Aditi runs a handicraft shop in Bapu bazar in Jaipur. She makes beautiful necklaces using colourful beads which she keeps in a potli. Today she prepared 19 necklaces but could not make the 20th necklace as she had no yellow beads left. She counted the beads and found that there were 8 red, 6 green and 14 blue beads remaining in her potli. Her little daughter Dulari requested for a bead. Aditi decides to take out one bead from her potli for Dulari.

(a) Find the probability that she draws a green bead.

(b) Find the probability that the bead drawn by her is not green.

(c) Find the probability that she draws either a green or a blue bead.

(d) Find the probability that she draws neither a red nor a green bead.

(e) Which of the following is an impossible event?
(i) The bead drawn is not red
(ii) The bead drawn is neither red nor blue
(iii) The bead drawn is either red or green or blue.
(vi) The bead drawn is yellow.

One day, during games period four friends A, B, C and D planned to play game using number cards. They prepared 20 numbered cards with labelled 1 to 20 and then they put all the number cards in the empty chalk box available in the classroom. In this game, every friend was asked to pick the card randomly and after each draw, card was replaced back in the chalk box.

(i) Find the probability, first boy pick the card and he get the card with an even number?

(ii) If the card drawn in first case is replaced,  and the second boy draws a card.  What is the probability getting a prime number?

(iii) If the card drawn, is not replaced in the second draw, what is the probability that he got a multiple of 3 greater than 4?

(iv) For a sure event A, P(A) = ?

(v) If all cards drawn are replaced then what is the probability of getting a multiple of 3 and 5?

One day Rahul visited park along with his friend. There he saw a game of change that consists of spinning an arrow (as shown in below figure) that comes to rest pointing at one of the numbers 1, 2, 3, 4, 5, 6, 7, 8 and these are equally likely outcomes.

(a) Find the probability that the arrow will point at 2.

(b) Find the probability that the arrow will point at an even number.

(c) Find the probability that the arrow will point at a prime number.

(d) Find the probability that the arrow will point at a number divisible by 3.

(e) Find the probability that the arrow will point at a number greater than 2.

Aditya went to shop to purchase a child's game along with his friend. He selected one child's game which has 8 triangles of which 3 are blue and rest are red, and 10 squares of which 6 are blue and rest are red. While checking the game, one piece is lost at random.

(a) How many triangles are of red colour and how many squares are of red colour?

(b) Find the probability that lost piece is square.

(c) Find the probability that lost piece is triangle.

(d) Find the probability that lost piece is square of blue color.

(e) Find the probability that lost piece is triangle of red color.

Diwali Fest is an annual South Asian arts & culture festival, produced by the Diwali Celebration Society. In the Diwali fest, a game is played which is like that, there is a spinner on which some numbers are written. The numbers on spinner are 2, 5, 7, 9, 12, 16. Depending on the condition of stall owner, if a particular number comes, than a die will be thrown.

(i) What is the probablity, that spinner stops on an even number?

(ii) A player will get a special prize, if spinner stops on a perfect square:

(iii) If the player gets a chance to throw, a dice, what is the probability of getting a multiple of 2 on dice?

(iv) If a dice is thrown, what is the probability of getting a number less than 4?

(v) An event whose probability to occur is 1. Then this type of event is called:

The Class teacher of Class X preparing result analysis of a student. She compares the marks of a student obtained in Class IX (2018-19) and Class X (2019-20) using the double bar graph as shown below:

(i) In which subject has the performance improved the most?

(ii) In which subject has the performance deteriorated?

(iii) In which subject is the performance at par?

(iv) What is the difference in Maths Subject?

(v) What is the percentage of marks obtained by a student in Class X (2019-20)?

A Mathematics teacher asks students to collect the marks of Mathematics in Half yearly exam. She instructed to all the students to prepare frequency disctribution table using the data collected. Ram collected the following marks (out of 50) obtained in Mathematics by 60 students of Class IX
21, 10, 30, 22, 33, 5, 37, 12, 25, 42, 15, 39, 26, 32, 18, 27, 28, 19, 29, 35, 31, 24, 36, 18, 20, 38, 22, 44, 16, 24, 10, 27, 39, 28, 49, 29, 32, 23, 31, 21, 34, 22, 23, 36, 24, 36, 33, 47, 48, 50, 39, 20, 7, 16, 36, 45, 47, 30, 22, 17.


(i) How many students scored more than 20 but less than 30?

(ii) How many students scored less than 20 marks?

(iii) How many students scored more than 60% marks?

(iv) What is the class size of the classes?

(v) What is the class mark of the class interval 30 – 40?

A group of students decided to make a project on Statistics. They are collecting the heights (in cm) of their 51 girls of Class IX-A, B and C of their school. After collecting the data, they arranged the data in the following frequency distribution table form:

Based on the information, answer the following questions:
(a) The class interval with highest frequency is :

(b) What is the width of the class?

(c) How many students of the height 150 cm and below are there?

(d) How many students of the height 145 cm and above are there?

(e) How many students of the height more than 145 cm but less than 155 are there?

The COVID-19 pandemic, also known as the coronavirus pandemic, is an ongoing pandemic of coronavirus disease 2019 (COVID-19) caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). It was first identified in December 2019 in Wuhan, China.
During survey, the ages of 80 patients infected by COVID and admitted in the one of the City hospital were recorded and the collected data is represneted in the less than cumulative frequency distribution table.

Based on the information, answer the following questions:

(a) The class interval with highest frequency is: 

(b) Which age group was affected the least?

(c) Which are group was affected the most?

(d) How many patients of the age 45 years and above were admitted?

(e) How many patients of the age 35 years and less were admitted?

Anil is a Mathematics teacher in Hyderabad. After Periodic test 3, he asks students to collect the Mathematics marks of all the students of Class IX- A, B and C. A student is able to collect marks from some students. Rekha scored least mark 6 in the class and Ram scored highest marks 59 in the class. He prepares the frequency distribution table using the collected marks and draws Histogram using the table as shown in adjoining figure.

(a) What is the width of the class?

(b) What is the total number of students in Histogram?

(c)  How many students scored 50% and above marks?

(d) How many students scored less than 50% marks?

(e)  What is the range of the collected marks?

Mathematics teacher of a school took her 9th standard students to show Gol Gumbaz. It was a part of their Educational trip. The teacher had interest in history as well. She narrated the facts of Gol Gumbaz to students. Gol Gumbaz is the tomb of king Muhammad Adil Shah, Adil Shah Dynasty. Construction of the tomb, located in Vijayapura , Karnataka, India, was started in 1626 and completed in 1656. It reaches up to 51 meters in height while the giant dome has an external diameter of 44 meters, making it one of the largest domes ever built. At each of the four corners of the cube is a dome shaped octagonal tower seven stories high with a staircase inside.

(a) What is the total surface area of a cuboid?

(b) What is the curved surface area of hemispherical dome ?

(c) What is the height of the cubodial part ?

d) What is the circumference of the base of the dome ?

(v) The total surface area of a hemispherical dome having radius 7 cm is:

Mathematics teacher of a school took his 10th standard students to show Taj Mahal. It was a part of their Educational trip. The teacher had interest in history as well. He narrated the facts of Taj Mahal to the students. Then the teacher said in this monument one can find combination of solid figures. There are 4 pillars which are cylindrical in shape. Also, 2 domes at the back side which are hemispherical. 1 big domes at the centre. It is the finest example of the symmetry. (Use π = 22/7)
(i)How much cloth material will be required to cover 2 small domes each of radius 4.2 metres?

(ii) Write the formula to find the volume of one pillar (including hemispherical dome) :

(iii) The volume of the hemispherical dome at the centre if base radius is 7 m is :

(iv) What is the lateral surface area of all 4 pillars if height of the each pillar is 14 m and base radius is 1.4 m (without dome)?

(v) The cost of polishing all the four pillars if the cost of 1 m² is Rs. 270, will be :

Mathematics teacher of a school took his 10th standard students to show Taj Mahal. It was a part of their Educational trip. The teacher had interest in history as well. He narrated the facts of Taj Mahal to the students. Then the teacher said in this monument one can find combination of solid figures. There are 4 pillars which are cylindrical in shape. Also, 2 domes at the back side which are hemispherical. 1 big domes at the centre. It is the finest example of the symmetry. (Use π = 22/7)
(i)How much cloth material will be required to cover 2 small domes each of radius 4.2 metres?

(ii) Write the formula to find the volume of one pillar (including hemispherical dome) :

(iii) The volume of the hemispherical dome at the centre if base radius is 7 m is :

(iv) What is the lateral surface area of all 4 pillars if height of the each pillar is 14 m and base radius is 1.4 m (without dome)?

