M_2Y_Straight Line_Class Test - 6.docx

Vidyamandir Classes

Agawam Corporate Heights, 3rd Floor, Plot No. A - 7, Netaji S!hash Pla"e, Pitam Pra, #elhi - $$%%3& Phone' %$$-&())$$*%-*3. %$$-&())$$*%-*3. Fa+ ' )()))*(3

#ate '

0est 4 Straight /ines

A0HS

Name '

5at"h

MCQ with One Correct Option

1.

2.

If two vertice verticess of of an equila equilatera terall trian triangle gle have have integ integral ral coordi coordinat nates es then then the third third vert vertex ex will have have (a)

integral coordinates

(b)

coordinates which are rational

(c)

at least one coordinate irrational

(d)

coordinates which are irrational

If the line line segm segment ent joinin joining g (2,3) (2,3) and ( 1, 2) is divide divided d inte interna rnall! ll! in the ratio ratio 3 " # b! b! the line line x $ 2! % & then then & is #1

(

3)

31

'

'

'

'

(a) 3.

.

(c)

(d)

*he *he +ol +olar ar coor coordi dina nate tess of of the the vert vertic ices es of a tri trian angl glee are are ( , ) (3, (3, π-2) and (3 ,π-). *hen the triangle is (a)

#.

(b)

right angled

(b)

isosceles

(c )

equilateral

collinear  

(d)

none of these

*he +oints ints (a, b $ c) , (b, c $ a) , (c, a$ b) are (a)

vertices of an equilateral triangle

(b)

(c)

conc!clic

(d)

none of these x ! + =1 a  b *he *he inc incen entr tree of of the the tria triang ngle le form formed ed b! the the axe axess and and the the lin linee is

  ab   a + b +

 a , b       2 2   (a)

ab

,

      a +  b + ab   ab

(b)

  ab    a + b + a 2 + b 2

 a , b       3 3   (c)

,

    2 2   a + b + a +  b   ab

(d)

π .

In the

the coordinate coordinatess of / are ∆/0, the

( , ) / / % 2 ,

∠/0 %

3 and the middle +oint of /0 has the

coordinates (2 .) *he centroid of the triangle

 1  ,   2

3   2

(a) '.

 (  ,  3

     

 # +     3

      3  

1

(b)

3 1   ,   3    

(c)

(d)

none of these

*he coordi coordinat nates es of three three consec consecuti utive ve vertice verticess of of a +aralle +arallelog logram ram are (1 , 3), 3), (1,2) (1,2) and (2 , ) ) *he coordi coordinat nates es of the fourth vertex are (a)

( , #)

(b)

)Straight /ine 0est 0est 11 02

(# , )

(c )

( 2 , )

(d)

$

none of these

Vidyamandir Classes .

*he area of the +entagon whose vertices are (# , 1) (3 , ), ( , 1), (3 , 3) and (3 ,) is (a)

3 unit 2

(b)

)Straight /ine 0est 11 02

 unit 2 (c)

12 unit2

(d)

none of these

)

Vidyamandir Classes .

 +oint moves in the x! +lane such that the sum of the distances from two mutuall! +er+endicular liens is alwa!s equal to 3. *he area enclosed b! the locus of the +oint is.  unit 2  unit 2 2 (a) 1 unit 2 (b) (c)

1.

(d)

none of these

4et  % (1,2), / % (3 , #) and let 0 % (x , !) be a +oint such that (x  1) (x3)$ (!2) (! #) % If ar ( ∆/0) % 1 then maximum number of +ositions of 0 in the x! +lane is (a)

11.

12.

2

(b)

#

(c)



(d)

*he +oints (α, β), (γ ,δ) , (α ,δ) and ( γ  , β)ta&en in order , where

none of these

α , β, γ , δ are different real numbers, are

(a)

collinear

(b)

vertices of a square

(c)

vertices of a rhombus

(d)

conc!clic

*he diagonals of a +arallelogram 5678 are along the lines x $ 3! % # and x  2! % '. *hen 5678 must be a

13.

