MULTIPLE CHOICE QUESTIONS (Part b)
1).What is the Equation of the line whose X-intercept is 4 and whose Yintercept is 9 ? a).
x y 1 9 4
b).
x y 36 9 4
x y 36 4 9 x y 1 4 9
c).
d).
Sol: Correct option is (d) The Equation of the line whose X-intercept is 4 and whose Yintercept is 9 is
x y 1 4 9
2).What is the equation of the straight line on which the length of the perpendicular from the origin (0,0) is P and this perpendicular makes an angle with the +ve X-axis ? a).x cos( )+y sin( )=
1 P
1 P c). x cos( )+y sin( )=P
b). x sin( )+y cos( )=
d). x sin( )+y cos( )=P Sol: Correct option is (c) The equation of the straight line on which the length of the
perpendicular from the origin (0,0) is P and this perpendicular makes an angle with the +ve X-axis is x cos( )+y sin( )=P 3).What is the equation of the straight line on which the length of the perpendicular from the origin (0,0) is 4 and this perpendicular makes an angle 45 with the +ve X-axis ? a).x +y =4 2 4 2 c). x +y= 4 b). x+y= d). x +y = 2 Sol: Correct option is (a) The equation of the straight line on which the length of the perpendicular from the origin (0,0) is 4 and this perpendicular makes an angle 45 with the +ve X-axis is x cos( 45 )+y sin( 45 )=4
x y =4 2 2
x +y =4 2
4).What is the equation of the straight line passing through ( x1 , y1 ) and making an angle with the positive direction of x-axis and also R is the distance of the point (x,y) on the line from the point ( x1 , y1 ) ?
a).
x x1 y y1 =R sin cos x x1 y y1 =R cos sin x x1 y y1 1 = sin cos R x x1 y y1 1 cos sin R
b).
c).
d).
Sol: Correct option is (b) The equation of the straight line passing through ( x1 , y1 ) and making an angle with the positive direction of x-axis and also R is the distance of the point (x,y) on the line from the point ( x1 , y1 ) is
x x1 y y1 =R cos sin
5).What is the equation of the straight line passing through (4,7) and making an angle 45 with the positive direction of x-axis and also the the distance of the point (x,y) on the line from the point (4,7) is 5 ? a).
x4 y7 =5 sin 45 cos 45 x x1 y y1 1 cos 45 sin 45 5 x x1 y y1 1 = sin 45 cos 45 5 x x1 y y1 =5 cos 45 sin 45
b).
c).
d).
Sol: Correct option is (d) The equation of the straight line passing through (4,7) and making an angle 45 with the positive direction of x-axis and if the distance of the point (x,y) on the line from the point (4,7) is 5 is
x x1 y y1 =5 cos 45 sin 45
6).What are the coordinates of the intersection point P(x’,y’) of the two lines having equations a1 x b1 y c1 0 and a2 x b2 y c2 0 ? a). x’=
b1c2 b2 c1 c a c a ,y’= 1 2 2 1 a1b2 a2b1 a1b2 a2b1
b1c2 b2 c1 c a c a ,y’= 1 2 2 1 a1b2 a2b1 a1b2 a2b1
b). x’=
c). x’=
b1c2 b2 c1 c a c a ,y’= 1 2 2 1 a1b2 a2b1 a1b2 a2b1
b1c2 b2 c1 c a c a ,y’= 1 2 2 1 a1b2 a2b1 a1b2 a2b1
d). x’=
Sol: Correct option is (a) The coordinates of the intersection point P(x’,y’) of the two lines having equations a1 x b1 y c1 0 and a2 x b2 y c2 0 are x’=
b1c2 b2 c1 a1b2 a2b1
c1a2 c2 a1 a1b2 a2b1
y’=
7).What are the coordinates of the intersection point P(x’,y’) of the two lines having equations 2x+5y=19 and 3x-y=3 ? a). x’=3,y’=1 b). x’=3,y’=2 c). x’=1,y’=4 d). x’=2,y’=3 Sol: Correct option is (d) The coordinates of the intersection point P(x’,y’) of the two lines
having equations 2x+5y=19 and 3x-y=3 are These equations can be written as 2x+5y-19=0 and 3x-y-3=0
x’=
5 (3) (1) (19) 15 19 34 = = =2 2 (1) 3 5 2 15 17 (19) 3 (3) 2 57 6 51 = = =3 2 (1) 3 5 2 15 17
Also y’=
8).What is the condition of the concurrency of the three lines a1 x b1 y c1 0 , a2 x b2 y c2 0 and a3 x b3 y c3 0 ?
