Chapter 3

Class 8 Mathematics · 140 questions · 135 with answers

Solved examples

example-1Multiple choice

Which of the following is the square of an odd number?

  • (a)256
  • (b)361
  • (c)144
  • (d)400
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Solution : Correct answer is (b).

example-2Multiple choice

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

  • (a)192
  • (b)172
  • (c)182
  • (d)162
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Solution : Correct answer is (a).

example-3Multiple choice

How many natural numbers lie between 182 and 192?

  • (a)30
  • (b)37
  • (c)35
  • (d)36
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Solution : Correct answer is (d).

example-4Multiple choice

Which of the following is not a perfect square?

  • (a)361
  • (b)1156
  • (c)1128
  • (d)1681
Show solution

Solution : Correct answer is (c).

example-5Multiple choice

A perfect square can never have the following digit at ones place.

  • (a)1
  • (b)6
  • (c)5
  • (d)3
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Solution : Correct answer is (d).

example-6Multiple choice

The value of 176 + 2401 is

  • (a)14
  • (b)15
  • (c)16
  • (d)17
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Solution : Correct answer is (b). ( 176 + 2401 = 176 + 49 = 225 = 15)

example-7Multiple choice

Given that 5625 =75, the value of 0.5625 + 56.25 is:

  • (a)82.5
  • (b)0.75
  • (c)8.25
  • (d)75.05
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Solution : Correct answer is (c). If ( 5625 = 75, then 0.5625 = 0.75 and 56.25 = 7.5) In examples 8 to 14, fill in the blanks to make the statements true.

example-8Fill in the blanks

There are __________ perfect squares between 1 and 50.

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Solution :6

example-9Fill in the blanks

The cube of 100 will have __________ zeroes.

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Solution :6

example-10Fill in the blanks

The square of 6.1 is ____________.

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Solution : 37.21 Squaring a number and taking a square root are inverse operations. What other inverse operations do you know? When the factors of a perfect square are written in order from the least to greatest, what do you notice? Why do you think numbers such as 4, 9, 16, ... are called perfect squares? Suppose you list the factors of a perfect square. Why is one factor square root and not the other factors?

example-11Fill in the blanks

The cube of 0.3 is ____________.

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Solution : 0.027 here are some ways to tell whether a number is a square number. If we can find a division sentence for a number so that the quotient is equal to the divisor, the number is a square number. For example, 16 ÷ 4 = 4, so 16 is a square number. ↑ ↑ ↑quotient dividend divisor We can also use factoring. Factors of a number occur in pairs. These are the dimensions of a rectangle. 1 and 16 are factors of 16 16 unit 2 unit 2 and 8 factors is of 16 4 is factor of 16 It occurs 4 unit

example-12Fill in the blanks

682 will have __________ at the units place.

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Solution : 4

example-13Fill in the blanks

The positive square root of a number x is denoted by __________.

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Solution : x

example-14Fill in the blanks

The least number to be multiplied with 9 to make it a perfect cube is _______________.

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Solution : 3 In examples 15 to 19, state whether the statements are true (T) or false (F)

example-15Short answer

The square of 0.4 is 0.16.

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Solution : True

example-16Short answer

The cube root of 729 is 8.

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Solution : False

example-17Short answer

There are 21 natural numbers between 102 and 112.

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Solution: False

example-18Short answer

The sum of first 7 odd natural numbers is 49.

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Solution : True

example-19Short answer

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

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Solution : True

example-20Short answer

Express 36 as a sum of successive odd natural numbers.

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Solution : 1+3+5+7+9+11 = 36 A rectangle is a quadrilateral with 4 right angles. A square also has 4 right angles. 1cm A rectangle with base 4 cm and height 1cm is the same as a rectangle with base 1cm and height 4 cm. 4cm These two rectangles are congruent. Is every square a rectangle? Is every rectangle a square? Copy this diagram on grid paper. Then estimate the value of 7 to one decimal place.

example-21Short answer

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

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Solution : Prime factorisation of 90 is 2 90 3 45 3 15 5 5 90 = 2 × 3 × 3 × 5 The prime factors 2 and 5 do not occur in pairs. Therefore, 90 is not a perfect square.

example-22Short answer

Check whether 1728 is a perfect cube by using prime factorisation.