(a)

rectangle

(b)

square

(c)

c!clic quadrilateral

(d)

rhombus

*he coordinates of the four vertices of a quadrilateral are (2 , #) (1 , 2) (1 , 2) and (2 , #) ta&en in order. *he equation of the line +assing through the vertex (1 , 2) and dividing the quadrilateral in two equals areas is (a)

1#.

x $ 1% 

(b)

x$!%1

(c)

x ! $ 3 % 

(d)

none of these

*he equation of the straight line which +asses through the +oint (# , 3) such that the +ortion of the line  between the axes is divided internall! b! the +oint in the ratio  " 3 is

1.

(a)

x2! $  % 

(b)

x $2! % 2#

(c)

2x $ ! $ 3 %

(d)

none of these

*he equation of the straight line which bisects the interce+ts made b! the axes on the lines x $ ! % 2 and 2x $ 3! %  is (a)

1.

2x % 3

(b)

! %1

(c)

2! % 3

(d)

x %1

*he equation of a straight line +assing through the +oint (2 , 3) and ma&ing interce+ts of equal length on the axes is (a)

1'.

2x $ ! $ 1% 

(b)

x!%

(c)

*he foot of the +er+endicular on the line 3x $ ! %

x !$%

(d)

λ drawn from the origin is

none of these 0. If the line cuts the xaxis

and !axis at  and / res+ectivel! then /0 " 0 is (a) 1.

1 "3

(b)

(c)

1"

(d)

"1

*he distance of the line 2x  3! % # form the +oint (1 , 1) in the direction of the line x $ ! % 1 is 1 2

(a) 1.

3"1

2

( 2 (b)

(c)

(d)

9one of these

*he four sides of a quadrilateral are given b! the equation x!(x  2) (! 3) %  *he equation of the line  +arallel to x  #! %  that divided the quadrilateral in two equal areas is (a)

x  #! $  %  (b)

)Straight /ine 0est 11 02

x  #!  % 

(c)

#! % x $1

(d)

3

#! $ 1% x

Vidyamandir Classes 2.

*he coordinates of two consecutive vectors  and / of a regular hexagon /0:;< are (1 , ) and (2 , ) res+ectivel!. *he equation of the diagonal 0; is

3x + ! = # (a) 21.

x + 3! = #

3! + # =  (b)

x$

(c)

(d)

none of these

/0 is an isosceles triangle in which a is (1 , ), ∠ % 2π-3 / % c and / is along the xaxis If /0 3 %#

the equation of the line /0 is

3x + ! = 3

3! = 3 (a) 22.

23.

(b)

(c)

x$! %

(d)

none of these

*he gra+h of the function cos x . cos (x $ 2)  cos 2 (x $ 1) is a (a)

straight line +assing through the +oint (,  sin 21) with slo+e 2

(b)

straight lie +assing through the origin

(c)

+arabola with vertex (1 ,  sin 21)

(d)

straight line +assing through the +oint ( π-2,  sin21) and +arallel to the xaxis.

If the +oints (2 ,) (1 , 1- √3) and (cos θ , sin θ) are collinear then the number of values of (a)

2#.

x $

3



(b)

1

(c)

2

(d)

θ ∈ = , 2π> is

infinite

*he limiting +osition of the +oint of intersection of the lines 3x$ #! % 1 and (1 $ c) x$ 3c 2! % 2 as c tends to 1 is (a)

2.

( , #)

(b)

( ,#)

(c)

(# ,)

(d)

none of these

*he coordinates of the +oint on the xaxis which is equidistant from the +oints (3 ,#) and (2 ,) are

 # ,       (   (a) 2.

(2 ,)

(23 , )

(c)

(d)

none of these

(d)

none of these

*he distance between the lines 3x$ #! %  and x $ ! $ 1 %  is 3

33

33

1

1

(

(a) 2'.

(b)

(b)

(c)

If a vertex of an equilateral triangle is the origin and the side o++osite to it has the equation x $ ! % 1 then the coordinates of the triangle is

  2  ,  3  

 1 , 1       3 3   (a) 2.

2    

3

 2 , 2       3 3  

   

(b)

(c)

(d)

none of these

*he equations of the three sides of a triangle are x % 2 , ! $ 1 %  and x $ 2! % # the coordinates of the circumcentre of the triangle are (a)

2.

