Class X - Maths - Case Based Questions

Quadratic Equation

I. Raj and Ajay are very close friends. Both the families decide to go to Ranikhet by their own cars. Raj’s car travels at a speed of x km/h while Ajay’s car travels 5 km/h faster than Raj’s car. Raj took 4 hours more than Ajay to complete the journey of 400 km.

1. What will be the distance covered by Ajay’s car in two hours?

(a) 2(x + 5) km (b) (x – 5) km (c) 2(x + 10) km (d) (2x + 5) km

2. Which of the following quadratic equations describes the speed of Raj’s car?

(a) x² – 5x – 500 = 0 (b) x² + 4x – 400 = 0 (c) x² + 5x – 500 = 0 (d) x² – 4x + 400 = 0

3. The roots of the quadratic equation which describe the speed of Raj’s car are

(a) 15, – 20 (b) 20, – 15 (c) 20, – 25 (d) 25, – 25

4. Which of the following quadratic equations has 2 as a root?

(a) x² – 4x + 5 = 0 (b) x² + 3x – 12 = 0 (c) 2x² – 7x + 6 = 0 (d) 3x² – 6x – 2 = 0

5. The positive root of √(3x² + 6) = 9 is

(a) 5 (b) –5 (c) 3 (d) –3

II. The speed of a motor boat is 20 km/hr. For covering the distance of 15 km the boat took 1 hour more for upstream than downstream.

1. Let speed of the stream be x km/hr, then speed of the motorboat in upstream will be

(a) 20 km/hr (b) (20 + x) km/hr (c) (20 – x) km/hr (d) 2 km/hr

2. What is the relation between speed, distance and time?

(a) speed = (distance)/time (b) distance = (speed)/time (c) time = speed × distance (d) speed = distance × time

3. Which is the correct quadratic equation for the speed of the stream?

(a) x² + 30x − 200 = 0 (b) x² + 20x − 400 = 0 (c) x² + 30x − 400 = 0 (d) x² – 20x − 400 = 0

4. What is the speed of stream?

(a) 20 km/hour (b) 10 km/hour (c) 15 km/hour (d) 25 km/hour

5. How much time boat took in downstream?

(a) 90 minutes (b) 15 minutes (c) 30 minutes (d) 45 minutes

III. Water Distribution System: Delhi Jal Board (DJB) is the main body of the Delhi Government which supplies drinking water in the National Capital Territory of Delhi. Distribution system is well knit and properly planned. Maintenance of underground pipe and hose system is also performed at regular interval of time. Many rivers and canals are inter-connected in order to ensure un-interrupted water supply. It has been meeting the needs of potable water for more than 16 million people. It ensures availability of 50 gallons per capita per day of pure and filtered water with the help of efficient network of water treatment plants and pumping stations. In our locality, DJB constructed two big reservoir labelled as Reservoir–A and Reservoir–B.

Reservoir–A: In order to fill it, department uses two pipes of different diameter.

Reservoir–B: Department uses two taps to store water in this reservoir.

Refer to Reservoir-A

1. Two pipes running together can fill the reservoir in 100/9 minutes. If one pipe takes 5 minutes more than the other to fill the reservoir, the time in which each pipe alone would fill the reservoir is

(a) 10 min, 12 min (b) 25 min, 20 min (c) 15 min, 18 min (d) 22 min, 28 min

2. Two pipes running together can fill a reservoir in 6 minutes. If one pipe takes 5 minutes more than the other to fill the reservoir, the time in which each pipe would fill the reservoir separately is

(a) 8 min, 6 min (b) 10 min, 15 min (c) 12 min, 16 min (d) 16 min, 18 min

Refer to Reservoir-B

3. Two water taps together can fill a reservoir in 75/8 hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the reservoir separately. The time in which each tap can separately fill the reservoir will be

(a) 15 hrs, 25 hrs (b) 20 hrs, 22 hrs (c) 14 hrs, 18 hrs (d) 18 hrs, 16 hrs

4. Two taps running together can fill the reservoir in 40/13 minutes. If one tap takes 3 minutes more than the other to fill it, how many minutes each tap would take to fill the reservoir

