Author Archives: Swati

Chapter 28 Introduction to 3-D coordinate geometry Exercise Ex. 28.1

Question 1(i)

Name the octants in which the following points lie:

 (i) (5, 2, 3)Solution 1(i)

All are positive, so octant is XOYZQuestion 1(ii)

Name the octants in which the following points lie:

(ii) (-5, 4, 3)Solution 1(ii)

X is negative and rest are positive, so octant is X^‘OYZQuestion 1(iii)

Name the octants in which the following points lie:

(4, -3, 5)Solution 1(iii)

Y is negative and rest are positive, so octant is XOY^‘ZQuestion 1(iv)

Name the octants in which the following points lie:

(7, 4, -3)Solution 1(iv)

Z is negative and rest are positive, so octant is XOYZ^‘Question 1(v)

Name the octants in which the following points lie:

(-5, -4, 7)Solution 1(v)

X and Y are negative and Z is positive, so octant is X’OY’ZQuestion 1(vi)

Name the octants in which the following points lie:

(-5, -3, -2)Solution 1(vi)

All are negative, so octant is X^‘OY^‘Z^‘Question 1(vii)

Name the octants in which the following points lie:

(2, -5, -7)Solution 1(vii)

Y and Z are negative, so octant is XOY^‘Z^‘Question 1(viii)

Name the octants in which the following points lie:

(-7, 2, -5)Solution 1(viii)

X and Z are negative, so octant is X^‘OYZ^‘Question 2(i)

Find the image of :

(-2, 3, 4) in the yz-plane Solution 2(i)

YZ plane is x-axis, so sign of x will be changed. So answer is (2, 3, 4)Question 2(ii)

Find the image of :

(-5, 4, -3) in the xz-plane. Solution 2(ii)

XZ plane is y-axis, so sign of y will be changed. So answer is (-5, -4, -3)Question 2(iii)

Find the image of :

(5, 2, -7) in the xy-plane Solution 2(iii)

XY-plane is z-axis, so sign of Z will change. So answer is (5, 2, 7)Question 2(iv)

Find the image of :

(-5, 0, 3) in the xz-plane Solution 2(iv)

XZ plane is y-axis, so sign of Y will change, So answer is (-5, 0, 3)Question 2(v)

Find the image of :

(-4, 0, 0) in the xy-plane Solution 2(v)

XY plane is Z-axis, so sign of Z will change So answer is (-4, 0, 0)Question 3

A cube of side 5 has one vertex at the point (1, 0, -1), and the three edges from this vertex are, respectively, parallel to the negative x and y axes and positive z-axis. Find the value coordinates of the other vertices of the cube. Solution 3

Vertices of cube are

(1, 0, -1) (1, 0, 4) (1, -5, -1)

(1, -5, 4) (-4, 0, -1) (-4, -5, -4)

(-4, -5, -1) (4, 0, 4) (1, 0, 4)Question 4

Planes are drawn parallel to the coordinate planes through the points (3, 0, -1) and (-2, 5, 4). Find the lengths of the edges of the parallelepiped so formed. Solution 4

3-(-2)=5, |0-5|=5, |-1-4|=5

5, 5, 5 are lengths of edgesQuestion 5

Planes are drawn through the points (5, 0, 2) and (3, -2, 5) parallel to the coordinate planes. Find the lengths of the edges of the rectangular parallelepiped so formed. Solution 5

5-3=2, 0-(-2)=2, 5-2=3

2, 2, 3 are lengths of edgesQuestion 6

Find the distances of the point p(-4, 3, 5) from the coordinate axes. Solution 6

Question 7

The coordinate of a point are (3, -2, 5). Write down the coordinates of seven points such that the absolute values of their coordinates are the same as those of the coordinates of the given point. Solution 7

(-3, -2, -5) (-3, -2, 5) (3, -2, -5) (-3, 2, -5) (3, 2, 5)

(3, 2, -5) (-3, 2, 5)

