Two friends Richa and Sohan have some savings in their piggy bank. They decided to count the total coins they both had. After counting they find that they have fifty \(\begin{equation} ₹ \end{equation} \) 1 coins, forty eight \(\begin{equation} ₹ \end{equation} \) 2 coins, thirty six \(\begin{equation} ₹ \end{equation} \) 5 coins, twenty eight \(\begin{equation} ₹ \end{equation} \)10 coins and eight \(\begin{equation} ₹ \end{equation} \) 20 coins. Now, they said to Nisha, their another friends, to choose a coin randomly.
Find the probability that the coin chosen is

(i)  \(\begin{equation} ₹ \end{equation} \)5 coin

(ii) \(\begin{equation} ₹ \end{equation} \) 20 coin

(iii) not a  \(\begin{equation} ₹ \end{equation} \) 10 coin

(iv) of denomination of atleast  \(\begin{equation} ₹ \end{equation} \)10. 

(v) of denomination of atmost \(\begin{equation} ₹ \end{equation} \) 5.

In a play zone, Nishtha is playing claw crane game which consists of 58 teddy bears, 42 pokemons, 36 tigers and 64 monkeys. Nishtha picks a puppet at random. Now, find the probability of getting

(i) a tiger 

(ii) a monkey

(iii) a teddy bear

(iv) not a monkey 

(v) not a pokemon 

Rohit wants to distribute chocolates in his class on his birthday. The chocolates are of three types: Milk chocolate, White chocolate and Dark chocolate. If the total number of students in the class is 54 and everyone gets a chocolate, then answer the following questions.

(i) If the probability of distributing milk chocolates is 1/3, then the number of milk chocolates Rohit has, is

(ii) If the probability of distributing dark chocolates is 4/9, then the number of dark chocolates Rohit has, is

(iii) The probability of distributing white chocolates is

(iv) The probability of distributing both milk and white chocolates is

(v) The probability of distributing all the chocolates is

In a party, some children decided to play musical chair game. In the game the person playing the music has been advised to stop the music at any time in the interval of 3 mins after he start the music in each turn. On the basis of the given information, answer the following questions.
(i) What is the probability that the music will stop within first 30 sees after starting?

(ii) The probability that the music will stop within 45 sees after starting is

(iii) The probability that the music will stop after 2 mins after starting is

(iv) The probability that the music will not stop within first 60 sees after starting is

(v) The probability that the music will stop within first 82 sees after starting is

Three persons toss 3 coins simultaneously and note the outcomes. Then, they ask few questions to one another. Help them in finding the answers of the following questions.

(i) The probability of getting atmost one tail is

(ii) The probability of getting exactly 1 head is

(iii) The probability of getting exactly 3 tails is 

(iv) The probability of getting atmost 3 heads is 

(v) The probability of getting atleast two heads is

An agency has decided to install customised playground equipments at various colony parks. For that they decided to study the age-group of children playing in a park of the particular colony. The classification of children according to their ages, playing in a park is shown in the following table

Based on the above information, answer the following questions.
(i) The maximum number of children are of the age-group

(ii) The lower limit of the modal class is

(iii) Frequency of the class succeeding the modal class is

(iv) The mode of the ages of children playing in the park is

(v) If mean and mode of the ages of children playing in the park are same, then median will be equal to

As the demand for the products grew, a manufacturing company decided to hire more employees. For which they want to know the mean time required to complete the work for a worker. The following table shows the frequency distribution of the time required for each worker to complete a work.

 

Based on the above information, answer the following questions.
(i) The class mark of the class 25-29 is

(ii) If x_i's denotes the class marks and f_i's denotes the corresponding frequencies for the given data, then the value of \(\sum x_{i} f_{i}\) equals to

(iii) The mean time required to complete the work for a worker is

(iv) If a worker works for 8 hrs in a day, then approximate time required to complete the work for a worker is

(v) The measure of central tendency is

On a particular day, National Highway Authority ofIndia (NHAI) checked the toll tax collection of a particular toll plaza in Rajasthan.

The following table shows the toll tax paid by drivers and the number of vehicles on that particular day.

Based on the above information, answer the following questions.
(i) If A is taken as assumed mean, then the possible value of A is

(ii) If x_i's denotes the class marks and f_i's denotes the deviation of assumed mean (A) from x_i's, then the minimum value of |d_i| is

(iii) The mean of toll tax received. by NHAI by assumed mean method is

(iv) The mean of toll tax received by NHAI by direct method is

(v) The average toll tax received by NHAI in a day, from that particular toll plaza, is

Transport department of a city wants to buy some Electric buses for the city. For which they wants to analyse the distance travelled by existing public transport buses in a day.

The following data shows the distance travelled by 60 existing public transport buses in a day.

Based on the above information, answer the following questions.
(i) The upper limit of a class and lower limit of its succeeding class is differ by

(ii) The median class is

(iii) The cumulative frequency of the class preceding the median class is

(iv) The median of the distance travelled is

(v) If the mode of the distance travelled is 223.78 km, then mean of the distance travelled by the bus is

A group of 71 people visited to a museum on a certain day. The following table shows their ages.

Based on the aboxe information, answer the following questions.
(i) If true class limits have been decided by making the classes of interval 10, then first class must be

(ii) The median class for the given data will be

(iii) The cumulative frequency of class preceding the median class is

(iv) The median age of the persons visited the museum is

(v) If the price of a ticket for the age group 30-40 is Rs 30, then the total amount spent by this age group is

Arun a 10^th standard student makes a project on corona virus in science for an exhibition in his school. In this project, he picks a sphere which has volume 38808 cm³ and 11 cylindrical shapes, each of volume 1540 cm³ with length 10 cm.

Based on the above information, answer the following questions.
(i) Diameter of the base of the cylinder is

(ii) Diameter of the sphere is

(iii) Total volume of the shape formed is

(iv) Curved surface area of the one cylindrical shape is

(v) Total area covered by cylindrical shapes on the surface of sphere is

Ajay is a Class X student. His class teacher Mrs Kiran arranged a historical trip to great Stupa of Sanchi. She explained that Stupa of Sanchi is great example of architecture in India. Its base part is cylindrical in shape. The dome of this stupa is hemispherical in shape, known as Anda. It also contains a cubical shape part called Hermika at the top. Path around Anda is known as Pradakshina Path.

Based on the above information, answer the following questions.
(i) Find the lateral surface area of the Hermika, if the side of cubical part is 8 m.

(ii) The diameter and height of the cylindrical base part are respectively 42 m and 12 m. If the volume of each brick used is 0.01 m³, then find the number of bricks used to make the cylindrical base.

(iii) If the diameter of the Anda is 42 m, then the volume of the Anda is

(iv) The radius of the Pradakshina path is 25 m. If Buddhist priest walks 14 rounds on this path, then find the distance covered by the priest.

(v) The curved surface area of the Anda is

One day Rinku was going home from school, saw a carpenter working on wood. He found that he is carving out a cone of same height and same diameter from a cylinder. The height of the cylinder is 24 ern and base radius is 7 cm. While watching this, some questions came into Rinkus mind. Help Rinku to find the answer of the following questions.

(i) After carving out cone from the cylinder,

(ii) Find the slant height of the conical cavity so formed.

(iii) The curved surface area of the conical cavity so formed is

(iv) External curved surface area of the cylinder is

(v) Volume of conical cavity is

To make the learning process more interesting, creative and innovative, Amayras class teacher brings clay in the classroom, to teach the topic - Surface Areas and Volumes. With clay, she forms a cylinder of radius 6 ern and height 8 cm. Then she moulds the cylinder into a sphere and asks some questions to students.

(i) The radius of the sphere so formed is 

(ii) The volume of the sphere so formed is

(iii) Find the ratio of the volume of sphere to the volume of cylinder.

(iv) Total surface area of the cylinder is 

(v) During the conversion of a solid from one shape to another the volume of new shape will 

A carpenter used to make and sell different kinds of wooden pen stands like rectangular, cuboidal, cylindrical, conical. Aarav went to his shop and asked him to make a pen stand as explained below. Pen stand must be of the cuboidal shape with three conical depressions, which can hold 3 pens. The dimensions of the cuboidal part must be 20 cm x 15 cm x 5 cm and the ffrlog radius and depth of each conical depression must be 0.6 cm and 2.1 cm respectively.

Based on the above information, answer the following questions.
(i) The volume of the cuboidal part is

(ii) Total volume of conical depressions is 

(iii) Volume of the wood used in the entire stand is

(iv) Total surface area of cone of radius r is given by

(v) If the cost of wood used is Rs 5 per cm³, then the total cost of making the pen stand is

In a pathology lab, a culture test has been conducted. In the test, the number of bacteria taken into consideration in various samples is all3-digit numbers that are divisible by 7, taken in order.

On the basis of above information, answer the following questions.
(i) How many bacteria are considered in the fifth sample?

(ii) How many samples should be taken into consideration?