(# , )

(b)

(2 , 1)

(c)

( ,#)

(d)

none of these

4 is a variable line such that the algebraic sum of the distances of the +oints (1 , 1) (2 ,) and ( , 2) from the line is equal to ?ero. *he lien 4 will alwa!s +ass through (a)

3.

(1 , 1)

(b)

(2 , 1)

(c)

( 1 , 2)

(d)

none of these

/0 is an equilateral triangle such that the vertices / and 0 lie on two +arallel lines at a distance . If  lies  between the +arallel , lines at a distance # from one of them then the length of a side of the equilateral triangle is

)Straight /ine 0est 11 02

&

Vidyamandir Classes

(a)



(b)

)Straight /ine 0est 11 02



# '

3

3 (c)

(d)

(

none of these

Vidyamandir Classes  +′

31.

are the +er+endicular from the origin u+on the lines xsec θ $ ! cosec

If + and

θ

% a and x cos θ  ! sin

(d)

none of these

θ

% acos2θ res+ectivel! then # +

2

+ +′ 2 = a 2

 +

(a) 32.

+ # +′ 2 = a 2

(b)

 +

2

+ +′ 2 = a 2

(c)

4et the +er+endiculars form an! +oint on the line 2x $ 11! %  u+on the lines 2#x$ '! % 2 and #x  3! % 2  +′

have the lengths + and

res+ectivel!, *hen  + =  +′

2 + =  +′

(a) 33.

2

(b)

 + = 2 +′

(c)

(d)

none of these

If 5 (1 $ 1- √2, 2 $1-√2) be an! +oint on a line then the range of values of t for which the +oint 5 lies between the +arallel lines x $2! % 1 and 2x$ #! % 1 is

−

# 2 (



(d)

none of these

AaAB1

(d)

none of these

If the +oint (a , a) falls between the lines A x $ !A % 2 then (a)

##.

(b)

AaA%2

If  (sin then

(b)

AaA%1

α , 1 √2) and / (1- √2 ,

cos

(c)

α) ,  π ≤ α ≤ π , are two +oints on the same side of the line x  ! %

α belongs to the interval  − π , π ∪  π 3π       # #÷  # , #÷      

 − π , π     # #÷   

(a)

(b)

 π , 3π     # #÷    (c) #.

#.

(d)

*he straight lines 41 ≡ #x3! $ 2 %  , 42 % 3x $#!  # %  and 4 3 ≡ x  '! $  %  (a)

form a right angled triangle

(b)

form a right angled isosceles triangle

(c)

are concurrent

(d)

none of these

*he equation of the bisector of the acute angle between the liens 2x! $ # %  and x 2! % 1 is (a)

#'.

none of these

x$! $%

(b)

x! $1 %

(c)

xt %3

(d)

none of these

*he equation of bisector of that angle between the liens x$ ! % 3 and 2x  ! % 2 which contains the +oint (1 ,1) is (a)

(√  2√2) x $ (√ $ 2√2) ! % 3 √  2√2

(b)

(√ $ 2√2) x $ ( √  √2) ! % 3 √ $ 2√2

(c)

3x % 1

x + 2y + 3 = 0

I

II

(d)

none of these

(0 , 0)

III

2x – 3y =1 IV

#.

*wo lines 2x  3! % 1 and x $ 2! $ 3 %  divide the x! +lane in four com+artments

which are named as shown in the figure. 0onsider the locations of the +oints (2 , 1) (3 ,2) and (1 ,  2) Ce get (a) (c) #.

∈ ID ( 1 , 2) ∈ II (2 , 1)

∈ III

(b)

(3 , 2)

(d)

none of these

If the lines !  x %  , 3x $ #! % 1 and ! % mx $ 3 are concurrent then the value of m is 1 ( ( (a)

1 (b)

)Straight /ine 0est 11 02

1

(c)

(d)



none of these

Vidyamandir Classes .

If the +oint (cos θ , sin

θ) does not fall in that angle between the liens ! % Ax  1A in which the origin lines

then θ belongs to

 π , 3π     2 2÷    (a) 1.

 − π , π     2 2÷    (b)

(c)

( , π)

(d)

none of these

x $ ! %1

(d)

none of these

*he +oints (1 , 1) and (1 , 1 ) are s!mmetrical about the line (a)

!$x%

(b)

)Straight /ine 0est 11 02

!%x

(c)

*

Vidyamandir Classes 2.