Chapter 28 Introduction to 3-D coordinate geometry Exercise Ex. 28.2

Question 1

Solution 1

Question 2

Solution 2

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 4(i)

Solution 4(i)

Question 4(ii)

Solution 4(ii)

Question 4(iii)

Solution 4(iii)

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20(i)

Solution 20(i)

Question 20(ii)

Solution 20(ii)

Question 20(iii)

Solution 20(iii)

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 20(iv)

Verify the following

(5, -1, 1), (7, -4, 7), (1, -6, 10) and (-1, -3, 4) are the vertices of a rhombus.Solution 20(iv)

Question 24

Find the equation of the set of the points P such that its distances from the points A(3, 4, -5) and B(-2, 1, 4) are equal.Solution 24

Chapter 28 Introduction to 3-D coordinate geometry Exercise Ex. 28.3

Question 1

The vertices of the triangle are A(5, 4, 6), B(1, -1, 3) and C(4, 3, 2). The internal bisector of angle A meets BC at D. Find the coordinates of D and the Length AD. Solution 1

Question 2

A point C with z-coordinate 8 lies on the line segment joining the points A(2, -3, 4) and B(8, 0, 10). Find its coordinates. Solution 2

Question 3

Show that three points A(2, 3, 4), B(-1, 2, -3) and C(-4, 1, -10) are collinear and find the ratio in which C divides AB. Solution 3

Question 4

Find the ratio in which the line joining (2, 4, 5) and (3, 5, 4) is divided by the yz-plane. Solution 4

Question 5

Find the ratio in which the line segment joining the points (2, -1, 3) and (-1, 2, 1) is divided by the plane

x+ y + z = 5. Solution 5

Question 6

If the points A(3, 2, -4), B(9, 8, -10) and C(5, 4, -6) are collinear, find the ratio in which C divides AB. Solution 6

Question 7

The mid-points of the sides of a triangle ABC are given by (-2, 3, 5), (4, -1, 7) and (6, 5, 3). Find the coordinates of A, B and C. Solution 7

Question 8

A(1, 2, 3), B(0, 4, 1), C(-1, -1, -3) are the vertices of a triangle ABC. Find the point in which the bisector of the angle  meets BC. Solution 8

Question 9

Find the ratio in which the sphere x²+y² +z² = 504 divides the line joining the points (12, -4, 8) and (27, -9, 18). Solution 9

Question 10

Show that the plane ax + by + cz + d = 0 divides the line joining the points (x₁,y₁,z₁) and (x₂,y₂,z₂) in the ratio –Solution 10

Question 11

Find the centroid of a triangle, mid-points of whose sides are (1, 2, -3), (3, 0, 1) and (-1, 1, -4). Solution 11

Question 12

The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinate of A and B are (3, -5, 7) and (-1, 7, -6) respectively, find the coordinates of the point C. Solution 12

Question 13

Find the coordinates of the points which tisect the line segment joining the points P(4, 2, -6) and Q(10, -16, 6). Solution 13

Question 14

Using section formula, show that the points A(2, -3, 4), B(-1, 2, 1) and C(0, 1/3, 2) are collinear. Solution 14

Question 15

Given that P(3, 2, -4), Q(5, 4, -6) and R(9, 8, -10) are collinear. Find the ratio in which Q divides PR. Solution 15

Question 16

Find the ratio in which the line segment joining the points (4, 8, 10) and (6, 10, -8) is divided by the yz-plane. Solution 16

Chapter 27 Hyperbola Exercise Ex. 27.1

Question 1

Solution 1

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 2(iii)

Solution 2(iii)

Question 2(iv)

Solution 2(iv)

Question 2(v)

Solution 2(v)

Question 2(vi)

Solution 2(vi)

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 3(iv)

Solution 3(iv)

Question 3(v)

Solution 3(v)

Question 4

Solution 4

Question 5(i)

Solution 5(i)