(iii) Find the total number of bacteria in the first 10 samples.

(iv) How many bacteria are there in the 7^th sample from the last?

(v) The number of bacteria in 50^th sample is

In a class the teacher asks every student to write an example of A.P. Two friends Geeta and Madhuri writes their progressions as -5, -2, 1,4, ... and 187, 184, 181, .... respectively. Now, the teacher asks various students of the class the following questions on these two progressions. Help students to find the answers of the questions.

(i) Find the 34^th term of the progression written by Madhuri.

(ii) Find the sum of common difference of the two progressions.

(iii) Find the 19^th term of the progression written by Geeta.

(iv) Find the sum of first 10 terms of the progression written by Geeta.

(v) Which term of the two progressions will have the same value?

Meenas mother start a new shoe shop. To display the shoes, she put 3 pairs of shoes in 1^st row,S pairs in 2^nd row, 7 pairs in 3^rd row and so on.

On the basis of above information, answer the following questions.
(i) If she puts a total of 120 pairs of shoes, then the number of rows required are

(ii) Difference of pairs of shoes in 17^th row and 10^th row is

(iii) On next day, she arranges x pairs of shoes in 15 rows, then x =

(iv) Find the pairs of shoes in 30^th row.

(v) The total number of pairs of shoes in 5^th and 8^th row is

Anuj gets pocket money from his father everyday. Out of the pocket money, he saves Rs 2.75 on first day, Rs 3 on second day, Rs 3.25 on third day and so on.
On the basis of above information, answer the following questions .

(i) What is the amount saved by Anuj on 14^th day?

(ii) What is the total amount saved by Anuj in 8 days?

(iii) What is the amount saved by Anuj on 30^th day?

(iv) What is the total amount saved by him in the month of June, if he starts savings from 1^st June?

(v) On which day, he save tens times as much as he saved on day-I?

In a board game, the number of sea shells in various cells forms an A.P. If the number of sea shells in the 3^rd and 11^th cell together is 68 and number of shells in 11^th cell is 24 more than that of 3^rd cell, then answer the following
questions based on this data.
(i) What is the difference between the number of sea shells in the 19^th and 20^th cells?

(ii) How many sea shells are there in the first cell?

(iii) How many total sea shells are there in first 13 cells?

(iv) Altogether, how many sea shells are there in the first 5 cells?

(v) What is the sum of number of sea shells in the 7^th and 9^th cell?

In a park, four poles are standing at positions A, B, C and D around the fountain such that the cloth joining the poles AB, BC, CD and DA touches the fountain at P, Q, Rand S respectively as shown in the figure.

Based on the above information, answer the following questions.
(i) If 0 is the centre of the circular fountain, then \(\angle\)OSA = 

(ii) Which of the following is correct?

(iii) If DR = 7 cm and AD = 11 ern, then AP =

(iv) If O is the centre of the fountain, with \(\angle\)QCR = 60°, then \(\angle\)QOR

(v) Which of the following is correct?

Smita always finds it confusing with the concepts of tangent and secant of a circle. But this time she has determined herself to get concepts easier. So, she started listing down the differences between tangent and secant of a circle along with their relation. Here, some points in question form are listed by Smita in her notes. Try answering them to clear your concepts also.

(i) A line that intersects a circle exactly at two points is called

(ii) Number of tangents that can be drawn on a circle is

(iii) Number of tangents that can be drawn to a circle from a point not on it, is

(iv) Number of secants that can be drawn to a circle from a point on it is

(v) A line that touches a circle at only one point is called

A backyard is in the shape of a triangle with right angle at B, AB = 6 m and BC = 8 m. A pit was dig inside it such that it touches the walls AC, BC and AB at P, Q and R respectively such that AP = x m.

Based on the above information, answer the following questions.
(i) The value of AR =

(ii) The value of BQ =

(iii) The value of CQ =

(iv) Which of the following is correct?

(v) Radius of the pit is

In a maths class, the teacher draws two circles that touch each other externally at point K with centres A and B and radii 5 em and 4 em respectively as shown in the figure.

Based on the above information, answer the following questions.
(i) The value of PA =

(ii) The value of BQ =

(iii) The value of PK =

(iv) The value of QY =

(v) Which of the following is true?

Prem did an activity on tangents drawn to a circle from an external point using 2 straws and a nail for maths project as shown in figure.

Based on the above information, answer the following questions.
(i) Number of tangents that can be drawn to a circle from an external point is

(ii) On the basis of which of the following congruency criterion,\(\Delta \mathrm{OAP} \cong \Delta \mathrm{OBP} ?\)

(iii) If \(\angle\)AOB = 150°, then \(\angle\)APB =

(iv) If \(\angle\)APB = 40°, then \(\angle\)BAO =

(v) If \(\angle\)ABO = 45°, then which of the following is correct option?

There are two temples on each bank of a river. One temple is 50 m high. A man, who is standing on the top of 50 m high temple, observed from the top that angle of depression of the top and foot of other temple are 30° and 60° respectively. (Take \(\sqrt{3}\) = 1.73)

Based on the above information, answer the following questions.
(i) Measure of \(\angle\)ADF is equal to 

(ii) Measure of \(\angle\)ACB is equal to

(iii) Width of the river is 

(iv) Height of the other temple is

(v) Angle of depression is always

There are two windows in a house. First window is at the height of 2 m above the ground and other window is 4 m vertically above the lower window. Ankit and Radha are sitting inside the two windows at points G and F respectively. At an instant, the angles of elevation of a balloon from these windows are observed to be 60° and 30° as shown below

Based on the above information, answer the following questions.
(i) Who is more closer to the balloon?

(ii) Value of DF is equal to

(iii) Value of h is

(iv) Height of the balloon from the ground is

(v) If the balloon is moving towards the building, then both angle of elevation will

A circus artist is climbing through a 15 m long rope which is highly stretched and tied from the top of a vertical pole to the ground as shown below. Based on the above information, answer the following questions.

(i) Find the height of the pole, if angle made by rope to the ground level is 45°.

(ii) If the angle made by the rope to the ground level is 45°, then find the distance between artist and pole at ground level.

(iii) Find the height of the pole if the angle made by the rope to the ground level is 30°.

(iv) If the angle made by the rope to the ground level is 30° and 3 m rope is broken, then find the height of the pole

(v) Which mathematical concept is used here?

There is fire incident in the house. The house door is locked so, the fireman is trying to enter the house from the window. He places the ladder against the wall such that its top reaches the window as shown in the figure .

Based on. the above information, answer the following questions.
(i) If window is 6 m above the ground and angle made by the foot ofladder to the ground is 30°, then length of the ladder is

(ii) If fireman place the ladder 5 m away from the wall and angle of elevation is observed to be 30°, then length of the ladder is

(iii) If fireman places the ladder 2.5 m away from the wall and angle of elevation is observed to be 60°, then find the height of the window. (Take \(\sqrt{3}\) = 1.73)

(iv) If the height of the window is 8 m above the ground and angle of elevation is observed to be 45°, then horizontal distance between the foot of ladder and wall is

(v) If the fireman gets a 9 m long ladder and window is at 6 m height, then how far should the ladder be placed?

An electrician has to repair an electric fault on the pole of height of8 m. He needs to reach a point 2 m below the top of the pole to undertake the repair work.

Based on the above information, answer the following questions.
(i) Length of BD is

(ii) What should be the length of ladder, so that it makes an angle of 60° with the ground?

(iii) The distance between the foot ofladder and pole is

(iv) What will be the measure of \(\angle\)BCD when BD and CD are equal?

(v) Find the measure of \(\angle\)DBC.

Three friends - Anshu, Vijay and Vishal are playing hide and seek in a park. Anshu and Vijay hide in the shrubs and Vishal have to find both of them. If the positions of three friends are at A, Band C respectively as shown in the figure and forms a right angled triangle such that AB = 9 m, BC = 3\(\sqrt{3}\) m and \(\angle\)B = 90°, then answer the following questions.

(i) The measure of \(\angle\)A is 

(ii) The measure of  \(\angle\)C is

(iii) The length of AC is 

(iv) cos2A =

(v) sin \(\left(\frac{C}{2}\right)\) =

Two aeroplanes leave an airport, one after the other. After moving on runway, one flies due North and other flies due South. The speed of two aeroplanes is 400 km/hr and 500 km/hr respectively. Considering PQ as runway and A and B are any two points in the path followed by two planes, then answer the following questions.

(i) Find \(\tan \theta ; \text { if } \angle A P Q=\theta\)

(ii) Find cot B

(iii) Find tanA.

(iv) Find secA.

(v) Find cosecB.

Anita, a student of class 10^th, has to made a project on 'Introduction to Trigonometry' She decides to make a bird house which is triangular in shape. She uses cardboard to make the bird house as shown in the figure. Considering the front side of bird house as right angled triangle PQR, right angled at R, answer the following questions.