a1 a). a2 a 3
a1 b). a2 a 3
a2 b2 b3
b1 b2 b3
a3 c2 =1 c3
c1 c2 =0 c3
a1 c). a2 a 3 a1 d). a2 a 3
a2 b2 b3 b1 b2 b3
a3 c2 =0 c3 c1 c2 =1 c3
Sol: Correct option is (b) The condition of the concurrency of the three lines a1 x b1 y c1 0 , a2 x b2 y c2 0 and a3 x b3 y c3 0 is
a1 a2 a 3
b1 b2 b3
c1 c2 =0 c3
9).What is the condition of the concurrency of the three lines 2 x 5 y 8 0 , 3 x y 11 0 and 7 x 9 y 4 0 ?
2 3 7 a). 3 1 11 =1 7 9 4
2 3 7 b). 3 1 11 =0 7 9 4
2 5 8 c). 3 1 11 =0 7 9 4
2 5 8 d). 3 1 11 =1 7 9 4
Sol: Correct option is (c) The condition of the concurrency of the three lines 2 x 5 y 8 0 , 3 x y 11 0 and 7 x 9 y 4 0 is
2 5 8 3 1 11 =0 7 9 4
10).What is the angle between the two lines having equations a1 x b1 y c1 0 , a2 x b2 y c2 0 ?
a b ab a). tan 1 2 1 1 2 a1a2 b1b2
a b ab b). sin 1 2 1 1 2 a1a2 b1b2 a b ab c). cos1 2 1 1 2 a1a2 b1b2 a b ab d). tan 1 2 2 1 1 a1a2 b1b2
Sol: Correct option is (a) The angle between the two lines having equations a1 x b1 y c1 0 , a2 x b2 y c2 0 is
= tan 1
a2b1 a1b2 a1a2 b1b2
11).Which of the following statement is true if the two lines a1 x b1 y c1 0 and a2 x b2 y c2 0 are parallel ? a). a1a2 b1b2 b).
a1 b1 a2 b2
c). a1b1 a2b2 d).
a1 a2 1 b2 b1
Sol: Correct option is (b) If the two lines a1 x b1 y c1 0 and a2 x b2 y c2 0 are parallel then their slope are equal
a1 a2 b1 b2 a1 b1 a2 b2
12).What is the value of k if the two lines 15x+ky=34 and 5x+3y=11 are parallel ? a).21 b). 3 c). 11 d). 9 Sol: Correct option is (d) If the two lines 15x+ky=34 and 5x+3y=11 are parallel then their slope are equal
15 5 k 3
15 3 =9 5
k
13).Which of the following statement is true if the two lines a1 x b1 y c1 0 and a2 x b2 y c2 0 are perpendicular ? a). a1b1 a2b2 =0 b). a1b2 b1a2 =0 c). a1a2 b1b2 =0 d). a1a2 b1b2 =0 Sol: Correct option is (c) If the two lines a1 x b1 y c1 0 and a2 x b2 y c2 0 are perpendicular Then the product of their slope is -1
a1 a2 1 b1 b2
a1a2 b1b2 =0
14).What is the value of k if the two lines 2x+ky=13 and 6x+4y=12 are perpendicular ? a). -1 b). 3 c). -2 d). -3 Sol: Correct option is (d) If the two lines 2x+ky=13 and 6x+4y=12 are perpendicular Then the product of their slope is -1
2 6 1 k 4
k 3
15).Which of the following condition must be true if the two lines a1 x b1 y c1 0 and a2 x b2 y c2 0 are coincident ?
a).
a1 b1 c1 a2 b2 c2
b).
b1 c1 b2 c2 a1 c1 a2 c2
a1 b1 a2 b2
c).
d).