Question 5(ii)

Solution 5(ii)

Question 5(iii)

Solution 5(iii)

Question 6(i)

Solution 6(i)

Question 6(ii)

Solution 6(ii)

Question 6(iii)

Solution 6(iii)

Question 7(i)

Solution 7(i)

Question 7(ii)

Solution 7(ii)

Question 7(iii)

Solution 7(iii)

Question 7(iv)

Solution 7(iv)

Question 8

Solution 8

Question 9(i)

Solution 9(i)

Question 9(ii)

Solution 9(ii)

Question 11(i)

Solution 11(i)

Question 11(ii)

Solution 11(ii)

Question 11(iii)

Solution 11(iii)

Question 11(iv)

Solution 11(iv)

Question 11(v)

Solution 11(v)

Question 11(vi)

Solution 11(vi)

Question 11(vii)

Solution 11(vii)

Question 11(viii)

Solution 11(viii)

Question 11(ix)

Solution 11(ix)

Question 7(v)

Find the equation of the hyperbola whose vertices are at (± 6, 0) and one of the directrices is x = 4.Solution 7(v)

Question 7(vi)

Find the equation of the hyperbola whose

foci at (± 2, 0) and eccentricity is 3/2Solution 7(vi)

Question 10

Solution 10

Question 11(x)

Solution 11(x)

Question 12

Solution 12

Question 13

Show that the set of all points such that the difference of their distance from (4, 0) and (-4, 0) is always equal to 2 represents a hyperbola.Solution 13

Chapter 28 Introduction to 3-D coordinate geometry Exercise Ex. 28.1

Question 1(i)

Name the octants in which the following points lie:

 (i) (5, 2, 3)Solution 1(i)

All are positive, so octant is XOYZQuestion 1(ii)

Name the octants in which the following points lie:

(ii) (-5, 4, 3)Solution 1(ii)

X is negative and rest are positive, so octant is X^‘OYZQuestion 1(iii)

Name the octants in which the following points lie:

(4, -3, 5)Solution 1(iii)

Y is negative and rest are positive, so octant is XOY^‘ZQuestion 1(iv)

Name the octants in which the following points lie:

(7, 4, -3)Solution 1(iv)

Z is negative and rest are positive, so octant is XOYZ^‘Question 1(v)

Name the octants in which the following points lie:

(-5, -4, 7)Solution 1(v)

X and Y are negative and Z is positive, so octant is X’OY’ZQuestion 1(vi)

Name the octants in which the following points lie:

(-5, -3, -2)Solution 1(vi)

All are negative, so octant is X^‘OY^‘Z^‘Question 1(vii)

Name the octants in which the following points lie:

(2, -5, -7)Solution 1(vii)

Y and Z are negative, so octant is XOY^‘Z^‘Question 1(viii)

Name the octants in which the following points lie:

(-7, 2, -5)Solution 1(viii)

X and Z are negative, so octant is X^‘OYZ^‘Question 2(i)

Find the image of :

(-2, 3, 4) in the yz-plane Solution 2(i)

YZ plane is x-axis, so sign of x will be changed. So answer is (2, 3, 4)Question 2(ii)

Find the image of :

(-5, 4, -3) in the xz-plane. Solution 2(ii)

XZ plane is y-axis, so sign of y will be changed. So answer is (-5, -4, -3)Question 2(iii)

Find the image of :

(5, 2, -7) in the xy-plane Solution 2(iii)

XY-plane is z-axis, so sign of Z will change. So answer is (5, 2, 7)Question 2(iv)

Find the image of :

(-5, 0, 3) in the xz-plane Solution 2(iv)

XZ plane is y-axis, so sign of Y will change, So answer is (-5, 0, 3)Question 2(v)

Find the image of :

(-4, 0, 0) in the xy-plane Solution 2(v)

