EM-SSLC Worksheets - Questions - John P A

5) A hemisphere is carved from a solid cone of radius 15cm and height 20 cm.

a) What is the slant height of the cone?
b) What is the radius of the hemisphere ?
c) Calculate the surface area of the hemisphere
d) Calculate the volume of the hemisphere

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2020-21 Academic year Worksheets

Mathematics X
ഘന പ ൾ

Concepts

a) The basic measurement of a sphere is its radius. Surface area and volume can be calculated using
radius of the sphere.

b) If r is the radius of the sphere , its surface area is 4πr2

c) Volume of the sphere with radius r is 4 πr3
3

d) Volume of the hemisphere is 2 πr3
3

e) Curved surface area of the hemisphere = 2πr2. Total surface area of the hemisphere = 3πr2

1) Total surface area of a solid sphere is 100 sq.cm . It splits up into two hemispheres .

a) What is the curved surface area of the hemisphere ?
b) What is the total surface area of the hemisphere?

2) A cylindrical vessel of radius 4cm contains water to the height 10cm. A small solid sphere of radius 2 cm is
immersed in it.

a) Find the volume of the sphere.
b) Calculate the level to which water rises.
c) A child said: If a sphere of double the radius is immersed the level also rises twice. Is this statement

true? Justify

3) A cone of height 16cm and maximum size is carved from a solid sphere of radius 10cm.

a) calculate the radius of the cone?
b) Find the slant height of the cone?
c) Calculate the volume of the cone?

1

4) Ratio of the volume of two cones is 64 : 27. The sum of the radii is 21cm
a) What is the ratio of their radii?
b) Calculate the radius of each cone?
c) What is the ratio of the surface area of the cones?

5) Radius of a cone and s hemisphre are equal. Volume of cone and hemisphere are equal.
a) What is the ratio of the heights?
b) If the radius of the sphere is 10cm then what is its height?
c) If the radius of the sphere is 2cm then what is its curved surface area ?
d) Find the ratio of the surface area of the cone and hemisphere.
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2020-21 Academic year Worksheets

Mathematics X
Solids

1) A solid is made by fixing a hemisphere in the base of a cone . The height of the cone is 12 cm. Cone and
hemisphere have smae base area.Total height of the solid is 17cm.

a) What is the radius of the hemisphe?
b) What is the slant height of the cone?
c) Calculate the surface area of the solid.
d) Calculate the volume of the solid.
2) Hemispheres are fixed at the ends of a cylinder.Radius of the hemispheres and cylinder are equal.Total
height of the solid is 19 cm.If the common radius is 7 cm then
a) What is the height of the cylinder?
b) Calculate the total surface area of the solid.
c) Calculate the volume of the solid.
3) A cone is fixed at one end and a hemisphere at other end of a cylinder. Common radius is 4.2 cm and total
height is 23.2 cm.
a) What is the height of the cone?
b) Calculate the volume of the cone
c) Caculate the volume of the cone.
d) Calculate the volume of the cylinder.
e) Find the total volume of the combination.

1

4) A cone of radius 10cm and 10cm height 8 cm is divided equally by a plane parallel to the base half of its
height.
a) What is the radius of the small cone taken from the big cone?
b) Calculate the volume of the small cone .
c) What is the volume of the remaining part ?ബാ ിവ ഭാഗ ിെ വ ാ ം കണ ാ ക
d) What is the ratio of the volume of two parts .

5) A sphere is fixed inside a cone in such a way that the sphere touches the lateral face and base of the
cone.The base radius of the cone is 6 cm and height 8 cm
a) What is the slant height of the cone?
b) What is the radius of the sphere ?
c) What is the volume of the sphere?
d) Calculate the volume of the cone.
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09

Geometry and
Algebra

ജ്യാമിതിയും
ബീജഗണിതവും

2020-21 Academic year Worksheets

Mathematics X
Geometry and Algebra

Concepts

⋆ In this unit we discuss the relation between the coordinates of points in a geometric figure
algebraically.

⋆ The relation between the coordinates of points on a line is the equation of the line. A line will have
an equation.

⋆ The coordinates of two points on a line and the coordinates of any point on that line in between the
given points are related to eachother by a ratio.

⋆ The slope of a line is the measure of its inclination.

1) ABCD is a parallelogram . If A(1, 1), B(4, 2), C(6, 7)then

a) Write the difference between x coordinates of A and B
b) Write the difference between y coordinates of A and B
c) Write the coordinates of D.
2) In the figure P QRS is a parallelogram .If P (0, 3), P S = 4, Q(5.4)then

a) Write the coordinates of S 1
b) Write the coordinates of R
c) Find the length of the sides.

3) P (1, 4)the mid point of the side AB ,Q(2, 3) is the mid point of side BC , R(5, 6)is the mid point of side
AC
a) Draw a suitable diagram representing the position of points
b) Write the coordinates of B
c) Write the coordinates of C
d) Write the coordinates of A

4) (1, 5), (4, 1), (4, 5) are the mid points of the sides of a triangle.
a) Draw a diagram and mark the mid points
b) Find the coordinates of the vertices of the triangle.
c) What kind of triangle is this ?
d) Calculate the area of this triangle.

5) In the parallelogram ABCD the line AB and line CD are parallel to xaxis. A(1, 5) .Length of these
parallel sides is 6cm and the area of the parallelogram is 48.
a) What is the distance between the parallel sides AB and CD
b) Write the coordinates of B
c) If x coordinates of C is 9, then what is the y coordinates of C. Write the point C .
d) Write the coordinates of D
e) Find the perimetre of the parallelogram.
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2020-21 Academic year Worksheets

Mathematics X
Geometry and Algebra

Concepts

In the parallelogram ABCD, A(x1, y1), B(x2, y2), C(x3, y3), D(x4, y4) then x3 = x2 + x4 −
x1, y3 = y2 + y4 − y1.
Using this the vertices of a triangle can be determined by using the mid points of the sides.