(i) If \(\angle P Q R=\theta, \text { then } \cos \theta=\)

(ii) The value of sec \(\theta\) =

(iii) The value of \(\frac{\tan \theta}{1+\tan ^{2} \theta}=\)

(iv) The value of \(\cot ^{2} \theta-\operatorname{cosec}^{2} \theta=\) 

(v) The value of \(\sin ^{2} \theta+\cos ^{2} \theta=\)

Ritu's daughter is feeling so hungry and so thought to eat something. She looked into the fridge and found some bread pieces. She decided to make a sandwich. She cut the piece of bread diagonally and found that it forms a
righ.t angled triangle with sides 4 cm, 4\(\sqrt{3}\) cm and 8 cm.

On the basis of above information, answer the following questions.
(i) The value of \(\angle\)M = 

(ii) The value of \(\angle\)K =

(iii) Find the value of tanM.

(iv) sec²M - 1 =

(v) The value of \(\frac{\tan ^{2} 45^{\circ}-1}{\tan ^{2} 45^{\circ}+1}\) is

Aanya and her father go to meet her friend Juhi for a party. When they reached to [uhi's place, Aanya saw the roof of the house, which is triangular in shape. If she imagined the dimensions of the roof as given in the figure, then answer the following questions.

(i) If D is the mid point of AC, then BD =

(ii) Measure of \(\angle\)A =

(iii) Measure of \(\angle\)C =

(iv) Find the value of sinA + cosC.

(v) Find the value of tan²C + tan² A.

The Chief Minister of Delhi launched the, 'Switch Delhi: an electric vehicle mass awareness campaign in the National Capital. The government has also issued tenders for setting up 100 charging stations across the city. Each station will have five charging points. For demo charging station is set up along a straight line and has charging points at \(A\left(\frac{-7}{3}, 0\right), B\left(0, \frac{7}{4}\right)\), C(3, 4), D(7, 7) and E(x, y). Also, the distance between C and E is 10 units.

Based on the above information, answer the following questions.
(i) The distance DE is

(ii) The value of x + y is 

(iii) Which of the following is true?

(iv) The ratio in which B divides AC is

(v) Which of the following equations is satisfied by the given points?

Alia and Shagun are friends living on the same street in Patel Nagar. Shaguns house is at the intersection of one street with another street on which there is a library. They both study in the same school and that is not far from Shagun's house. Suppose the school is situated at the point 0, i.e., the origin, Alia's house is at A. Shaguns house is at B and library is at C. Based on the above information, answer the following questions.

(i) How far is Alia's house from Shaguns house?

(ii) How far is the library from Shaguns house?

(iii) How far is the library from Alia's house?

(iv) Which of the following is true?

(v) How far is the school from Alia's house than Shaguns house?

A person is riding his bike on a straight road towards East from his college to city A and then to city B. At some point in between city A and city B, he suddenly realises that there is not enough petrol for the journey. Also, there is no petrol pump on the road between these two cities.

Based on the above information, answer the following questions.
(i) The value of y is equal to

(ii) The value of x is equal to

(iii) If M is any point exactly in between city A and city B, then coordinates of M are

(iv) The ratio in which A divides the line segment joining the points O and M is

(v) If the person analyse the petrol at the point M(the mid point of AB), then what should be his decision?

Satellite image of a colony is shown below. In this view, a particular house is pointed out by a flag, which is situated at the point of intersection of x and y-axes. If we go 2 em east and 3 em north from the house, then we reach to a Grocery store. If we go 4 em west and 6 em south from the house, then we reach to a Electrician's shop. If we go 6 em east and 8 em south from the house, then we reach to a food cart. If we go 6 em west and 8 em north from the house, then we reach to a bus stand.

Based on the above information, answer the following questions.
(i) The distance between grocery store and food cart is

(ii) The distance of the bus stand from the house is

(iii) If the grocery store and electrician's shop lie on a line, the ratio of distance of house from grocery store to that from electrician's shop, is

(iv) The ratio of distances of house from bus stand to food cart is

(v) The coordinates of positions of bus stand, grocery store, food cart and electrician's shop form a

A round clock is traced on a graph paper as shown below. The boundary intersect the coordinate axis at a distance of 4/3 units from origin.

Based on the above information, answer the following questions .
(i) Circle intersect the positive y-axis at 

(ii) The centre of circle is the

(iii) The radius of the circle is

(iv) The area of the circle is

(v) If \(\left(1, \frac{\sqrt{7}}{3}\right)\) is one of the ends of a diameter, then its other end is

In a classroom, students were playing with some pieces of cardboard as shown below.

All of a sudden, teacher entered into classroom. She told students to arrange all pieces. On seeing this beautiful image, she observed that \(\Delta\) ADH is right angled triangle, which contains.

(i) right triangles ABJ and IGH.
(ii) quadrilateral GFJI
(iii) squares JKLM and LCBK
(iv) rectangles MLEF and LCDE.
After observation, she ask certain questions to students. Help them to answer these questions.
(i) If an insect (small ant) walks 24 m from H to F, then walks 6 m to reach at M, then walks 4 m to reach at L and finally crossing K, reached at J. Find the distance between initial and final position of insect.

(ii) If m, n and r are the sides of right triangle ABJ, then which of the following can be correct?

(iii) If \(\Delta\)ABJ ~ \(\Delta\)ADH, then which similarity criterion is used here?

(iv) If  \(\angle\)ABJ = 90° and B, J are mid points of sides AD and AH respectively and BJ || DH, then which of the following option is false?

(v) If \(\Delta\)PQR is right triangle with QM \(\perp\) PR, then which of the following is not correct?

An aeroplane leaves an airport and flies due north at a speed of 1200km /hr. At the same time, another aeroplane leaves the same station and flies due west at the speed of 1500 km/hr as shown below. After \(1 \frac{1}{2}\) hr both the aeroplanes reaches at point P and Q respectively.

(i) Distance travelled by aeroplane towards north after \(1 \frac{1}{2}\) hr is

(ii) Distance travelled by aeroplane towards west after  \(1 \frac{1}{2}\) hr is

(iii) In the given figure,\(\angle\)POQ is 

(iv) Distance between aeroplanes after \(1 \frac{1}{2}\) hr is

(v) Area of \(\Delta\)POQ is

Rohit's father is a mathematician. One day he gave Rohit an activity to measure the height of building. Rohit accepted the challenge and placed a mirror on ground level to determine the height of building. He is standing at a certain distance so that he can see the top of the building reflected from mirror. Rohit eye level is at 1.8m above ground. The distance of Rohit from mirror and that of building from mirror are 1.5 m and 2.5 m respectively.

Based on the above information, answer the following questions.
(i) Two similar triangles formed in the above figure is

(ii) Which criterion of similarity is applied here?

(iii) Height of the building is

(iv) In \(\Delta\)ABM, if LBAM = 30°, then LMCD is equal to 

(v) If \(\Delta\)ABM and \(\Delta\)CDMare similar where CD = 6 ern, MD = 8 cm and BM = 24 ern, then AB is equal to 

Meenal was trying to find the height of tower near his house. She is using the properties of similar triangles. The height of Meenal's house is 20 m. When Meenal's house casts a shadow of 10m long on the ground, at the same time, tower casts a shadow of 50 m long and Arun's house casts a shadow of 20 m long on the ground as shown below.

Based on the above information, answer the following questions.
(i) What is the height of tower?

(ii) What will be the length of shadow of tower when Meenal's house casts a shadow of 15 m? 

(iii) Height of Aruns house is 

(iv) If tower casts a shadow of 40 rn, then find the length of shadow of Arun's house 

(v) If tower casts a shadow of 40 m, then what will be the length of shadow of Meenal's house? 

In the backyard of house, Shikha has some empty space in the shape of a \(\Delta\)PQR. She decided to make it a garden. She divided the whole space into three parts by making boundaries AB and CD using bricks to grow flowers and
vegetables where ABIICDIIQR as shown in figure.

Based on the above information, answer the following questions.
(i) The length of AB is

(ii) The length of CD is 

(iii) Area of whole empty land is 

(iv) Area of \(\Delta\)PAB is 

(v) Area of \(\Delta\)PCD is

In a pathology lab, a culture test has been conducted. In the test, the number of bacteria taken into consideration in various samples is all3-digit numbers that are divisible by 7, taken in order.

On the basis of above information, answer the following questions.
(i) How many bacteria are considered in the fifth sample?

(ii) How many samples should be taken into consideration?

(iii) Find the total number of bacteria in the first 10 samples.

(iv) How many bacteria are there in the 7^th sample from the last?

(v) The number of bacteria in 50^th sample is

In a class the teacher asks every student to write an example of A.P. Two friends Geeta and Madhuri writes their progressions as -5, -2, 1,4, ... and 187, 184, 181, .... respectively. Now, the teacher asks various students of the class the following questions on these two progressions. Help students to find the answers of the questions.

(i) Find the 34^th term of the progression written by Madhuri.

(ii) Find the sum of common difference of the two progressions.