(a) 12 min, 15 min (b) 6 min, 9 min (c) 18 min, 14 min (d) 5 min, 8 min

5. If two tapes function simultaneously, reservoir will be filled in 12 hours. One tap fills the reservoir 10 hours faster than the other. The time that the second tap takes to fill the reservoir is given by

(a) 25 hrs (b) 28 hrs (c) 30 hrs (d) 32 hrs

IV. A Hill Station: In the last summer, I enjoyed a tour to a hill station at Shimla. I was accompanied by my five friends and enjoyed the natural beauties of mountains, rivers, streams, forests etc. The beginning of the tour was the most adventurous itself! How amazingly my group win the bet! Actually, the story is that my two friends along with me preferred train to go to Shimla, but other three were forcing for a car or a bus. At last the consensus was reached and we were divided ourselves in two groups of 3 each and started for Shimla at the same time. It was decided that the group who reach the destination first, would be declared as the winner, and runner up the group have to bear the expanses of the tour. I named my group, ‘Group A’ while the second group was named as ‘Group B’. Luckily we reached Shimla 1 hour before the Group-B and enjoyed the trip for absolutely FREE!! How thrilling it was the tour!

Refer to Group-A

1. An express train takes 1 hour less than a passenger train to travel 132 km between Delhi and Shimla (without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is 11 km/hr more than that of the passenger train, the average speeds of the two trains will be

(a) 33 km/h, 44 km/hr (b) 40 km/h, 45 km/h (c) 30 km/h, 38 km/h (d) 42 km/h, 62 km/h

2. An express train makes a run of 240 km at a certain speed. Another train whose speed is 12 km/hr less takes an hour longer to make the same trip. The speed of the express train will be

(a) 60 km/h (b) 50 km/h (c) 65 km/h (d) 48 km/h

3. A journey of 192 km from Delhi to Shimla takes 2 hours less by a super-fast train than that by an ordinary passenger train. If the average speed of the slower train is 16 km/hr less than that of the faster train, average speed of super fast train is

(a) 50 km/h (b) 48 km/h (c) 55 km/h (d) 60 km/h

Refer to Group-B

4. A deluxe bus takes 3 hours less than a ordinary bus for a journey of 600 km. If the speed of the ordinary bus is 10 km/hr less than that of the deluxe bus, the speeds of the two buses will be

(a) 35 km/h, 42 km/h (b) 42 km/h, 52 km/h (c) 40 km/h, 50 km/h (d) 30 km/h, 58 km/h

5. A bus travels a distance of 300 km at a uniform speed. If the speed of the bus is increased by 5 km an hour, the journey would have taken two hours less. The original speed of the bus will be

(a) 20 km/h (b) 15 km/h (c) 22 km/h (d) 25 km/h

Arithmetic Progression

Answer

I. Your friend Veer wants to participate in a 200 m race. Presently, he can run 200 m in 51 seconds and during each day practice it takes him 2 seconds less. He wants to do in 31 seconds.

1. Which of the following terms are in AP for the given situation?

(a) 51, 53, 55, … (b) 51, 49, 47, … (c) –51, –53, –55, … (d) 51, 55, 59, …

2. What is the minimum number of days he needs to practice till his goal is achieved?

(a) 10 (b) 12 (c) 11 (d) 9

3. Which of the following term is not in the AP of the above given situation?

(a) 41 (b) 30 (c) 37 (d) 39

4. If n^th term of an AP is given by an = 2n + 3 then common difference of an AP is

(a) 2 (b) 3 (c) 5 (d) 1

5. The value of x, for which 2x, x + 10, 3x + 2 are three consecutive terms of an AP is

(a) 6 (b) – 6 (c) 18 (d) –18

II. India is competitive manufacturing location due to the low cost of manpower and strong technical and engineering capabilities contributing to higher quality production runs. The production of TV sets in a factory increases uniformly by a fixed number every year. It produced 16000 sets in 6th year and 22600 in 9th year

1. The production during first year is

(a) 3000 TV sets (b) 5000 TV sets (c) 7000 TV sets (d) 10000 TV sets

2. The production during 8th year is

(a) 10500 (b) 11900 (c) 12500 (d) 20400

3. The production during first 3 years is

(a) 12800 (b) 19300 (c) 21600 (d) 25200

4. In which year, the production is 29,200?

(a) 10th year (b) 12th year (c) 15th year (d) 18th year

5. The difference of the production during 7th year and 4th year is

(a) 6600 (b) 6800 (c) 5400 (d) 7200

III. Pollution—A Major Problem: One of the major serious problems that the world is facing today is the environmental pollution. Common types of pollution include light, noise, water and air pollution.