(v) The cost of polishing all the four pillars if the cost of 1 m² is Rs. 270, will be :

Mathematics teacher of a school took her 9th standard students to show Red fort. It was a part of their Educational trip. The teacher had interest in history as well. She narrated the facts of Red fort to students. Then the teacher said in this monument one can find combination of solid figures. There are 2 pillars which are cylindrical in shape. Also 2 domes at the corners which are hemispherical.7 smaller domes at the centre. Flag hoisting ceremony on Independence Day takes place near these domes.

(i) How much cloth material will be required to cover 2 big domes each of radius 2.5 metres?

(ii) Write the formula to find the volume of a cylindrical pillar:

(iii) Find the lateral surface area of two pillars if height of the pillar is 7 m and radius of the base is 1.4 m.

(iv) How much is the volume of a hemisphere if the radius of the base is 3.5 m?

(v) What is the ratio of sum of volumes of two hemispheres of radius 1 cm each to the volume of a sphere of radius 2 cm?

An architect's planned design for a room with dimensions of 8.6 m, 5.4 m and 4 m respectively. He also planned to make 4 windows with blue colour and 2 doors with brown wood colour. The room needs to be painted with Asian paint of Green colour except for the floor and square tiles were used for flooring as shown in the below figure:

(i) The total area of the four walls is :

(ii) If the area of windows and doors is 22 sq m.  The area of the walls to be painted

(iii) What is the area of the tiles to be used for flooring?

(iv) The total area of the room is (including windows and doors):

(v) What is the volume of the air in the room?

In the below given layout, the design and measurements has been made such that area of two bedrooms and Kitchen together is 95 sq. m.

(i) The area of two bedrooms and kitchen are respectively equal to

(ii) Find the length of the outer boundary of the layout.

(iii) The pair of linear equation in two variables formed from the statements are
(a) x + y = 13, x + y = 9
(b) 2x + y = 13, x + y = 9
(c) x + y = 13, 2x + y = 9
(d) None of the above
(iv) Which is the solution satisfying both the equations formed in (iii)?

(v) Find the area of each bedroom.

Deepak bought 3 notebooks and 2 pens for Rs. 80. His friend Ram said that price of each notebook could be Rs. 25. Then three notebooks would cost Rs.75, the two pens would cost Rs.5 and each pen could be for Rs. 2.50. Another friend Ajay felt that Rs. 2.50 for one pen was too little. It should be at least Rs. 16. Then the price of each notebook would also be Rs.16.

Lohith also bought the same types of notebooks and pens as Aditya. He paid 110 for 4 notebooks and 3 pens. Later, Deepak guess the cost of one pen is Rs. 10 and Lohith guess the cost of one notebook is Rs. 30.
(i) Form the pair of linear equations in two variables from this situation by taking cost of one notebook as Rs. x and cost of one pen as Rs. y.
(a) 3x + 2y = 80 and 4x + 3y = 110
(b) 2x + 3y = 80 and 3x + 4y = 110
(c) x + y = 80 and x + y = 110
(d) 3x + 2y = 110 and 4x + 3y = 80
(ii) Which is the solution satisfying both the equations formed in (i)?

(iii) Find the cost of one pen?

(iv) Find the total cost if they will purchase the same type of 15 notebooks and 12 pens.

(v) Find whose estimation is correct in the given statement.

Prime Minister's National Relief Fund (also called PMNRF in short) is the fund raised to provide support for people affected by natural and man-made disasters. Natural disasters that are covered under this include flood, cyclone, earthquake etc. Man-made disasters that are included are major accidents, acid attacks, riots, etc.

Two friends Sita and Gita, together contributed Rs. 200 towards Prime Minister's Relief Fund. Answer the following :
(a) Which out of the following is not the linear equation in two variables ?

(b) How to represent the above situation in linear equations in two variables ?

(c) If Sita contributed Rs. 76, then how much was contributed by Gita ?

(d) If both contributed equally, then how much is contributed by each?

(e) Which is the standard form of linear equations x = – 5 ?

Sanjay bought 5 notebooks and 2 pens for Rs. 120. He told to guess the cost of each notebook and pen to his friends Mohan and Anil. Sanjay has given the clue that both the costs are positive integers and divisible by 5 such that the cost of a notebook is greater than that of a pen.

Now, Mohan and Anil tried to guess.
Mohan said that price of each notebook could be Rs. 18. Then five notebooks would cost Rs.90, the two pens would cost Rs.30 and each pen could be for Rs. 15. Anil felt that Rs. 18 for one notebook was too little. It should be at least Rs. 20. Then the price of each pen would also be Rs.10.
(i) Form the linear equations in two variables from this situation by taking cost of one notebook as Rs. x and cost of one pen as Rs. y.

(ii) Which is the solution of the equations formed in (i)?

(c) If the cost of one notebook is Rs. 15 and cost of one pen is 10, then find the total amount.

(d) If the cost of one notebook is twice the cost of one pen, then find the cost of one pen?

(e) Which is the standard form of linear equations y = 4 ?

On his birthday, Manoj planned that this time he celebrates his birthday in a small orphanage centre. He bought apples to give to children and adults working there. Manoj donated 2 apples to each children and 3 apples to each adult working there along with birthday cake. He distributed 60 total apples.

(a) How to represent the above situation in linear equations in two variables by taking the number of children as 'x' and the number of adults as 'y'?

(b) If the number of children is 15, then find the number of adults?

(c)  If the number of adults is 12, then find the number of children?

(d) Find the value of b, if x = 5, y = 0 is a solution of the equation 3x + 5y = b.

(e) Which is the standard form of linear equations in two variables: y - x = 5?

Students of a school are standing in rows and columns in their playground for a drill practice. A, B, C and D are the positions of four students as shown in the figure.

(a) What are the coordinates of A and B respectively?

(b) What are the coordinates of C and D respectively?

(c) What is the distance between B and D?

(d) What is the distance between A and C?

(e) What are the coordinates of the point of intersection of AC and BD?

Aditya is a Class IX student residing in a village. One day, he went to a city Hospital along with his grandfather for general checkup. From there he visited three places - School, Library and Police Station. After returning to his village, he plotted a graph by taking Hospital as origin and marked three places on the graph as per his direction of movement and distance. The graph is shown below:

(i) What are the coordinates of School?

(ii) What are the coordinates of Police Station?

(iii) Distance between school and police station:

(iv) What are the coordinates of Library?

(v) In which quadrant the point (-1, 4) lies?
 

Kumar has a rectangular sketch, which he needs to draw on a coloured paper of length and breadth 32 units and 16 units respectively, using a plotter. Plotter is a device which is attached to a computer like a printer It is used for drawing complicated sketches. Plotter accepts only positive coordinates where the point (0, 0) is the left-bottom corner of the paper. The sketch ABCD needs to be centrally aligned on the paper.
(a) What are the coordinates of A and B respectively?

 

(b) What are the coordinates of A and B respectively?

(c) The coordinates of point O in the sketch -2 is

(d) The point on the y-axis ( in sketch 2) which is equidistant from the points B and C is 

(e) The point on the x-axis ( in sketch 2) which is equidistant from the points C and D is

The Class IX students of a secondary school in Krishinagar have been allotted a rectangular plot of land for their gardening activity. Sapling of Gulmohar are planted on the boundary at a distance of 1m from each other. There is a lawn PQRS in the ground as shown in below figure.
(a) What are the coordinates of C, taking A as origin?

(b) What are the coordinates of R, taking A as origin?

(c) Side of lawn is :

(d) Shape of lawn is :

(e) Area of lawn is :

Student of class IX are on visit of Sansad Bhawan. Teacher assign them the activity of observe and take some pictures to analyses the seating arrangement between variuos MP and speaker based on coordiate geometry. The staff tour guide explained various facts related to Math's of Sansad Bhawan to the students, students were surprised when teacher ask them you need to apply coordiante geometry on the seating arrangement of MP's and speaker.
Calcualte the following reger to the below image and graph. Answer the following questions:


Answer the following refer to the above image and graph:
(i) What are the coordinates of postion 'F'?

(ii) What are the coordinates of position 'D'?

(iii) What are the coordinates of position 'H'?

(iv) In which quadrant, the point 'C' lie?

(v) Find the perpendicular distance of the point E from the y-axis.

If \(\sqrt { x } \) is an irrational number, then x is:

In the polynomial \(1-\sqrt{11} x,\) the coefficient of x is:

The points (-5,2) and (2,-5) lie in the:

Graph of linear equation 2x+by+c=0, a≠0, b≠0 cuts x-axis and y-axis respectively at the points:

'Lines are parallel if they do not intersect' is stated in the form of:

Which of the following is a rational number?

