In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many people like both coffee and tea?

chapter 1 sets exercise 1.6 cbse ncert 11th mathematics

In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many people like both coffee and tea?

C = set of people who like coffee

T =set of people who like tea

n(C)=37

n(T) =52

 because it is given that each person likes at least one of the two drinks

n(C union T)=70 

n(C union T) = n(C) +n(T) - n(C intersection T)

70 = 37 +52 -n(C intersection T)

n(C intersection T) = 37+52 -70 =19

 

19 persons like both coffee and tea.

 

8.In a committee, 50 people speak French, 20 speak Spanish and 10 speak both
Spanish and French. How many speak at least one of these two languages? 

F = set of people who speak french

S= set of people who speak spanish

n(F)=50

n(S)=20

n( F intersection S ) =10

n( F union S ) = n(F) +n(S) - n( F intersection S )

= 50 +20 -10 =60

60 people speak at least one of these two languages.

=================================================

 

 chapter 1 miscellaneous sets ncert cbse

 16. In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C. If 14 people liked products A and B, 12 people liked products C and A, 14 people liked products B and C and 8 liked all the three products. Find how many liked product C only.

solution

15. In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T,  26 read newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find:
(i) the number of people who read at least one of the newspapers.
(ii) the number of people who read exactly one newspaper

solution

 

14. In a group of students, 100 students know Hindi, 50 know English and 25 know both. Each of the students knows either Hindi or English. How many students are there in the group? 

solution

 

 

13. In a survey of 600 students in a school, 150 students were found to be taking tea and 225 taking coffee, 100 were taking both tea and coffee. Find how many
students were taking neither tea nor coffee?

solution

 
ncert cbse 11th mathematics chapter 1 sets exercise 1.6

8.In a committee, 50 people speak French, 20 speak Spanish and 10 speak both
Spanish and French. How many speak at least one of these two languages? 

solution 

In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many people like both coffee and tea?

solution

 

 

11th cbse ncert chapter 2 relations and functions miscellaneous exercise

 

12. Let A = {9,10,11,12,13} and let f : A → N be defined by f (n) = the highest prime factor of n. Find the range of f.

solution

 

11. Let f be the subset of Z × Z defined by f = {(ab, a + b) : a, b ∈ Z}. Is f a
function from Z to  Z? Justify your answer.

solution

 

10. Let A ={1,2,3,4}, B = {1,5,9,11,15,16} and f = {(1,5), (2,9), (3,1), (4,5), (2,11)}
Are the following true?
(i) f is a relation from A to B
(ii) f is a function from A to B.
Justify your answer in each case.

solution 

 

9. Let R be a relation from N to N defined by

 R = {(a, b) : a, b ∈ N and a = (b^2) }. 

Are the following true?
(i) (a,a) ∈ R, for all a ∈ N
(ii) (a,b) ∈ R, implies (b,a) ∈ R
(iii) (a,b) ∈ R, (b,c) ∈ R implies (a,c) ∈ R.

 solution 

 

 8. Let f = {(1,1), (2,3), (0,–1), (–1, –3)} be a function from Z to Z defined by
f(x) = ax + b, for some integers a, b. Determine a, b.

solution  

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If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus.

ncert cbse 11th mathematics chapter 11 conic section 

miscellaneous

 If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus

 

radius at the opening is 20/2 = 10

Choose the parabola as y^2 = 4ax

(5,10) lies on the parabola so that

10^2 = 4a(5)

or a = 100/20

a = 5

Therefore focus and centre are at  of 5 cm apart

In other words the focus is the midpoint of the given diameter.

 

2. An arch is in the form of a parabola with its axis vertical. The arch is 10 m high
and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola?