In a school, student’s thoughts of planting trees in and around the school to reduce noise pollution and air pollution.

Condition I: It was decided that the number of trees that each section of each class will plant be the same as the class in which they are studying, e.g. a section of class I will plant 1 tree a section of class II will plant 2 trees and so on a section of class XII will plant 12 trees.

Condition II: It was decided that the number of trees that each section of each class will plant be the double of the class in which they are studying, e.g. a section of class I will plant 2 trees, a section of class II will plant 4 trees and so on a section of class XII will plant 24 trees.

Refer to Condition I

1. The AP formed by sequence i.e. number of plants by students is

(a) 0, 1, 2, 3, …, 12 (b) 1, 2, 3, 4, …, 12 (c) 0, 1, 2, 3, …, 15 (d) 1, 2, 3, 4, …, 15

2. If there are two sections of each class, how many trees will be planted by the students?

(a) 126 (b) 152 (c) 156 (d) 184

3. If there are three sections of each class, how many trees will be planted by the students?

(a) 234 (b) 260 (c) 310 (d) 326

Refer to Condition II

4. If there are two sections of each class, how many trees will be planted by the students?

(a) 422 (b) 312 (c) 360 (d) 540

5. If there are three sections of each class, how many trees will be planted by the students?

(a) 468 (b) 590 (c) 710 (d) 620

IV. Your elder brother wants to buy a car and plans to take loan from a bank for his car. He repays his total loan of ` 1,18,000 by paying every month starting with the first instalment of ` 1000. If he increases the instalment by ` 100 every month, answer the following:

1. The amount paid by him in 30th instalment is

(a) ₹ 3900 (b) ₹ 3500 (c) ₹ 3700 (d) ₹ 3600

2. The total amount paid by him upto 30 instalments is

(a) ₹ 37000 (b) ₹ 73500 (c) ₹ 75300 (d) ₹ 75000

3. What amount does he still have to pay after 30th instalment?

(a) ₹ 45500 (b) ₹ 49000 (c) ₹ 44500 (d) ₹ 54000

4. If total instalments are 40, then amount paid in the last instalment is

(a) ₹ 4900 (b) ₹ 3900 (c) ₹ 5900 (d) ₹ 9400

5. The ratio of the 1st instalment to the last instalment is

(a) 1 : 49 (b) 10 : 49 (c) 10 : 39 (d) 39 : 10

Circles

Answers

I. A Ferris Wheel (or a big wheel in the United Kingdom) is an amusement ride consisting of a rotating upright wheel with multiple passenger-carrying components (commonly referred to as passenger cars, cabins, tubs, capsules, gondolas, or pods) attached to the rim in such a way that as the wheel turns, they are kept upright, usually by gravity. After taking a ride in Ferris Wheel, Aarti came out from the crowd and was observing her friends who were enjoying the ride. She was curious about the different angles and measures that the wheel will form. She forms the figure as given below.

1. In the given figure, ∠ROQ is

(a) 60° (b) 100° (c) 150° (d) 90°

2. ∠RQP equals

(a) 75° (b) 60° (c) 30° (d) 90°

3. ∠RSQ equals

(a) 60° (b) 75° (c) 100° (d) 30°

4. ∠ORP equals

(a) 90° (b) 75° (c) 100° (d) 60°

5. If ∠P = 60° and OQ = OR = 5 m, then the length of PQ is

(a) 5 m (b) 12 m (c) 5√3 m (d) None of these

II. Varun has been selected by his School to design logo for Sports Day T-shirts for students and staff . The logo design is given as in the figure and he is working on the fonts and different colours according to the theme. In the given figure, a circle with centre O is inscribed in a ΔABC, such that it touches the sides AB, BC and CA at points D, E and F respectively. The lengths of sides AB, BC and CA are 12 cm, 8 cm and 10 cm respectively.