If the polynomial p(x) is divided by (x+3), then the remainder will be:

If the points A(2,0), B(- 6,0) and C(3,a-3) lie on the x-axis, then the value of a is:

The sum of the ages of Apala and Meenu is 48.Write a linear equation in two variables to represent the statement.

Pythagoras was a student of

If \(\sqrt { 3 } =1.732\) and \(\sqrt { 2 } =1.414\) , the value of \(\frac { 1 }{ \sqrt { 3 } -\sqrt { 2 } } \) is:

For what value of a, is the polynomial \(x^3+2x^2-3ax-8\) divisible by x-4?

The distance of the point (1,0) from O is:

Any point on the line y = 3x is of the form:

John Playfair was a 

The value of \({ \left( 243 \right) }^{ \frac { 1 }{ 3 } }\) is equal to:

(x+2) is a factor of \(2x^3+5x^2-x-k.\) The value of k is:

The points (-5,2) and (2,-5) lie in the:

The maximum number of points that lie on the graph of a linear equation in two variables is:

Two interesting lines cannot be parallel to the same line, is started in the form of:

The simplified value of \({ \left( 81 \right) }^{ -1/4 }\times \sqrt [ 4 ]{ 81 } \) is:

The value of p for which x+p is a factor of \(x^2+px+3-p \) is:

The line of intersection of IV and I quadrants is

Which of the following ordered pairs is a solution of the equation x-2y=6?

Euclid stated that things which are equal to the same thing are equal to one another in the form of:

What is the number of outcomes when a coin is tossed?

What is the number of outcomes when a cubical dice is thrown?

Probability of impossible event is:

A coin is tossed 200 times.The head appears 79 times.The probability of a tail is:

When a coin is tossed 500 times, the following outcomes were recorded:
Head: 235 times, Tail: 265 times
If now a coin is tossed once again, the probability of getting a Tail is

Statistics is branch of

When the information is gathered from a source which already had the information stored, the data obtained is called

Class mark = 

In the distribution, the frequency of the class 0 - 5 is
0,3,2,5,8,10,13,5,6,6,14,0.

The width of the class interval 70.5 - 75.5 is

Identify the wrong statement of the following:

The number of edges of a cube are

If the edges of a cuboid are l, b and h respectively, then the total surface area of the cuboid is

The area of the four walls of a room is 300 m². Its length and height are 15 m and 6 m respectively. Find its breadth. 

The area of the four walls of a room is 80 cm² and its height is 4 m. Then, the perimeter of the floor of the room is

Base of a triangle = 

The side of an isosceles right triangle of hypotenuse \(5\sqrt { 2 } \) cm is

The area of an equilateral triangle is \(16\sqrt { 3 } \) m².Its perimeter (in meters) is

The diagonals of a rhombus are 10 cm and 8 cm.Its area is

Heron's formula is

Bisector of an angle divides it in to_______equal parts

If a 60^o angle is bisected twice,what will be measure of each that is constructed?

In the figure below,PR is the perpendicular bisectors of a line segment AB=16 cm.Is PA=PB true?

Construct ∠POY=30 ^o. using compass and ruler.

Construct an angle of 15^o

The centre of a circle lies

Equal chords of a circle subtend equal angles at

Given a circle with centre O and smallest chord AB is of length 6 cm and the longest chord CD of the circle is of length 10 cm, then the radius of the circle is:

In figure, if OA = 5 cm, AB = 8 cm and OD丄AB then CD is equal to:

Equal chords of a circle are equidistant from

The areas of a parallelogram and a triangle are equal and they lie on the same base. If the altitude of the parallelogram is 2 cm, then the altitude of triangle is

In the figure, the area of parallelogram PQRS is:

In the following figure, find the area of quad. ABCD.

Two parallelograms are on equal bases and between the same parallels. The ratio of their areas is

If a rectangle and a square stand on the same base and between the same parallels, then the ratio of their areas is

Each angle of a rectangle is

If in a quadrilateral, two pairs of adjacent sides are equal, then it is called a

In the following figure the measure of \(\angle \)DAB is 
 

The angles of a quadrilateral are in the ratio 2 : 3 : 6 : 7. The largest angle of the quadrilateral is

Two consecutive angles of a parallelogram are in the ratio 1 : 3, then what will be the smaller angles?

'Tri' means

Two circles are congruent.If the radius of one circle is 3cm, what is the radius of the other circle?

Two circles are congruent.If the radius of one circle is 1cm, then the diameter of the other circle is

ΔABC ≅ ΔPQR.If AB = 5cm, ㄥB = 40⁰and ㄥA = 80⁰, then which of the following is true?

In two triangles ABC and DEF, ㄥA = ㄥD, ㄥB = ㄥE and AB = EF, then are the two triangles congruent? If yes, by which congruency rule?

An obtuse angle

An right angle

The angle of supplementary to \(90^{ 0 }\)+\(9^{ 0 }\) is

Which of the following is not a pair of complementary angles? 

The length of the common perpendiculars at different points on parallel lines is the same and is called

The number of line segments determined by three collinear points is:

Two planes intersect each other to form a:

The thing which coincide with one another are:

How many lines can be passed through two distinct points?

Write the number of dimension(s) of a surface.

The equation \(x+\sqrt{2}=0\) has

The condition that the equation ax+by+c=0 represents a linear equation in two variables is:

Write a, b, c for the equation 2x=5

The equation y=3 in two variables can be written as:

The sum of the ages of Apala and Meenu is 48.Write a linear equation in two variables to represent the statement.

The famous mathematician associated with the problem of describing the position of a point in a plane was

Which of the following is an example of a geometrical line?

If x is negative and y is negative, then the point (x,y) lies in

The line of intersection of I and II quadrants is

The line of intersection of IV and I quadrants is

The expansion for \((x-y)^2\) is
\((x-y)^2=x^2-2xy+y^2\)  is an algebraic identity

The number 0 is called a

Which of the following is a polynomial in one variable?

Which of the following is a trinomial in x?

Degree of the polynomial \(4x^4+0x^3+0x^5+5x+7\) is:

Every rational number is:

Two rational numbers between \(\frac { 2 }{ 3 } \) and \(\frac { 5 }{ 3 } \)are:

A terminating decimal is:

A number is an irrational if and only if its decimal representation is:

Which of the following is a rational number?

The rational number between -1/5 and -2/5 is

Two rational numbers between \(\frac { 2 }{ 3 } \) and \(\frac { 5 }{ 3 } \)are:

The decimal expansion of \(\sqrt { 2 } \) is

The value of \(\sqrt [ 4 ]{ { \left( 64 \right) }^{ -2 } } \) is

\(\sqrt [ 3 ]{ \frac { 54 }{ 250 } } \) equals:

Prove that: \({ \left( \frac { { x }^{ { a }^{ 2 } } }{ { x }^{ { b }^{ 2 } } } \right) }^{ \frac { 1 }{ a+b } }.{ \left( \frac { { x }^{ { b }^{ 2 } } }{ { x }^{ { c }^{ 2 } } } \right) }^{ \frac { 1 }{ b+c } }.{ \left( \frac { { x }^{ { c }^{ 2 } } }{ { x }^{ { a }^{ 2 } } } \right) }^{ \frac { 1 }{ c+a } }=1\)

Geetha told her classmate Radha that "\(\sqrt { \frac { \left( \sqrt { 2 } -1 \right) }{ \left( \sqrt { 2 } +1 \right) } } \) is an irrational number." Radha replied that "you are wrong" and further claimed that "If there is a number 'x' such that x³ is an irrational number, then x⁵ is also irrational". Geetha said, "No Radha, you are wrong". Radha took some time and after verification accepted her mistakes and thanked Geetha for pointing out these mistakes. 
(i) Justify both the statements.
(ii) What value is depicted from this question?

In the figure, AD = AE, BD = EC. Prove that \(\triangle\)ABC is an isosceles triangle.

In \(\triangle\)ABC, if AB is the greatest side, then prove that LC > 60°.

AD and BC are equal perpendiculars to a line segment AB (see figure).

(i) Show that CD bisects AB.
(ii) Which mathematical concept is used in this problem?
(iii) What is its value?