(iii) Find the 19^th term of the progression written by Geeta.

(iv) Find the sum of first 10 terms of the progression written by Geeta.

(v) Which term of the two progressions will have the same value?

Meenas mother start a new shoe shop. To display the shoes, she put 3 pairs of shoes in 1^st row,S pairs in 2^nd row, 7 pairs in 3^rd row and so on.

On the basis of above information, answer the following questions.
(i) If she puts a total of 120 pairs of shoes, then the number of rows required are

(ii) Difference of pairs of shoes in 17^th row and 10^th row is

(iii) On next day, she arranges x pairs of shoes in 15 rows, then x =

(iv) Find the pairs of shoes in 30^th row.

(v) The total number of pairs of shoes in 5^th and 8^th row is

Anuj gets pocket money from his father everyday. Out of the pocket money, he saves Rs 2.75 on first day, Rs 3 on second day, Rs 3.25 on third day and so on.
On the basis of above information, answer the following questions .

(i) What is the amount saved by Anuj on 14^th day?

(ii) What is the total amount saved by Anuj in 8 days?

(iii) What is the amount saved by Anuj on 30^th day?

(iv) What is the total amount saved by him in the month of June, if he starts savings from 1^st June?

(v) On which day, he save tens times as much as he saved on day-I?

In a board game, the number of sea shells in various cells forms an A.P. If the number of sea shells in the 3^rd and 11^th cell together is 68 and number of shells in 11^th cell is 24 more than that of 3^rd cell, then answer the following
questions based on this data.
(i) What is the difference between the number of sea shells in the 19^th and 20^th cells?

(ii) How many sea shells are there in the first cell?

(iii) How many total sea shells are there in first 13 cells?

(iv) Altogether, how many sea shells are there in the first 5 cells?

(v) What is the sum of number of sea shells in the 7^th and 9^th cell?

A quadratic equation can be defined as an equation of degree 2. This means that the highest exponent of the polynomial in it is 2. The standard form of a quadratic equation is ax²+ bx + c = 0, where a, b, and c are real numbers and \(a \neq 0\) Every quadratic equation has two roots depending on the nature of its discriminant, D = b2 - 4ac.Based on the above information, answer the following questions.
(i) Which of the following quadratic equation have no real roots?

(ii) Which of the following quadratic equation have rational roots?

(iii) Which of the following quadratic equation have irrational roots?

(iv) Which of the following quadratic equations have equal roots?

(v) Which of the following quadratic equations has two distinct real roots?

In our daily life we use quadratic formula as for calculating areas, determining a product's profit or formulating the speed of an object and many more.
Based on the above information, answer the following questions.
(i) If the roots of the quadratic equation are 2, -3, then its equation is

(ii) If one root of the quadratic equation 2x² + kx + 1 = 0 is -1/2, then k =

(iii) Which of the following quadratic equations, has equal and opposite roots?

(iv) Which of the following quadratic equations can be represented as (x - 2)² + 19 = 0?

(v) If one root of a qua drraattiic equation is \(\frac{1+\sqrt{5}}{7}\),then I.ts other root is

Quadratic equations started around 3000 B.C. with the Babylonians. They were one of the world's first civilisation, and came up with some great ideas like agriculture, irrigation and writing. There were many reasons why Babylonians needed to solve quadratic equations. For example to know what amount of crop you can grow on the square field;
Based on the above information, represent the following questions in the form of quadratic equation.
(i) The sum of squares of two consecutive integers is 650.

(ii) The sum of two numbers is 15 and the sum of their reciprocals is 3/10.

(iii) Two numbers differ by 3 and their product is 504.

(iv) A natural number whose square diminished by 84 is thrice of 8 more of given number.

(v) A natural number when increased by 12, equals 160 times its reciprocal.

Amit is preparing for his upcoming semester exam. For this, he has to practice the chapter of Quadratic Equations. So he started with factorization method. Let two linear factors of \(a x^{2}+b x+c \text { be }(p x+q) \text { and }(r x+s)\)
\(\therefore a x^{2}+b x+c=(p x+q)(r x+s)=p r x^{2}+(p s+q r) x+q s .\)
Now, factorize each of the following quadratic equations and find the roots.
(i) 6x² + x - 2 = 0

(ii) 2x²-+ x - 300 = 0

(iii) x²-  8x + 16 = 0

(iv) 6x²-  13x + 5 = 0

(v) 100x²- 20x + 1 = 0

If p(x) is a quadratic polynomial i.e., p(x) = ax²- + bx + c, \(a \neq 0\), then p(x) = 0 is called a quadratic equation. Now, answer the following questions.
(i) Which of the following is correct about the quadratic equation ax²- + bx + c = 0 ?

(ii) The degree of a quadratic equation is

(iii) Which of the following is a quadratic equation?

(iv) Which of the following is incorrect about the quadratic equation ax²- + bx + c = 0 ?

(v) Which of the following is not a method of finding solutions of the given quadratic equation?

A part of monthly hostel charges in a college is fixed and the remaining depends on the number of days one has taken food in the mess. When a student Anu takes food for 25 days, she has to pay Rs 4500 as hostel charges, whereas another student Bindu who takes food for 30 days, has to pay Rs 5200 as hostel charges.

Considering the fixed charges per month by Rs x and the cost of food per day by Rs y, then answer the following questions.
(i) Represent algebraically the situation faced by both Anu and Bindu.

(ii) The system of linear equations, represented by above situations has

(iii) The cost of food per day is

(iv) The fixed charges per month for the hostel is

(v) If Bindu takes food for 20 days, then what amount she has to pay?

From Bengaluru bus stand, if Riddhima buys 2 tickets to Malleswaram and 3 tickets to Yeswanthpur, then total cost is Rs 46; but if she buys 3 tickets to Malleswaram and 5 tickets to Yeswanthpur, then total cost is Rs 74.

Consider the fares from Bengaluru to Malleswaram and that to Yeswanthpur as Rs x and Rs y respectively and answer the following questions.
(i) 1^st situation can be represented algebraically as

(ii) 2^nd situation can be represented algebraically as

(iii), Fare from Ben~aluru to Malleswaram is

(iv) Fare from Bengaluru to Yeswanthpur is

(v) The system oflinear equations represented by both situations has

Points A and B representing Chandigarh and Kurukshetra respectively are almost 90 km apart from each other on the highway. A car starts from Chandigarh and another from Kurukshetra at the same time. If these cars go in the same direction, they meet in 9 hours and if these cars go in opposite direction they meet in 9/7 hours. Let X and Ybe two cars starting from points A and B respectively and their speed be x km/hr and y km/hr respectively.

Then, answer the following questions.
(i) When both cars move in the same direction, then the situation can be represented algebraically as

(ii) When both cars move in opposite direction, then the situation can be represented algebraically as

(iii) Speed of car X is

(iv) Speed of car Y is

(v) If speed of car X and car Y, each is increased by 10 km/hr, and cars are moving in opposite direction, then after how much time they will meet?

Mr Manoj Jindal arranged a lunch party for some of his friends. The expense of the lunch are partly constant and partly proportional to the number of guests. The expenses amount to Rs 650 for 7 guests and Rs 970 for 11 guests .

Denote the constant expense by Rs x and proportional expense per person by Rs y and answer the following questions.
(i) Represent both the situations algebraically.

(ii) Proportional expense for each person is

(iii) The fixed (or constant) expense for the party is

(iv) If there would be 15 guests at the lunch party, then what amount Mr Jindal has to pay?

(v) The system of linear equations representing both the situations will have

In a office, 8 men and 12 women together can finish a piece of work in 10 days, while 6 men and 8 women together can finish it in 14 days. Let one day's work of a man be l/x and one day's work of a woman be 1/y.

Based on the above information, answer the following questions.
(i) 1^st situation can be represented algebraically as

(ii) 2^nd situation can be represented algebraically as

(iii) One woman alone can finish the work in

(iv) One man alone can finish the work in

(v) If 14 men and 28 women work together, then in what time, the work will be completed?

ABC construction company got the contract of making speed humps on roads. Speed humps are parabolic in shape and prevents overspeeding, mini mise accidents and gives a chance for pedestrians to cross the road. The mathematical representation of a speed hump is shown in the given graph.

Based on the above information, answer the following questions.
(i) The polynomial represented by the graph can be _______polynomial.

(ii) The zeroes of the polynomial represented by the graph are

(iii) The sum of zeroes of the polynomial represented by the graph are

(iv) If a and β are the zeroes of the polynomial represented by the graph such that \(\beta>\alpha, \text { then }|8 \alpha+\beta|=\)

(v) The expression of the polynomial represented by the graph is

While playing in garden, Sahiba saw a honeycomb and asked her mother what is that. She replied that it's a honeycomb made by honey bees to store honey. Also, she told her that the shape of the honeycomb formed is parabolic. The mathematical representation of the honeycomb structure is shown in the graph.

Based on the above information, answer the following questions.
(i) Graph of a quadratic polynomial is in___________shape.