Choosing the parabola as (x^2)  = -4ay open downwards

because base is 5m wide at the base, (5/2) =  2.5m to each side  at 10 m depth

we can assume that (2.5,-10) lies on the parabola  (x^2)  = -4ay

so that (2.5) ^2 = -4 * a *(-10)

6.25 = 40 a

a =6.25 /40

resubstitute in (x^2)  = -4(6.25 /40)y

to get  (x^2)  = ( - 0.625 ) y

(x^2)  = ( - 0.625 )y

at 2 m distant, let (x,-2)  lie on (x^2)  = -40y

 (x^2)  = ( - 0.625 )(-2)

(x^2)  = 1.25

 (x^2)  = 125 / 100

 (x^2)  = 5 / 4

x =(sqrt(5)) / 2

Required width  = 2 * [(sqrt(5)) / 2] = sqrt(5)

Required width =2.23 m approximately

 

 

=================================================

ncert cbse 11th mathematics chapter 11 conic section 

miscellaneous

 If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus

solution

2. An arch is in the form of a parabola with its axis vertical. The arch is 10 m high
and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola?

solution 

 

ncert cbse chapter 10 straight lines miscellaneous exercise

24.  A person standing at the junction (crossing) of two straight paths represented by the equations 2x – 3y + 4 = 0 and 3x + 4y – 5 = 0 wants to reach the path whose equation is 6x – 7y + 8 = 0 in the least time. Find equation of the path that he should follow.

solution

 

22. A ray of light passing through the point (1, 2) reflects on the x-axis at point A and the reflected ray passes through the point (5, 3). Find the coordinates of A.

 

solution 

 21. Find equation of the line which is equidistant from parallel lines 9x + 6y – 7 = 0 and 3x + 2y + 6 =0

solution

 

19. If the lines y = 3x +1 and 2y = x + 3 are equally inclined to the line y = mx + 4, find the value of m.

solution

18.Find the image of the point (3, 8) with respect to the line x +3y = 7 assuming the line to be a plane mirror.

solution 

 

17. The hypotenuse of a right angled triangle has its ends at the points (1, 3) and (– 4, 1). Find an equation of the legs (perpendicular sides) of the triangle 

 solution

 14. In what ratio, the line joining (–1, 1) and (5, 7) is divided by the 

line x + y = 4 ?

solution

12.Find the equation of the line passing through the point of intersection of the lines 4x + 7y – 3 = 0 and 2x – 3y + 1 = 0 that has equal intercepts on the axes

solution

 

11. Find the equation of the lines through the point (3, 2) which make an angle of 45 degrees with the line x – 2y = 3.

solution

 

8. Find the area of the triangle formed by the lines y – x = 0, x + y = 0 

and x – k = 0

solution

9. Find the value of p so that the three lines 3x + y – 2 = 0, px + 2 y – 3 = 0 and
2x – y – 3 = 0 may intersect at one point.

 solution

6. Find the equation of the line parallel to y-axis and drawn through the point of
intersection of the lines x – 7y + 5 = 0 and 3x + y = 0.

solution

 

4. What are the points on the y-axis whose distance from the line
[x/3] + [y/4]=1 is 4 units.

solution

 

 3. Find the equations of the lines, which cut-off intercepts on the axes whose sum
and product are 1 and – 6, respectively.

solution

 2. Find the values of θ and p, if the equation x cos θ + y sin θ = p is the normal form of the line [sqrt(3)] x + y + 2 = 0.

solution 

 

 Find the values of k for which the line 

(k–3) x – (4 –( k^ 2) ) y + (k^2) –7k + 6 = 0 is

(a) Parallel to the x-axis,
(b) Parallel to the y-axis,
(c) Passing through the origin.

 solution  

 

ncert cbse chapter 10 exercise 10.3

  17. In the triangle ABC with vertices A (2, 3), B (4, –1) and C (1, 2), find the equation and length of altitude from the vertex A.

solution

 

14. Find the coordinates of the foot of perpendicular from the point (–1, 3) to the
line 3x – 4y – 16 = 0.

solution

 

 13. Find the equation of the right bisector of the line segment joining the points

 (3, 4) and (–1, 2).

solution

 

10. The line through the points (h, 3) and (4, 1) intersects the line 

7 x − 9 y − 19 = 0 at right angle. Find the value of h. 

solution

 

 

8. Find equation of the line perpendicular to the line x – 7y + 5 = 0 and having
x intercept 3.

solution  

 

exercise 10.2

19. Point R (h, k) divides a line segment between the axes in the ratio 1: 2. Find
equation of the line.

solution

17.The owner of a milk store finds that, he can sell 980 litres of milk each week at Rs 14/litre and 1220 litres of milk each week at Rs 16/litre. Assuming a linear
relationship between selling price and demand, how many litres could he sell
weekly at Rs 17/litre?