1. The length of AD is

(a) 7 cm (b) 8 cm (c) 5 cm (d) 9 cm

2. The length of BE is

(a) 8 cm (b) 5 cm (c) 2 cm (d) 9 cm

3. The length of CF is

(a) 20 cm (b) 5 cm (c) 2 cm (d) 3 cm

4. If the radius of the circle is 4 cm, then the area of triangle OAB is

(a) 20 sq cm (b) 36 sq cm (c) 24 sq cm (d) 48 sq cm

5. The area of triangle ABC is

(a) 50 sq cm (b) 60 sq cm (c) 100 sq cm (d) 90 sq cm

Constructions

Answers

I. The management of a school decided to arouse interest of their students in Mathematics. So they want to construct some geometrical shapes in one corner of the school premises. They showed a rough sketch of a right triangular structure on a plane sheet of paper with sides AB = 6 m, BC = 8 m and ∠B = 90°. The diagram shows a perpendicular from the vertex B to the front side AC. They want to build a circular wall through B, C and D but they had certain problems in doing so. So they called on some students of class X to solve this problem. They made some suggestions.

1. To find centre of the circle, the students made some suggestions which are as follows:

(a) Draw perpendicular BD on AC

(b) Draw perpendicular bisectors of BC and CD.

(d) All of the above

2. Referring to the above, what will be the length of AD?

(a) 3.6 m (b) 3.8 m (c) 4.8 m (d) 5.6 m

3. Referring to the above, what is the length of perpendicular drawn on side AC from vertex B?

(a) 2.6 m (b) 3.0 m (c) 4.8 m (d) 4.0 m

4. Referring to the above, the length of tangent AE is

(a) 10 m (b) 8 m (c) 12 m (d) 6 m

5. Referring to the above, sum of angles ∠BAE and ∠BOE is

(a) 120° (b) 180° (c) 90° (d) 60°

II. The construction of a road is in progress. A road already exists through a forest that goes over a circular lake. The engineer wants to build another road through the forest that connects this road, but does not go through the lake.

As it turns out, the road the engineer will be building and the road it will connect to both represent characteristics of a circle that have their own name. The road/bridge that already exists is called a secant of the circular lake, and the road the engineer is going to build is called the tangent of the circular lake.

1. Refer to the above, if PT = 12 km and PA = 9 km, then the length of existing bridge is

(a) 7 km (b) 9 km (c) 12 km (d) 16 km

2. Refer to the above if the length of existing bridge is 5 km and the length of the existing road outside the lake is 4 km, then the length of the road under construction is

(a) 4 km (b) 6 km (c) 10 km (d) 14 km

3. Refer to the question (2) if the road under construction, PT is 6 km and it is inclined at an angle of 30° to the line joining the centre, the radius of the lake is

(a) 3√3 km (b) 4√3 km (c) 2√3 km (d) 5√3 km

4. Refer to the question (3) above, the circumference of the lake is

(a) 2√3 π km (b) 3√3 π km (c) 4√3 π km (d) 5√3 π km

5. Refer to the question (3) above, the area of the lake is

(a) 12π km² (b) 16π km² (c) 18π km² (d) 9π km²

Some Applications of Trigonometry

Answers

I. Application of Trigonometry—Height of Tree/Tower: Mr. Suresh is an electrician. He receives a call regarding a fault on a pole from three different colonies A, B and C. He reaches one-by-one to each colony to repair that fault. He needs to reach a point 1.3 m below the top of each pole to undertake the repair work. Observe the following diagrams.

Refer to Diagram A

1. What should be the length of ladder DQ that enable him to reach the required position if the height of the pole is 4 m?

(a) 5√3/7 m (b) 9√3/5 m (c) 7√2/5 m (d) 4√3/5 m

2. What is the distance of the point where the ladder is placed on the ground if the height of pole is 4 m?

(a) 2.5 m (b) 3.8 m (c) 1.56 m (d) 5.3 m

Refer to Diagram B

3. Given that the length of ladder is 4√2 m. What is height of pole?

(a) 4.5 m (b) 4√5 m (c) 5√5 m (d) 5.3 m

4. The distance of the point where the ladder lies on the ground is

(a) 3√5 m (b) 4√2 m (c) 4 m (d) 4√7 m

Refer to Diagram C

5. The angle of elevation of reaching point of ladder at pole, i.e., H, if the height of the pole is 8.3 m and the distance GF is 7√3 m, is

(a) 30° (b) 60° (c) 45° (d) None of these

II. A group of students of class X visited India Gate on an educational trip. The teacher and students had interest in history as well. The teacher narrated that India Gate, official name Delhi Memorial, originally called All-India War Memorial, monumental sandstone arch in New Delhi, dedicated to the troops of British India who died in wars fought between 1914 and 1919.The teacher also said that India Gate, which is located at the eastern end of the Rajpath (formerly called the Kingsway), is about 138 feet (42 metres) in height.