Find three rational numbers \(\frac { 3 }{ 5 } \) and \(\frac { 7 }{ 8 } \)

Find your rational numbers between \(\frac { 1 }{ 5 } \) and \(\frac { 1 }{ 6 } \)  

Express \(0.15\overline { 9 } \) in \(\frac { p }{ q } \) , where p and q are integers and \(q\neq 0\)

(i) express 2.4 \(\overline { 248 } \) in the form of \(p\over q\) where \(p\over q\) is in its lowest form.
(ii) What is the value of p?
(iii) what is the value of q?
(iv) what name can be given to the number p?
(v) what name can be given to the number q?
(vi) Apala declares that p and q are co-prime. IS she correct? If so, which value of Apala is depicted by her declaration?
(vii) Which mathematical concept has been converted in this problem?
(viii) Write the formulae used in the solution.

Find the value of 'p' if \({ 5 }^{ p-3 }\times { 3 }^{ 2p-8 }=225\)

Find six rational numbers between 3 and 4.

Write the following in decimal form and say what kind of decimal expansion each has:
\(\frac { 3 }{ 13 } \)

Express \(2.\overline { 93 } \) in the form of \(\frac { p }{ q } \), where p and q are integers, \(q\neq 0\)

Simplify: \((\sqrt{x})^{-\frac{2}{3}}\sqrt{y^{4}}\div\sqrt{(xy)^{-\frac{1}{2}}}\)

Simplify: \(\left( \sqrt { 3 } +1 \right) \left( 1-\sqrt { 12 } \right) +\frac { 9 }{ \left( \sqrt { 3 } +\sqrt { 12 } \right) } \)

Find two rational numbers between 3/4 and 5/9.

Write three rational numbers between -2/5 and -1/5.

Express \(18.\overline{48} \) in the form of p/q, where p and q are integers, \(q\neq 0\) .

Simplify \(\sqrt { 2 } (\sqrt { 6 } -\sqrt { 8 } )+\sqrt { 3 } (\sqrt { 27 } -\sqrt { 6 } )\)

If x = 5 and y = 2, find the value of \((i)\quad \left( { x }^{ y }+{ y }^{ y } \right) \ (ii)\ { \left( { x }^{ y }+{ y }^{ y } \right) }^{ -1 }\)

A survey of 500 families was conducted to know their opinion about a particular detergent powder. If 375 families liked the detergent powder and the remaining families disliked it, find the probability that a family chosen at random
(i) likes the detergent powder
(ii) does not like it.

Out of the past 250 consecutive days, its weather forecasts were correct 175 times.
(i) What is the probability that on a given day it was correct?
(ii) What is the probability that it was not correct on a given day?

A bag contains 5 red balls, 8 white balls, 4 green balls and 7 black balls. If one ball is drawn at random, find the probability that it is
(i) black
(ii) not green.

Cards marked with numbers 2 to 101 are placed in a box and mixed thoroughly. One card is drawn from this box. Find the probability that the number on the card is
(a) a number less than 14
(b) a number which is a perfect square
(c) a prime number less than 20

At a hospital, a doctor compiled the following data about 400 patients whom he could cure of hepatitis:

Another case of hepatitis is reported. What is the probability that this patient will be cured in
(i) less than 2 months?
(ii) 1 month or more but not more than 3 months?

Draw a histogram representing the following frequency distribution:

Draw a frequency polygon to represent the following information:

Draw a histogram and frequency polygon of the following data:

The mean of 10,12,18,13,x and 17 is 15. Find the value of x.

Arithmetic mean of terms 21,16,24,x,29,15 is 23. Find this value of x.

Two cubes of side 6 cm each, are joined end to end. Find the surface area of the resulting cuboid.

The length of a hall is 20 m and width 16 m. The sum of the areas of the floor and the flat roof is equal to the sum of the areas of the four walls. Find the height of the hall.

The dimensions of a rectangular box are in the ratio of 2 : 3 : 4 and the difference between the cost of covering it with sheet of paper at the rates of Rs 8 and Rs 9.50 per m² is Rs 1248. Find the dimensions of the box.

A cylindrical vessel, without lid, has to be tin-coated including both of its sides. If the radius of its base is \(\frac { 1 }{ 2 } m\) and its height is 1.4 m, calculate the cost of tin-coating at the rate of Rs 50 per 1000 cm \(\left( Use\pi =3.14 \right) \)

The radius and vertical height of a cone are 5 cm and 12 cm respectively. Find the curved surface area.

Find the area of a right-angled \(\Delta \)ABC, right angled at B in which AB = 24 metre and BC = 10 metre.

lf the area of an equilateral triangle is \(81\sqrt { 3 } \)cm², find its perimeter.

The base of an isosceles triangle measures 24 cm and its area is 60 cm², Find its perimeter.

An isosceles triangle has perimeter 30 m and each of the equal sides is 12 cm.Find area of the triangle.

Sides of a triangle are in the ratio 13:14:15 and its perimeter is 84 cm.Find its area.

Construct an equilateral triangle with one side 6 cm.

Construct an equilateral triangle LMN, one of whose sides is 5 cm. Bisect \(\angle M\) of the triangle.

Draw a line segment AB = 5 cm. From the point A draw a line segment AD = 6 cm making an angle of 60°. Draw perpendicular bisector of AD.

(i) Construct a ΔABC in which AB = 5.8cm, BC + CA = 8.4cm and B = 60⁰
(ii) Measure AC
(iii) Measure BC
(iv) Is ACV + BC = 8.4cm?
(v) Meenu says that ㄥACB = 84⁰ .Verify by measurement.Can you say that Meenu is right?Which value is depicted by Meenu's statement?

(i) Construct a triangle ABC in which BC = 5 cm, ㄥB = 45°and AB - AC= 2.8cm.
(ii) Measure AB.
(iii) Measure AC
(iv) Verify that AB - AC = 2.8 cm.
(v)Hari comments that ㄥACB = 112°. Is he true? Which value is depicted by comment of Hari?

Out of the past 250 consecutive days, its weather forecasts were correct 175 times.
(i) What is the probability that on a given day it was correct?
(ii) What is the probability that it was not correct on a given day?

A bag contains 5 red balls, 8 white balls, 4 green balls and 7 black balls. If one ball is drawn at random, find the probability that it is
(i) black
(ii) not green.

A die is rolled 25 times and outcomes are recorded as under:

It is thrown one more time. Find the probability of getting
(a) an even number
(b) a multiple of 3
(c) a prime number.

The percentages of marks obtained by a student in examination are given below:

Find the probability that the student gets
(i) a first class i.e. at least 60% marks
(ii) a distinction i.e. 75% or above
(iii) marks between 70% and 80%

An insurance company selected 1600 drivers at random in a particular city to find a relationship between age and number of accidents. The data obtained are given in the following table:

Find the number of drivers
(a) in the age of 25-40 years and has more than 2 accidents in the year.
(b) in the age above 40 years and has accidents more than 1 but less than 3.

The class marks of a frequency distribution are 104, 114, 124, 134, 144, 154, 164. Find the class size and class intervals.

The following data gives the number (in thousands) of applicants registered with an Employment Exchange during 2005-2010.

Construct a bar graph to represent the above data.

The marks scored by 750 students in examination are given in the form a frequency distribution table.

Draw a histogram to represent the above data.

The following table gives the performance of 90 students in a mathematics test of 100 marks.

Represent the given information with the help of a histogram.

Draw a frequency polygon to represent the following information:

A cast-iron pipe has an external diameter of 75 mm. If it is 4.2 m long, find the area of the outer surface.   \(\left[ Assume\pi =\frac { 22 }{ 7 } \right] \)      

Find the curved surface area of a closed cylindrical petrol storage tank that is 3.8 m in diameter and 4.9 m in height.

The diameter of roller 1.5 m long is 84 cm. If it takes 100 revolutions to level a playground, find the cost of levelling this ground at the rate of 50 paise per square metre.

Find the ratio of the curved surface areas of two cones if their diameters of the bases are equal and slant heights are in the ratio 4 : 3.

A hemispherical bowl made of steel is of 1 cm thickness. The inner radius of the bowl is 6 cm. Find the total surface area of the bowl, in terms of \(\pi .\)

Find the area of a right-angled \(\Delta \)ABC, right angled at B in which AB = 24 metre and BC = 10 metre.

lf the area of an equilateral triangle is \(81\sqrt { 3 } \)cm², find its perimeter.

An isosceles triangle has perimeter 30 m and each of the equal sides is 12 cm.Find area of the triangle.

Sides of a triangle are in the ratio 13:14:15 and its perimeter is 84 cm.Find its area.

The perimeter of a triangle field is 300 cm and its sides are in the ratio 5:12:13.Find the length of the perpendicular from the opposite vertex to the side whose length is 130 cm.

Construct an equilateral triangle with one side 6 cm.