solution

13. Find equation of the line passing through the point (2, 2) and cutting off intercepts on the axes whose sum is 9.  

solution 

 

12.Find the equation of a line that cuts off equal intercepts on the coordinate axes and passes through the point (2, 3).

 solution

 

11.A line perpendicular to the line segment joining the points (1, 0) and (2, 3) divides it in the ratio 1: n. Find the equation of the line.   

 solution 

10. Find the equation of the line passing through (–3, 5) and perpendicular to the line through the points (2, 5) and (–3, 6). 

solution

 

9. The vertices of ∆ PQR are P (2, 1), Q (–2, 3) and R (4, 5). Find equation of the
median through the vertex R.

solution

 

 

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There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks and other organizations using tracking cookie and / or using other means

 

 

Point R (h, k) divides a line segment between the axes in the ratio 1: 2. Find equation of the line.

 ncert cbse chapter 10  straight lines exercise 10.2

19. Point R (h, k) divides a line segment between the axes in the ratio 1: 2. Find
equation of the line.

let the intercepts of the line be a and b so that the line cuts the x axis and y axis at

(a,0)  and (0,b) respectively.

using section formula with ratio 1:2 ,

 ( ( 0 + 2a )  / (1+2) ,  (1b+0) / (2+1) ) = (h,k) by given condition

 so

2a/3 = h  and b/3 = k

a =3h/2  ; b= 3k

use equation of intercept form

(x/a) + (y/b) = 1

[x / (3h/2)] + [y / (3k)] = 1

2kx + hy = 3hk

 

10. Find the equation of the line passing through (–3, 5) and perpendicular to the line through the points (2, 5) and (–3, 6). 

slope of the line passing through (2, 5) and (–3, 6)

is m1 = [6-5]  / [(-3) -2] = 1 /(-5)

 

condition for perpendicular lines is

m1 * m2 = (-1)

[1 /(-5)] * m2 = (-1)

so 

m2 = 5

is the slope of the required line.

required line passes through (-3,5) with slope 5

use point slope form to get the

equation of line as

y-5 =5[x+3] 

y-5 = 5x+15

5x-y+20 = 0

=================================================

ncert cbse chapter 10 straight lines miscellaneous exercise

24.  A person standing at the junction (crossing) of two straight paths represented by the equations 2x – 3y + 4 = 0 and 3x + 4y – 5 = 0 wants to reach the path whose equation is 6x – 7y + 8 = 0 in the least time. Find equation of the path that he should follow.

solution

 

22. A ray of light passing through the point (1, 2) reflects on the x-axis at point A and the reflected ray passes through the point (5, 3). Find the coordinates of A.

 

solution 

 21. Find equation of the line which is equidistant from parallel lines 9x + 6y – 7 = 0 and 3x + 2y + 6 =0

solution

 

19. If the lines y = 3x +1 and 2y = x + 3 are equally inclined to the line y = mx + 4, find the value of m.

solution

18.Find the image of the point (3, 8) with respect to the line x +3y = 7 assuming the line to be a plane mirror.

solution 

 

17. The hypotenuse of a right angled triangle has its ends at the points (1, 3) and (– 4, 1). Find an equation of the legs (perpendicular sides) of the triangle 

 solution

 14. In what ratio, the line joining (–1, 1) and (5, 7) is divided by the 

line x + y = 4 ?

solution

12.Find the equation of the line passing through the point of intersection of the lines 4x + 7y – 3 = 0 and 2x – 3y + 1 = 0 that has equal intercepts on the axes

solution

 

11. Find the equation of the lines through the point (3, 2) which make an angle of 45 degrees with the line x – 2y = 3.

solution

 

8. Find the area of the triangle formed by the lines y – x = 0, x + y = 0 

and x – k = 0

solution

9. Find the value of p so that the three lines 3x + y – 2 = 0, px + 2 y – 3 = 0 and
2x – y – 3 = 0 may intersect at one point.

 solution

6. Find the equation of the line parallel to y-axis and drawn through the point of
intersection of the lines x – 7y + 5 = 0 and 3x + y = 0.

