Integration for JEE Mains

 Integration for JEE Mains

Geometry Problem for Secondary Students

ABC is an isosceles triangle whose ∠C is right angle. If D is any point on AB, then prove that, ABC is an isosceles triangle whose ∠C is right angle. If D is any point on AB, then let us prove that, AD² + DB² = 2CD². ​

Integration for JEE Mains

 Integration for JEE Mains

Height and Distance Worksheet - CBSE and ICSE Class 10

**Height and Distance Worksheet**

**Instructions:**

1. Solve all the problems.

2. Show your work and calculations where necessary.

3. Circle or underline your final answers.

**Easy Problems:**

1. A flagpole stands vertically on the ground. If the angle of elevation to the top of the flagpole is 45 degrees and you are standing 20 meters away from the flagpole, find the height of the flagpole.

2. From the top of a building 30 meters high, the angle of depression of an object on the ground is 60 degrees. Find the distance of the object from the base of the building.

3. If the length of a shadow of a 10-meter pole is 8 meters, find the angle of elevation of the sun.

4. A ladder leans against a wall. The angle of elevation of the ladder is 60 degrees, and the ladder reaches a height of 10 meters on the wall. Find the length of the ladder.

5. Two ships are sailing towards each other. They spot each other when they are 1,000 meters apart, and the angle of elevation is 30 degrees from one ship to the other. Find the altitude of each ship.

**Medium Problems:**

6. A 15-meter ladder is leaning against a wall. If the ladder makes a 45-degree angle with the ground, how far is the bottom of the ladder from the wall?

7. A person standing 50 meters away from a tree finds that the angle of elevation to the top of the tree is 30 degrees. Find the height of the tree.

8. An observer on a cliff sees a boat in the sea below at an angle of depression of 45 degrees. If the cliff is 60 meters high, find the distance between the boat and the observer.

9. A 12-meter ladder is placed against a wall. If the top of the ladder slides down the wall at a rate of 2 meters per second, how fast is the bottom of the ladder moving away from the wall when the top of the ladder is 9 meters above the ground?

10. Two buildings are 100 meters apart. From the top of one building, the angle of elevation to the top of the other is 30 degrees. If the height of the first building is 40 meters, find the height of the second building.

**Difficult Problems:**

11. From a point 15 meters above the water surface, the angle of elevation of a cliff is 60 degrees. If the cliff is 15 meters from the shore, find the depth of the water.

12. An airplane is flying at an altitude of 5,000 meters. An observer on the ground sees the airplane at an angle of elevation of 30 degrees. Find the horizontal distance between the observer and the airplane.

13. Two towers are 100 meters apart. From the top of the first tower, the angle of elevation to the top of the second tower is 45 degrees. If the height of the first tower is 60 meters, find the height of the second tower.

14. A man stands on the top of a hill and sees a car coming towards him. If the angle of depression of the car is 30 degrees when it is 300 meters away, find the height of the hill.

15. A 20-meter ladder leans against a wall. If the ladder makes an angle of 75 degrees with the ground, find the height at which the ladder touches the wall.

**Answers:**

**Easy Problems:**

1. 20 meters

2. 30 meters

3. 60 degrees

4. 20 meters

5. Altitude of each ship is 500 meters.

**Medium Problems:**

6. 15 meters

7. 25 meters

8. 60 meters

9. 2 m/s

10. 20 meters

**Difficult Problems:**

11. 15 meters

12. 10,000 meters (10 km)

13. 60 meters

14. 150 meters

15. 5√3 meters

Worksheet on Direct and Inverse Variation ICSE Class 8

 Certainly! Here's a worksheet on the topic of Direct and Inverse Variation for 8th-grade students following the ICSE board curriculum. The problems are categorized into easy, medium, and difficult levels.