(ii) The expression of the polynomial represented by the graph is

(iii) Find the value of the polynomial represented by the graph when x = 6.

(iv) The sum of zeroes of the polynomial x² + 2x - 3 is

(v) If the sum of zeroes of polynomial at² + 5t + 3a is equal to their product, then find the value of a.

Just before the morning assembly a teacher of kindergarten school observes some clouds in the sky and so she cancels the assembly. She also observes that the clouds has a shape of the polynomial. The mathematical representation of a cloud is shown in the figure.

(i) Find the zeroes of the polynomial represented by the graph.

(ii) What will be the expression for the polynomial represented by the graph?

(iii) What will be the value of polynomial represented by the graph, when x = 3?

(iv) If a and \(\beta\) are the zeroes of the polynomial \(f(x)=x^{2}+2 x-8 \text { , then } \alpha^{4}+\beta^{4}=\)

(v) Find a quadratic polynomial where sum and product of its zeroes are 0,\(\sqrt (7)\) respectively.

Pankaj's father gave him some money to buy avocado from the market at the rate of p(x) = x² - 24x + 128. Let a , \(\beta\) are the zeroes of p(x).
Based on the above information, answer the following questions.

(i) Find the value of a and \(\beta\), where a < \(\beta\).

(ii) Find the value of \(\alpha\) + \(\beta\) + \(\alpha\)\(\beta\).

(iii) The value of p(2) is

(iv) If \(\alpha\) and \(\beta\) are zeroes of \(x^{2}+x-2, \text { then } \frac{1}{\alpha}+\frac{1}{\beta}=\)

(v) If sum of zeroes of \(q(x)=k x^{2}+2 x+3 k\) is equal to their product, then k =

Two friends Trisha and Rohan during their summer vacations went to Manali. They decided to go for trekking. While trekking they observes that the trekking path is in the shape of a parabola. The mathematical representation of the track is shown in the graph.

Based on the above information, answer the following questions.
(i) The zeroes of the polynomial whose graph is given, are

(ii) What will be the expression of the given polynomial p(x)?

(iii) Product of zeroes of the given polynomial is

(iv) The zeroes of the polynomial 9x² - 5 are

(v) If f(x) = x² - 13x + 1, then f(4) =

Srikanth has made a project on real numbers, where he finely explained the applicability of exponential laws and divisibility conditions on real numbers. He also included some assessment questions at the end of his project as listed below. Answer them.
(i) For what value of n, 4ⁿ ends in 0?

(ii) If a is a positive rational number and n is a positive integer greater than 1, then for what value of n, aⁿ is a rational number?

(iii) If x and yare two odd positive integers, then which of the following is true?

(iv) The statement 'One of every three consecutive positive integers is divisible by 3' is

(v) If n is any odd integer, then n2 - 1 is divisible by

Real numbers are extremely useful in everyday life. That is probably one of the main reasons we all learn how to count and add and subtract from a very young age. Real numbers help us to count and to measure out quantities of different items in various fields like retail, buying, catering, publishing etc. Every normal person uses real numbers in his daily life. After knowing the importance of real numbers, try and improve your knowledge about them by answering the following questions on real life based situations.
(i) Three people go for a morning walk together from the same place. Their steps measure 80 cm, 85 cm, and 90 cm respectively. What is the minimum distance travelled when they meet at first time after starting the walk assuming that their walking speed is same?

(ii) In a school Independence Day parade, a group of 594 students need to march behind a band of 189 members. The two groups have to march in the same number of columns. What is the maximum number of columns in which they can march?

(iii) Two tankers contain 768litres and 420 litres of fuel respectively. Find the maximum capacity of the container which can measure the fuel of either tanker exactly.

(iv) The dimensions of a room are 8 m 25 cm, 6 m 75 crn and 4 m 50 cm. Find the length of the largest measuring rod which can measure the dimensions of room exactly.

(v) Pens are sold in pack of 8 and notepads are sold in pack of 12. Find the least number of pack of each type that one should buy so that there are equal number of pens and notepads

In a classroom activity on real numbers, the students have to pick a number card from a pile and frame question on it if it is not a rational number for the rest of the class. The number cards picked up by first 5 students and their questions on the numbers for the rest of the class are as shown below. Answer them.
(i) Suraj picked up \(\sqrt{8}\) and his question was - Which of the following is true about \(\sqrt{8}\)?

(ii) Shreya picked up 'BONUS' and her question was - Which of the following is not irrational?

(iii) Ananya picked up \(\sqrt{5}\)  -.\(\sqrt{10}\) and her question was - \(\sqrt{5}\)  -.\(\sqrt{10}\) _________is number.

(iv) Suman picked up \(\frac{1}{\sqrt{5}}\) and her question was - \(\frac{1}{\sqrt{5}}\) is __________ number.

(v) Preethi picked up \(\sqrt{6}\) and her question was - Which of the following is not irrational?

Decimal form of rational numbers can be classified into two types.
(i) Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form \(\frac{p}{\sqrt{q}}\) where p and q are co-prime and the prime faetorisation of q is of the form 2ⁿ·5^m, where n, mare non-negative integers and vice-versa.
(ii) Let x = \(\frac{p}{\sqrt{q}}\) be a rational number, such that the prime faetorisation of q is not of the form 2ⁿ 5^m, where n and m are non-negative integers. Then x has a non-terminating repeating decimal expansion.
(i) Which of the following rational numbers have a terminating decimal expansion?

(ii) 23/(2³ x 5²) =

(iii) 441/(2² x 5⁷ x 7²) is a_________decimal.

(iv) For which of the following value(s) of p, 251/(2³ x p²) is a non-terminating recurring decimal?

(v) 241/(2⁵ x 5³) is a _________decimal.

HCF and LCM are widely used in number system especially in real numbers in finding relationship between different numbers and their general forms. Also, product of two positive integers is equal to the product of their HCF and LCM. Based on the above information answer the following questions.
(i) If two positive integers x and yare expressible in terms of primes as x = p2q3 and y = p3 q, then which of the following is true?

(ii) A boy with collection of marbles realizes that if he makes a group of 5 or 6 marbles, there are always two marbles left, then which of the following is correct if the number of marbles is p?

(iii) Find the largest possible positive integer that will divide 398, 436 and 542 leaving remainder 7, 11, 15 respectively.

(iv) Find the least positive integer which on adding 1 is exactly divisible by 126 and 600.

(v) If A, Band C are three rational numbers such that 85C - 340A :::109, 425A + 85B = 146, then the sum of A, B and C is divisible by

If ax² + bx + c , a≠0 is factorizable into product of two linear factors, then roots of ax² + bx + c = 0 can be found by equating each factor to

If p, q, r, s, t are the terms of an A.P. with common difference -1 the relation between p and t is

From a point A, the length of a tangent to a circle is 8cm and distance of A from the circle is 10cm. The length of the diameter of the circle is

In which of the following ratios a line segment cannot be divided using ruler and compass?

If the height and length of the shadow of a man are the same, then the angle of elevation of the sun is

If the length of the rectangle is one more than the twice its width, and the area of the rectangle is 300 square meter. What is the measure of the width of the rectangle?

How many terms of AP 54, 51, 48… are required to give a sum of 513

The angle between two tangents drawn from an external point to a circle is 110°. The angle subtended at the centre by the segments joining the points of contact to the centre of circle is:

In the figure, P divides AB internally in the ratio

The ——– is the line drawn from the eye of an observer to the point in the object viewed by the observer

The same value of x satisfies the equations 4x + 5 = 0 and 4x² + (5 + 3p)x + 3p²=0, then p is

The nth term of the AP 9, 13, 17, 21, 25, ………….. is:

If radii of two concentric circles are 4 cm and 5 cm, then length of each chord of one circle which is tangent to the other circle, is

If TP and TQ are two tangents to a circle with centre O so that angle POQ = 110o then angle PTQ is equal to

In the following figure α is

The solution of x² + 4x + 4 = 0 is

The 8^th term of 117, 104, 91, 78, …….is.....