solution

 

4. What are the points on the y-axis whose distance from the line
[x/3] + [y/4]=1 is 4 units.

solution

 

 3. Find the equations of the lines, which cut-off intercepts on the axes whose sum
and product are 1 and – 6, respectively.

solution

 2. Find the values of θ and p, if the equation x cos θ + y sin θ = p is the normal form of the line [sqrt(3)] x + y + 2 = 0.

solution 

 

 Find the values of k for which the line 

(k–3) x – (4 –( k^ 2) ) y + (k^2) –7k + 6 = 0 is

(a) Parallel to the x-axis,
(b) Parallel to the y-axis,
(c) Passing through the origin.

 solution  

 

ncert cbse chapter 10 exercise 10.3

  17. In the triangle ABC with vertices A (2, 3), B (4, –1) and C (1, 2), find the equation and length of altitude from the vertex A.

solution

 

14. Find the coordinates of the foot of perpendicular from the point (–1, 3) to the
line 3x – 4y – 16 = 0.

solution

 

 13. Find the equation of the right bisector of the line segment joining the points

 (3, 4) and (–1, 2).

solution

 

10. The line through the points (h, 3) and (4, 1) intersects the line 

7 x − 9 y − 19 = 0 at right angle. Find the value of h. 

solution

 

 

8. Find equation of the line perpendicular to the line x – 7y + 5 = 0 and having
x intercept 3.

solution  

 

exercise 10.2

19. Point R (h, k) divides a line segment between the axes in the ratio 1: 2. Find
equation of the line.

solution

12.Find the equation of a line that cuts off equal intercepts on the coordinate axes and passes through the point (2, 3).

 solution

 

11.A line perpendicular to the line segment joining the points (1, 0) and (2, 3) divides it in the ratio 1: n. Find the equation of the line.   

 solution 

10. Find the equation of the line passing through (–3, 5) and perpendicular to the line through the points (2, 5) and (–3, 6). 

solution

 

9. The vertices of ∆ PQR are P (2, 1), Q (–2, 3) and R (4, 5). Find equation of the
median through the vertex R.

solution

 

 

disclaimer:
There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks and other organizations using tracking cookie and / or using other means 

 

17. In the triangle ABC with vertices A (2, 3), B (4, –1) and C (1, 2), find the equation and length of altitude from the vertex A.

 17. In the triangle ABC with vertices A (2, 3), B (4, –1) and C (1, 2), find the equation and length of altitude from the vertex A.

 

B (4, –1) and C (1, 2)

using two point form equation of BC is 

{(y-(-1)) / [2-(-1)]} = {(x-4) / (1-4)} 

[y+1] / 3 = [x-4] / (-3)

y+1 = -x+4

x+y-3 = 0--------------(1)

 

line perpendicular to ax+by+c=0 is of the form bx-ay+k=0

 

altitude through A is perpendicular to BC

so the form of the altitude is

x-y+k=0--------------------(2)

This passes through A(2,3)

so 

2-3+k =0

k = 1

so required altitude through A is

x - y +1 =0 using k = 1 in (2)

solving  equation of altitude with that of BC (1)

 

x -  y = (-1)

x + y = 3

solving 

2x = 2

x = 1

resubstitute

y =2

point of intersection is P(1,2)

A is (2,3)

 

using distance formula

length of the altitude is

sqrt[ {(2-1)^2} + {(3-2)^2}] 

=sqrt[2]

we can also use perpendicular distance formula to find the distance between

 A(2,3)  and the line BC { x + y = 3 } for finding the length of the altitude.

14. Find the coordinates of the foot of perpendicular from the point (–1, 3) to the
line 3x – 4y – 16 = 0.

Given line is 3x – 4y – 16 = 0

line perpendicular to ax+by+c=0 is of the form bx-ay+k=0

line perpendicular to given line is

-4x-3y+k = 0

This passes through  (–1, 3) if

-4(-1) - 3(3) + k =0

k =5

so perpendicular line is

-4x-3y+5 = 0

or 4x + 3y  = 5

given equation is

3x - 4y = 16

solving

[4x + 3y  = 5  ] *3

[3x - 4y = 16  ]  *4

 