**Direct and Inverse Variation Worksheet**

**Instructions:**

1. Solve all the problems.

2. Show your work and calculations where necessary.

3. Circle or underline your final answers.

**Easy Problems:**

1. If y varies directly with x, and y = 12 when x = 4, find the constant of variation (k).

2. If y varies inversely with x, and y = 10 when x = 5, find the constant of variation (k).

3. If y varies directly with x, and y = 25 when x = 5, find y when x = 8.

4. If y varies inversely with x, and y = 6 when x = 9, find y when x = 12.

5. If y varies directly with x, and y = 15 when x = 3, find x when y = 30.

**Medium Problems:**

6. The cost (C) of printing flyers is directly proportional to the number of flyers (n). If it costs $40 to print 200 flyers, find the cost to print 600 flyers.

7. The time (t) it takes to complete a task is inversely proportional to the number of workers (w). If it takes 8 hours for 6 workers to complete the task, how long will it take for 12 workers to finish the same task?

8. A car travels at a constant speed. If it covers 60 miles in 2 hours, how long will it take to cover 150 miles at the same speed?

9. The force (F) of attraction between two objects is directly proportional to the product of their masses (m1 and m2) and inversely proportional to the square of the distance (d) between them. If F = 12 when m1 = 4, m2 = 6, and d = 3, find F when m1 = 8, m2 = 9, and d = 5.

10. The pressure (P) in a closed container is inversely proportional to its volume (V). If P = 48 kPa when V = 4 liters, find the pressure when V = 10 liters.

**Difficult Problems:**

11. A car's fuel efficiency (miles per gallon) varies inversely with its speed (in miles per hour). If the car gets 30 miles per gallon at 60 mph, find the fuel efficiency at 70 mph.

12. The force (F) of gravity between two objects varies directly with the product of their masses (m1 and m2) and inversely with the square of the distance (d) between them. If F = 9.8 N when m1 = 5 kg, m2 = 10 kg, and d = 1 m, find F when m1 = 3 kg, m2 = 8 kg, and d = 2 m.

13. The time (t) it takes for a pendulum to complete one full swing varies directly with the square root of its length (L). If a pendulum takes 2 seconds to complete one swing when L = 9 meters, find the time it takes when L = 16 meters.

14. The resistance (R) in an electrical circuit is inversely proportional to the square of the current (I). If R = 25 ohms when I = 5 amperes, find R when I = 10 amperes.

15. The force (F) required to lift an object with a pulley system varies directly with the weight (W) of the object and inversely with the number (n) of supporting ropes. If F = 120 N when W = 600 N and n = 4, find F when W = 800 N and n = 6.

**Answers:**

**Easy Problems:**

1. k = 3

2. k = 50

3. y = 40

4. y = 4

5. x = 6

**Medium Problems:**

6. $150

7. 4 hours

8. 5 hours

9. F = 5.76

10. P = 19.2 kPa

**Difficult Problems:**

11. Fuel efficiency at 70 mph = 25 mpg

12. F = 4.35 N

13. Time = 3 seconds

14. R = 6.25 ohms

15. F = 160 N

Worksheet on Sets ICSE Class 8

 

**Sets Worksheet – ICSE – Class 8**

 

**Instructions:**

1. Answer all the questions.

2. Circle or underline your final answer.

3. Show your work or reasoning if required.

4. Answers to all the problems are given at the end. You should look at the solutions only after attempting all the problems.

 

 

**Questions:**

1. Define a "set" in your own words. Provide an example.

 

2. Classify the following into sets:

   a) The days of the week

   b) Even numbers less than 20

   c) Vowels in the English alphabet

 

3. List the elements of the set A = {2, 4, 6, 8, 10}. Also, find the cardinality of set A.

 

4. Create a set B with the first five prime numbers. Write it in the set-builder notation.

 

5. Determine whether the following statements are true or false:

   a) {1, 2, 3} ⊆ {1, 2, 3, 4, 5}

   b) {a, b, c} ⊇ {b, c, d}

   c) {2, 4, 6} ⊂ {1, 2, 3, 4, 5, 6}

 