Construct an equilateral triangle LMN, one of whose sides is 5 cm. Bisect \(\angle M\) of the triangle.

Draw a line segment AB = 5 cm. From the point A draw a line segment AD = 6 cm making an angle of 60°. Draw perpendicular bisector of AD.

(i) Construct a ΔABC in which AB = 5.8cm, BC + CA = 8.4cm and B = 60⁰
(ii) Measure AC
(iii) Measure BC
(iv) Is ACV + BC = 8.4cm?
(v) Meenu says that ㄥACB = 84⁰ .Verify by measurement.Can you say that Meenu is right?Which value is depicted by Meenu's statement?

(i) Construct a triangle ABC in which BC = 5 cm, ㄥB = 45°and AB - AC= 2.8cm.
(ii) Measure AB.
(iii) Measure AC
(iv) Verify that AB - AC = 2.8 cm.
(v)Hari comments that ㄥACB = 112°. Is he true? Which value is depicted by comment of Hari?

Show that each angle of a rectangle is a right angle.

"A diagonal of a parallelogram divides it into two congruent triangles:"  Prove it.

In a parallelogram, show that the angle bisectors of two adjacent angles intersect at right angles.

In triangle, ABC points M and N  on sides AB and AC respectively are taken so that AM = \(1\over2\) AB and AN = \(1\over4\) AC Prove that MN= \(1\over4\)BC

ABCD is a square and on the side DC, an equilateral triangle is constructed. Prove that:
(i) AE = BE
(ii) ㄥDAE=15°

ABCD is a quadrilateral and BD is one of its diagonals as shown in figure. Show that ABCD is a parallelogram and find its area.

 

In the given figure, AB 11DC. Show that ar(BDE) = ar(ACED).
     
 

Areas of triangles on the same bases and between the same parallels are equal in. Prove it.

In the given figure, ABED is a parallelogram in which DE = EC. Show that: area (ABF) = area (BEC).

Prove that the area of a rhombus is equal to half the rectangle contained by its diagonals.

In the figure, diameter AB and a chord AC have a 'common end point A. If the length of AB is 20 cm and of AC is 12 cm, how far is AC from the centre of the circle?

Prove that the perpendicular from the centre of a circle to a chord, bisects the chord.

Prove that the line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.

Bisector AD of  \(\angle BAC\) of \(\Delta ABC\) passes through the centre O of the circumcircle of
\(\Delta ABC\) Prove that AB = AC.

Find the angle marked as x in each of following figures where O is the centre of the circle:
(i) 

(ii)

(iii)

(iv)

(v)

In a rectangle ABCD, E is a point which bisects BC Prove that AE = ED

In figure OA = OB, OC = OD and \(\angle AOB=\angle COD\). Prove that AC = BD

In the given figure, if AB = FE, BC=ED, \(AB\bot BD\) and \(FE\bot EC\), then prove that
(i) \(\triangle ABD\cong \triangle FEC\)
(ii) \(AD\cong FC\)

In figure, \(\angle QPR=\angle PQR\) and M and N are respectively points on sides QR and PR of \(\triangle PQR\) , such that QM = PN. Prove that OP = OQ, where O is the point of intersecting of PM and QN.

Prove that the angles opposite to equal sides of a triangle are equal. Is the converse true?

Rays OA,OB, OC,OD, and OE have the common initial point O Show the \(\angle \)AOB+\(\angle \)BOC+\(\angle \)COD+\(\angle \)DOE+\(\angle \)EOA=\(360^{ 0 }\).Draw a ray OP opposite to ray OA.
​​

If two lines are perpendicular to the same line prove that they are parallel to each other.

If l,m,n are three lines such that l || m and n \(\bot \) l , then prove that n \(\bot \)m

In figure PQ || RS and T is any point as shown in the figure then show that
\(\angle PQT+\angle QTS+\angle RST=360^{ 0 }\)

In figure l || m , show that \(\angle 1+\angle 2-\angle 3=180^{ 0 }\)

Prove that the perpendicular from the centre of a circle to a chord, bisects the chord.

AB and CD are equal chords of a circle whose centre is O. When produced, these chords meet at E. Prove that  EB = ED and AE = CE.

In the given figure, ABCD is a cyclic quadrilateral whose diagonals intersect at P. If \(\angle DBC=70°\) and \(\angle BAC=30°\) , find \(\angle BCD\).
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ABCD is a cyclic quadrilateral with AD || BC Prove that AB = DC.

Prove that the opposite angles of an isosceles trapezium are supplementary.

ABCD is a quadrilateral and BD is one of its diagonals as shown in figure. Show that ABCD is a parallelogram and find its area.

 

In the figure, diagonals AC and BD of a trapezium ABCD with AB || CD intersect each other at O. Show that ar (\(\Delta \) AOD)= ar(\(\Delta \) BOC).

Prove that the area of a trapezium is equal to half of the product of its height and sum of parallel sides.

The diagonals of a parallelogram ABCD intersect at a point O. Through O, a line is drawn to intersect AD at P and BC at Q. Show that PQ divides the parallelogram into two parts of equal area.

\(\Delta \)ABC and \(\Delta \) ABD are two triangles on the same base AB. If line segment CD is bisected by AB at O, show that ar (\(\Delta \) ABC) = ar (\(\Delta \)ABD)

Solve the equation x -15 = 25 and state Euclid's Axiom used here.

In the given figure, if AB = CD, then prove that AC = BD. Also, write the Euclid's Axiom used for proving it.

In figure, AC = XD, C is the midpoint of AB and D is the midpoint of XY. Using an Euclid's axiom, show that AB = XY.

In the given figure, it is given that \(\angle \)1 = \(\angle \)4 and \(\angle \)3 = \(\angle \)2. By which Euclid's axiom, it can be shown that if \(\angle \)2 = \(\angle \)4, then \(\angle \)1 = \(\angle \)3.

In figure, C is the mid-point of AB and Dis the mid-point of AC. Prove that AD = \(1\over2\)  AB.

From the given figure, write the points whose
(a) ordinate = 0
(b) abscissa = 0
(c) abscissa = -3
(d) ordinate = 4

See figure and write the following:
(i) Coordinates of point P
(ii) Abscissa of point Q
(iii) The point identified by the coordinates (-4,4)
(iv) The point identified by the coordinates (-3,-6)

In figure, ABCD is a rectangle with length 6 cm and breadth 3 cm.O is the mid point of AB.Find the coordinates of A, B C and D.

Plot the following points on a graph sheet and join them in order B(-5,3), E(-3,-2), S(4,-2), T(1,3).Also mention the quadrants in which the points lie.

Plot the points A, B, C, D from the table:

and answer the following:
(a) Write the coordinates of A, B, C, D.
(b) Shade the triangle ABC.

Write the various coefficients in the following polynomials:
7

How many terms are there in the following polynomials?
3x² - 5x + 7

If \(x=-2\) is the root of the equation \(\sqrt { 2 } (x+p)=0\) and is also the zero the zero of the polynomial  \({ px }^{ 2 }+kx+2\sqrt { 2 } \) then find the value of k.

The polynomial \(p(x)=kx^3+9x^2+4x-8\) when divided by (x+3) leaves a remainder 10 (1-k). Find the value of k.

The polynomial \(p(x)=2{ x }^{ 3 }-3{ x }^{ 2 }+ax-3a+9\)when divided by x+1, leaves the remainder 16. Find the value of a. Also, find the remainder when p(x) is divided by x+2.

Draw the graph of linear equation 4x + 3y = 36. From the graph, find the value of y when x = 3 and value of x when y = 6.

Draw the graph of the equations x = 3 and 4x = 3y in the same graph.Find the area of the triangle formed by these two lines and the x-axis

If x is the number of hours a labourer is on work and y his wages in rupees then y = 4x + 3. Draw the work wages graph of this equation. From the graph, find the wages of a labourer who puts in 4 hours of work

A part of family budget on milk is constant and is fixed at Rs.500, while the other is variable and it depends on the need for milk at the rate of Rs.20 per litre.If extra milk taken is x litre and total expenditure on milk is Rs y, then write a linear equation for this problem. Draw its graph.

Draw the graph of \({2\over3}x-y=2\) and find the points where it cuts the co-ordinate axes.

Simplify: \(\frac { 1 }{ \sqrt { 3 } +\sqrt { 2 } } -\frac { 2 }{ \sqrt { 5 } -\sqrt { 3 } } -\frac { 3 }{ \sqrt { 2 } +\sqrt { 5 } } \)

If √2 = 1.414 and √3 = 1.732, then calculate \(\frac { 4 }{ 3\sqrt { 3 } -2\sqrt { 2 } } +\frac { 3 }{ 3\sqrt { 3 } -2\sqrt { 2 } } \)

If \(x=\frac { 1 }{ 3-2\sqrt { 2 } } \) and \(y=\frac { 1 }{ 3+2\sqrt { 2 } } \), then find the value of x + y + xy.