12x+9y =15

12x-16y=64

 

subtracting

25y =(-49) 

y = [(-49) / 25]

resubstitute and solve for x

3x -4 [(-49) / 25] = 16

3x =16 -[196/25]

3x =204/25

x =68/25

 

foot of the perpendicular is [(68/25) , (-49/25)]

 

 

=================================================

ncert cbse chapter 10 straight lines miscellaneous exercise

24.  A person standing at the junction (crossing) of two straight paths represented by the equations 2x – 3y + 4 = 0 and 3x + 4y – 5 = 0 wants to reach the path whose equation is 6x – 7y + 8 = 0 in the least time. Find equation of the path that he should follow.

solution

 

22. A ray of light passing through the point (1, 2) reflects on the x-axis at point A and the reflected ray passes through the point (5, 3). Find the coordinates of A.

 

solution 

 21. Find equation of the line which is equidistant from parallel lines 9x + 6y – 7 = 0 and 3x + 2y + 6 =0

solution

 

19. If the lines y = 3x +1 and 2y = x + 3 are equally inclined to the line y = mx + 4, find the value of m.

solution

18.Find the image of the point (3, 8) with respect to the line x +3y = 7 assuming the line to be a plane mirror.

solution 

 

17. The hypotenuse of a right angled triangle has its ends at the points (1, 3) and (– 4, 1). Find an equation of the legs (perpendicular sides) of the triangle 

 solution

 14. In what ratio, the line joining (–1, 1) and (5, 7) is divided by the 

line x + y = 4 ?

solution

12.Find the equation of the line passing through the point of intersection of the lines 4x + 7y – 3 = 0 and 2x – 3y + 1 = 0 that has equal intercepts on the axes

solution

 

11. Find the equation of the lines through the point (3, 2) which make an angle of 45 degrees with the line x – 2y = 3.

solution

 

8. Find the area of the triangle formed by the lines y – x = 0, x + y = 0 

and x – k = 0

solution

9. Find the value of p so that the three lines 3x + y – 2 = 0, px + 2 y – 3 = 0 and
2x – y – 3 = 0 may intersect at one point.

 solution

6. Find the equation of the line parallel to y-axis and drawn through the point of
intersection of the lines x – 7y + 5 = 0 and 3x + y = 0.

solution

 

4. What are the points on the y-axis whose distance from the line
[x/3] + [y/4]=1 is 4 units.

solution

 

 3. Find the equations of the lines, which cut-off intercepts on the axes whose sum
and product are 1 and – 6, respectively.

solution

 2. Find the values of θ and p, if the equation x cos θ + y sin θ = p is the normal form of the line [sqrt(3)] x + y + 2 = 0.

solution 

 

 Find the values of k for which the line 

(k–3) x – (4 –( k^ 2) ) y + (k^2) –7k + 6 = 0 is

(a) Parallel to the x-axis,
(b) Parallel to the y-axis,
(c) Passing through the origin.

 solution  

ncert cbse chapter 10 exercise 10.3

  17. In the triangle ABC with vertices A (2, 3), B (4, –1) and C (1, 2), find the equation and length of altitude from the vertex A.

solution

 

14. Find the coordinates of the foot of perpendicular from the point (–1, 3) to the
line 3x – 4y – 16 = 0.

solution

 

 

disclaimer:
There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks and other organizations using tracking cookie and / or using other means

 

2. Find the values of θ and p, if the equation x cos θ + y sin θ = p is the normal form of the line 3 x + y + 2 = 0.

 2. Find the values of θ and p, if the equation x cos θ + y sin θ = p is the normal form of the line [sqrt(3)] x + y + 2 = 0.