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Solution : Prime factorisation of 1728 is 1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 Since all prime factors can be grouped in triplets. Therefore, 1728 is a perfect cube. Apply Use square tiles. Make as many different rectangles as you can with area 28 square units. Draw your rectangles on grid paper. Is 28 a perfect square? Justify your answer.

example-23Short answer

Using distributive law, find the square of 43.

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Solution : 43 = 40 + 3 So 432 = (40 + 3)2 = (40 + 3) (40 + 3) = 40 (40 +3) + 3(40 + 3) = 40 × 40 + 40 × 3 + 3 × 40 + 3 × 3 = 1600 + 240 + 9 = 1849 So, 432 = 1849

example-24Short answer

Write a pythagorean triplet whose smallest number is 6.

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Solution : Smallest number is 6 2m = 6 or m = 3 m 2 + 1 = 32 + 1 = 9 + 1 = 10 m 2 – 1 = 32 – 1 = 9 – 1 = 8 So, the pythagorean triplet is 6, 8, 10. Here is one way to estimate the value of 20 : 25 is the square number closest to 20, but greater than 20. On grid paper, draw a square with area 25. 9 square units 7 square units Its side length: 25 = 5 4 square units 16 is the square number closest to 9 20, but less than 20. 7 Draw a square with area 16 Its side length: 16 = 4 Draw the squares so that they overlap. A square with area 20 lies between these two squares. Its side length 20 . 20 is between 16 and 25, but closer to 16. So, 20 is between 16 and 25 , but closer to 16 . So, 20 is between 4 and 5, but closer to 4. An estimate of 20 is 4.4 to one decimal place. 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. Understand the problem First find the length of a side. Then you can use the length of the side to find the perimeter – the length of the trim around the window. Make a Plan The length of a side, in cm, is the number that you multiply by itself to get 500. Find this number to the nearest tenth. Use guess and check to find 500 . Solve Because 5000 is between 222 (484) and 232 (529), the square root of 500 is between 22 and 23. The square root is between 22.3 and 22.4. To round to the nearest tenth, consider 22.35. 22.352 = 499.5225 low Guess 22.5 Guess 22.2 Guess 22.4 Guess 22.3 2 2 2 22.5 506.25 22.2 492.84 22.4 501.76 22.32 497.29 high low high low Square root is Square root is Square root is Square root is between 22 and between 22.2 between 22.2 between 22.3 22.5 and 22.5 and 22.4 and 22.4 The square root must be Guess greater than 22.35, so you 2 4 3 1 can round up. 22.0 22.1 22.2 22.3 22.4 22.5 22.6 22.7 22.8 To the nearest tenth, 500 is about 22.4. Now estimate the length around the window. The length of a side of the window to the nearest tenth of an inch is 22.4 inches. 4 × 22.4 = 89.6 (Perimeter = 4 × side) The trim is about 89.6 cm long. Look Back The length 90 cm divided by 4 is 22.5 cm. A 22.5 cm square has an area of 506 square cm, which is close to 500, so the answers are reasonable.

example-25Short answer

Using prime factorisation, find the cube root of 5832.

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Solution : The prime factorisation of 5832 is 2 5832 2 2916 2 1458 3 729 3 243 3 81 3 27 3 9 3 3 5832 = 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3 Therefore, 3 5832 = 3 2×2×2×3×3×3×3×3×3 =2×3×3 = 18 Take It Further a) Find the square root of each palindromic number. A palindromic number is a number that reads the same – forward and backward. (i) 121 (ii) 12 321 (iii) 1 234 321 (iv) 123 454 321 b) Continue the pattern. Write the next 4 palindromic numbers in the pattern and their square roots. Is 1 a square number? How can you tell? Suppose you know the area of a square. How can you find its perimeter? Suppose you know the perimeter of a square. How can you find its area?

example-26Short answer

Evaluate the square root of 22.09 by long division method.

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Solution : 4.7 4 22.09 87 609 0 Therefore, 22.09 = 4.7

example-27Short answer

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

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Solution : The least number divisible by 3, 4, 5 and 6 is their LCM. The LCM of 3, 4, 5 and 6 is 60. Now, 60 = 2 × 2 × 5 × 3. We see that prime factors 5 and 3 are not in pairs. Therefore 60 is not a perfect square. So, 60 should be multiplied by 5 × 3 = 15 to get a perfect square. Thus, the required least square number = 60 × 15 = 900.

example-28Short answer

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

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Solution : Let AC be the ladder. B 6m C Therefore, AC = 10m Let BC be the distance between the foot of the ladder and the wall. Therefore,BC = 6m ∆ABC forms a right angled triangle, right angled at B. By Pythagoras theorem, AC2 = AB2 + BC2 102 = AB2 + 62 or AB2 = 102 –62 = 100 – 36 = 64 or AB = 64 = 8m Hence, the wall is 8m high.

example-29Short answer

Find the length of a diagonal of a rectangle with dimensions 20m by 15m.