1. What is the angle of elevation if they are standing at a distance of 42 m away from the monument?

(a) 30° (b) 45° (c) 60° (d) 0

2. They want to see the tower at an angle of 60°. The distance where they should stand will be

(a) 25.24 m (b) 20.12 m (c) 42 m (d) 24.25 m

3. If the altitude of the Sun is at 60°, then the height of the vertical tower that will cast a shadow of length 20 m is

(a) 20√3 m (b) 20/√3 m (c) 15/√3 m (d) 15√3 m

4. The ratio of the length of a rod and its shadow is 1:1. The angle of elevation of the Sun is

(a) 30° (b) 45° (c) 60° (d) 90°

5. The angle formed by the line of sight with the horizontal when the object viewed is below the horizontal level is

(a) corresponding angle (b) angle of elevation (c) angle of depression (d) complete angle

III. A satellite flying at a height h is watching the top of the two tallest mountains in Uttarakhand and Karnataka, they are being Nanda Devi (height 7,816 m) and Mullayanagiri (height 1,930 m). The angles of depression from the satellite, to the top of Nanda Devi and Mullayanagiri are 30° and 60° respectively. If the distance between the peaks of two mountains is 1937 km, and the satellite is vertically above the mid-point of the distance between the two mountains.

1. The distance of the satellite from the top of Nanda Devi is

2. The distance of the satellite from the top of Mullayanagiri is

3. The distance of the satellite from the ground is

4. What is the angle of elevation if a man is standing at a distance of 7816 m away from Nanda Devi?

(a) 30° (b) 45° (c) 60° (d) 0°

5. If a mile stone very far away from, makes 45° to the top of Mullayangiri mountain. So, find the distance of this mile stone from the mountain.

Surface Areas and Volumes

Answers

I. Adventure camps are the perfect place for the children to practise decision making for themselves without parents and teachers guiding them every move. Some students of a school reached for adventure at Sakleshpur. At the camp, the waiters served some students with a welcome drink in a cylindrical glass while some students in a hemispherical cup whose dimensions are shown below. After that they went for a jungle trek. The jungle trek was enjoyable but tiring. As dusk fell, it was time to take shelter. Each group of four students was given a canvas of area 551 m2 . Each group had to make a conical tent to accommodate all the four students. Assuming that all the stitching and wasting incurred while cutting, would amount to 1 m2 , the students put the tents. The radius of the tent is 7 m.

1. The volume of cylindrical cup is

(a) 295.75 cm³ (b) 7415.5 cm³ (c) 384.88 cm³ (d) 404.25 cm³

2. The volume of hemispherical cup is

(a) 179.67 cm³ (b) 89.83 cm³ (c) 172.25 cm³ (d) 210.60 cm³

3. Which container had more juice and by how much?

(a) Hemispherical cup, 195 cm³ (b) Cylindrical glass, 207 cm³ (c) Hemispherical cup, 280.85 cm³ (d) Cylindrical glass, 314.42 cm³

4. The height of the conical tent prepared to accommodate four students is

(a) 18 m (b) 10 m (c) 24 m (d) 14 m

5. How much space on the ground is occupied by each student in the conical tent

(a) 54 m² (b) 38.5 m² (c) 86 m² (d) 24 m²

II. The Great Stupa at Sanchi is one of the oldest stone structures in India, and an important monument of Indian Architecture. It was originally commissioned by the emperor Ashoka in the 3rd century BCE. Its nucleus is a simple hemispherical brick structure built over the relics of the Buddha. It is a perfect example of combination of solid figures. A big hemispherical dome with a cuboidal structure mounted on it. (Take = 22/7)