If x = 2 + √3, then find the value of x²+\(\frac{1}{x^{2}}\)

If x = √2-1, then find the value of \((x-\frac{1}{x})^{3}\) .

ABCD is a parallelogram and line segments AX, CY bisects the angles A and C respectively. Show that AX II CY.

Prove that the opposite angles of an isosceles trapezium are supplementary.

Show that the bisectors of angles of a parallelogram enclose a rectangle.

In the figure, ABCD is a parallelogram. E and F are the mid-points of sides AB and CD respectively. Show that the line segments AF and EC trisect the diagonal BD.

In D ABC, D, E and F are respectively the mid-points of sides AB, BC and CA (see Fig.). Show that Δ ABC is divided into four congruent triangles by joining D, E and F.

In a rectangle ABCD, E is a point which bisects BC Prove that AE = ED

In the given figure, if AB = FE, BC=ED, \(AB\bot BD\) and \(FE\bot EC\), then prove that
(i) \(\triangle ABD\cong \triangle FEC\)
(ii) \(AD\cong FC\)

Prove that the medians of an equilateral triangle are equal.

In the given figure \(BL\bot AC,MC\bot LN\), AL = CN and BL = CM. Prove  that \(\triangle ABC\cong \triangle NML\)

In the figure below, ABC is a triangle in which AB = AC. X and Yare points on AB and AC such that AX = AY. Prove that \(\triangle ABY\cong \triangle ACX\) .

1500 families with 2 children were selected randomly, and the following data were recorded:

Compute the probability of a family, chosen at random, having
(i) 2 girls  (ii) 1 girl (iii) No girl
also, check whether the sum of these probabilities is 1.

In a particular section of Class IX, 40 students were asked about the months of their birth, the following graph was prepared for the data so obtained. Find the probability that a student of the class was born in August.

To know the opinion of the students about the subject statistics, a survey of 200 students was conducted. The data is recorded in the following table:

 Find the probability that a student chosen at random
(i) likes statistics,
(ii) does not like it

Eleven bags of wheat flour, each marked 5 kg, actually contained the following weights of flour (in kg):
4.97, 5.05 ,5.08 ,5.03 ,5.00 ,4.86, 5.08, 4.98 ,5.04, 5.07 ,5.00
Find the probability that any of these bags chosen at random contains more than 5 kg of flour.

In a cricket match, a batsman hits a boundary 6 times out of 30 balls she plays.Find the probability that she did not hit a boundary.

Give five examples of data that you can collect from your day-to-day life.

Classify the data as primary or secondary data.
(i) Number of students
(ii) Number of fans un our school.
(iii) Electricity bills of our house for last two years.
(iv) Election results obtained from television or newspaper.
(v) Literacy rate figures obtained from Educational Survey.

The blood groups of 30 students of Class VIII are recorded as follows:
A,B,O,O,AB,O,A,O,B,A,O,B, A,O,O,
A,AB,O,A,A,O,O,AB,B,A,O,B,A,B,O.
Represent this data in the form of a frequency distribution table. Which is the most common and which is the rarest, blood group among these students.

The length of 40 leaves of a plant are measured a correct one millimeter, and the obtained data is represented in the following table:

(i) Draw a histogram to represent the given data.
(ii) Is there any suitable graphical representation for the same data?
(iii) Is it correct to conclude that the maximum number of leaves are 153 mm long? Why?

The following table gives the distribution of students of two sections according to the marks obtained by them:

Represent the marks of the students of both the sections on the same graph by two frequency polygons. From the two polygons compare the performance of the two sections.

The length, breadth and height of a room are 5 m, 4 m respectively. Find the cost of white washing the walls of the room and the ceiling at the rate of Rs 7.50 per m².

The floor of a rectangular hall has a perimeter 250 m. If the cost of painting the four walls at the rate of Rs 10 per m² is Rs 15000, find the height of the hall. 

The paint in a certain container is sufficient to paint an area equal to 9.375 m². How many bricks of dimensions 22.5 cm \(\times\) 10 cm \(\times\) 7.5 cm can be painted out of this container?

The curved surface area of a right circular cylinder of height 14 cm is 88 cm². Find the diameter of the base of the cylinder.

It is required to make a closed cylindrical tank of height 1 m and base diameter 140 cm from a metal sheet. How many square meters of the sheet are required for the same?

Find the area of an equilateral triangle whose perimeter is 60 cm. (Using Heron's formula).

(a) Find the area of the triangle.

(b) Find the area of a triangle whose sides are 16 cm, 14 cm, nd 10 cm.
(c) The sides of a triangle are 7 cm, 12 cm, and 13 cm. Find its area. 
 d) Find the area of a triangle whose sides are 11 m, 60 m and 61 m.

The sides of a triangle are in the ratio of 25: 17: 12 and its perimeter is 1080 cm. Find its area.

The lengths of the sides of a triangle are in the ratio 3: 4: 5 and its perimeter is 144 cm. Find the area of the triangle.

Construct an equilateral triangle, given its side and justify the construction.

Construct angle of  \(52\frac { { 1 }^{ o } }{ 2 } \) ,using compass and ruler

Construct a triangle ABC such that BC= 6 cm, AB = 3 cm and median AD = 4.5 cm, Write steps of construction.

In the given figure PO \(\bot \) AB If x:y:z=1:3:5 then find the degree measure of x,y and z

If two lines intersect each other, then the vertically opposite angles are equal. prove it

In the given figure , two straight lines PQ and RS intersect each other at O
If \(\angle \)POT =\(75^{ 0 }\) Find the values of a,b,c

If two lines are perpendicular to the same line prove that they are parallel to each other.

If l,m,n are three lines such that l || m and n \(\bot \) l , then prove that n \(\bot \)m

In the given figure, if AB = CD, then prove that AC = BD. Also, write the Euclid's Axiom used for proving it.

In the given figure AB = BC and BX = BY. Show that AX = CY. State Euclid's Axiom used.

In figure, C is the mid-point of AB and Dis the mid-point of AC. Prove that AD = \(1\over2\)  AB.

In the fig, we have \(\angle1=\angle3 \ and\ \angle2=\angle4.\) Show that \(\angle A= \angle C.\) State which axiom you use here. Also give two more axioms other than the axioms used in the above situation.

In the fig., if \(OX=\frac{1}{2}XY,\ PX=\frac{1}{2}XZ\) and OX = PX, Show that XY = XZ. State which axiom you use here. Also give two more axioms other than the oxiom used in the above situation.

Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.

Draw different pairs of circles. How many points does each pair have in common? What is the maximum number of common points?

Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4 cm. Find the length of the common chord.

If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord.

A circular park of radius 20 m is situated in a colony. Three boys Ankur, Syed and David are sitting at equal distance on its boundary each having a toy telephone in his hands to talk each other. Find the length of the string of each phone.

In figure, ABCD is a parallelogram, \(AE\bot DC\)and \(CF\bot AD\). If AB = 16cm, AE = 8 cm and CF = 10 cm, find AD.

if the diagonals of a parallelogram are equal, then show that it is a rectangle.

Diagonal AC of a parallelogram ABCD bisects \(\angle A\) (see figure). Show that:
(i) it bisects \(\angle C\) also
(ii) ABCD is a rhombus.

ABCD is a parallelogram and APand CQ are perpendiculars from vertices A and C on diagonal BD respectively. Show that:
\((i)\Delta APB\cong \Delta CQD\)
\(\\ (ii)AP=CQ\)

l and m are two parallel lines intersected by another pair of parallel lines p and q (see figure). Show that \(\triangle ABC\cong \triangle CDA\)

Show that the angles of an equilateral triangle are 60° each.

In figure if AB \(\parallel \) CD .EF \(\bot \) CD and \(\angle \) AGE,\(\angle \) GEF and \(\angle \) FGE

Prove that bisectors of pair of vertically opposite angles are in the same straight line.

Express the linear equation 7=2x in the form ax+by+c=0 and also write the values of a, b and c.

Find the value of 'm' if (-m, 3) is a solution of equation 4x+9y-3=0

Determine the point on the graph of the equation 2x+5y=20 where x-coordinate is\({5\over2}\) times its ordinate.