Sol: Correct option is (a) If the two lines a1 x b1 y c1 0 and a2 x b2 y c2 0 are coincident then the condition which must be true is
a1 b1 c1 a2 b2 c2
16).What is the value of k if the two lines 2x+ky=13 and 4x+12y=26 are coincident ? a). 4 b). 6 c). 2 d). -6 Sol: Correct option is (b) If the two lines 2x+ky=13 and 4x+12y=26 are coincident then
2 k 13 4 12 26
k=6
17).Which of the condition must be true if the two lines a1 x b1 y c1 0 and
a2 x b2 y c2 0 intersect ?
a).
a1 c1 a2 c2
b1 c1 b2 c2 a1 b1 a2 b2
b).
c).
d).
a1 b1 a2 b2
Sol: Correct option is (c) If the two lines a1 x b1 y c1 0 and a2 x b2 y c2 0 are intersecting then the condition which must be true is
a1 b1 a2 b2
18). Which of the condition must be true if the two lines 9x+ky=12 and 3x+y=18 intersect ? a). k 3 b). k -3 c). k 6 d). k 5 Sol: Correct option is (a) If the two lines if the two lines 9x+ky=12 and 3x+y=18 intersect then the condition which must be true is
9 k 3 1
k3
19).What is the length of the perpendicular P from the point ( x1 , y1 ) to the line ax+by+c=0 a). P
ax1 by1 c 2
ax1 by1 c a b ax1 by1 a 2 b2 ax1 by1 c a 2 b2
b). P
c). P
d). P
Sol: Correct option is (d) The length of the perpendicular P from the point ( x1 , y1 ) to the line ax+by+c=0 is
P
ax1 by1 c a 2 b2
20).What is the length of the perpendicular P from the point (2,5) to the line 3x+4y+16=0 ? a). P=3 b). P=2 c). P=1 d). P=5 Sol: Correct option is (b) The length of the perpendicular P from the point (2,5) to the line 3x+4y+16=0 is
P
3 2 4 5 16 3 3
2 2
=
10 =2 5
21).What is the perpendicular distance P between two parallel lines a x b y c1 0 and a x b y c2 0 ? a). P b). P
c2 c1 a2 b2 c2 c1
a2 b2
c2 c1 a2 b2 c2 c1 a 2 b2
c). P
d). P
Sol: Correct option is (c) The perpendicular distance P between two parallel lines a x b y c1 0 and a x b y c2 0 is
P
c2 c1 a2 b2
22).What is the perpendicular distance P between two parallel lines 5x+12y=78 and 5x+12y=104 ? a). P=2 b). P=5 c). P=1 d). P=3 Sol: Correct option is (a) The perpendicular distance P between two parallel lines 5x+12y=78 and 5x+12y=104 is
P
104 78 5 12
2 2
26 =2 13
23).What is the equation of any line passing through the point of intersection of the two lines a1 x b1 y c1 0 and a2 x b2 y c2 0 ? a). a1 x b1 y c1 0 b).
a1 x b1 y c1 =0 a2 x b2 y c2
c). a2 x b2 y c2 0 d). a1 x b1 y c1 0 +k( a2 x b2 y c2 ) 0 Sol: Correct option is (d) The equation of any line passing through the point of intersection of the two lines a1 x b1 y c1 0 and a2 x b2 y c2 0 is
a1 x b1 y c1 0 +k( a2 x b2 y c2 ) 0
Where k is a constant.
24).What is the length of the perpendicular P from the origin on the line ax+by+c=0 and c>0 ? a).
c a b c a b2
2
b).
c).
c a b2
2
d).
c a b
Sol: Correct option is (b) The length of the perpendicular P from the origin on the line ax+by+c=0 is P=
c a b2
2
25).What is the equation of the bisectors of the angle made by the lines having equations a1 x b1 y c1 0 and a2 x b2 y c2 0 respectively ? a).
a1 x b1 y c1 a2 b2 a1 x b1 y c1 a2 b2
2 2
a2 x b2 y c2 a2 b2 a2 x b2 y c2 a2 2 b2 2
b).
c).
a1 x b1 y c1 a2 b2
2 2
a2 x b2 y c2 a2 2 b2 2
a2 x b2 y c2 a2 2 b2 2
d).