6. Find the union of sets P = {1, 2, 3, 4, 5} and Q = {4, 5, 6, 7}. Write the result in roster form.

 

7. Calculate the intersection of sets X = {a, b, c} and Y = {b, c, d}. Write the result in set-builder notation.

 

8. Consider two sets: M = {1, 2, 3, 4} and N = {3, 4, 5, 6}. Find the difference M - N.

 

9. Solve the following set equation for set Z: Z ∩ {2, 3, 4} = {3, 4}. Write the result in roster form.

 

10. Given the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, find the complement of the set V = {2, 4, 6, 8}.

 

**Additional Challenging Questions:**

 

11. Let A = {1, 2, 3, 4, 5, 6} and B = {4, 5, 6, 7, 8, 9}. Find A ∪ B, A ∩ B, and A - B.

 

12. Consider a universal set U = {x | x is a positive integer less than 10}. If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, find A ∩ B and A ∪ B.

 

13. Let U = {a, b, c, d, e, f, g} be the universal set. If A = {a, b, c, d} and B = {b, c, e, f}, find A ∪ B and A' (complement of A).

 

14. Define three sets A, B, and C as follows:

    A = {x | x is a multiple of 2 and 3}

    B = {x | x is a multiple of 2 and 5}

    C = {x | x is a multiple of 3 and 5}

   Find A ∩ B, A ∪ C, and B ∩ C.

 

15. Let U be the set of all students in a school, A be the set of students who play chess, and B be the set of students who play cricket. If there are 120 students in total, 60 play chess, and 80 play cricket, how many students play both chess and cricket?

 

16. Consider the set P = {x | x is a prime number less than 20} and the set Q = {x | x is an odd number less than 20}. Find P ∩ Q.

 

17. Let A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7}. Find the symmetric difference of sets A and B.

 

18. Determine whether the following statement is true or false: For any two sets A and B, A ∪ B = B ∪ A.

**Answers:**

 

1. A set is a collection of distinct objects or elements. Example: Set of even numbers less than 10 = {2, 4, 6, 8}.

 

2.

   a) Set of days of the week = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}

   b) Set of even numbers less than 20 = {2, 4, 6, 8, 10, 12, 14, 16, 18}

   c) Set of vowels in the English alphabet = {a, e, i, o, u}

 

3. Set A = {2, 4, 6, 8, 10}, Cardinality of A = 5.

 

4. Set B = {2, 3, 5, 7}. In set-builder notation: B = {x | x is a prime number and 1 < x < 10}.

 

5.

   a) True

   b) False

   c) True

 

6. P ∪ Q = {1, 2, 3, 4, 5, 6, 7}.

 

7. X ∩ Y = {b, c}. In set-builder notation: {x | x is an element of X and x is an element of Y}.

 

8. M - N = {1, 2}. (Elements in M but not in N)

 

9. Z = {3, 4}.

 

10. U - V = {1, 3, 5, 7, 9}.

 

**Answers to Additional Questions:**

 

11. A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A ∩ B = {4, 5, 6}, A - B = {1, 2, 3}

 

12. A ∩ B = {3, 4}, A ∪ B = {1, 2, 3, 4, 5, 6}

 

13. A ∪ B = {a, b, c, d, e, f}, A' = {e, f, g}

 

14. A ∩ B = {x | x is a multiple of 2, 3, and 5}, A ∪ C = {x | x is a multiple of 2 or 3 or 5}, B ∩ C = {x | x is a multiple of 3 and 5}

 

15. Students who play both chess and cricket = 20 students.

 

16. P ∩ Q = {3, 5, 7, 11, 13, 17, 19}

 

17. Symmetric difference of sets A and B = {1, 2, 6, 7}

 

18. True. The union of two sets is commutative, so A ∪ B = B ∪ A.

 

Problem based on Direct and Inverse Variation

35 cattle can graze on a field for 18 days. After 10 days, 15 cattle are move to a different field. For how long can the remaining cattle graze on the field? b) 14.5 days c) 14 days • d) 12 days a) 15 days ​ 

Answer:

If one increases the number of cattle, then the number of days will decrease to graze the same field.