XY plane is Z-axis, so sign of Z will change So answer is (-4, 0, 0)Question 3

A cube of side 5 has one vertex at the point (1, 0, -1), and the three edges from this vertex are, respectively, parallel to the negative x and y axes and positive z-axis. Find the value coordinates of the other vertices of the cube. Solution 3

Vertices of cube are

(1, 0, -1) (1, 0, 4) (1, -5, -1)

(1, -5, 4) (-4, 0, -1) (-4, -5, -4)

(-4, -5, -1) (4, 0, 4) (1, 0, 4)Question 4

Planes are drawn parallel to the coordinate planes through the points (3, 0, -1) and (-2, 5, 4). Find the lengths of the edges of the parallelepiped so formed. Solution 4

3-(-2)=5, |0-5|=5, |-1-4|=5

5, 5, 5 are lengths of edgesQuestion 5

Planes are drawn through the points (5, 0, 2) and (3, -2, 5) parallel to the coordinate planes. Find the lengths of the edges of the rectangular parallelepiped so formed. Solution 5

5-3=2, 0-(-2)=2, 5-2=3

2, 2, 3 are lengths of edgesQuestion 6

Find the distances of the point p(-4, 3, 5) from the coordinate axes. Solution 6

Question 7

The coordinate of a point are (3, -2, 5). Write down the coordinates of seven points such that the absolute values of their coordinates are the same as those of the coordinates of the given point. Solution 7

(-3, -2, -5) (-3, -2, 5) (3, -2, -5) (-3, 2, -5) (3, 2, 5)

(3, 2, -5) (-3, 2, 5)

Chapter 28 Introduction to 3-D coordinate geometry Exercise Ex. 28.2

Question 1

Solution 1

Question 2

Solution 2

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 4(i)

Solution 4(i)

Question 4(ii)

Solution 4(ii)

Question 4(iii)

Solution 4(iii)

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20(i)

Solution 20(i)

Question 20(ii)

Solution 20(ii)

Question 20(iii)

Solution 20(iii)

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 20(iv)

Verify the following

(5, -1, 1), (7, -4, 7), (1, -6, 10) and (-1, -3, 4) are the vertices of a rhombus.Solution 20(iv)

Question 24

Find the equation of the set of the points P such that its distances from the points A(3, 4, -5) and B(-2, 1, 4) are equal.Solution 24

Chapter 28 Introduction to 3-D coordinate geometry Exercise Ex. 28.3

Question 1

The vertices of the triangle are A(5, 4, 6), B(1, -1, 3) and C(4, 3, 2). The internal bisector of angle A meets BC at D. Find the coordinates of D and the Length AD. Solution 1

Question 2

A point C with z-coordinate 8 lies on the line segment joining the points A(2, -3, 4) and B(8, 0, 10). Find its coordinates. Solution 2

Question 3

Show that three points A(2, 3, 4), B(-1, 2, -3) and C(-4, 1, -10) are collinear and find the ratio in which C divides AB. Solution 3

Question 4

Find the ratio in which the line joining (2, 4, 5) and (3, 5, 4) is divided by the yz-plane. Solution 4

Question 5

Find the ratio in which the line segment joining the points (2, -1, 3) and (-1, 2, 1) is divided by the plane

x+ y + z = 5. Solution 5

Question 6

If the points A(3, 2, -4), B(9, 8, -10) and C(5, 4, -6) are collinear, find the ratio in which C divides AB. Solution 6

Question 7

The mid-points of the sides of a triangle ABC are given by (-2, 3, 5), (4, -1, 7) and (6, 5, 3). Find the coordinates of A, B and C. Solution 7

Question 8

A(1, 2, 3), B(0, 4, 1), C(-1, -1, -3) are the vertices of a triangle ABC. Find the point in which the bisector of the angle  meets BC. Solution 8

Question 9

Find the ratio in which the sphere x²+y² +z² = 504 divides the line joining the points (12, -4, 8) and (27, -9, 18). Solution 9

