Extra Curriculum Support

Sec A

1. Given that, 4096 = 64, the value of 4096 + 40.96 is:

(a) 60.4 (b) 70.4 (c) 74 (d) 64.4

2. The value of 248+52+144

(a) 13 (b) 16 (c) 14 (d) 12

3. The square of 39 is:

(a) 1521 (b) 1500 (c) 378 (d) none of these

4. The smallest number by which 396 must be multiplied so that the product becomes a perfect square is:

(a) 11 (b) 5 (c) 2 (d) 3

5. A side of square is 32, then the length of its diagonal is:

(a) 32 cm (b) 18 cm (c) 3 cm (d) 6 cm

7. How many natural numbers lie between 182 and 192 ?

(a) 37 (b) 30 (c) 36 (d) 35

8. Find the square of 25.

9. Verify whether 784 is a perfect square or not.

10. Write the next three perfect squares after 121.

11. Find the square root of 144.

Sec B

1. Find the square root of 1764 by the Prime Factorisation Method.

2. Find the square root of 529 by Division method.

3. Find the square root of 324 using the prime factorization method.

4. If the area of a square is 625 cm², find the length of its side.

5. A number has its square root equal to 15. What is the number? Also, verify your answer.

6. Find the smallest number that must be added to 86 to make it a perfect square.

Sec C

1. Is 2352 a perfect square? If not, find the smallest multiple of 2352 which is a perfect square. Find the square root of the new number.

2. Find the least number which must be subtracted from 4000 so as to get a perfect square. Also find the square root of the perfect square so obtained.

3. Find the least number which must be added to 525 so as to get a perfect square. Also find the square root of the perfect square so obtained.

4. Find the square root of 1024 using long division.

5. A gardener wants to plant trees in a square-shaped garden such that each side has 12 trees. How many trees are planted in total?

6. Determine whether 15625 is a perfect square or not using prime factorization.

7. A square room has a diagonal length of 8√2 m. Find the side length of the room.

8. A square field has an area of 8100 m². How many meters of fencing is required to enclose the field?

SecD

1. Find the smallest number by which 192 must be multiplied so that it becomes a perfect square. Also, find the square root of the resulting number.

2. The sum of two consecutive perfect squares is 130. Find the two numbers.

3. A school is constructing a square-shaped playground with an area of 4900 m². If fencing costs ₹10 per meter, calculate the total cost of fencing the playground.

4. A staircase has 10 steps, and the height of each step increases as per the square of its position number. For example, the first step is 1 cm high, the second is 4 cm high, the third is 9 cm high, and so on. Find the total height of the staircase after 10 steps.

5. Prove that the difference between the squares of any two consecutive numbers is equal to the sum of those numbers. Use an example to demonstrate.

Problem 1

A group of students is raising funds to support an underprivileged school. They decide to sell square-shaped handmade greeting cards. Each card has an area of 121 square centimeters. The students plan to make 100 cards.

(a) What is the side length of each card?

(b) Why is it important to help those who are less fortunate?

(c) How can small efforts, like making and selling greeting cards, make a big difference in someone's life?

Problem 2

A neighborhood association decides to clean up a square park that has an area of 1,024 square meters. They want to involve children in the cleanup drive to teach them the value of maintaining cleanliness in public spaces.

(a) What is the side length of the park?

(b) Why is it important to keep public spaces clean?

(c) How can involving young people in such activities help in promoting a culture of cleanliness?

Q1

Consider the sequence of perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ...

(a) Observe the difference between consecutive perfect squares (e.g., 4−1, 9−4, etc.). What pattern do you notice?

(b) Based on the pattern you observed, what will be the difference between the square of the next number and the square of the current number?

(c) Can you generalize the relationship between the squares of consecutive integers? How can this be expressed algebraically?

Q2

You know that 144 = 12 and 169=13.

Without using a calculator, estimate the value of 150. Justify your answer.

Use a more precise method to refine your estimate (e.g., averaging, successive approximations). What is your refined estimate for 150?

How might this method of estimation be useful in real-life situations where quick mental calculations are needed?

Q3

You are given that a + b = 20 and a - b = 4.

Solve these two equations simultaneously to find the values of a and b.

Verify your solution by substituting the values of a and b back into the original equations.

What does this problem illustrate about the relationship between addition and subtraction of square roots? How might this be applied in more complex algebraic situations?