1) The mid point of the side AB of triangle ABC is P (1, 1).The mid point of the side BC is (5, 4), mid point
of the side AC is (7, 4)

a) Find the coordinate of the vertex A.
b) Find the coordinate of the vetex B
c) Find the coordiante of the vertex C.

2) There is a line joining the points A(x1, y1), B(x2, y2). The point P (x, y) divides AB in the ratio m : n.

a) Establish the relation x = x1 + m (x2 − x1)
m+n

b) Establish the relation y = y1 + m (y2 − y1)
m+n

3) P (x, y) is a point on the line AB in between A and B which divides AB in the ratio 2 : 3.If A(1, 3) and
B(3, 4) then

a) Find the x coordinates of P
b) Find the y coordinates of P

4) There is a line jpining the points A(x1, y1) and B(x2, y2). P (x, y) is the mid point of AB.

Prove that the coordinates of P are (x1 +x2 ) , (y1 +y2 )
2 2
Find the coordinates of the mid point of the line joining A(−3, 5) and B(7, 7)

5) In triangle ABC, A(−3, 2), B(1, 5), C(3, −4)then

a) Find the coordinates of the mid point of AB
b) Find the coordiantes of the mid point of BC
c) Find the coordinates of the mid point of AC

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2020-21 Academic year Worksheets

Mathematics X
Geometry and Algebra

Concepts

⋆ The line joining vertex to the mid point of the opposite side is called median.There are three medians
in a triangle.

⋆ The point of intersection of the medians is called centroid of the triangle.It is denoted by G.
⋆ The centroid of a triangle divides the median in the ratio 2 : 1 from the vertex.

1) In triangle ABC, A(0, 0), B(4, 0), C(2, 10)
AB, BC, AC has the mid points P, Q, Rrespectively
a) Write the coordinates of P, Q, R
b) Find the coordinates of the point which divides CP in the ratio 2 : 1.
c) Find the coordinates of the point which divides AQ in the ratio2 : 1
d) Find the coordinates of the point which divides BR in the ratio 2 : 1

2) The line joining A(−1, 3), B(4, 1) intersect y axis at P
a) What is the x coordinate of P ?
b) Find AP : BP
c) Find the coordinates of P .

3) Triangle ABC is an equilateral triangle.A(1, 0), B(5, 0)

a) Find side of the triangle
b) What is the coordinates of the mid point of AB
c) Find the coordinates of C
d) Find the coordinates of the mid point of the centroid .
4) In triangle ABC , CA = CB, A(1, 4), B(9, 4), Altitude from C is 6
a) Find the coordinates of the mid point of AB
b) Find the coordinates of C
c) Find the coordinates of the centriod of the t1riangle.
5) There is a line AB joining (1, 6) and B(5, 2).Find the coordinates of the points which divides AB into three
equal parts.

2020-21 Academic year Worksheets

Mathematics X
Geometry and Algebra

Concepts

⋆ The line joining vertex to the mid point of the opposite side is called median.There are three medians
in a triangle.

⋆ The point of intersection of the medians is called centroid of the triangle.It is denoted by G.
⋆ The centroid of a triangle divides the median in the ratio 2 : 1 from the vertex.

1) ABCD is a parallelogram.If A(1, 2), B(4, y), C(x, 6), D(3, 5)then

a) What is the x coordinate of the mid point of the diagonal BD.
b) Write the coordinates of C
c) Write the y coordinate of the diagonal AC
d) Write the coordiantes of B.

2) The centre of a circle is (2, −3),AB is the diametre of the circle.If B(4, −3) then

a) What is the radius of the circle?
b) Write the coordinates of the end A of the diametre
c) CD is another diametre perpendicular to AB. Write the coordiantes of C and D
d) What is the area of ACBD?

3) In the line AB, A(−2, −2), B(2, −4).P is a point in between A and B.

Length of AP is 3 of the length of AB
7

a) What is the ratio at which P divides AB

b) Write the coordinates of P

c) Find the coordinates of the mid point of AB

1

4) Two vertices of △ABCare on x axis .If A(1, 3)then

a) Write the coordiantes of B and C from the figure
b) Find the point at which the median from A intersect BC
c) Find the coordinates of the centroid of the triangle.
5) A(4, 2), B(6, 5), C(1, 4)are the vertices of a triangle. P is a point on AB, Q is a point on BC
a) If AP : BP = 1 : 2then write the coordinates of P
b) If Q is the mid point of BC write the coordinates of Q
c) Join these points suitably to get three equal area triangles by dividing the area of triangle ABC.

2

2020-21 Academic year Worksheets

Mathematics X
Geometry and Algebra

Concepts

⋆ The inclination of a line with x axis is very important for higher level learning coordinate geometry.
There is a slope to a line. It is the measure of inclination. But mere inclination cannot determine a
perticular line. There are more more one line with same slope.

⋆ Let A(x1, y1), B(x2, y2)be two lines. Slope of this line is .y2 −y1

x2 −x1

⋆ The tan measure of the angle made by a line with the positive direction of x axis is also the slope

of this line

1) In the figure you can see a horizontal plane and an inclined plane.A is walking from A to B along the inclined
plane .

a) What is the horizontal distance and vertical distance from A to B

b) If BC = 1 then what is DE
AC 2 AE

c) If AG = 20then what is F G?

d) Write your inference on this concept.

2) Three lines passes through a point on x axis .

a) What is the slope of the line AD?

b) What is the slope of the line AC? 1

c) What is the slope of the line AB?

d) Write the coordinates of two points on AC

3) Draw x axis and y axis (rough diagram), mark the points A(4, 3) and B(12, 7)
a) What is the slope of this line?
b) Write the coordinates of another point on this line?
c) How many lines are there having the same slop?