[sqrt(3)] x + y + 2 = 0

[sqrt(3)] x + y = (-2)  --------------(1)

we require the RHS to be no negative.

so multiply by (-1)

-[sqrt(3)]x +(-1)y = 2

a = { -[sqrt(3)] } , b = (-1)

sqrt [(a^2) +(b^2)] = sqrt[3+1]  =sqrt[4] = 2

so divide each term in (1) with 2

 [ - {sqrt(3)} /2 ]x +(-1/2)y =1

use x cos θ + y sin θ = p 

to get 

cos θ =[ - {sqrt(3)} /2 ]

 sin θ = (-1/2)

so  θ is in the third quadrant

θ = pi + ( pi /6 ) =[7 pi/6]

using RHS

p =1

 Find the values of k for which the line 

(k–3) x – (4 –( k^ 2) ) y + (k^2) –7k + 6 = 0 is

(a) Parallel to the x-axis,
(b) Parallel to the y-axis,
(c) Passing through the origin.

slope of the given line is 

m = [k-3] / [ 4 –( k^ 2)  ]

(a)

When parallel to x axis

slope = 0

hence [k-3] / [ 4 –( k^ 2)  ] = 0

k-3 = 0

k = 3

(b) Parallel to the y-axis,

 

When parallel to y axis

slope = infinity

hence [k-3] / [ 4 –( k^ 2)  ] = infinity

hence  

[ 4 –( k^ 2)  ] = 0

( k^ 2) = 4

k = 2 or (-2)

 

(c) Passing through the origin.

when line passes through the origin (0,0) 

the (k–3) x – (4 –( k^ 2) ) y + (k^2) –7k + 6 = 0 is satisfied by (0,0)

 0 - 0 + (k^2) –7k + 6 = 0

(k^2) –7k + 6 = 0

[k-6][k-1]  = 0

k =6 , k = 1

=================================================

ncert cbse chapter 10 straight lines miscellaneous exercise

24.  A person standing at the junction (crossing) of two straight paths represented by the equations 2x – 3y + 4 = 0 and 3x + 4y – 5 = 0 wants to reach the path whose equation is 6x – 7y + 8 = 0 in the least time. Find equation of the path that he should follow.

solution

 

22. A ray of light passing through the point (1, 2) reflects on the x-axis at point A and the reflected ray passes through the point (5, 3). Find the coordinates of A.

 

solution 

 21. Find equation of the line which is equidistant from parallel lines 9x + 6y – 7 = 0 and 3x + 2y + 6 =0

solution

 

19. If the lines y = 3x +1 and 2y = x + 3 are equally inclined to the line y = mx + 4, find the value of m.

solution

18.Find the image of the point (3, 8) with respect to the line x +3y = 7 assuming the line to be a plane mirror.

solution 

 

17. The hypotenuse of a right angled triangle has its ends at the points (1, 3) and (– 4, 1). Find an equation of the legs (perpendicular sides) of the triangle 

 solution

 14. In what ratio, the line joining (–1, 1) and (5, 7) is divided by the 

line x + y = 4 ?

solution

12.Find the equation of the line passing through the point of intersection of the lines 4x + 7y – 3 = 0 and 2x – 3y + 1 = 0 that has equal intercepts on the axes

solution

 

11. Find the equation of the lines through the point (3, 2) which make an angle of 45 degrees with the line x – 2y = 3.

solution

 

8. Find the area of the triangle formed by the lines y – x = 0, x + y = 0 

and x – k = 0

solution

9. Find the value of p so that the three lines 3x + y – 2 = 0, px + 2 y – 3 = 0 and
2x – y – 3 = 0 may intersect at one point.

 solution

6. Find the equation of the line parallel to y-axis and drawn through the point of
intersection of the lines x – 7y + 5 = 0 and 3x + y = 0.

solution

 

4. What are the points on the y-axis whose distance from the line
[x/3] + [y/4]=1 is 4 units.

solution

 

 3. Find the equations of the lines, which cut-off intercepts on the axes whose sum
and product are 1 and – 6, respectively.

solution

 2. Find the values of θ and p, if the equation x cos θ + y sin θ = p is the normal form of the line [sqrt(3)] x + y + 2 = 0.

solution 

 

 Find the values of k for which the line 

(k–3) x – (4 –( k^ 2) ) y + (k^2) –7k + 6 = 0 is

(a) Parallel to the x-axis,
(b) Parallel to the y-axis,
(c) Passing through the origin.

 solution 

ncert cbse chapter 9 sequences and series miscellaneous exercise

 

 32.

150 workers were engaged to finish a job in a certain number of days. 4 workers dropped out on the second day. 4 more workers dropped out on the third day and so on. It took 8 more days to finish the work. Find the number of days in which the work was finished

solution

 

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