a1 x b1 y c1 a2 b2
2 2
Sol: Correct option is (c) The equation of the bisectors of the angle made by the lines having equations a1 x b1 y c1 0 and a2 x b2 y c2 0 are
a1 x b1 y c1 a2 2 b2 2 a2 x b2 y c2 a2 2 b2 2
a2 x b2 y c2 a2 2 b2 2 a2 x b2 y c2 a2 2 b2 2
And
a1 x b1 y c1 a2 2 b2 2
Equations are
a1 x b1 y c1 a2 b2
2 2
26).A company manufactures 20 units of good for Rs. 70 and 90 units for Rs. 280. What is the cost of manufacturing 60 units supposing that the cost curve is a straight line ? a). Rs. 190 b). Rs. 280 c). Rs. 210 d). Rs. 150 Sol: Correct option is (a) Let x be the number of units manufactured and y be the cost in Rs. for manufacturing x units,then as the cost curve is a straight line
y=mx+c
Where c is a constant Also 70=20x+c and 280=90x+c
x=3 and c=10 y=3x+10
Cost of manufacturing 60 units is Rs. 190
27).Function f maps every given number to a number that is 20 more than thrice of the given number.What is the image of 5 ? a). 5 b). 10 c). 25 d).35 Sol: Correct option is (d) f:x 3x+20
Image of 5 =3(5)+20=35
28).Function f maps every given number to a number that is 7 more than the given number.What is the preimage of 19 ? a). 51 b). 15 c). 12 d).35 Sol: Correct option is (c) f:x x+7
As preimage is 19
x+7=19 x=12
29).What is the image of point (3,4) if the line of reflection is X-axis ?
a).(3,-4) b). (3,4) c). (-3,4) d). (-3,-4) Sol: Correct option is (a) If the line of reflection is X-axis then x’=x and y’= -y where (x’,y’) are the coordinates of the image. 30).What is the image of point (3,4) if the line of reflection is Y-axis ? a).(3,-4) b). (3,4) c). (-3,4) d). (-3,-4) Sol: Correct option is (c) If the line of reflection is Y-axis then x’= -x and y’=y where (x’,y’) are the coordinates of the image. 31).The length of the line PQ is 10 units.What is the length of the line AB which is the image of the line PQ and the line of reflection is X-axis ? a). 8 units b). 1 unit c). 5 units d).10 units Sol: Correct option is (d) The length of the image AB is same as the length of the line PQ
Length of AB = Length of PQ= 10 units
32). The image of point P(2,5) is Q(x,y) where the line of reflection is Y-axis. What is the image R(x’,y’) of the point Q if now the line of reflection is X- axis? a).(-2,-5) b). (-2,5) c). (2,-5) d). (2,5) Sol: Correct option is (a) The image of point P(2,5) is Q(x,y) where the line of reflection is Y-axis
x= -2 and y=5
The coordinates (x,y) of Q are (-2,5)
Also The image of point Q(-2,5) is R(x’,y’) where the line of reflection is X-axis
x’=-2 and y’= -5 The coordinates (x’,y’) of R are (-2,-5)
33).The image of the PQR is ABC where the line of reflection is X-axis.Which of the following is true ? a). ABC PQR b). Perimeter of ABC =Perimeter of PQR c). Area of ABC =Area of PQR d). All of the above are true. Sol: Correct option is (d) If the image of the PQR is ABC where the line of reflection is X-axis then
ABC PQR and Perimeter of ABC =Perimeter of PQR and
Area of ABC =Area of PQR 34). The coordinates of the point P are (x,y).The point is displaced by the vector t x i t y j .What are the coordinates (x’,y’)of the new point ? a).(x- t x ,y- t y ) b). (x+ t x ,y+ t y ) c). ( t x , t y ) d). (x,y) Sol: Correct option is (b) As the point is displaced by vector t x i t y j
x’= x+ t x and y’=y+ t y
35). The coordinates of the point P are (5,9).The point is displaced by the vector 3 i 7 j .What are the coordinates (x,y)of the new point ? a).(7,3) b). (3,7) c). (16,8) d). (8,16) Sol: Correct option is (d) As the point is displaced by vector 3 i 7 j
x= 5+3=8 and y=9+7=16
36). The coordinates of the point P are (x,y).The point is rotated by the the angle about origin in the counter-clockwise direction. .What are the coordinates (x’,y’)of the new point ?
a). ( x cos y sin , x sin y cos ) b). ( x cos y sin , x sin y cos ) c). ( x cos y sin , x sin y cos ) d). ( x cos y sin , x sin y cos ) Sol: Correct option is (a) As the point is rotated by the angle about origin in counterclockwise direction
x’= x cos y sin and y’= x sin y cos
37). The coordinates of the point P are (7,9).The point is rotated by the the angle 90 about origin in the counter-clockwise direction. .What are the coordinates (x,y)of the new point ? a). ( 9, 7) b). (9,-7) c). (-9,7) d). (9,7) Sol: Correct option is (c) As the point is rotated by the angle 90 about origin in counterclockwise direction
x= x cos y sin = 7 cos 90 9 sin 90 = -9
and y= x sin y cos = 7 sin 90 9 cos 90 =7 38). The coordinates of the point P in the coordinate system XOY are (x,y). The coordinate system is displaced by the vector t x i t y j .What are the new coordinates (x’,y’)of the point ?