4) Consider the points A(2, 3), B(3, 4), C(4, 5)
a) Find the slope of the line passing through A(2, 3) and B(3, 4)
b) Find the slope of the line passing through B(3, 4) and C(4, 5)
c) Are these points on a line? How can we realize it.
d) Write the coordinates of one more point on the line?

5) Consider the points A(2, 0), B(−6, −2), C(−4, −4), D(4, −2)
a) Find the slope pf the lines AB and CD
b) Find the slope of the line AD and BC
c) Is ABCD a parallelogram ?Explain

2

2020-21 Academic year Worksheets

Mathematics X
Geometry and Algebra

Concepts

⋆ The inclination of a line with x axis is very important for higher level learning coordinate geometry.
There is a slope to a line. It is the measure of inclination. But mere inclination cannot determine a
perticular line. There are more more one line with same slope.

⋆ Let A(x1, y1), B(x2, y2)be two lines. Slope of this line is .y2 −y1

x2 −x1

⋆ The tan measure of the angle made by a line with the positive direction of x axis is also the slope

of this line

1) Consider the pairs in the order (1, 3), (2, 5), (3, 7), (4, 9) · · · . These are the coordinates of points

a) Write the sequence of x coordiantea and y coordinates seperately.
b) Take the term of the first sequence as x and write the algebraic form of the second sequence .
c) Take two points and find the slope of the line joining these points
d) Prove that these are the points on a line.

2) Consider the points A(2, −3), B(−5, 1), C(7, −1), D(0, 3)

a) Find the slope of the line AB
b) Find the slope of the line CD
c) Are these points the vertices of a parallelogram ?

3) A(1, −2), B(x, 4) are the points on a line of slope x.

a) Find x
b) Write the coordinates of another point on this line
c) Find the point at which the line cut x axis
d) Find the point at which the line cut y axis

1

4) A(−4, 2), B(2, 6), C(8, 5), D(9, −7)are the vertices of a quadrilateral
a) Find the coordinates of the mid point of the sides.
b) Prove that the quadrilateral formed by the mid points is a parallelogram
c) Find the coordinaters of the point where the diagonals intersect.

5) A(−4, 3), B(7, 3), C(5, 1), D(−2, 1) are the vertices of a quadrilateral.
a) Find the slope of the sides ABand CD
b) Prove that ABCD is an isosceles trapezium
c) Find the area of the trapezium
d) Calculate the perimetre of the trapezium

2

2020-21 Academic year Worksheets

Mathematics X
Geometry and Algebra

Concepts

a) The relation between the coordinates of points in a line which is common in all points is the
characteristic property of that line.The relation between x and y coordinates is named as the
equation of the line.

b) y coordinates of all points in x axis is 0. The equation y = 0 is considered as the equation of x
axis.

c) The y coordinates of all points on a line parallel to x axis are equal. For example in the line parallel
to x axis and passing through (0, 5) , y coordinates of all points are 5. Therefore y = 5 is the
equation of this line.

d) x coordinates of all points on y axis is 0. Therefore x = 0 is considered as the equation of y axis.
e) x coordinates of all points on a line parallel to y axis are equal. For example , in the line parallel to

y axis and passing through 5, 0), the x coordinates of all points are 5 . x = 5 is the equation of
this line.
f) Sometimes it is difficult to find the linear relation between the coordinates for getting the sequation.
We can use the measure slope for writing the equation successfully.
g) There are three types of lines. The line parallel to x axis, line parallel to y axis and inclined lines .
The inclination may be either to the left or right . The line inclined to the right has positive slope,
the line inclined to the left have negative slope.

1) A line is drawn parallel to x axis and passing through (0, 4).Another line is drawn parallel to y axis and
passing through (4, 0).
a) Write the equation of these lines.
b) What are the coordinates of points intersecting these lines?
c) There is a line passing through the intersecting point and the origin. Write the equation of this line

2) (1, 5), (2, 7), (3, 9), (4, 11) are the points on a line.
a) Write the property common in all points
b) Find the slope of this line
c) Take (x, y) as a point on this line and write the equation of the line using slope and another given
point

1

3) The sum of the x coordinates and y coordinates of points on a line is 0. Write the equation of this line
a) Write the equation of this line.
b) Find the point where this line cut x axis .
c) Find the coordiantes of the point where this line cut y axis
d) Calculate the slope of the line.

4) Two measures are necessary for finding the equation of a line.Knowing the slope of a line and and a point
through which the line passes we can write the equation of the line easily.
The slope of a line is 3. The line passes through (1, 2).
a) If (x, y) is a point on this line write the statement connecting the slope and the given point (1, 2)
b) Write the equation of this line.
c) At what point the line cut x axis ?
d) At what point the line cut y axis ?

5) Consider the lines x = 3, x = 7, y = 4, y = 8
a) Draw a rough diagram showing the lines
b) Write the name of the geometric figure enclosing these lines .
c) Write the coordinates of the vertices.
d) Calculate area and perimetre

2

2020-21 Academic year Worksheets

Mathematics X
Geometry and Algebra

Concepts

a) The relation between the coordinates of points in a line which is common in all points is the
characteristic property of that line.The relation between x and y coordinates is named as the
equation of the line.

b) y coordinates of all points in x axis is 0. The equation y = 0 is considered as the equation of x
axis.

c) The y coordinates of all points on a line parallel to x axis are equal. For example in the line parallel
to x axis and passing through (0, 5) , y coordinates of all points are 5. Therefore y = 5 is the
equation of this line.

d) x coordinates of all points on y axis is 0. Therefore x = 0 is considered as the equation of y axis.
e) x coordinates of all points on a line parallel to y axis are equal. For example , in the line parallel to

y axis and passing through (5, 0), the x coordinates of all points are 5 . x = 5 is the equation of
this line.
f) Sometimes it is difficult to find the linear relation between the coordinates for getting the sequation.
We can use the measure slope for writing the equation successfully.
g) There are three types of lines. The line parallel to x axis, line parallel to y axis and inclined lines .
The inclination may be either to the left or right . The line inclined to the right has positive slope,
the line inclined to the left have negative slope.