*he equation of the line segment / is ! % x . If  and / lie on the same side of the line mirror 2x! %1 , the image of / has the equation. (a)

x $ ! %2

(b)

x $ ! %

(c)

'x  ! %

(d)

none of these

3.

4et 5 % (1, 1) and 6 % (3 , 2) *he +oint 7 on the xaxis such that 57 $ 76 is the minimum is  ( ,    1 ,   3 ÷ 3 ÷       (a) (b) (c) (3 , ) (d) none of these

#.

If a ra! travelling along the line x % 1 gets reflected from the line x $ ! % 1 then the equation of the line along which the reflected ra! travels is (a)

!%

(b)

x!%1

(c)

x%

(d)

none of these

3 2

.

*he +oint 5 (2, 1) is shifted b!

 +arallel to the line x$ ! % 1, in the direction of increasing ordinate to

reach 6. *he image of 6 b! the line x $ ! %1 is (a) .

(, 2)

(b)

(1 ,#)

(3 , #)

(d)

4et  % (1 , ) and / % (2 , 1) *he line / turns about  through an angle /′ /′ new +osition of / is  *hen  has the coordinate

 3+   2

3

,

− 1  ÷ 2 ÷  

3−   2

3

(a)

3

,

(3 , 2)

π- in the cloc&wise sense and the + 1  ÷ 2 ÷  

3

(b)

1−   2

3 1 + 3  , ÷ 2 ÷  

(c) '.

(c)

(d)

none of these

 line has interce+ts a , b on the coordinate axes. If the axes are rotated about the origin through an angle

α

then the line has interce+ts + , q on the new +osition of the axes res+ectivel! *hen 1 1 1 1 1 1 1 1  p 2

+

q2

=

a2

+

b2

 p 2

(a)

q2

=

a2

−

b2

(b) 1  p

2

+

1 a

2

=

1 q

2

+

1 b2

(c) .

−

(d)

none of these

*wo +oints  and / move on the xaxis and the !axis res+ectivel! such that the distance between the two  +oints is alwa!s the same. *he locus of the middle +oint of / is (a)

.

a straight line

(b)

a +air of straight line

(c)

a circle

(d)

none of these

*hree vertices of a quadrilateral in order are (, 1) , (', 2) and (1 ,) if the area of the quadrilateral is # unit

2

then the locus of the fourth vertex has the equation.

.

(a)

x '! % 1

(b)

x '! $ 1 % 

(c)

(x '!) 2 $1#(x '!)1 % 

(d)

none of these

 variable line through the +oint (a, b) cuts the axes of reference at  and / res+ectivel! *he lines through  and / +arallel to the !axis and the xaxis res+ectivel! meet at 5. *hen the locus of 5 has the equation

)Straight /ine 0est

$% 11 02

Vidyamandir Classes  x a (a)

y

 x

b

b

+ =1 (b)

y

a

a

 x

+ =1

b

b

y

 x

+ =1

(c)

)Straight /ine 0est

(d)

$$ 11 02

a

+ =1 y

Vidyamandir Classes )Straight /ine 0est Answers 1.

(c)

2.

(a)

3.

(c)

#.

(b)

.

(d)

.

(b)

'.

(b)

.

(a)

.

(a)

1.

(b)

11.

(d)

12.

(d)

13.

(c)

1#.

(a)

1.

(b)

1.

(c)

1'.

(d)

1.

(a)

1.

(a)

2.

(c)

21.

(a)

22.

(d)

23.

(b)

2#.

(a)

2.

(d)

2.

(b)

2'.

(a)

2.

(a)

2.

(a)

3.

(c)

31.

(a)

32.

(b)

33.

(a)

3#.

(d)

3.

(a)

3.

(b)

3'.

(c)

3.

(a)

3.

(c)

#.

(a)

#1.

(b)

#2.

(a)

#3.

(c)

##.

(a)

#.

(c)

#.

(b)

#'.

(a)

#.

(a)

#.

(c)

.

(b)

1.

(b)

2.

(c)

3.

(a)

#.

(a)

.

(d)

.

(a)

'.

(a)

.

(c)

.

(c)

.

(c)

)Straight /ine 0est

$) 11 02