1. The volume of the hemispherical dome if the height of the dome is 21 m, is

(a) 19404 cu. m (b) 2000 cu. m (c) 15000 cu. m (d) 19000 cu. M

2. The formula to find the volume of sphere is

(a) 2/3 πr³ (b) 4/3 πr³ (c) 4 πr² (d) 2 πr²

3. The cloth required to cover the hemispherical dome if the radius of its base is 14 m is

(a) 1222 sq. m (b) 1232 sq. m (c) 1200 sq. m (d) 1400 sq. m

4. The total surface area of the combined figure, i.e. hemispherical dome with radius 14 m and cuboidal shaped top with dimensions 8 m × 6 m × 4 m is

(a) 1200 sq. m (b) 1232 sq. m (c) 1392 sq.m (d) 1932 sq. m

5. The volume of the cuboidal shaped top with dimensions mentioned in question 4, is

(a) 182.45 m³ (b) 282.45 m³ (c) 292 m³ (d) 192 m³

Statistics

Answers

Student-Teacher Ratio: Student-teacher ratio expresses the relationship between the number of students enrolled in a school and the number of teachers in that school. It is important for a number of reasons. For example, it can be an indicator of the amount of individual attention any child is likely to receive, keeping in mind that not all class size are going to be the same. The following distribution gives the state-wise student-teacher ratio in higher secondary schools of India (28 states and 7 UTs only).

1. In order to find the mean by direct method, we use the formula

2. The mean of the above data is

(a) 29.2 (b) 30.5 (c) 38.3 (d) 40.1

3. The formula for assumed mean method to find the mean is

4. The sum of class marks of 25-30 and 45-50 is

(a) 62 (b) 70 (c) 75 (d) 85

5. The sum of the upper and lower limits of modal class is

(a) 55 (b) 65 (c) 85 (d) 75

II. Females’ Literacy: The education of women helps to remove the social stigma that surrounds it. It is the key to eliminating social evils such as female infanticide, dowry, child marriage, harassment, etc. This will not just help the women of today but of the future generations who can live in a world where gender equality exists which ultimately raises the literacy rate. The following distribution shows the number of literate females in the age group 0 to 60 years of a particular area.

1. The class marks of class 40-50 is

(a) 70 (b) 90 (c) 10 (d) 45

2. The number of literate females whose ages are between 20 years and 50 years is

3. The modal class of the above distribution is

(a) 0-10 (b) 10-20 (c) 20-30 (d) 30-40

4. The number of literate females whose ages are less than 40 years is

(a) 1450 (b) 2350 (c) 3100 (d) 3700

5. The upper limit of modal class is

(a) 10 (b) 20 (c) 30 (d) 60

III. 100 m Race

A stopwatch was used to find the time that it took a group of students to run 100 m

1. The estimated mean time taken by a student to finish the race is

(a) 54 (b) 63 (c) 43 (d) 50

2. What will be the upper limit of the modal class?

(a) 20 (b) 40 (c) 60 (d) 80

3. The construction of cumulative frequency table is useful in determining the

(a) mean (b) median (c) mode (d) All of the above

4. The sum of lower limits of median class and modal class is

(a) 60 (b) 100 (c) 80 (d) 140

5. How many students finished the race within 1 minute?

(a) 18 (b) 37 (c) 31 (d) 8

III. COVID-19 Pandemic: The COVID-19 pandemic, also known as coronavirus pandemic, is an ongoing pandemic of coronavirus disease caused by the transmission of severe acute respiratory syndrome coronavirus 2 among humans. The following tables shows the age distribution of case admitted during a day in two different hospitals.

Refer to Table 1

1. The average age for which maximum cases occurred is

(a) 32.24 years (b) 34.36 years (c) 35.91 years (d) 42.24 years

2. The upper limit of modal class is

(a) 15 (b) 25 (c) 35 (d) 45

3. The mean of the given data is

(a) 26.2 (b) 32.4 (c) 33.5 (d) 35.4

Refer to Table 2

4. The mode of the given data is

(a) 41.4 (b) 48.2 (c) 55.3 (d) 64.6

5. The median of the given data is

(a) 32.7 (b) 40.2 (c) 42.3 (d) 48.6

Class X – Maths – Case Based Questions