1) A line cut x axis at (12, 0) , cut y axis at (0, 5).
a) What is the slope of this line ?
b) Write the equation of this line.
c) Calculate the area of the triangle formed by the line and coordinate axes
d) What is the length of the line in between coordinate axes ?
e) What is the perpendicular diatance from origin to the line?

2) Consider the equation 2x + 3y = 6
a) What is the point at which the line cut x axis ?
b) What is the point at which the line cut y axis ?
c) what is the slope of this line?
d) How does the slope is related to the coefficients of x and y in the equation?

1

3) Consider the equation 3x + 2y = 6
a) What is the slope of this line?
b) Write the equation of another line having the same slope . Suggest a method to write the equation of
lines parallel to the given line.
c) Write the equation of the line parallel to the line 3x + 2y = 6 and passing through (1, 1)

4) Consider the equation x + y + 1 = 0
a) Find the point at which the line cut x axis.
b) Find the point at which the line cut y axis .
c) Find the slope of the line
d) Calculate the area of the triangle enclosed by the line with the coordinate axes .
e) What is the length of median of the triangle from the origin.

5) A line makes 45◦ with the positive direction of x axis .
a) What is the slope of this line?
b) The line passes through (3, 5). Write the equation of this line
c) At what point the line cut x axis ?
d) At what ppoint the line cut y axis .

2

2020-21 Academic year Worksheets

Mathematics X
Geometry and Algebra

Concepts

a) The relation between the coordinates of points in a line which is common in all points is the
characteristic property of that line.The relation between x and y coordinates is named as the
equation of the line.

b) y coordinates of all points in x axis is 0. The equation y = 0 is considered as the equation of x
axis.

c) The y coordinates of all points on a line parallel to x axis are equal. For example in the line parallel
to x axis and passing through (0, 5) , y coordinates of all points are 5. Therefore y = 5 is the
equation of this line.

d) x coordinates of all points on y axis is 0. Therefore x = 0 is considered as the equation of y axis.
e) x coordinates of all points on a line parallel to y axis are equal. For example , in the line parallel to

y axis and passing through (5, 0), the x coordinates of all points are 5 . x = 5 is the equation of
this line.
f) Sometimes it is difficult to find the linear relation between the coordinates for getting the sequation.
We can use the measure slope for writing the equation successfully.
g) There are three types of lines. The line parallel to x axis, line parallel to y axis and inclined lines .
The inclination may be either to the left or right . The line inclined to the right has positive slope,
the line inclined to the left have negative slope.

1) Consider the numbers (1, 2), (5, 7)
a) What is the slope of the line passing through these points ?
b) Write the equation of this line
c) At what point the line cut x axis ?
d) At what point the line cut y axis ?

2) A line cut the coordinate axea at (4, 0) and (0, 4)
a) What is the slope of this line ?
b) What is the angle made by this line with the positive direction of x axis ?
c) Write the equation of this line.
d) What is the perpendicular distance from origin to the line ?

1

3) A line passes through (4, 0) and (0, 5)

a) What is the slope of this line?

b) Show that the equation of this line is x + y = 1.
4 5

c) What is the equation of the line passing through (a, 0) and (0, b)?

4) Consider the points (6, 0) and (0, 6)

a) What are the coordinates of the mid point of the line joining these points?
b) The line passing through the mid point makes 60◦ angle with the positive direction of x axis.What is

the slope of this line?

c) Write the equation of this line.

d) At what point this line cut y axis ?വരy

5) A line passes through (a, 0) and (0, b).

a) If the mid point of the line in between the coordinate axes is (x1, y1) then show that the equation of
y
this line is x + y1 = 2
x1

b) If this line passes through (6, 0) and (0, 6)then write the equation of the line .

2

2020-21 Academic year Worksheets

Mathematics X
Geometry and Algebra

Concepts

a) The relation between the coordinates of points in a line which is common in all points is the
characteristic property of that line.The relation between x and y coordinates is named as the
equation of the line.

b) y coordinates of all points in x axis is 0. The equation y = 0 is considered as the equation of x
axis.

c) The y coordinates of all points on a line parallel to x axis are equal. For example in the line parallel
to x axis and passing through (0, 5) , y coordinates of all points are 5. Therefore y = 5 is the
equation of this line.

d) x coordinates of all points on y axis is 0. Therefore x = 0 is considered as the equation of y axis.

e) x coordinates of all points on a line parallel to y axis are equal. For example , in the line parallel to
y axis and passing through (5, 0), the x coordinates of all points are 5 . x = 5 is the equation of
this line.

f) Sometimes it is difficult to find the linear relation between the coordinates for getting the sequation.
We can use the measure slope for writing the equation successfully.

g) There are three types of lines. The line parallel to x axis, line parallel to y axis and inclined lines .
The inclination may be either to the left or right . The line inclined to the right has positive slope,
the line inclined to the left have negative slope.

1) There are some cows and hens in a farm. Total number of heads is 15 and total number of legs is 50

a) If the number of cows is x and number of hens is y then form equations

b) These are the equation of lines. Find the intersecting point of these lines .

c) Calculate the number of cows and hens പ െട എ ം േകാഴിക െട എ ംഎ ക

2) Following are the equation of lines
x − y = −1, x − y = 1, x + y = 7, x + y = 3

a) Make the pairs of parallel lines
b) Find the intersecting points of non parallel sides
c) Prove that the quadrilateral enclosed by the lines is a rectangle .
d) Calculate the area of the rectangle .