In fig., two circles with centres A and B touch each other externally at k. The length of PQ (in cm) is

In drawing triangle ABC, it is given that AB = 3 cm, BC = 2 cm and AC = 6 cm. It is not possible to draw the triangle as:

The figure shows the observation of point C from point A. The angle of depression from A is:

The roots of quadratic equation x² – 9 = 0 are

The first and last terms of an AP are 1 and 11. If the sum of all its terms is 36, then the number of terms will be

in figure , if ㄥAOB = 125^o, then ㄥCOD is equal to

In the given figure, AC: CB is

An observer 1.5 m tall is 28.5 m away from a tower. The angle of elevation of the top of the tower from his eyes is 45°. The height of the tower is

If  1/2 is a root of the equation x² + kx-5/4 = 0 then the other root of the quadratic equation is

A tree in each year grows 4cm less than it grew in previous year. If it grew 1 metre in the first year, in how many years will it have ceased growing and what will be its height then,

The angle between two tangents drawn from an external point to a circle is 110°. The angle subtended at the centre by the segments joining the points of contact to the centre of circle is:

PT and PS are tangents drawn to a circle, with cantre C, from a point P. If ∠TPS = 50° , then the measure of ΔTCS is

If the height and length of the shadow of a man are the same, then the angle of elevation of the sun is

The roots of quadratic equation x² – 9 = 0 are

Ramesh’s salary in February 2008 is Rs. 10,000. If he’s promised an increase of Rs. 1000 every year, what would be his salary in Feb 2011

In figure, AB is a chord of the circle and AOC is its diameter such that ∠ACB = 50°. If AT is the tangent to the circle at the point A, then ∠BAT is equal to

Given a triangle with side AB = 8 cm. To get a line segment AB’ = 3/4 of AB, it is required to divide the line segment AB in the ratio:

Consider a ship with a right triangular mast. If the base of the mast is 10 m long, and the angle that the mast makes with the base is 60°, then what area of cloth is used to make the mast?

The roots of quadratic equation x² – 9 = 0 are

Which term of the A.P. 1, 4, 7 … is 88?

 PQ is a tangent drawn from a point P to a circle with centre O and QOR is a diameter of the circle such that ∠POR=120°, then ∠OPQ is

If four sides of a quadrilateral ABCD are tangential to a circle, then

Find AB in the given figure

The equation 4x² = 4x has following solution/solutions

If p, q, r, s, t are the terms of an A.P. with common difference -1 the relation between p and t is

A circle can pass through

Which of the following relation would hold true for the sides of the similar triangles in the given diagram?

\(\frac { A'B }{ AB } =\frac { A'C }{ AC } =\frac { BC' }{ BC } =\frac { 3 }{ 4 } \)
\(\frac { A'B }{ AB } =\frac { A'C' }{ AC } =\frac { BC' }{ BC' } =\frac { 3 }{ 4 } \)
\(\frac { A'B }{ AB } =\frac { A'C' }{ AC } =\frac { BC' }{ BC } =\frac { 4 }{ 3 } \)
\(\frac { A'B }{ AB } =\frac { A'C' }{ AC } =\frac { BC }{ BC' } =\frac { 4 }{ 2 } \)

If the angle of elevation of a cloud from a point 100 metres above a lake is 30° and the angle of depression of its reflection in the lake is 60°, then the height of the cloud above the lake is

In the triangles PQR and NLM, angle M will be

Consider a constellation of 3 stars A, B and C forming a right triangle with angle ABC = 90° and angle BAC = 30° . If the distance between star A and B is 3√3 x 1013 km, then how much time does light take to travel from star C to B with a speed of 3 x 108 m/s?

A golf ball has diameter equal to 4.2 cm. Its surface has 200 dimples each of radius 2 mm. Assuming that the dimples are hemispherical, total surface area which is exposed to the surroundings is

A rational number can be expressed as a terminating decimal if its denominator has factors

To construct a triangle similar to a given triangle as per given scale factor which may be __________ than or may be __________ than 1.

In the triangles PQR and NLM, angle M will be

Consider a constellation of 3 stars A, B and C forming a right triangle with angle ABC = 90° and angle BAC = 30° . If the distance between star A and B is 3√3 x 1013 km, then how much time does light take to travel from star C to B with a speed of 3 x 108 m/s?

A golf ball has diameter equal to 4.2 cm. Its surface has 200 dimples each of radius 2 mm. Assuming that the dimples are hemispherical, total surface area which is exposed to the surroundings is

A rational number can be expressed as a terminating decimal if its denominator has factors

To construct a triangle similar to a given triangle as per given scale factor which may be __________ than or may be __________ than 1.

In the figure, AE is the bisector of exterior angle CAD meeting BC produced in E. If AB = 10 cm, AC = 6 cm and BC = 12 cm, then CE is equal to

In the above fig Q and α respectively are

A bucket is in the form of a frustum of a cone ad holds 28.490 liters of water. The radii of the top and bottom are 28cm and 21cm respectively. Find the height of the bucket.

Which of the following rational numbers has a denominator that can be expressed as a product of powers of 2 and 5?

Find the next term of the series \(\sqrt { 2 } ,\sqrt { 8 } ,\sqrt { 18 } ,\sqrt { 32 } ....\)

If four sides of a quadrilateral ABCD are tangential to a circle, then

Find AB in the given figure

In the figure, the shape of a solid copper piece (made of two pieces) with dimensions as shown. The face ABCDEFA has uniform cross section. Assume that the angles at A, B, C, D, E and F are right angles. Calculate the volume of the piece.

The prime factorization of 184 is

The sum of all the angles of a triangle is________________

In the triangles PQR and NLM, angle M will be

The angle formed by the line of sight with the horizontal, when the point being viewed is above the horizontal level is called:

If the curved surface area of a right circular cylinder is 1760 cm2 and its radius is 10 cm, then what is its height?

What is the HCF of 1076 and 584

A Segment AB is divided at point P such that \(\frac { PB }{ AB } =\frac { 3 }{ 7 } \) then find the radio AP : PB.

PT and PS are tangents drawn to a circle, with cantre C, from a point P. If ∠TPS = 50° , then the measure of ΔTCS is

The length of shadow of a tower on the plane ground is √3 times the height of the tower. The angle of elevation of sun is :

If the radius of the base of a right circular cylinder is halved keeping the height same, then the ratio of the volume of the cylinder thus obtained to the volume of original cylinder is

There is a circular path around a sport field. Sonia takes 18min to drive one around of the field, while Ravi takes 12min for the same. Suppose they both start at the same point at the same time, and go in the same direction, after how minutes will they meet again at the starting point?

The difference of any two sides of a triangle is always __________ than the third side.

Given a triangle with side AB = 8 cm. To get a line segment AB’ = 3/4 of AB, it is required to divide the line segment AB in the ratio:

If sun’s elevation is 60° then a pole of height 6 m will cast a shadow of length

What is \(1-\sqrt { 3 } \)?

A school has three sections of Class 10. They need to have enough books in the class library so that they can be distributed equally in the three sections . What is the minimum number of books required if the number of students in section A , B and C are 30, 32 and 36 respectively?

The sum of any two sides of a triangle is always _____________ than the third side.

In the triangles PQR and NLM, angle M will be

The horizontal distance between two towers is 140 m. The angle of elevation of the top of the first tower when seen from the top of the second tower is 30°. If the height of the second tower is 60 m then, the height of the first tower is

A sequence a₁, a₂, a₃,.......... a_n, a_n+1,..... is called an A.P. If there exists constant d such that

In the figure, if AB, AC and line I are tangents to the circle and semi-perimeter of  \(\triangle APQ\)= 14 cm, then AC = ___________cm.

When are the two triangles said to be similar?

To divide a line segment AB in the ratio 4:7, a ray AX is drawn first such that ∠BAX is an acute angle and then points  are located at equal distances on the ray AX and the point B is joined to

An observer 1.5 m tall is 28.5 m away from a tower. The angle of elevation of the top of the tower from his eyes is 45°. The height of the tower is

If H and h be the heights of two cylinders, then the ratio of curved surface areas of two cylinders with equal radii is

‘a’ and ‘b’ are two prime numbers . What is their HCF

The sum of all the angles of a triangle is________________

In the figure, P divides AB internally in the ratio

Write the HCF of the smallest composite number and the smallest even number

In reduced scale-factor, the geometric figure to be constructed is .______________in size.

Three points A, B and C are collinear, if any one of the following takes place:
........... + AB = CB

A right circular cylinder of radius r cm and height h cm (h>2r) just enclosed a sphere of diameter

Amit starts his exercise regime with 25 push ups on Monday. He plans to increase 5 push ups every following Monday. How many push ups will he be doing on the 3rd Monday since he started?

Which term of the A.P 10,8,6,… will be the first negative term?

An AP has first term -3 and a common difference -1. Find the 3rd term of the A.

Find the fifth term of an A.P whose first term is -1 and common difference is -3.

If for an A.P s_n= + 3n What is the n^th term?

The roots of quadratic equation x² – 9 = 0 are

If x = -2 is a root of equation x² – 4x + K = 0 then value of K is

Which of the following is not a quadratic equation?