Questions

1. Which of the following will have 1 at its units place?

(a) 192 (b) 172 (c) 182 (d) 162

2. How many natural numbers lie between 182 and 192?

(a) 30 (b) 37 (c) 35 (d) 36

3. Which of the following is not a perfect square?

(a) 361 (b) 1156 (c) 1128 (d) 1681

4. If one member of a pythagorean triplet is 2m, then the other two members are:

(a) m,m2+1

(b) m2+1,m2−1

(c) m2,m2−1

(d) m2,m+1

5. Which letter best represents the location of 25 on a number line?

(a) A (b) B (c) C (d) D

Questions

1. The value of 176+2401 is ?

2. The square of 6.1 is ?

Questions

1. The square of 0.4 is 0.16.

2. There are 21 natural numbers between 102 and 112.

3. The square root of a perfect square of n digits will have n2 digits if n is even.

Questions

1. Discuss whether 9.5 is a good first guess for 75.

2. Check whether 90 is a perfect square or not by using prime factorisation.

3. Using distributive law, find the square of 43.

4. Write a pythagorean triplet whose smallest number is 6.

5. A couple wants to install a square glass window that has an area of 500 square cm. Calculate the length of each side and the length of trim needed to the nearest tenth of cm.

6. A ladder 10m long rests against a vertical wall. If the foot of the ladder is 6m away from the wall and the ladder just reaches the top of the wall, how high is the wall?

7. Find the smallest perfect square divisible by 3, 4, 5 and 6.

8. A student said that since the square roots of a certain number are 1.5 and –1.5, the number must be their product, –2.25. What error did the student make?

9. During a mass drill exercise, 6250 students of different schools are arranged in rows such that the number of students in each row is equal to the number of rows. In doing so, the instructor finds out that 9 children are left out. Find the number of children in each row of the square.

10. A General wishes to draw up his 7500 soldiers in the form of a square. After arranging, he found out that some of them are left out. How many soldiers were left out?

11. A king wanted to reward his advisor, a wise man of the kingdom. So he asked the wiseman to name his own reward. The wiseman thanked the king but said that he would ask only for some gold coins each day for a month. The coins were to be counted out in a pattern of one coin for the first day, 3 coins for the second day, 5 coins for the third day and so on for 30 days. Without making calculations, find how many coins will the advisor get in that month?

Q1

Ravi has a rectangular garden in his backyard. He wants to create a square flower bed inside the garden. The area of the rectangular garden is 576 square meters, and the flower bed will occupy exactly one-fourth of the garden's area.

(a) What is the area of the square flower bed?

(b) Find the side length of the square flower bed.

(c) If the rectangular garden was actually a square, what would be the side length of the garden?

(d) How does the side length of the flower bed compare to the side length of the square garden (if it was a square)?

Sol 1

(a) The area of the flower bed is one-fourth of the area of the rectangular garden:

Area of flower bed = 14 × Area of garden

Substitute the given area of the garden:

Area of flower bed = 14 × 576 = 144 m2.

So, the area of the square flower bed is 144 square meters.

(b) Side length of flower bed = √ Area of flower bed = √144 = 12 m ​ So, the side length of the square flower bed is 12 meters.

(c) Area of square garden = sidelength2

To find the side length of the square garden, we take the square root of the area:

Side length of square garden = 576 = 24 m

So, if the garden were a square, the side length would be 24 meters.

(d) Comparing the two:

Side length of flower bed=12m

Side length of square garden=24m

So, the side length of the flower bed is half the side length of the square garden.

Q2

A community park is square-shaped and has an area of 625 square meters. The park management decides to build a square-shaped pond inside the park, such that the side of the pond is half the side of the park.

(a) Calculate the side length of the square park.

(b) Determine the side length of the pond.

(c) bCalculate the area of the pond.

(d) What is the difference in area between the park and the pond?

Sol 2

(a) Side length of park = √Area of park = √625 = 25 meters

So, the side length of the square park is 25 meters.

(b) Side length of pond= 12 × Side length of park = 12 × 25 = 12.5 meters

So, the side length of the pond is 12.5 meters.

(c) Area of pond = 12.52 = 12.5 × 12.5 = 156 .25square meters

So, the area of the pond is 156.25 square meters.

(d) Difference in area = Area of park − Area of pond = 625 − 156.25 = 468.75 square meters

So, the difference in area between the park and the pond is 468.75 square meters.