3) Three lines x = 3, y = 4, 4x + 3y = 36make a closed figure

a) Suggest a suitable name to this figure
b) What are the vertices of this figure
c) Calculate the area of the figure
d) What is the radius of the circle passing through the vertices

1

4) Consider the points A(3, 4) and B(−1, 2)
a) All points equidiatant from A and B is on a line.Write the equation of this line
b) Write the equation of the line passing through A(3, 4)and B(−1, 2)
c) Using these lines find the mid point of AB

5) Consider the lines x + y = 4, x + y = −2, x − y = 2, x − y = −2
a) What are the parallel lines
b) Fint the intersecting point of non parallel lines (draw rough figure )
c) Prove that these points make a square .

2

2020-21 Academic year Worksheets

Mathematics X
Geometry and algebra

Concepts
⋆ The equation of the circle with centre at the origin and radius r is x2 + y2 = r2
⋆ If the centre is (a, b)and radiusr then equation of the circle is (x − a)2 + (y − b)2 = r2

1) There is a circle with centre at the origin and radius 4
a) What are the points where the circle cut the axes?
b) If P (x, y) is a point on the circle then write the equation of the circel.
√√
c) Check whether (2 2, 2 2) a point on this cicle.
√√
d) If (2 2, 2 2) is a point on the circle what are the other points where the circle cut the axes?

2) Following are the equations aof circle
x2 + y2 = 1, x2 + y2 = 2, x2 + y2 = 3, x2 + y2 = 9, x2 + y2 = 17
a) Find the centre and radius of each circle
b) Find the coordinates of the point where the circle cut the axes
c) Calculate the area of the square joining these points
d) Calculate the perimetre of the square
√

3) There is a semicircle with diametre on x axis and centre ar the origin.One end of the diametre is (2 5, 0)

a) What are the points where the semicircle cut the axes
b) What is the side of the square ABCD
c) What are the coordinates of the vertices of the square?

1

4) The circle x2 + y2 = 3touches the sides of a square.

a) What are the points where the circle cut the axes?
b) What are the vertices of the square ?
c) Calculate the area and perimetre
5) AP is the tangent from an exterior point A to the circle x2 + y2 = 36.∠OAP = 30◦

a) What is the radius of the circle ?
b) What are the points the circle cut the axes ?
c) Find the coordinates of A
d) What is the length of tangent?

2

2020-21 Academic year Worksheets

Mathematics X
ജ ാമിതി ം ബീജഗണിത ം

Concepts

⋆ The equation of the circle with centre at the origin and radius r is x2 + y2 = r2
⋆ If the centre is (a, b)and radius r then equation of the circle is (x − a)2 + (y − b)2 = r2

1) Centre of a circle is (3, 2) and radius 1.
a) Write the equation of the circle.
b) What are the coordinates of the end points of the diametre parallel to x axis ?
c) What are the coordinates of the ends of the diametre parallel to y axis.
d) Find the area and perimetre of the square formed by joining the ends of these two perpendicular
diametres .

2) Consider the circles (x − 3)2 + (y − 4)2 = 1, (x + 3)2 + (y − 4)2 = 1, (x + 3)2 + (y + 4)2 = 1,
(x − 3)2 + (y + 4)2 = 1
a) With the help of coordiante axes draw a rough diagram
b) Write the equation of line joining the centres
c) Suggest a suitable name to the geometric figure formed by enclosed by these lines .
d) Calculate the area of the quadrilateral so fomed .

3) The end points of the diametre of a circle are (−3, 7) and (7, 13)
a) Find the coordinates of its centre.
b) What is the diametre of the circle?
c) Write the equation of the circle.

4) The end points of the diametre of a circle are (−1, 2), (3, 2)
a) Write the coordinates of the centre
b) Write the equation of the circle
c) What is the point where the circle touches x axis?
d) What are the points where the circle cut y axis?

1

5) The end points of the diametre of a circle are (4, 2), (2, 4)
a) Find the centre and radius of the circle .
b) Write the equation of the circle.
c) If the tangent through (4, 2)to the circle make 45◦ with the positive direction of x axis. Write
the equation of tangent .
d) At what point the tangent intersect x axis ?

1

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2020-21 Academic year Worksheets

Mathematics X
Geometry and Algebra

Concepts

⋆ The equation of the circle with centre at the origin and radius r is x2 + y2 = r2
⋆ If the centre is (a, b)and radius r then equation of the circle is (x − a)2 + (y − b)2 = r2

1) Consider the equation of the circle x2 + y2 − 6x − 10y − 15 = 0
a) write this equation in the form (x − a)2 + (y − b)2 = r2
b) Write the centre and radius of this circle.
c) Write coordinates of four points in this circle.

2) Consider the equation of the circle x2 + y2 − 6x − 10y + 9 = 0
a) Write this equation in the form (x − a)2 + (y − b)2 = r2
b) Write the centre and radius of this circle.
c) Write the coordinates of points at which the circle touches the x axis
d) Write the coordinates of the end points of the diametres parallel to x axis and parallel to y
axis

3) Consider the quation x2 + y2 + 6x + 8y = 0
a) Write the equation in the form (x − a)2 + (y − b)2 = r2
b) Write the centre and radius of the circle.
c) What are the points where the circle intersect the coordinate axes ?
d) What is the distance between centre and origin of coordinates ?

4) The centre of a circle is (3, 0) and radius 6
a) Write the equation of the circle .
b) What are the points the circle intersect the axes ?
c) Write the coordinates of another point of this circle.
d) Is (, 5) a point of this circle. How can we realize it .