The condition for equation ax² + bx + c = 0 to be quadratic is

Solve 9x²= 36

If x = 1 is a root of equation x² – Kx + 5 = 0 then value of K is

Solve 9x²= 36

The roots of quadratic equation x² – 9 = 0 are

The roots of quadratic equation ax² + bx + c = 0 is given by

If 4 is a root of the equation x² + 3x + k = 0 , then k is

If the perimeter and area of a circle are numerically equal, then the radius of the circle is

If the area of a circle is equal to sum of the areas of two circles of diameters 10 cm and 24 cm, then the diameter of the larger circle (in cm) is

The ratio of radii of two circles is in the ratio of 1:5. Calculate the ratio of their perimeters

The area of a sector of a circle of radius 5 cm is 5 cm2. The angle contained by the sector will be

The inner circumference of a circular track is 440m, and the track is 14m wide. The cost of levelling the track at 25 paise/m2 will be

In the given figure, PA and PB are tangents from P to a circle with centre O. If ∠AOB = 130°, then find ∠APB.

in figure , if ㄥAOB = 125^o, then ㄥCOD is equal to

The length of the tangent drawn from a point 8 cm away from the centre of a circle, of radius 6 cm, is :

Number of tangents, that can be drawn to a circle, parallel to a given chord is

A circle can pass through

A motor boat whose speed is 24km/h in still water takes 1 hour more to go 32 km upstream than no return downstream to the same spot. Find the speed of the steam.

If the pth terms of an AP is \(\frac{1}{q}\) and the qth term is \(\frac{1}{p}\), show that the sum of pq terms is \(\frac{1}{2}\) (pq + 1).

Check graphically, whether the following pair of linear equations is consistent. If yes, solve it graphically.
2x-y=0, x+y=0

Solve the following pair of linear equations graphically:
2x + 3y = 12 and x - y = 1.
Find the area of the region bounded by the two lines representing the above equations and Y-axis.

Using quadratic formula, solve the following quadratic equation for x: 
p²x²+(p²-q²)x-q²=0

Find the roots of the following quadratic equation (if they exist) by the method of completing square \(5x^{2} - 6x-2 = 0\) .

Find the numbers of terms in A.P.: 5, 12, 19, 28, ....., 159.

In the given figure, TBP and TCQ are tangents to the circle whose centre isO.Also \(\angle PBA=60^0\ and \ \angle ACQ=70^0.\)Determine \(\angle BAC\ and \ \angle BTC.\)

In the given figure, if AB = AC, then prove that BC = 2CE.

Check whether the following are quadratic equations: x² – 2x = (–2) (3 – x)

Check whether the following are quadratic equations: (2x – 1)(x – 3) = (x + 5)(x – 1)

Find the roots of the following quadratic equation, if they exist, by the method of completing the square: 2x²+x+4=0

Out of a group of children, \({7\over 2}\) times the square root of the number are creative, the two remaining ones are visionary. What is the total number of children ? How many persons in the group are creative?
Write one-one characteristics each of creativity and vision.

Solve for x : \(\sqrt { 6x+7 } -(2x-7)=0\)

If \(\triangle ABC\sim \triangle DFE\), \(\angle A={ 30 }^{ ° }\) , \(\angle C={ 50 }^{ ° }\) , AB = 5 cm, AC = 8 cm and DF = 75 cm, then find DE and \(\angle F\)

If the areas of two similar triangles are respectively 81 cm² and 49 cm²  Find the ratio of their corresponding medians.

If the lengths of the diagonals of rhombus are 16 cm and 12 cm. Then, find the length of the sides of the rhombus.

E and F are points on the sides PQ and PR respectively of a ΔPQR. For the following case, state whether EF || QR. PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm

A guy wire attached to a vertical pole of height 18 m is 24 m long and has a stake attached to the other end. How far from the base of the pole should the stake be driven so that the wire will be taut?

If sinA=\(\frac {12}{13}\) , what is the value of cos A?

If \(\sqrt { 3 } \sin { \theta } =\cos { \theta } \), find the value of \(\frac { \tan { \theta } (1+\cot { \theta } ) }{ \sin { \theta } +\cos { \theta } } .\)

Find the value of \((\sin { { 30 }^{ 0 } } +\cos { { 30 }^{ 0 } } )-(\sin { { 60 }^{ 0 } } +\cos { { 60 }^{ 0 } } ).\)

Evaluate \({ \left( \frac { \sin { { 25 }^{ 0 } } }{ \cos { { 65 }^{ 0 } } } \right) }^{ 2 }+{ \left( \frac { \tan { { 65 }^{ 0 } } }{ \cot { { 25 }^{ 0 } } } \right) }^{ 2 }-2\cos ^{ 2 }{ { 45 }^{ 0 } } .\)

Prove that \(\frac { \sec ^{ 2 }{ \theta } -\sin ^{ 2 }{ \theta } }{ \tan ^{ 2 }{ \theta } } =1+\cot ^{ 2 }{ \theta } -\cos ^{ 2 }{ \theta } \)

Find the mean of the following distribution by direct method.

Using assumed mean method find the mean of the following frequency distribution.

Find the mode of given data.

On sports day of a school, agewise participation of students is shown in the following distribution:

Find the mode of the data. Also, find missing frequencies when sum of frequencies is 181. 

The following table gives the literacy rate (in %) of 25 cities.

Find the median class and modal class. 

Five years ago, Jacob's age was seven times that of his son. After five years, the age of Jacob will be three times that of his son. Represent this situation algebraically and graphically.

Solve graphically the following pair of equations.
2x-y+3=0 and 3x-5y+1=0

Two straight paths are represented by the lines 7x-5y=3 and 21x-15y=5. Check whether the paths cross each other.

Two numbers are in the ratio 5:6. If 8 is subtracted from each of the numbers, the ratio becomes 4:5. Find the numbers.

In \(\triangle ABC,\quad \angle C=5\angle B=3(\angle A+\angle B)\),  find all angles of \(\triangle ABC\) .

Find the degree of the following polynomial
(i) \(7y^{ 5 }+6y^{ 2 }-1\)
(ii) \(\frac { y^{ 4 }+3y^{ 2 }+y }{ y } \)

If 2 is a zero of polynomial f(x)=ax²-3(a-1)x-1, then find the value of a.

Find the zeroes of quadratic polynomial y²+92y+1920.

If α and β are the zeroes of the polynomial 2y²+7y+5, then find the value of α+β+αβ.

Find the quadratic polynomial, whose sum of zeroes is 8 and their products is 12. Then, find the zeroes of the polynomial.

A tent is in the shape of a cylinder surmounted by a conical top.If the height and diameter of the cylindrical part are 2.1m and 4m respectively, and the slant height of the top is 2.8m, find the area of the canvas used for making the tent.Find the cost of the canvas of the tent at the rate of Rs.500 per m².Also find the volume air enclosed in the tent.

A juice seller serves his customers using a glass as shown in figure.The inner diameter of the cylindrical glass is 5cm, but the bottom of the glass has a hemispherical portion raised which reduces the capacity of the glass.If the height of the glass is 10cm, find the apparent capacity of the glass and its actual capacity.[\(\pi\)=3.14]

A solid right circular cone of diameter 14 cm and height 8 cm is melted to form a hollow sphere. If the external diameter of the sphere is 10 cm, find the internal diameter of the sphere.

Water flows out through a circular pipe whose internal radius is 1 cm, at the rate of 80 cm/second into an empty cylindrical tank, the radius of whose base is 40 cm. By how much will the level of water rise in the tank in half an hour?

Metal spheres, each of radius 2 cm are packed into a rectangular box of internal dimensions 16 cm x 8 cm x 8 cm. When 16 spheres are packed the box is filled with preservative liquid. Find the volume of the liquid. [Use \(\pi\) = 3.14]

Show that the square of an odd positive integer is of the form 8m + 1, where m is some whole number.

Use Euclid's division algorithm to the HCF of the following three numbers.
(i) 441, 567 and 693
(ii) 1620, 1725 and 255

If the LCM of 26 and 91 is 182. find their HCF.

If the HCF of 150 and 100 is 50, find the LCM of 150 and 100.

Without actually performing the long division, state whether \(\frac{543}{225}\) has a terminating decimal expansion or non-terminating recurring decimal expansion.

Find the zeroes of quadratic polynomial y²+92y+1920.

If zeroes α and β of a polynomial x²-7x+k are such that α-β=1, then find the value of k.

If α and β are the zeroes of the quadratic polynomial f(x)=3x²-5x-2, then evaluate α³+β³.

If α and β are zeroes of the quadratic polynomial p(x)=6x²+x-1, then find the value of \(\frac { \alpha }{ \beta } +\frac { \alpha }{ \alpha } +2\left( \frac { 1 }{ \alpha } +\frac { 1 }{ \beta } \right) +3\alpha \beta \)

On dividing polynomial p(x) by 3x+1, the quotient is 2x-3 and the remainder is -2. Find p(x).

Solve graphically, the pair of linear equations x-y=-1 and 2x+y-10=0. Also, find the vertices of the triangle formed by these lines and X-axis.

Solve graphically, the pair of linear equations 3x+y-11=0,x-y-1=0. Also, find the vertices of the triangle formed by these lines and Y-axis.

Two straight paths are represented by the lines 7x-5y=3 and 21x-15y=5. Check whether the paths cross each other.

The sum of the digits of a two-digit number is 8 and the difference between the number and that formed by reversing the digits is 18. Find the number.

The area of a rectangle gets reduced by 80 sq units, if its length is reduced by 5 units and the breadth is increased by 2 units. If we increase the length by 10 units and decrease the breadth by 5 units, then the area is increased by 50 q units. Find the length and the breadth of the rectangle.

If the lengths of the diagonals of rhombus are 16 cm and 12 cm. Then, find the length of the sides of the rhombus.

For going to city B from city A, there is a route via city C such that \(AC\bot CB\) , AC = 2x km and CB = 2 (x + 7) km. It is proposed to construct a 26 km highway, which directly connects the two cities A and B. Find how much distance will be saved in reaching city B from city A after the construction on the highway?

Find the third side of a right angled triangle whose hypotenuse is of length p cm, one side of length q cm and p - q = 1.

Find the value of unknown variables, if \(\triangle ABC\) and \(\triangle PQR\)are similar.
      

In \(\triangle PQR\) and \(\triangle MST\) , \(\angle P={ 55 }^{ ° }\), \(\angle Q={ 25 }^{ ° }\), \(\angle M={ 100 }^{ ° }\) and \(\angle S={ 25 }^{ ° }\). Is \(\triangle QPR\sim \triangle TSM\) ? Why?

If sin C=\(\frac {15}{17}\) , find the value of sin A.

In a \(\triangle ABC,\angle B={ 90 }^{ 0 }\) .  If AB=2 cm and AC=3 cm, find the value of sin A,

If 17 cosA = 8, find 15 cosecA - 8 sec B.

Find the value of
(i) \(\sin { \theta } \cos { \theta } \)   for \(\theta ={ 30 }^{ 0 }\)
(ii) \(3\tan ^{ 2 }{ { 45 }^{ 0 } } +2\sin { { 45 }^{ 0 } } \cos { { 45 }^{ 0 } } \)

If \(\sin { \theta } -\cos { \theta } =0,(0\le \theta \le { 90 }^{ 0 })\) find the value of \(\theta \) .

Calculate the mean of the scores of 20 students in a Mathematics test.

Find the mode of given data.

Compute the median marks for the following data.

If the coordinates of the point of intersection of less than ogive and more than ogive is (12.5,20) then find the value of median.

A class teacher has the following absentee record of 40 students of a class for the whole term. Find the mean number of days a student was absent.

A bucket is raised from a well by means of a rope which is wound round a wheel of diameter 77 cm. If bucket ascends in 1 minute 28 seconds with a uniform speed of 1.1 m/s, then calculate the number of complete revolutions the wheel makes in raising the bucket.

In fig., PQRS is a square lawn with side PQ = 42 metres. Two circular flower beds are there on the sides PS and QR with centre at O, the intersection of its diagonals. Find the total area of the two flowers beds (shaped parts).

The long and short hands of a clock are 6 cm and 3 cm respectively. Find the sum of distance travelled by their tips in a day.

A bus has wheels which are 112 cm in diameter. How many complete revolutions does each wheel make in 20 minutes, when the bus is travelling at a speed of 66 km/h?

A road which is 7 m wide surrounds a circular track whose circumference is 352 m. Find the area of the road.

Find the points of trisection of the line segment determined by (7,5) and (16,-1).

Find the coordinates of the points of trisection (i.e., points dividing in three equal parts) of the line segment joining the points A(2, -2) and B(-7, 4)

Find the coordinates of the points P,Q and R which divide the line segment joining A(5,4) and B(11,6) into four equal parts.

In what ratio does the point (-4,6) divide the line segment joining the point A(-6,10) and B(3,-8)?

Find the value of p for which the points (3,6), (7,p) and (-5,2) are collinear.

A coin is tossed. If it results in a head a coin is tossed, otherwise a die is thrown. Describe the following events:
(i) A = getting atleast one head  
(ii) B = getting an even number
(iii) C = getting a tail  
(iv) D = getting a tail and an odd number

20 cards numbered 1, 2, 3, ...., 20 are put in a box and mixed thoroughly. Shashi draws a cards from the box. Find the probability that the number on the card is

(i) odd    (ii) even  (iii) a prime

(iv) divisible by 3    (v) divisible by 3 and 2 both.

At a fete cards bearing numbers 1 to 500, one on each card, are put in a box. Each player selects one card at random and that card is not replaced. If the selected card bears a number which is a perfect square of an even number the player wins prize.

(i) What is the probability that the first player wins a prize?

(ii) The second player wins prize, if the first has not won.

A letter is at drawn at random from the word 'MATHEMATICS'.Find the probability of drawing each of the different letters in the given word.

A box of 24 solar cells contain 8 defective cells.One cell is drawn at random.What is the probability that the cell is not defective and it is not replaced and a second cell is selected at random from the rest, what is the probability that second cell is defective?

A bird is sitting on the top of a tree, which is 80 m high. The angle of elevation of the bird, from a point on the ground is 45^o . The bird flies away from the point of observation horizontally and remains at a constant height. After 2 seconds, the angle of elevation of the bird from the point of observation becomes 30^o . Find the speed of flying of the bird.

At the foot of a mountain, the elevation of its summit is 45^o . After ascending 1000 m towards the mountain up a slope of 30^o inclination, the elevation is found to be 60^o . Find the height of the mountain.

A path separates two walls. A ladder leaning against one wall rests at a point on the path. It reaches a height of 90 m on the wall and makes an angle of 60^o with the ground. If while resting at the same point on the path, it were made to lean against the other wall, it would have made an angle of 45^o with the ground. Find the height it would have reached on the second wall.

Two stations due south of a leaning tower which leans towards north are at distances a and b from its foot. If \(\alpha\) and \(\beta\) be the elevation  of the top of the tower from these stations, prove that its inclination \(\theta\) to the horizontal is given by \(cot\theta =\frac { b\quad cot\alpha -a\quad cot\beta }{ b-a } \).

From an aeroplane vertically above a straight horizontal plane, the angles of depression of two consecutive kilometre stones on the opposite sided of the aeroplane are found to be \(\alpha\) and \(\beta\), show that the height of the aeroplane is \(\frac { tan\alpha tan\beta }{ tan\alpha +tan\beta } \).

Draw an isosceles triangle ABC in which AB = AC = 6cm and BC = 5cm. Construct a triangle PQR similar to \(\triangle ABC\) in which PQ = 8cm.  Also justify the construction.

Draw a \(\Delta ABC\) with sides BC = 6cm, AB = 5cm and \(\angle ABC=60°\), Then, construct a triangle whose sides are 3/4 of the corresponding sides of the \(\Delta ABC\)

Draw two equal circles with centres A and B and distance between A and B is 6 cm. Construct a pair of tangents from centres A and B to each other. Measure the lengths of the tangents. What type of 31. figure is enclosed by these four tangents?

Draw a circle of radius 3.5 cm. Take a point T out side the circle at a distance of 7 cm from the centre and construct a pair of tangents from this point T to the circle and justify your construction.

A number when divided by 53 gives 34 as quotient and 21 as remainder. Find the number.

The product of two consecutive positive integers is divisible by 2. Is this statement true or false? Give reason.

Use the Euclid's division algorithm to find the HCF of
(i) 650 and 1170
(ii) 870 and 225

If two positive integers p and q can be expressed as p = ab² and q = a³b; where a,b being prime numbers, find the LCM (p,q)

Find the HCF and LCM of 60, 84 and 108 by using the prime factorisation method.

A wooden article was made by scooping out a hemisphere from each end of a solid cylinder.If the height of the cylinder is 20cm and radius of the base is 3.5cm, find the total surface area of the article.

A juice seller serves his customers using a glass as shown in figure.The inner diameter of the cylindrical glass is 5cm, but the bottom of the glass has a hemispherical portion raised which reduces the capacity of the glass.If the height of the glass is 10cm, find the apparent capacity of the glass and its actual capacity.[\(\pi\)=3.14]

A solid right circular cone of diameter 14 cm and height 8 cm is melted to form a hollow sphere. If the external diameter of the sphere is 10 cm, find the internal diameter of the sphere.

A tent consists of a frustum of cone, surmounted by a cone. If the diameter of the upper and lower circular ends of the frustum are 14 m and 26 m respectively, the height of the frustum is 8 m and the slant height of the surmounted conical portion is 12 m, find the area of canvas required to make the tent. (Assume that the radii of the upper circular end of the frustum and the base of surmounted conical portion are equal.)

A lead pencil consists of a cylinder of wood with solid cylinder of graphite filled into it. The diameter of the pencil is 7 mm; the diameter of the graphite is 1 mm and the length of the pencil is 10 cm. Calculate the weight of the whole pencil if the specific gravity of the wood is 0.7 g/cm³ and that of the graphite is 2.1 g/cm³ .

In figure, the sides AB, BC and CA of triangle ABC touch a circle with centre O and radius r at P, Q and R respectively.Prove that
(i) AB + CQ = AC + BQ
(ii)area (\(\Delta\)ABC) = \(1\over2\) (perimeter of \(\Delta\)ABC) x r

Two circles touch each other externally at C.AB and CD are two common tangents.If D lies on AB such that CD=6cm, then find AB.