If we increase the number of days, then less cattle will be needed to graze the same field.

So the variables, cattle and number of days are inversely proportional.

Now, if we consider the variables cattle and field size ( keeping the number of days fixed), it is easy to see that they are directly proportional. ( as if you increase the field size, then the number of cattle must be increased to graze in the same day and vice versa)

Let the corresponding quantities for cattle, days and field size be C, D and F.

Then C ∝  ......(i)

⇒C = k , where k is some non-zero constant.

For the initial data, C= 35 and F = 1 ( taking field size as 1 unit ) and D = 18

∴

Now in 1 day, 35 cattle will graze

unit of the field.

Thus in 10 days, they will graze,

unit of the field.

Thus remaining unit of field to be grazed

As 15 cattle were removed, remaining cattle

So the question, boils down to finding the number of days (D) in which 20 cattle (C) will graze

unit (F) of the field.

From (i), C = k

⇒ 

⇒   days

(c) is the correct option

Bayes' Theorem Problem from ISC 2023 Maths Paper

In a company, 15% of the employees are graduates and 85% of the employees are non-graduates. as per the annual report of the company, 80% of the graduate employees and 10% of the non-graduate employees are in the administrative positions. find the probability that an employee selected at random from those working in administrative positions will be a graduate. 

Answer:

Step-by-step explanation:

Let G be the event that the selected employee is a graduate, and NG be the event that the selected employee is non-graduate.

Clearly, G and NG forms a mutually exclusive and exhaustive set of events.

Further, let A be the event that the selected employee works in administrative office.

According to the problem, we have to find, the selected employee is a graduate given he/she works in the administrative position P(G/A).

By Bayes' theorem,

Now, Probability of an employee to be graduate =

Probability of an employee to be non-graduate =

Probability of an employee working in administrative office given he is a graduate =

( as 80% of the graduate employee works in the administrative positions)

Probability of an employee working in administrative office given he is a non-graduate =

( as 10% of the non-graduate employee works in the administrative positions)

Substituting the values in

, we get,

Prove that cos pi/11 + cos 3pi/11 + cos 5pi/11 + cos 7pi/11 + cos 9pi/11...

A elegant solution for a difficult trigonometric problem using complex numbers. Many problem of trigonometry and even algebra can be beautifully solved using complex numbers. The use of different formulas from trigonometry can be avoided. #CBSE #ISC #IIT #ISI #wbjee #wbchse #jeeadvanced Prove that cos pi/11 + cos 3pi/11 + cos 5pi/11 + cos 7pi/11 + cos 9pi/11 =1/2

Integrate max { x+|x|,x-[x] } from -n to n JEE Main and Advance and Indian Statistical Institute B.Math & B.Stat

Integrate max { x+|x|,x-[x] } from -n to n Difficult Integration for JEE Main and Advance and Indian Statistical Institute Involving Limits Integration for Indian Statistical Institute B.Math & B.Stat : Integration, JEE Main and Advance, WBJEE #CBSE #ISC #wbchse #HS #jeeadvanced

Combinations SN Dey Solved

1. An executive committee of 6 is to be formed from 4 ladies and 7 gentlemen. In how many ways can this be formed when the committee contains (i) only 2 lady members, (ii) at least 2 lady members?

2. Find the number of committees of 5 members that can be formed from 6 gentlemen and 4 ladies if each committee has at least one lady and two gentlemen.

 

3. A committee of 5 is to be formed from six ladies and four gentlemen. In how many ways this can be done so that the committee contains (i) exactly two ladies, (ii) at least two ladies, (iii) at most two ladies?