1

√
5) The vertices of a triangle are (4, 2), (8, 2), (6, 2 3 + 2).

a) What kind of triangle is this ?
b) Find the centre of its circumcircle.
c) What is the radius of its circumcircle?

1

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10

Polynomials

ബഹുപദങ്ങൾ

2020-21 Academic year Worksheets

Mathematics X
ബ പദ ൾ

Concepts

⋆ If the polynomial p(x) is the product of the polynomials q(x) and r(x)then we say that the
polynomilals q(x) and r(x) are the factors of the polynomial p(x)

⋆ If the first degree polynomial (x − a) is a factor of the polynomial p(x) then p(a) = 0: that is a is
a solution of the equation p(x) = 0

⋆ If the polynomial p(x) can be split into first degree factors as p(x) = (x − a1)(x − a2)(x −
a3) · · · (x − an) then a1, a2, a3 · · · anare the solutions of the equation p(x) = 0.

⋆ In the second degree polynomial p(x) if p(a) = 0 then x − a will be the factor of p(x).

1) The sides of a rectangle are (x − 3) and (x + 1)
a) Find the area a(x)
b) If x = 4then what is its area ?
c) If x = 0 is it possible to get a rectangle ? Why?
d) What is the condition for x to get a rectangle?

2) Sides of a rectangular box are x − 1, x + 2 and x + 3.
a) Calculate the volume v(x).
b) If x = 2 then what is its volume?
c) Is it possible to make the box for x = 1
d) What is the condition for x to make the recctangular box?

3) The difference between breadth ,length and length of diagonal of a rectangle differ by 1
a) If breadth is x what is its length and diagonal of the rectangle?
b) Make an equation connecting breadth ,length and diagonal in the form p(x) = 0
c) Find the solution of the equation p(x) = 0
d) Find the length , breadth and diagonal.

4) Consider the polynomial p(x) = x2 − 7x + 12
a) Write p(x) = (x − a)(x − b).
b) Write p(x)as the product of two first degree polynomials.
c) Find the solution of the equation p(x) = 0

5) Consider the polynomial p(x) = x3 − 4x2 + 2x + k
a) If x is a factor then find x.
b) x − 1 is a first degree factor of p(x)then what is k?
c) Use k for becoming x − 1 a factor and write the polynomial
d) Is (x + 1 a factor of this polynomial .

1

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2020-21 Academic year Worksheets

Mathematics X
ബ പദ ൾ

Concepts

⋆ If the polynomial p(x) is the product of the polynomials q(x) and r(x)then we say that the
polynomilals q(x) and r(x) are the factors of the polynomial p(x)

⋆ If the first degree polynomial (x − a) is a factor of the polynomial p(x) then p(a) = 0: that is a is
a solution of the equation p(x) = 0

⋆ If the polynomial p(x) can be split into first degree factors as p(x) = (x − a1)(x − a2)(x −
a3) · · · (x − an) then a1, a2, a3 · · · anare the solutions of the equation p(x) = 0.

⋆ In the second degree polynomial p(x) if p(a) = 0 then x − a will be the factor of p(x).
If p(−a) = 0 then x + a will be a factor.

1) Consider the polynomial p(x) = x2 − 8x + 12
a) If p(x) = (x − a)(x − b)then what is a + b and ab
b) Find a, b and write p(x) as the product of two first degree factors
c) Find the solution of the equation p(x) = 0

2) If p(x) = x3 − 4x2 + 6x − kthen
a) Find k such that x − 1 a factor of p(x)
b) Write the polynomial . Is (x + 1) a factor of p(x)
c) What is the speciality of the coefficients of p(x) having x − 1 a factor
d) Write three polynomials having x − 1 a factor

3) Consider the polynomials p(x) = x3 + 1 , q(x) = x3 + x2 + x + 1
a) Find p(−1) and q(−1)
b) What is the factor common to both the polynomials
c) Find r(x) = p(x) + q(x)
d) what is the first degree factor of r(x)

4) x2 − 1 is the factor of p(x) = a3 + bx2 + cx + d
a) Find p(1), p(−1)
b) Show that a = −c, b = −d
c) Write a polynomial having x2 − 1 a factor

5) If p(x) = x3 − 8 then
a) Check whether x − 2 a factor of p(x)
b) Write a first degree factor of x3 − 27
c) What is the second degree factor of x3 − 27

1

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2020-21 Academic year Worksheets

Mathematics X
Polynomials

Concepts

⋆ If the polynomial p(x) is the product of the polynomials q(x) and r(x)then we say that the
polynomilals q(x) and r(x) are the factors of the polynomial p(x)

⋆ If the first degree polynomial (x − a) is a factor of the polynomial p(x) then p(a) = 0: that is a is
a solution of the equation p(x) = 0

⋆ If the polynomial p(x) can be split into first degree factors as p(x) = (x − a1)(x − a2)(x −
a3) · · · (x − an) then a1, a2, a3 · · · anare the solutions of the equation p(x) = 0.

⋆ In the second degree polynomial p(x) if p(a) = 0 then x − a will be the factor of p(x).
If p(−a) = 0 then x + a will be a factor.

1) Consider the polynomial p(x) = 3x2 + 4x + 1
a) Write p(x)as the product of two first degree factors
b) Find the solution of the equation p(x) = 0

2) Consider the equation p(x) = x3 + 4x2 + x − 7
a) Check whetherx − 1 a factor of this polynomial or not
b) If not what should be subtracted from p(x) to get another polynomial q(x) in which x − 1 is a factor
c) Write q(x) as the product of three first degree factors
d) Write the solution of the equation q(x) = 0.