Draw the graph of the linear equation \(y={2\over3}x+{1\over3}\).Check from the graph that (7, 5) is a solution of the linear equation

Solve for x:
\(3x-12+{3\over7}x=2(x-1)\)
What type of graph is it in two dimensions?

In which quadrant, can a point have:
(i) abscissa equal to its ordinate
(ii) ordinate equal in magnitude to abscissa
(iii) ordinate equal and opposite of abscissa
(iv) abscissa twice that of the ordinate.

From the given figure, write the points whose
(a) ordinate = 0
(b) abscissa = 0
(c) abscissa = -3
(d) ordinate = 4

From the given figure, write
(i) The coordinates of the points B and F
(ii) The abscissae of points A and C
(iii) The ordinates of the points A and C.
(iv) The perpendicular distance of the point G from the x-axis.

See figure and write the following:
(i) Coordinates of point P
(ii) Abscissa of point Q
(iii) The point identified by the coordinates (-4,4)
(iv) The point identified by the coordinates (-3,-6)

In figure, \(\triangle ABC\) and \(\triangle ABD\) are equilateral triangles.Find the coordinates of points C and D.

Find the remainder when \(x^3+3x^2+3x+1\) is divided by \(x\)

Factorise \(12x^2-7x+1\)

Use suitable identities to find the following products: \(\left( { y }^{ 2 }+\frac { 3 }{ 2 } \right) \left( { y }^{ 2 }-\frac { 3 }{ 2 } \right) ​​\)

Write the various coefficients in the following polynomials:
x⁷ - 3x⁵ + 4

How many terms are there in the following polynomials?
t - 7t² + 5 - t³

Find three rational numbers \(\frac { 3 }{ 5 } \) and \(\frac { 7 }{ 8 } \)

Find two irrational numbers between 2 and 2.5.

Evaluate:\(\frac { 40 }{ 2\sqrt { 10 } +\sqrt { 20 } +\sqrt { 40 } -2\sqrt { 5 } } \) when it is given that \(\sqrt { 10 } =3.162\)

Simplify: \(\frac { 7+3\sqrt { 5 } }{ 3+\sqrt { 5 } } -\frac { 7-3\sqrt { 5 } }{ 3-\sqrt { 5 } }\)

Prove that \(\frac { 1 }{ 3+\sqrt { 7 } } +\frac { 1 }{ \sqrt { 7 } +\sqrt { 5 } } +\frac { 1 }{ \sqrt { 5 } +\sqrt { 3 } } +\frac { 1 }{ \sqrt { 3 } +1 } =1\)

Which of the following statements are true and which are false? Give reasons for your answers:
(i) Only one line can pass through a single point.
(ii) There are an infinite number of lines which pass through two distinct points.
(iii) A terminated line can be produced indefinitely on both the sides.
(iv) If two circles are equal, then their radii are Equal.

Consider two 'postulates' given below:
(i) Given any two distinct points A and B, there exists a third point C which is in between A and B.
(ii) There exist at least three points that are not on the same line.
Do these postulates contain any undefined terms? Are these postulates consistent? Do they follow from Euclid's postulates? Explain.

Point C is called a mid-point of line segment AB. Prove that every line segment has one and only one mid-point.

In figure, if AC = BD, then prove that AB = CD.

(i) Why is Axiom 5, in the list of Euclid's axioms, considered a 'universal truth'?  (Note that the question is not about the fifth postulate).
(ii) How would you rewrite Eulid's fifth postulate so that it would be easier to understand?

Express the following linear equation in the form ax+by+c=0 and indicate the values of a, b and c in each case:
\(x-{y\over 5}-10=0\)

Express the following linear equation in the form ax+by+c=0 and indicate the values of a, b and c in each case:
-2x+3y=6

Which one of the following options is true and why?
y = 3x + 5 hs
(i) a unique solution
(ii) only two solutions
(iii) infinitely many solutions

Check which of the following are solutions of equation x-2y = 4 and which are not:
(0,2)

Check which of the following are solutions of equation x-2y = 4 and which are not:
(2,0)

Write the answer of each of the following questions:
(i) What is the name of horizontal and the vertical lines drawn to determine the position of any point in the Cartesian plane?
(ii) What is the name of each part of the plane formed by these two lines?
(iii) Write the name of the point where these two lines interesect

How will you describe the position of a table lamp on your study table another person?

(Street Plan):A city has two main roads cross each other at the centre of the city. These two roads are along the North-South direction and East-West direction.All the other streets of the city run parallel to these roads and are 200 m apart.There are 5 streets in each direction.Using 1 cm = 200 m, draw a model of the city on your notebook.Represent the roads/streets by single line.

There are many cross-streets in your model.A particular cross-street is made by two streets, one running in the North-South direction and another in the East- West direction.Each cross-street is referred to in the following manner: If the 2nd street running in the North-South direction and 5th in the East-West direction meet at some crossing, then we will call this cross-street (2, 5).Using this convention, find:
(i) how many cross-streets can be referred to as (4,3)?
(ii) how many cross-streets can be referred to as (3,4)?

In which quadrant do the given points lie?(-6,2)

In which quadrant do the given points lie?(-5,-4)

Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
\({ 4x }^{ 2 }-3x+7\)

Write the coefficients of x² in the following:
2 + x² + x

Give one example each of a binomial of degree 35, and of a monomial of degree 100.

Write the degree of the following polynomials:
5x³ + 4x² + 7x

Classify the following as linear, quadratic and cubic polynomials:
r²

Is zero a rational number?can you write it in the form \(\frac { p }{ q } \),where p and q are integers and \(q\neq 0\)?

Find six rational numbers between 3 and 4.

Write the following in decimal form and say what kind of decimal expansion each has:
\(\frac { 3 }{ 13 } \)

Write the following in decimal form and say what kind of decimal expansion each has:
\(\frac { 2 }{ 11 } \)

Express the following in the form p/q, where p and q are integers and \(q\neq 0\)
\(0.\overline { 001 } \)

In a cricket match, a batsman hits boundary in 20% of the balls he played. Find the probability that he did not hit a boundary.

On one page of a telephone directory, there were 200 telephone numbers. The frequency distribution of their unit place digit (for example, in the number 25828573, the unit place digit is 3) is given in the table below:

 Without looking at the page, the pencil is placed on one of these numbers, i.e., the number is chosen at random. What is the probability that the digit in its unit place is 6?

The record of a weather station shows that out of the past 250 consecutive days, its weather forecasts were correct 175 times:
(i) What is the probability that on a given day it was correct?
(ii) What is the probability that it was not correct on a given day?

Three coins are tossed simultaneously 200 times with the following frequencies of different outcomes

Compute the probability of getting
(i) less than 2 heads
(ii) 3 heads.

A bag has 3 red and 7 black balls. One ball is taken out of the bag. Find the probability that it is a
(i) red ball
(ii) blackball.

The marks obtained out of 75 by 30 students of a class in an examination are given below:
42,21,50,37,42,37,38,42,49,52,38,53,57,47,29,59,61,33,17,17,39,44,42,39,14,7,27,19,54,51
Prepare a frequency distribution table in which the size of class intervals is the same and one class intervalis 0-10.

The marks obtained by 40 students of class IX in an examination are given below:
12,8,18,8,6,16,12,5,
23,2,10,20,12,9,7,6,
5,3,5,13,21,13,15,20,
24,1,7,16,21,13,23,18,
7,3,18,17,16,16,23,12
Represent the data in the form of a freqency distribyuting using 15-20 (20 not included) as one of the class intervals.

Prepare a frequency table from the data follows:

Consider the marks, out of 100, obtained by 51 students of a class in a test:

Draw a frequency polygon corresponding to this frequency distribution table.

Find the median of the following data 15,28,72,56,44,32,31,43 and 51. If 32 is replaced by 23, find the new median.

The edge of a cube is 10.5 mm. Find its total surface area in cm². 

If the length of the diagonal of a cube is \(6\sqrt { 3 } \) cm, find the edge of the cube.

The length, breadth, and height of a cuboid are 15 cm, 10 cm, and 20 cm. Find the surface area of the cuboid.

The floor of a rectangular hall has a perimeter of 250 m and its length and breadth are in the ratio of 13: 12. If the cost of painting the four walls and ceiling at the rate of Rs. 5 per m² is Rs.27000, find the height of the hall.

Find the area of an equilateral triangle of side 10 cm.

Find the area of an isosceles triangle, whose equal sides are of length 15 cm each and third side is 12 cm.

The unequal side of an isosceles triangle is 6 cm and its perimeter is 24 cm. Find its area.

Find the area of a triangle whose sides are 6.5 cm. 7 cm and 7.5 cm.