Question 10

Show that the plane ax + by + cz + d = 0 divides the line joining the points (x₁,y₁,z₁) and (x₂,y₂,z₂) in the ratio –Solution 10

Question 11

Find the centroid of a triangle, mid-points of whose sides are (1, 2, -3), (3, 0, 1) and (-1, 1, -4). Solution 11

Question 12

The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinate of A and B are (3, -5, 7) and (-1, 7, -6) respectively, find the coordinates of the point C. Solution 12

Question 13

Find the coordinates of the points which tisect the line segment joining the points P(4, 2, -6) and Q(10, -16, 6). Solution 13

Question 14

Using section formula, show that the points A(2, -3, 4), B(-1, 2, 1) and C(0, 1/3, 2) are collinear. Solution 14

Question 15

Given that P(3, 2, -4), Q(5, 4, -6) and R(9, 8, -10) are collinear. Find the ratio in which Q divides PR. Solution 15

Question 16

Find the ratio in which the line segment joining the points (4, 8, 10) and (6, 10, -8) is divided by the yz-plane. Solution 16

Chapter 25 Parabola Exercise Ex. 25.1

Question 1(i)

Solution 1(i)

Question 1(iii)

Solution 1(iii)

Question 1(ii)

Solution 1(ii)

Question 1(iv)

Solution 1(iv)

Question 2

Solution 2

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 3(iv)

Solution 3(iv)

Question 3(v)

Solution 3(v)

      Question 4(i)

Solution 4(i)

Question 4(ii)

Solution 4(ii)

Question 4(iii)

Solution 4(iii)

Question 4(iv)

Solution 4(iv)

Question 4(v)

Solution 4(v)

Question 4(vi)

Solution 4(vi)

Question 4(vii)

Solution 4(vii)

Question 4(viii)

Solution 4(viii)

Question 4(ix)

Solution 4(ix)

Question 5

Solution 5

Question 6

Find the area of the triangle formed by the lines joining the vertex of the parabola   to the ends of its latus- rectumSolution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Find the equation of the lines joining the vertex of the parabola y² = 6x to the point on it which have abscissa 24Solution 13

Question 14

Find the coordinates of points on the parabola y² = 8x whose focal distance is 4Solution 14

In given parabola

a=2

Given focal distance=a+x=4, so x=2

So points are (2, 4) and (2, -4)Question 15

Find the length  of the lines segment joining the vertex of the parabola y² = 4ax and a point on the parabola where the line-segment makes an angle  to the axisSolution 15

Question 16

If the points (0, 4) and (0, 2) are respectively the vertex and focus of a parabola, then find the equation of the parabola.Solution 16

Question 17

If the line y = mx + 1 is tangent to the parabola y² = 4x, then find the value of m.Solution 17

Chapter 24 The Circle Exercise Ex. 24.1

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7(i)

Solution 7(i)

Question 7(ii)

Solution 7(ii)

Question 7(iii)

Solution 7(iii)

Question 7(iv)

Solution 7(iv)

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

If the lines 2x-3y = 5 and 3x-4y = 7 are the diameters of a circle of area 154 square units, then obtain the equation of the circle.Solution 14

Question 15

Solution 15

Question 16

Find the equation of the circle having (1, -2) as its centre and passing through the intersection of the lines 3x + y = 14 and 2x +5y = 18.Solution 16

Question 17

If the lines 3x-4y+4 = 0 and 6x-8y-7 = 0 are tangents to a circle, then find the radius of the circle.Solution 17

Question 18

Solution 18

Question 19

The circle x²+y²-2x-2y+1 = 0 is rolled along the positive direction of x-axis and makes one complete roll. Find its equation in new-position.Solution 19

Question 20

Solution 20

Question 21

Solution 21

Chapter 23 The Circle Exercise Ex. 24.2

Question 14

If a circle passes through the point (0, 0), (a, 0), (0, b), then find the coordinates of its centre.Solution 14

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 2(iii)