3) Consider the second degree polynomial x2 − 20x + 91.This is the area of a rectangle with the sides the
first degee polynomials.
a) What are the sides of this rectangle.
b) What is the condition for getting a rectangle .
c) What is the polynomial representing the perimetre of the rectangle.

4) When a first degree polynomial is added to x3 + 2x2 we get a polynomial p(x) such that x2 − 1 is a factor.
a) Write two first degee factors of p(x)
b) What is the first degree factor to be added?
c) What are the solutions of the equation p(x) = 0

5) Consider the equation p(x) = x2 + 4x + k
a) If k = 0 write the first degree factors of p(x)
b) If k = 4 what are the facors of this polynomial ?
c) What is the condition for k to get two first degree factors of p(x)?

1

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2020-21 Academic year Worksheets

Mathematics X
Polynomials

Concepts

⋆ If the polynomial p(x) is the product of the polynomials q(x) and r(x)then we say that the
polynomilals q(x) and r(x) are the factors of the polynomial p(x)

⋆ If the first degree polynomial (x − a) is a factor of the polynomial p(x) then p(a) = 0: that is a is
a solution of the equation p(x) = 0

⋆ If the polynomial p(x) can be split into first degree factors as p(x) = (x − a1)(x − a2)(x −
a3) · · · (x − an) then a1, a2, a3 · · · anare the solutions of the equation p(x) = 0.

⋆ In the second degree polynomial p(x) if p(a) = 0 then x − a will be the factor of p(x).
If p(−a) = 0 then x + a will be a factor.

1) Write a polynomial p(x) such that p(1) = p(−2) = p(0) = 0
a) What are the first degree factors of p(x)
b) What should be added to p(x) to get a polynomial in which x + 1 is a factor?

2) Consider the polynomial p(x) = 4x2 − 16x + 15
a) Write p(x) in the form (x − a)(x − b) and find a + b, ab
b) Find a − b
c) Write p(x)as the product of two first degree factors
d) Find the solution of the equation p(x) = 0

3) Consider the polynomial p(x) = xn + 1
a) What are the numbers suitable for n getting x + 1 a factor of p(x)?
b) Can x − 1 a factor of p(x) for any n ?
c) Can x2 − 1 a factor of p(x)?

4) Consider the polynomial p(x) = x2 + 6x + k
a) If k = 0then what are the first degree factors of p(x)?
b) What is the value of k to get two equal first degree factors ?
c) What are the values of k not for occuring a first degree factor to this polynomial?
d) If k = 8 what are the first degree factors of p(x)?

5) Consider the polynomial p(x) = ax2 − 2bx + c
a) If x − 1 is a factor of p(x) prove that a, b, c are in an arithmetic sequence.
b) Write twp polynomials in the form ax2 − 2bx + c such that a, b, c are in an arithmetic sequence .
c) If x2 − 1 is a factor of p(x) then what is a + b ?

1

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2020-21 Academic year Worksheets

Mathematics X
ബ പദ ൾ

Concepts

⋆ If the polynomial p(x) is the product of the polynomials q(x) and r(x)then we say that the
polynomilals q(x) and r(x) are the factors of the polynomial p(x)

⋆ If the first degree polynomial (x − a) is a factor of the polynomial p(x) then p(a) = 0: that is a is
a solution of the equation p(x) = 0

⋆ If the polynomial p(x) can be split into first degree factors as p(x) = (x − a1)(x − a2)(x −
a3) · · · (x − an) then a1, a2, a3 · · · anare the solutions of the equation p(x) = 0.

⋆ In the second degree polynomial p(x) if p(a) = 0 then x − a will be the factor of p(x).
If p(−a) = 0 then x + a will be a factor.

1) Consider the polynomial p(x) = x3 + 4x2 + x − 6

a) Find p(1).Is x − 1 a factor of this polynomial ?
b) What is the quotient when p(x) is divided by x − 1?
c) Write the quotient as the product of two first degree factors
d) Find the solution of the equation p(x) = 0

or

Write x3+4x2+x−6 as the product of three first degree factors .Also solve the equation x3+4x2+x−6 =
0.

2) Consider the polynomial p(x) = ax4 + bx3 + cx2 + dx + e

a) Write a fourth degree polynomial having x − 1 a factor
b) If x + 1 is a factor what is the relation between the coefficients of p(x)
c) Using these conditions write a fourth degree polynomial having x2 − 1 as a factor.

3) In the second degree polynomial p(x), p( 1 ) = 0, p( 1 ) = 0, 4 is the constant term
2 3

a) If p(x) = ax2 + bx + 4 then find a, b

b) Write the polynomial . What should be added to this polynomial to get another polynomial having x−1

a factor

4) Consider the polynomial x2 + kx + 6

a) If x − 1 is a factor then what is k.
b) Write the other first degree factor.
c) Find the solution of x2 − 7x + 6 = 0

5) Consider the polynomial p(x) = (x + 1)(x + 2)(x + 3) + k

a) If p(−1) = 10then what is k?
b) Using k value write the polynomial
c) Write the polynomial having factor x − 1 by changing the constant term only.

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Statistics

സ്ഥിതിവിവരക്കണക്ക്

2020-21 Academic year Worksheets

Mathematics X
Statistics

Concepts

a) Statistics is the study of numerical data collected from a group , its analysis and interpretation .
b) Mean is an average calculated by dividing the sum of the numerical data by the number of

observations.It touches each and every data collected from the group.
c) The middle observation of the arrangement in the increasing or decreasing order is called

median.Comparing to mean it touches the observations comes in the middle.
d) We use the concept of arithmetic sequence for calculating median .