a).(x- t x ,y- t y ) b). (x+ t x ,y+ t y ) c). ( t x , t y ) d). (x,y) Sol: Correct option is (a) As the coordinate system is displaced by vector t x i t y j
x’= x- t x and y’=y- t y
39). The coordinates of the point P in the coordinate system XOY are (11,3). The coordinate system is displaced by the vector 6 i 2 j .What are the new coordinates (x,y)of the point ? a).(1,5) b). (5,1) c). (17,5) d). (5,17) Sol: Correct option is (b) As the coordinate system is displaced by vector t x i t y j
x= 11-6=5 and y=3-2=1
40). The coordinates of the point P in the coordinate system XOY are (x,y).The coordinate system is rotated by the the angle about origin in the clockwise direction. .What are the coordinates (x’,y’)of the new point ?
a). ( x cos y sin , x sin y cos ) b). ( x sin y cos , x cos y sin ) c). ( x cos y sin , x sin y cos ) d). ( x cos y sin , x sin y cos ) Sol: Correct option is (d) As the coordinate system is rotated by the angle about origin in clockwise direction
x’= x cos y sin and y’= x sin y cos
41). The coordinates of the point P in the coordinate system XOY are (5,9).The coordinate system is rotated by the the angle 90 about origin in the clockwise direction. .What are the coordinates (x,y)of the new point ? a). (-9,5) b). (9,5) c). (9,-5) d). (-9,-5) Sol: Correct option is (c) As the coordinate system is rotated by the angle about origin in clockwise direction
x’= x cos y sin = 5 cos 90 9 sin 90 =9
and y’= x sin y cos = 5sin 90 9 cos 90 = -5 42).What is the total surface area of the cylinder whose diameter of the base d is equal to the height of the cylinder ?
a). 3 d 3 b).
3 d3 2 2 d3 3
c).
d).
3 d2 2
Sol: Correct option is (b) Diameter = d
Radius=r=
d 2
Also height h=d
Total surface area = 2 r(h+r)= 2
d d d 3d 3 (d+ )=2 = d3 2 2 2 2 2
43). What is the height of the cylinder whose ratio of curved surface area to the total surface area is 1:3 and the curved surface area of cylinder is 4 square units ? a).1 b). 3 c). 7 d). 5 Sol: Correct option is (a) Curved surface area= 2 rh Total surface area= 2 r ( h r ) Ratio= 2 rh h 1 = = 2 r (h r ) (h r ) 3
3h=h+r 2h=r
r=2h
Also 2 rh =4
rh=2 (2h)h=2
2 h 2 =2 h=1
44). The radii of two cylinders are in the ratio of 2:3 and their heights are in the ratio of 9:7.What is the ratio of their volumes ? a). 9:2 b). 7:4 c). 3:7 d). 4:7 Sol: Correct option is (d) Let the cylinders have radii r1 and r2 and the heights are h1 and h2
r1 2 h 9 = and 1 = r2 3 h2 7 Let the volume of cylinders be V1 and V2
V1 = r12 h1 and V2 = r2 2 h2
V1 r12 h1 2 9 4 = 2 = ( )2 . = V2 r2 h2 3 7 7 45). The radii of two cylinders are in the ratio of 1:5 and their heights are in the ratio of 2:3.What is the ratio of their curved surface area ?
a). 1:3 b). 15:2 c). 2:15 d). 13:12 Sol: Correct option is (c) Let the cylinders have radii r1 and r2 and the heights are h1 and h2
r1 1 h 2 = and 1 = r2 5 h2 3 Let the curved surface area of cylinders be S1 and S 2
S1 = 2 r1 h1 and S 2 = 2 r2 h2
S1 2 r1h1 1 2 2 = = . = S 2 2 r2 h2 5 3 15
46).A conical vessel has a capacity of 30 L of water.Its height is H and radius is R.How much milk can be contained in a vessel in cylindrical form having the same dimensions as that of the cone ? a).20 L b). 90 L c). 30H L d). 30R L Sol: Correct option is (b) The capacity of the cylindrical vessel having the same dimensions is thrice as the capacity of the conical vessel
Volume of cylindrical vessel=30 3=90 L