 

4. In a cricket team of 14 players 6 are bowlers. How many different teams of 11 players can be selected keeping at least 4 bowlers in the team?

 

5. A box contains 12 lamps of which 5 are defective. In how many ways can a sample of 6 be selected at random from the box so as to include at most 3 defective lamps?

 

6. An examinee is required to answer 6 questions out of 12 questions which are divided into two groups each containing 6 questions, and he is not permitted to answer more than 4 questions from any group. In how many ways can he answer 6 questions?

 

7. A question paper contains 10 questions, which are divided into two groups each containing 5 questions. A candidate is asked to answer 6 questions only, and to choose at least 2 questions from each group. In how many different ways can the candidate make up his choice?

 

8. In how many ways can a team of 11 cricketers be chosen from 9 batsmen and 6 bowlers to give a majority of batsman if at least 3 bowlers are to be included?

 

9. The Indian Cricket Eleven is to be selected out of fifteen players, five of them are bowlers. In how many ways the team can be selected so that the team contains at least three bowlers?

10. How many combinations can be formed of eight counters marked 1, 2, 3, 4, 5, 6, 7, 8 taking them 4 at a time, there being at least one odd and one even counter in each combination?

 

11. Find the number of permutations of the letters of the words FORECAST and MILKY taking 5 at a time of which 3 letters from the first word and 2 from the second.

 

12. In how many ways can the crew of an eight-oared boat be arranged if 2 of the crew can row only on the stroke side and 1 can row only on the bow side?

 

13. Of the 17 articles, 12 are alike and the remaining 5 are different. Find the number of combinations, if 13 articles are taken at a time.

 

14. Out of 3n given things 2n are alike and the rest are different. Show that a selection of 2n things can be made from these 3n things in 2" different ways.

 

15. Show that there are 136 ways of selecting 4 letters from the word EXAMINATION.

 

16. Find the total number of ways of selecting 5 letters from the letters of the word INDEPENDENT.

 

17. (i) Find the number of combinations in the letters of the word STATISTICS taken 4 at a time.

      (ii) Find the number of permutations in the letters of the word PROPORTION taken 4 at a time.

 

18. How many different numbers of 4 digits can be formed with the digits 1, 1, 2, 2, 3, 4?

 

19. (i) From 4 apples, 5 oranges mangoes, how many selections of fruits can be made, taking at least one of each kind if the fruits of the same kind are of different shapes?

(ii) In how many ways can one or more fruits be selected from 4 apples, 5 oranges and 3 mangoes, if the fruits of the same kind be of the same shape?

20. Find the total number of combinations taking at least one green ball and one blue ball, from 5 different green balls, 4 different blue balls and 3 different red balls.

 

21. How many different algebraic quantities can be formed by combining a, b, c, d, e with + and - signs, all the letters taken together?

 

22. There are n points in space, no four of which are in the in the same place with the exception of m points, all of which are in the same plane. How many planes can be formed by joining them?

 23. n1, n2 and n3, points are given on the sides BC, CA and AB respectively of the triangle ABC. Find the number of triangles formed by taking these given points as vertices of a triangle.

 

24. A man has 7 relatives, 4 of them are ladies and 3 gentlemen; his wife has also 7 relatives, 3 of them are ladies and 4 are gentlemen. In how many ways can they invite dinner party of 3 ladies and 3 gentlemen so that there are 3 of the man's relatives and 3 of the wife's relatives?

 

25. Eighteen guests have to be seated, half on each side of long table. Four particular guests desire to sit on one particular side and three others on the other side Determine the number of ways in which the arrangements can be made. 

Permutations SN Dey Solved XI Maths

Solved problems from the book of SN Dey Class XI WBCHSE, Permutations [ 4 and 5 marks only]

Problem from Geometry - Circles

In the figure given below, ABD is a right-angled triangle at B. Taking AB as diameter, a circle has been drawn intersecting AD at F. Prove that the tangent drawn at point F bisects BD.

Solution: