 2011] Statement-1 : The ratio PR : RQ equals 2 2 : 5 Because Statement-2 : In any triangle, bisector of an angle divides the triangle into two similar triangles. (A) Statement-1 is true, statement-2 is true; statement-2 is a correct explanation for statement-1. (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 false. (D) Statement-1 is false, statement-2 is true. A (0, 0)

[JEE-2008]

R

0

L

1

y

:y

x

–x

:2

0

Consider the lines given by

L2

2.

P L1 = x + 3y – 5 = 0 L3 : y + 2 = 0 Q L2 = 3x – ky – 1 = 0 (1,–2) (–2,–2) 3s L3 = 5x + 2y – 12 = 0 Match the statements / Expression in Column-I with the statements / Expressions in Column-II and indicate your answer by darkening the appropriate bubbles in the 4 x 4 matrix given in OMR. Column-I Column-II (A) L1 , L2, L3 are concurrent, if (P) k = – 9 6 (B) One of L1 , L2, L3 is parallel to at least one of the other two, if (Q) k = – 5 5 (C) L1, L2 , L3 form a triangle, if (R) k = 6 (D) L1, L2, L3 do not form a triangle, if (S) k = 5

3.

The locus of the orthocenter of the triangle formed by the lines (1 + p)x – py + p(1 + p) = 0, (1 + q)x – qy + q(1 + q) = 0 and y = 0, where p q, is (A) a hyperbola (B) a parabola (C) an ellipse (D) a straight line

4.

A straight line L through the point (3, –2) is inclined at an angle 60° to the line x-axis, then the equation of L is

3x  y  1. If L also intersect the [JEE-2011]

(A) y  3x  2 – 3 3  0 (C)

3y – x  3  2 3  0

(B) y – 3x  2  3 3  0 (D)

3y  x – 3  2 3  0

5.

For a > b > c > 0, the distance between (1, 1) and the point of Intersection of the lines ax + by + c = 0 and bx + ay + c = 0 is less than 2 2 . Then [JEE- Adv. 2013] (A) a + b – c > 0 (B) a – b + c < 0 (C) a – b + c > 0 (D) a + b – c < 0

6.

For a point P in the plane, let d1(P) be the distance of the point (P) from the liens x – y = 0 and x + y = 0 respectively. The area of the region R consisting of all points P lying in the first quadrant of the plane and satisfying 2 d1(P) + d2(P) 4, is [JEE- Adv. 2014]

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IIT JEE MAINS & ADVANCED MATHEMATICS BY OM SIR

ANSWER KEY EXERCISE # 1 Q.No.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

Ans.

C

B

C

D

A

D

A

D

C

B

D

A

C

A

A

A

B

B

D

B

A

A

Q.No.

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

Ans.

B

B

A

B

A

A

B

C

A

B

B

A

D

A

B

D

B

A

A

A

D

A

45. False 52. (5/2, 1/4)

46. False 53. 2

47. False 54. 2

48. 1

49. 7

51. –2

50. (p, q)

EXERCISE # 2 (PART-A) Q.No.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Ans.

D

A

C

A

B

A

B

B

D

B

A

A

C

C

A

A

Q.No.

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

Ans.

B

B

B

D

B

A

D

B

A

A

A

A

D

A

A

A

(PART-B) Q.No.

33

Ans.

B,C

34

35

A,B,C A,B,C,D

36

37

38

39

40

41

42

43

B,D

A,C

B,D

A,B

A,C

A,B

A,C

A,B

(PART-C) 44. A S; B  P ; C  Q; D  R

45. A P; B  Q ; C  R; D  S

46. A R, S; B  P, Q; C  S; D  Q

47. A R; B  P ; C  S; D  Q

48. A S; B  R ; C  P; D  Q

EXERCISE # 3 JEE MAINS

1 3 11 2 21 3

2 2 12 2 22 2

3 4 13 2 23 2

4 2 14 4 24 30

5 2 15 2 25 C

6 2 16 4 26 D

2 2

3 4

4 B

5 A,C

6 6

7 2 17 2 27 C

8 3 18 3,4 28 D

9 1 19 2 29 C

10 2 20 4 30 B

JEE Advanced

1 C

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