1) Atmospheric tempereature of seven days in a week are given below .
26◦C, 28◦C, 25◦C, 30◦C, 27◦C, 26◦C, 25◦C

a) Write the numbers in the ascending order.
b) Calculate the mean of the temperatures.
c) What is the median temperature?
d) How many days are having temperature less than median temperature?
e) How many temperatures are there below median temperature?
2) Consider th counting numbers from 1 to 100.
a) How many multiples of 7are there below 100?
b) Calculate the mean of the multiples of 7 below 100.
c) What is the median of the multiples of 7 belw 100?
d) How many multiples are there more are median in this collection ?
3) The marks obtained in ten class tests are given below

14, 17, 11, 19, 15, 17, 13, 10, 14, 18
a) Calculate the mean of the marks .
b) What are the marks comes in the middle if the marks are arranged in the increasing order?
c) What is the median mark ?
d) How many class tests are there scoring mark above median mark?
4) Consider the arithmetic sequence 7, 10, 13 · · ·
a) How many terms are there below 100?
b) Which term comes in the middle?
c) Calculate the mean of the numbers in the sequence below 100
d) CXalculate the median of numbers in the sequence below100
e) What is the relation between mean and median?

1

5) The algebraic form of an arithmetic sequence is 3n + 2
a) Write the sequence
b) Calculate the mean of first 20 terms.
c) Calculate the median of the first 20 numbers of this sequence number of this sequence .
d) What is the relation between mean and median.

1

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2020-21 Academic year Worksheets

Mathematics X
Statistics

Concepts

a) Statistics is the study of numerical data collected from a group , its analysis and interpretation .
b) Mean is an average calculated by dividing the sum of the numerical data by the number of

observations.It touches each and every data collected from the group.
c) The middle observation of the arrangement in the increasing or decreasing order is called

median.Comparing to mean it touches the observations comes in the middle.
d) We use the concept of arithmetic sequence for calculating median .

1) Consider a group of numbers in an arithmetic sequence
a) What is the general form of its algebra ?
b) Calculate the mean of these numbers
c) Find the median .
d) Are mean and median equal?Write a statement about the result.

2) The scores of 40 students in a quiz are given below

a) Calculate the total score in the class
b) Calculate mean score
c) Find the median of the scores
d) How many students are there above median score ?
3) The weights of 12 members of a team are given below

a) Prepare a table for calculating the median
b) What is the median of the weights ?
c) How many members are having medsian weight and below?
d) How many members are there above median weight?

1

4) The daily wages of 200 workers in a factory are given below .

a) Prepare the table for calculating the median.
b) Find the median wage.
c) How many workers are getting median wage and below ?
d) How many workers are getting median wage and above ?
5) Answer the following questions.
a) Find the mean of 100 odd numbers .
b) Find the median of first 100odd numbers
c) What is the mean of first n even numbers ?
d) What is the median of first n even numbers ?

1

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2020-21 Academic year Worksheets

Mathematics X
Statistics

Concepts

a) Statistics is the study of numerical data collected from a group , its analysis and interpretation .
b) Mean is an average calculated by dividing the sum of the numerical data by the number of

observations.It touches each and every data collected from the group.
c) The middle observation of the arrangement in the increasing or decreasing order is called

median.Comparing to mean it touches the observations comes in the middle.
d) We use the concept of arithmetic sequence for calculating median .
1) The marks obtained by 39 students of a clsss are given below .

a) What is meant by class and number of students in each class?
b) Preapare a table showing the number of marks below the upper limit of each class
c) If the marks are arranged as above which student comes in the middle?
d) Name the class in which the mark of 20rd student comes .
e) How many students are there in this class ?
f) Make sub classes in the median class showning equal sharing of marks to the children of the class.What

is the middle mark of each sub class?
g) If the marks in the subclasses are in an arithmetic sequence then what is the mark obtained by the

student comes in the middle.
h) Calculate median.
2) The marks obtained by 47 students in a class are given below .

1

a) Preapare a table for calculating the median
b) In which class 24 th mark comes ?
c) What are the assumptions for calculating median?
d) What is the mark obtained by 16 th student as per our assumption?
e) Calculate the median .
3) (TBQ) The table given below shows the daily wages of workers in a factory .

a) Prepare a table for calculating the median.
b) In which calss 21 st wage comes?
c) What are the assumpptions for calculating median.
d) What is the wage of 14 th worker in the arrangement?
e) Calculate median

1

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2020-21 Academic year Worksheets

Mathematics X
statistics

Concepts
a) Statistics is the study of numerical data collected from a group , its analysis and interpretation .
b) Mean is an average calculated by dividing the sum of the numerical data by the number of

observations.It touches each and every data collected from the group.
c) The middle observation of the arrangement in the increasing or decreasing order is called

median.Comparing to mean it touches the observations comes in the middle.
d) We use the concept of arithmetic sequence for calculating median .
1) The table shows the consumption of elecricity in houses in a region.

a) If the houses are arranged according to the increasing order of consumption which house comes in
the middle .In which class it belongs to?

b) If the consumptions in the median class are in an arithmetic sequence what is the consumption of 17
th house?

c) What are the consumption amount of the houses comes in the middle?
d) Calculate the median consumption of electricity?
2) The table gives the distribution of marks obtained by the students in a class in an examination

1

a) Prepare a table for calculating the median.
b) In which calss the middle marks occur
c) What will be the mark of 13th student according to the assumption of calculating median .
d) What are the marks comes in the middle?
e) Calculate median .
3) (SSLC 2019 Model )
The table shows the marks obtained by the students in a class.

a) If the scores are arranged in the ascending order what are the positions of scores comes in the
middle.

b) What is the mark of 15 th student?
c) Calculate the median.
4) (SSLC 2020 )
marks obtained by students in a class are given below

a) If the scores are arranged in the increasing order what is the position of themark comes in the middle.
b) What is the mark obtained by 12 th student?
c) Calculate median.

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