Find the area of a triangle two sides of which are 8 cm and 11 cm and the perimeter is 32 cm.

Find the length of a chord of a circle which is at a distance of 4 cm from the centre of the circle with radius 5 cm.

Two concentric circles are with centre O. A, B, C, D are the points of intersection with a line. If AD = 12 cm and BC = 8 cm, find the length of AB, CD, AC and BD.

Two circles of radii 10 cm and 8 cm intersect and the length of the common chord is 12 cm. Find the distance between their centres.

In the figure, O is the centre of the circle. Arc BCD subtends an angle of 140° at the centre. BC is produced to P and CD is joined. Find measure of \(\angle DCP\).

In the figure, \(\angle AOB= 90°\)  and, \(\angle ABC= 30°\) then find the measure of  \(\angle CAO\).

In the figure, PQRS is a parallelogram with PQ = 12 cm, altitudes corresponding to PQ and SP are respectively 8 cm and 10 cm. Find SP.

In a parallelogram ABCD, AB = 8 cm. The altitudes corresponding to sides AB and AD are respectively 4 cm and 5 cm. Find measure of AD.

In the given figure, ABC and DBC are triangles on the same base and between parallel lines I and m. If AB = 3 cm, BC = 5 cm, \(\angle A=90°\), find area of \(\Delta \) DBC.

In the figure, ar( \(\Delta\)ABE) = 50 cm². Find the area of the parallelogram ABCD. Give reasons.

In the figure, ABCD is a parallelogram. AB = 12 cm, DM = 6 cm and BN = 9 cm. Find the length of AD.

The angles A, B, C and D of a quadrilateral have measures in the ratio 2 : 4 : 5 : 7. Find the measures of these angles. What type of quadrilateral is it? Give reasons.

In a parallelogram PQRS, if \(\angle \)QRS=2x, \(\angle \)PQS=4x, and \(\angle \)PSQ=4x, find the angles of the parallelogram.

If an angle of a parallelogram in two-third of its adjacent angle then find the measure of all the angles,

In the figure, ABCD is a parallelogram in which AB is produced to E so that AB = BE
(a) Prove that ED bisects BC
(b) If AD = 10 cm, find OB.

In \(\Delta \) ABC, D, E and F are midpoints of sides AB, BC and CA. If AB = 6 cm,BC = 7.2 cm and AC = 7.8 crn find the perimeter of  \(\Delta \)DEF.

ABC is an isosceles triangle in which altitudes BE and CF are drawn to equal sides AC and AB respectively (see figure). Show that these altitudes are equal.

In ΔABC, if ㄥA = 50° and ㄥB = 60°, determine the shortest and the longest side of the triangle.

In a ΔDEF, if ㄥD = 30°, ㄥE = 60° then which side of the triangle is longest and which side is shortest?

In ΔABC, ㄥA = 60°, ㄥB = 40°, which side of this triangle is the smallest? Give reasons for your answer.

In ΔPQR, ㄥP = 100° and ㄥR = 60°, which side of the triangle is the longest. Give reasons for your answer.

Find the supplement of \(\frac { 4 }{ 3 } \) of right angle

If (3x-\(58^{ 0 }\)) and (x+\(38^{ 0 }\)) are supplementary angles,find x and the angles.

In figure \(\angle \) DOB =\(87^{ 0 }\) and \(\angle \) COA =\(82^{ 0 }\) If \(\angle \)BOA \(35^{ 0 }\) and Find \(\angle \) COB and \(\angle \)COD

An exterior angle of a triangle is \(115^{ 0 }\) and one of the interior opposite angles is \(35^{ 0 }\).Find the other two angles of the triangle.

find the angles of a triangle PQR if \(\angle p-\angle q=45^{ 0 }\) and \(\angle Q-\angle R=30^{ 0 }\)

Consider the following statement: There exists a pair of straight lines that are everywhere equidistant from one another. Is this statement a

State any two Eulis's axioms.

State any two Euclid's axioms.

In the given figure AC = DC, CB = CE, Show that AB = DE.

Write Euclid's axiom to support this.

In the given figure, we have AB=AD and AC=AD. Prove that AB=AC. State the Euclid's axiom to support this.

Write  the following as an equation in two variables:
5y=2

Express y in terms of x, it being given that 3x+y-9=0.Check whether the points (3,0) and (2, 2) lie on the equation

Find three different solutions for the equation 6x-8y+32=0

Given the point (1, 2), can you give the equation of a line on which it lies? How many such equations are there?

Draw the graph of the linear equation y = m.x + c for m = 2 and c = 1. Read from the graph the value of y when x = \(3\over 2\)

In which quadrant do the given point lie? (2,-1)

In which quadrant do the given point lie?(-1,7)

In which quadrant do the given point lie? (-4,-5)

In which quadrant do the given point lie? (-3,5)

In which quadrant do the given point lie? (4,-1)

Write the coefficient of x³ of the following polynomials:
\(1+{ x }^{ 2 }+{ x }^{ 4 }-x-{ x }^{ 3 }\)

Write the coefficient of x³ of the following polynomials:
\(4x+{ x }^{ 3 }+{ x }^{ 6 }-{ 7x }^{ 2 }\)

Find the degree of the polynomials given below:
​​​​​​​\(2-{ y }^{ 2 }-{ y }^{ 3 }+{ 2y }^{ 8 }\)

Point out which of the following polynomials are monomials, binomials or trinomials?
\({ x }^{ 3 }+2x-3\)

Find a zero of the polynomial p(x)=2x+1

Find two rational numbers between 0.1 and 0.2.

Express \(0.\overline { 001 } \) as a rational number in the form p/q, where p and q are integers and \(q\neq 0\)

Represent \(0.\overline { 237 } \) in the form p/q, where p and q are integers and \(q\neq 0\)

Simplify: 17².17⁵

Simplify: (5²)^-7

The minimum probability of an event is

What is the number of outcomes when a coin is tossed?

What is the number of outcomes when a cubical dice is thrown?

An experiment has two outcomes E and F P(E)+P(F) is equal to:

Which of the following cannot be the experiment probability of an event?

The word 'Latum' is a

Statistics is branch of

'Number of absentees in each day in your class for a month' form

When the information is gathered from a source which already had the information stored, the data obtained is called

The upper limit of the class 36 - 40 is

Which of the following is a plane figure?

The total surface area of a cube of side a is

The lateral surface area of a cuboid of length l, breadth b and height h is

The dimensions of a box are 1 m, 80 cm and 50 cm. The area of its four walls is

The area of the four walls of a room is 300 m². Its length and height are 15 m and 6 m respectively. Find its breadth. 

Area of a triangle = 

Area of a triangle having base 6 cm and altitude 8 cm is

Area of a triangle is 60 cm².Its base is 15 cm.Its altitude is

The side of an isosceles right triangle of hypotenuse \(5\sqrt { 2 } \) cm is

Side of an equilateral triangle is 4 cm. Its area is

Bisector of an angle divides it in to_______equal parts

If a 60^o angle is bisected twice,what will be measure of each that is constructed?

In the figure below,PR is the perpendicular bisectors of a line segment AB=16 cm.Is PA=PB true?

Construct ∠POY=30 ^o. using compass and ruler.

Construct an angle of 15^o

The path traced by the tip of the second's hand is a

The shape of the coin of RS 1 is

The wheels of a vehicle are in

The longest chord of a circle is called

The centre of a circle lies

Area of a triangle is equal to

In the following figure, find the area of quad. ABCD.

If length of the diagonal of a square is 8 cm, then its area will be

If P, Q, Rand S are the midpoints of a rectangle of area 36 sq. cm, then PQRS is a parallelogram of area

Which of the following figures lie on the same base and between the same parallels?

How many sides does a quadrilateral have?

If only one pair of opposite sides of a quadrilateral are parallel, then the quadrilateral is a 

Which of the following is false?

Which of the following is not true?

A blackboard is

A closed figure formed by three intersecting lines is called

A triangle has

The symbol for congruence is

The symbol for correspondence is

Two circles are congruent.If the radius of one circle is 3cm, what is the radius of the other circle?

The minimum number of points required to draw a line is 

How many types of angles are formed between the edges of plane surfaces? 

Two angles whose sum is \(90^{ 0 }\) are called

The sum of two complimentary angles is 

An angle which is greater than \(90^{ 0 }\) and less than \(180^{ 0 }\) is called 

In Indus Valley Civilisation (about 300 b.C), The brick used for construction work were having dimension in the ration

Pythagoras was a student of

Number of dimension(s) a surface:

How many numbers of lines do pass through two distinct points?

The number of lines that can pass through a given point is: