West Bengal Council of Higher Secondary Education New Mathematics Syllabus Semester System XI and XII
For session 2024 onwards.
Problem for #jeemains #WBJEE #CBSE #ISC #mathematics
Let m, n, p and q be four positive integers such that m+n+p+q = 200. If S = (-1)^m+(-1)^n+(-1)^p+(-1)^q, then what is the number of possible values of S? #algebra #ProblemSolving #schoolmathematics
**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
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
**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.
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
Thus in 10 days, they will graze,
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
From (i), C = k
⇒
⇒ days
(c) is the correct option