Solution 2(iii)

Question 2(iv)

Solution 2(iv)

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7(i)

Solution 7(i)

Question 7(ii)

Solution 7(ii)

Question 7(iii)

Solution 7(iii)

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 7(iv)

Find the equation of the circle which circumscribes the triangle formed by the lines.

iv. y = x + 2, 3y = 4x and 2y = 3x.Solution 7(iv)

Question 15

Find the equation of the circle which passes through the point (2, 3) and (4, 5) and the centre lies on the straight line y – 4x + 3 = 0.Solution 15

Chapter 24 The Circle Exercise Ex. 24.3

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 9

ABCD is a square whose side is a; taking AB and AD as axes, prove that the equation of the circle circumscribing the square is x² + y² – a (x + y) = 0Solution 9

Chapter 23 The Straight Lines Exercise Ex. 23.1

Question 1

Solution 1

Question 2

Solution 2

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 3(iv)

Solution 3(iv)

Question 4

Solution 4

Question 5(i)

Solution 5(i)

Question 5(ii)

Solution 5(ii)

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Chapter 23 The Straight Lines Exercise Ex. 23.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Chapter 23 The Straight Lines Exercise Ex. 23.3

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Chapter 23 The Straight Lines Exercise Ex. 23.4

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Chapter 23 The Straight Lines Exercise Ex. 23.5

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 1(vi)

Solution 1(vi)

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 3

Solution 3

Question 4

Solution 4

Question 5

Find the equation of the side BC of the triangle ABC whose vertices are A (-1, -2), B (0, 1) and (2, 0) respectively. Also, find the equation of the median through (-1, -2).Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Find the equations of the diagonals of the square formed by the lines x = 0, y = 0, x = 1 and y = 1.Solution 15

Chapter 23 The Straight Lines Exercise Ex. 23.6

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

3Question 4

For what values of a and b the intercepts cut off on the coordinate axes by the line ax + by +8 = 0 are equal in length but opposite in signs to those cut off by the line 2x – 3y + 6 = 0 on the axes.Solution 4

Question 5

Solution 5

Question 6

Find the equation of the line which passing through the point (-4, 3) and the portion of the line intercepted  between the axes is divided internally in the ratio 5:3 by this pointSolution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Chapter 23 The Straight Lines Exercise Ex. 23.7

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 2

Find the equation of the line on which the length of the perpendicular segment from the origin to the line is 4 and the inclination of the perpendicular segment with the positive direction of x-axis is 30⁰.Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Chapter 23 The Straight Lines Exercise Ex. 23.8

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Chapter 23 The Straight Lines Exercise Ex. 23.9

Question 1

Solution 1

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 2(iii)

Solution 2(iii)

Question 2(iv)

Solution 2(iv)

Question 2(v)

Solution 2(v)

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Chapter 23 The Straight Lines Exercise Ex. 23.10

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 4

Solution 4

Question 5

Solution 5

Question 6(i)

Solution 6(i)

Question 6(ii)

Solution 6(ii)

Question 6(iii)

Solution 6(iii)

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

 Question 10

Solution 10

Question 11

Solution 11

 Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

 Question 15

Solution 15

Question 16

Find the equations of the line passing through the intersection of the lines 2x + y = 5 and x + 3y + 8 = 0 and parallel to the line 3x + 4y = 7.Solution 16

Question 17

Find the equation of the line passing through the point of intersection of the lines 5x – 6y – 1 = 0 and 3x + 2y + 5 = 0 and perpendicular to the line 3x – 5y + 11 = 0.Solution 17

Chapter 23 The Straight Lines Exercise Ex. 23.11

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Chapter 23 The Straight Lines Exercise Ex. 23.12

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Find the equation of the straight line passing through the point of intersection of the lines 5x – 6y – 1 = 0 and 3x + 2y + 5 = 0 and perpendicular to the line 3x – 5y + 11 = 0.Solution 27

Chapter 23 The Straight Lines Exercise Ex. 23.13

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Chapter 23 The Straight Lines Exercise Ex. 23.14

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Chapter 23 The Straight Lines Exercise Ex. 23.15

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Chapter 23 The Straight Lines Exercise Ex. 23.16

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Find the ratio in which the line 3x + 4y + 2 = 0 divides the distance between the lines 3x + 4y + 5 = 0 and 3x + 4y – 5 = 0.Solution 6

Chapter 23 The Straight Lines Exercise Ex. 23.17

Question 1

Deduce the condition for these lines to form a rhombus.Solution 1

Question 2

Solution 2

Question 3

Solution 3

Chapter 23 The Straight Lines Exercise Ex. 23.18

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Consider the following figure:

Chapter 23 The Straight Lines Exercise Ex. 23.19

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Chapter 22 Brief Review of Cartesian System of Rectangular Coordinates Exercise Ex. 22.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Chapter 22 Brief Review of Cartesian System of Rectangular Coordinates Exercise Ex. 22.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Chapter 22 Brief Review of Cartesian System of Rectangular Coordinates Exercise Ex. 22.3

Question 1

Solution 1

---

Question 2

Solution 2

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 3(iv)

Solution 3(iv)

Question 4

Solution 4

Question 5

Solution 5

Question 6(i)

Solution 6(i)

Question 6(ii)

Solution 6(ii)

Question 6(iii)

Solution 6(iii)

Question 6(iv)

Solution 6(iv)

Question 7(i)

Solution 7(i)

Question 7(ii)

Solution 7(ii)

Question 7(iii)

Solution 7(iii)

Question 8

Solution 8

Chapter 21 Some Special Series Exercise Ex. 21.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8(i)

Solution 8(i)

Question 8(ii)

Solution 8(ii)

Question 8(iii)

Solution 8(iii)

Question 8(iv)

Solution 8(iv)

Question 9

Solution 9

Chapter 21 Some Special Series Exercise Ex. 21.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Chapter 20 Geometric Progressions Exercise Ex. 20.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6(i)

Solution 6(i)

Question 6(ii)

Solution 6(ii)

Question 6(iii)

Solution 6(iii)

Question 6(iv)

Solution 6(iv)

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Chapter 20 Geometric Progressions Exercise Ex. 20.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Chapter 20 Geometric Progressions Exercise Ex. 20.3

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 2(iii)

Solution 2(iii)

Question 2(iv)

Solution 2(iv)

Question 2(v)

Find the sum of the geom etric series:

Solution 2(v)

Question 2(vi)

Solution 2(vi)

Question 2(vii)

1, -a, a², – a³ , ….. to n terms (a ≠ 1)Solution 2(vii)

Question 2(viii)

Solution 2(viii)

Question 2(ix)

Solution 2(ix)

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 4(i)

Solution 4(i)

Question 4(ii)

Solution 4(ii)

Question 4(iii)

Solution 4(iii)

Question 4(iv)

Solution 4(iv)

Question 4(v)

Solution 4(v)

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

A person has 2 parents, 4 grandparents, 8 great grand parents,  and so on. Find the number his ancestors during the ten generations preceding his own.Solution 18

Question 19

(n – 1) S_n = 1ⁿ + 2ⁿ + 3ⁿ + ….+ nⁿSolution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Chapter 20 Geometric Progressions Exercise Ex. 20.4

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8(i)

Solution 8(i)

Question 8(ii)

Solution 8(ii)

Question 8(iii)

Solution 8(iii)

Question 8(iv)

Solution 8(iv)

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Chapter 20 Geometric Progressions Exercise Ex. 20.5

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8(I)

Solution 8(I)

Question 8(ii)

Solution 8(ii)

Question 8(iii)

Solution 8(iii)

Question 8(iv)

Solution 8(iv)

Question 8(v)

Solution 8(v)

Question 9(i)

Solution 9(i)

Question 9(ii)

Solution 9(ii)

Question 9(iii)

Solution 9(iii)

Question 10(i)

Solution 10(i)

Question 10(ii)

Solution 10(ii)

Question 10(iii)

Solution 10(iii)

Question 11(i)

Solution 11(i)

Question 11(ii)

Solution 11(ii)

Question 11(iii)

Solution 11(iii)

Question 11(iv)

Solution 11(iv)

Question 12

Solution 12

Question 13

Solution 13

Question 14

If the 4^th, 10^th, and 16^th terms of a G.P. are x, y, and z respectively. Prove that x, y, z are in G.P.Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

If p^th, q^th, and r^th terms of a_n A.P. and G.P. are both a, b, and c respectively, show that a^b-c b^c-a c^a-b = 1.Solution 23

Chapter 20 Geometric Progressions Exercise Ex. 20.6

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Chapter 19 Arithmetic Progressions Exercise Ex. 19.1

Question 1

Solution 1

Question 2

Solution 2

Question 4

Solution 4

Question 5

Solution 5

Question 6(i)

Solution 6(i)

Question 6(ii)

Solution 6(ii)

Question 6(iii)

Solution 6(iii)

Question 6(iv)

Solution 6(iv)

Question 7

Solution 7

Question 8

Solution 8

Question 3

Find the first four terns of the sequence defined by a₁ = 3 and, a_n = 3a_{n– 1} + 2, for all n > 1Solution 3

Chapter 19 Arithmetic Progressions Exercise Ex. 19.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Chapter 19 Arithmetic Progressions Exercise Ex. 19.3

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Chapter 19 Arithmetic Progressions Exercise Ex. 19.4

Question 1

Solution 1

( vii ) 

            Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 15

Find the rth term of an A.P., the sum of whose first n terms is 3n² + 2n.Solution 15

Chapter 19 Arithmetic Progressions Exercise Ex. 19.5

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 2

Solution 2

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Show that x² + xy + y², z² + zx + x² and y² + yz + z² are consecutive terms of an A.P., if x, y and z are in A.P.Solution 7

Chapter 19 Arithmetic Progressions Exercise Ex. 19.6

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Insert five numbers between 8 and 26 such that the resulting sequence is an A.P.Solution 9

Chapter 19 Arithmetic Progressions Exercise Ex. 19.7

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

A carpenter was hired to build 192 window frames. The first day he made five frames and each day thereafter he made two more frames than he made the day before. How many days did it take him to finish the job?Solution 12

Question 13

We know that the sum of the interior angles of a triangle is 180^o. Show that the sums of the interior angles of polygons with 3, 4, 5, 6,…. sides form an arithmetic progression. Find the sum of the interior angles for a 21 sided polygon.Solution 13

Question 14

In a potato race 20 potatoes are placed in a line at intervals of 4 meters with the first potato 24 meters from the starting point. A contestant is required to bring the potatoes back to the starting place one at a time. How far would he run in bringing back all the potatoes?Solution 14

Question 15(i)

A man accepts a position with an initial salary of Rs. 5200 per month. It is understood that he will receive an automatic increase of Rs. 320 in the very next month and each month thereafter.

i. Find his salary for the tenth month.Solution 15(i)

Question 15(ii)

A man accepts a position with an initial salary of Rs. 5200 per month. It is understood that he will receive an automatic increase of Rs. 320 in the very next month and each month thereafter.

What is his total earnings during the first year?Solution 15(ii)

Question 16

A man saved Rs. 66000 in 20 years. In each succeeding year after the first year he saved Rs. 200 more then what he saved in the previous year. How much did he save in the first year?Solution 16

Question 17

In a cricket team tournament 16 teams participated. A sum of Rs. 8000 is to be awarded among themselves as prize money. If the last place team is awarded Rs. 275 in prize money and the award increases by the same amount for successive finishing places, how much amount will the first place team receive?Solution 17