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Solution : Using Pythagoras theorem, we have Length of diagonal of the rectangle = (l 2 + b 2 ) units = (202 + 152 ) m = 400 + 225 m = 625 m = 25 m Hence, the length of diagonal is 25m. Work with a partner. You will need grid paper and 20 square tiles. Use the tiles to make as many different rectangles as you can with each area. 4 square units 12 square units 6 square units 16 square units 8 square units 20 square units

example-30Short answer

The area of a rectangular field whose length is twice its breadth is 2450 m2. Find the perimeter of the field.

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Solution : Let the breadth of the field be x metres. Then length of the field is 2x metres. Therefore, area of the rectangular field = length × breadth = (2x)(x) = (2x2) m 2 Given that area is 2450 m 2. Therefore, 2x2 = 2450 2450 x2 = x = 1225 or x = 35m Hence, breadth = 35m and length 35 × 2 = 70m Perimeter of the field = 2 (l + b) = 2(70+35)m = 2×105m = 210m

example-31Short answer

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.

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Solution : Total number of students = 6250 Number of students forming a square = 6250 – 9 = 6241 Thus, 6241 students form a big square which has number of rows equal to the number of students in each row. Let the number of students in each row be x, then the number of rows = x Therefore, x × x = 6241 or x= 6241 = 79 Hence, there are 79 students in each row of the square formed.

example-32Short answer

Find the least number that must be added to 1500 so as to get a perfect square. Also find the square root of the perfect square.

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Solution : 38 3 1500 68 600 We observe that 382 < 1500 < 392 Hence the number to be added = 392 – 1500 = 1521 – 1500 = 21 Therefore, the perfect square is 1500 + 21 = 1521 1521 = 39 Hence the required number is 21 and the square root is 39. Tsunamis, sometimes called tidal waves, move across deep oceans at high speeds with barely a ripple on the water surface. It is only when tsunamis hit shallow water that their energy moves them upward into a huge destructive force. The speed of a tsunami, in metre per second, can be found by the formula r = Tsunamis can be caused by 9.7344d , where d is the water depth earthquakes, volcanoes, in metre. Suppose the water depth is landslides, or meteorites. 6400m . How fast is the tsunami moving? The speed of a tsunami in km per hour can be found As the wave using r = 4.4944d where approaches the d is the water depth in beach, it slows, metre. Suppose the water builds in height and crashes on depth is 8100 metre shore a) How fast is the tsunami moving in km per hour? b) How long would it take a tsunami to travel 3000 km if the water depth was a consistent 3000 m?

example-33Short answer

Application of problem solving strategies • Find the smallest number by which 1620 must be divided to get a perfect square.

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Solution : Understand and Explore • What information is given in the question? – A number which is not a perfect square. • What are you trying to find? – The smallest number by which 1620 must be divided to get a perfect square. Plan a strategy • You have already learnt prime factorisation. Use it to find the product of prime factors of 1620. • Pair the prime factors to see if any factor is left unpaired. • This unpaired factor will be the smallest number that must be divided to get a perfect square. Solve Prime factorisation of 1620 is 2 1620 2 810 5 405 3 81 3 27 3 9 3 3 The product of prime factors = 2 × 2 × 5 × 3 × 3 × 3 × 3 Pair these prime factors = 2 × 2 ×5× 3 × 3×3 × 3 The factor 5 is left unpaired. Hence, the required smallest number is 5. Revise Divide 1620 by 5 and check if it is a perfect square. 1620 ÷ 5 = 324 We see that 324 is a perfect square, hence our answer is verified. 1. Find the square root of the number obtained in step IV. 2. Can you find the smallest number that can be multiplied to 1620 to get a perfect square? 3. Find the square root. In each of the questions, 1 to 24, write the correct answer from the given four options. 1. 196 is the square of (a) 11 (b) 12 (c) 14 (d) 16 2. Which of the following is a square of an even number? (a) 144 (b) 169 (c) 441 (d) 625 3. A number ending in 9 will have the units place of its square as (a) 3 (b) 9 (c) 1 (d) 6 A magic square is a square with numbers arranged so that the sum of the numbers in each row, column and diagonal is the same. Complete each magic square below. Use the numbers – 4, – 3, – 2, –1, 0, 1, 2, 3 and 4 to make a magic square with row, column and diagonal sums of 0. 4. Which of the following will have 4 at the units place? (a) 142 (b) 622 (c) 27 2 (d) 352 5. How many natural numbers lie between 52 and 62? (a) 9 (b) 10 (c) 11 (d) 12 6. Which of the following cannot be a perfect square? (a) 841 (b) 529 (c) 198 (d) All of the above 7. The one’s digit of the cube of 23 is (a) 6 (b) 7 (c) 3 (d) 9 8. A square board has an area of 144 square units. How long is each side of the board? (a) 11 units (b) 12 units (c) 13 units (d) 14 units 9. Which letter best represents the location of 25 on a number line? (a) A (b) B (c) C (d) D A B C D 0 1 2 3 4 5 6 7

Questions

Q1Short answer

or 9 at the units place ends in1.

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A |||||||||||||||||||||||| B | C |||||||||||| D |||||||| E |||||||||||||||||||||||| F ||||||| G |||| H |||||||| I |||||||||||||||| L |||||||| M |||||| N |||||||||||||| O |||||||||||| P |||| R |||||||||||| S ||||||||||||| T ||||||||||||||||||||| U ||||||| V |||| W | X | Y | Z |

Q2Short answer

or 8 at the units place ends in 4.

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2 letters |||||||||||||||| 20 3 letters ||||||||||||||| 18 4 letters ||||||||||||||| 18 5 letters ||||||| 08 6 letters |||||||| 09 more than 6 letters ||||||||||||||||||||||||||| 33 Crossword Answers Across 1. Pie Chart

Q3Short answer

or 7 at the units place ends in 9.

Q4Short answer

or 6 at the units place ends in 6.

Q5Long answer

at the units place ends in 5. • There are 2n natural numbers between the squares of numbers n and n+1. • A number ending in odd numbers of zeroes is not a perfect square. • The sum of first n odd natural numbers is given by n 2. • Three natural numbers a, b, c are said to form a pythagorean triplet if a2 + b2 = c 2. • For every natural number m > 1, 2m, m 2–1 and m 2 + 1 form a pythagorean triplet. • The square root of a number x is the number whose square is x. Positive square root of a number x is denoted by x. • The cube root of a number x is the number whose cube is x. It is denoted by 3 x . • Square root and cube root are the inverse operations of squares and cubes respectively. • If a perfect square is of n digits, then its square root will have n +1 digit if n is even or digit if n is odd. 2 • Cubes of the numbers ending with the digits 0, 1, 4, 5, 6 and 9 end with digits 0, 1, 4, 5, 6 and 9 respectively. Describe what is meant by a perfect square. Give an example. Explain how many square roots a positive number can have. How are these square roots different? SQUARE ROOTS A square root of a number n is a number m which, when multiplied by itself, equals n. The square roots of 16 are 4 and – 4 because 42 = 16 and (– 4)2 = 16. If m 2 = n, then m is a square root of n. Which type of number has an exact square root? Which type of number has an approximate square root? How can we use perfect squares to estimate a square root, such as 8 ? • Cube of the number ending in 2 ends in 8 and cube root of the number ending in 8 ends in 2. • Cube of the number ending in 3 ends in 7 and cube root of the number ending in 7 ends in 3. In examples 1 to 7, out of given four choices only one is correct. Write the correct answer.

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Five

Q8Long answer

unit twice. 4 unit Sixteen has 5 factors: 1, 2, 4, 8, 16 A factor that occurs twice Since there is an odd number of factors, is only written once in the one rectangle is a square. list of factors. The square has side length of 4 units. We say that 4 is a square root of 16. We write 4 = 16 When a number has an odd number of factors, it is a square number. Discuss whether 9.5 is a good first guess for 75 . Determine which square root or roots would have 7.5 as a good first guess.

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Event

Q9Long answer

square units Draw the rectangles on grid paper. For how many areas given above were you able to make a square? What is the side length of each square you made? How is the side length of a square related to its area? Compare your strategies and results with those of another pair of classmates. Find two areas greater than 20 square units for which you could use tiles to make a square. How do you know you could make a square for each of these areas?

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Whole

Q10Multiple choice

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

  • (a)m, m 2+1
  • (b)m 2+1, m 2–1
  • (c)m 2, m 2–1
  • (d)m 2, m+1
Show answer

One

Q11Multiple choice

The sum of successive odd numbers 1, 3, 5, 7, 9, 11, 13 and 15 is

  • (a)81
  • (b)64
  • (c)49
  • (d)36
Q12Multiple choice

The sum of first n odd natural numbers is

  • (a)2n+1
  • (b)n2
  • (c)n 2–1
  • (d)n 2+1
Show answer

(c) n 2–1

(a) 2n+1

(d) n 2+1

(b) n2

Q13Multiple choice

Which of the following numbers is a perfect cube?

  • (a)243
  • (b)216
  • (c)392
  • (d)8640
Show answer

(b) 216

Q14Multiple choice

The hypotenuse of a right triangle with its legs of lengths 3x × 4x is

  • (a)5x
  • (b)7x
  • (c)16x
  • (d)25x
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(a) 5x

Q15Multiple choice

The next two numbers in the number pattern 1, 4, 9, 16, 25 ... are

  • (a)35, 48
  • (b)36, 49
  • (c)36, 48
  • (d)35,49
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(b) 36, 49

Q16Multiple choice

Which among 432, 672, 522, 592 would end with digit 1?

  • (a)432
  • (b)672
  • (c)522
  • (d)592
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(d) 592

Q17Multiple choice

A perfect square can never have the following digit in its ones place.

  • (a)1
  • (b)8
  • (c)0
  • (d)6
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(b) 8

Q18Multiple choice

Which of the following numbers is not a perfect cube?

  • (a)216
  • (b)567
  • (c)125
  • (d)343
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(b) 567

Q19Multiple choice

3 1000 is equal to

  • (a)10
  • (b)100
  • (c)1
  • (d)None of these
Show answer

(a) 10

Q20Multiple choice

If m is the square of a natural number n, then n is

  • (a)the square of m
  • (b)greater than m
  • (c)equal to m
  • (d)m
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(d) m

Q21Multiple choice

A perfect square number having n digits where n is even will have square root with n n n +1

  • (a)n+1 digit
  • (b)digit
  • (c)digit
  • (d)digit 2 3 2
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(b) digit

Q22Multiple choice

If m is the cube root of n, then n is

  • (a)m3
  • (b)m
  • (c)
  • (d)3
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(a) m3

Q23Multiple choice

The value of 248 + 52 + 144 is

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

Q24Multiple choice

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

  • (a)74
  • (b)60.4
  • (c)64.4
  • (d)70.4 In questions 25 to 48, fill in the blanks to make the statements true.
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(d) 70.4 In questions 25 to 48, fill in the blanks to make the statements true.

Q25Fill in the blanks

There are _________ perfect squares between 1 and 100.

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8

Q26Fill in the blanks

There are _________ perfect cubes between 1 and 1000.

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8

Q27Fill in the blanks

The units digit in the square of 1294 is _________.

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6

Q28Fill in the blanks

The square of 500 will have _________ zeroes.

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4

Q29Fill in the blanks

There are _________ natural numbers between n 2 and (n + 1)2

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2n

Q30Fill in the blanks

The square root of 24025 will have _________ digits.

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3

Q31Fill in the blanks

The square of 5.5 is _________.

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30.25

Q32Fill in the blanks

The square root of 5.3 × 5.3 is _________.

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5.3

Q33Fill in the blanks

The cube of 100 will have _________ zeroes.

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6

Q34Fill in the blanks

1m2 = _________ cm2.

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10000

Q35Fill in the blanks

1m3 = _________ cm3.

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1000000

Q36Fill in the blanks

Ones digit in the cube of 38 is _________.

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2

Q37Fill in the blanks

The square of 0.7 is _________.

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0.49

Q38Fill in the blanks

The sum of first six odd natural numbers is _________.

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36

Q39Fill in the blanks

The digit at the ones place of 572 is _________.

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9

Q40Fill in the blanks

The sides of a right triangle whose hypotenuse is 17cm are _________ and _________.

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8, 15

Q41Fill in the blanks

1.96 = _________.

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1.4

Q42Fill in the blanks

(1.2)3 = _________.

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1.728

Q43Fill in the blanks

The cube of an odd number is always an _________ number.

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odd

Q44Fill in the blanks

The cube root of a number x is denoted by _________.

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3 x or x1/3

Q45Fill in the blanks

The least number by which 125 be multiplied to make it a perfect square is _____________.

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5

Q46Fill in the blanks

The least number by which 72 be multiplied to make it a perfect cube is _____________.

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2

Q47Fill in the blanks

The least number by which 72 be divided to make it a perfect cube is _____________.

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2

Q48Fill in the blanks

Cube of a number ending in 7 will end in the digit _______________. In questions 49 to 86, state whether the statements are true (T) or false (F).

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3

Q49Short answer

The square of 86 will have 6 at the units place.

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True

Q50Short answer

The sum of two perfect squares is a perfect square.

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False

Q51Short answer

The product of two perfect squares is a perfect square.

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True

Q52Short answer

There is no square number between 50 and 60.

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True

Q53Short answer

The square root of 1521 is 31.

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False

Q54Short answer

Each prime factor appears 3 times in its cube.

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True

Q55Short answer

The square of 2.8 is 78.4.

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False

Q56Short answer

The cube of 0.4 is 0.064.

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True

Q57Short answer

The square root of 0.9 is 0.3.

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False

Q58Short answer

The square of every natural number is always greater than the number itself.

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False

Q59Short answer

The cube root of 8000 is 200.

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False

Q60Short answer

There are five perfect cubes between 1 and 100.

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False

Q61Short answer

There are 200 natural numbers between 1002 and 1012.

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True

Q62Short answer

The sum of first n odd natural numbers is n 2.

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True

Q63Short answer

1000 is a perfect square.

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False

Q64Short answer

A perfect square can have 8 as its units digit.

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False

Q65Short answer

For every natural number m, (2m –1, 2m 2 –2m, 2m 2 – 2m + 1) is a pythagorean triplet.

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False

Q66Short answer

All numbers of a pythagorean triplet are odd.

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False

Q67Short answer

For an integer a, a3 is always greater than a2.

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False

Q68Short answer

If x and y are integers such that x2 > y2, then x3 > y3.

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False

Q69Short answer

Let x and y be natural numbers. If x divides y, then x3 divides y3.

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True

Q70Short answer

If a2 ends in 5, then a3 ends in 25.

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False

Q71Short answer

If a2 ends in 9, then a3 ends in 7. n +1

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False

Q72Short answer

The square root of a perfect square of n digits will have digits, 2 if n is odd.

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True

Q73Short answer

Square root of a number x is denoted by x.

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True

Q74Short answer

A number having 7 at its ones place will have 3 at the units place of its square. What’s the Error? 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?

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False

Q75Short answer

A number having 7 at its ones place will have 3 at the ones place of its cube.

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True

Q76Short answer

The cube of a one digit number cannot be a two digit number.

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False

Q77Short answer

Cube of an even number is odd.

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False

Q78Short answer

Cube of an odd number is even.

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False

Q79Short answer

Cube of an even number is even.

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True

Q80Short answer

Cube of an odd number is odd.

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True

Q81Short answer

999 is a perfect cube.

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False

Q82Short answer

363 × 81 is a perfect cube.

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False

Q83Short answer

Cube roots of 8 are +2 and –2.

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False

Q84Short answer

3 8 + 27 = 3 8 + 3 27 .

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False

Q85Short answer

There is no cube root of a negative integer.

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False

Q86Short answer

Square of a number is positive, so the cube of that number will also be positive. Solve the following questions.

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False

Q87Short answer

Write the first five square numbers.

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1, 4, 9, 16, 25

Q88Short answer

Write cubes of first three multiples of 3.

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27, 216, 729 90. 1+3+5+7+9+11+13+15+17

Q89Short answer

Show that 500 is not a perfect square.

Q90Short answer

Express 81 as the sum of first nine consecutive odd numbers.

Q91Multiple choice

Using prime factorisation, find which of the following are perfect squares.

  • (a)484
  • (b)11250
  • (c)841
  • (d)729
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(a) 484

(b) 11250

(c) 841

(d) 729

Q92Multiple choice

Using prime factorisation, find which of the following are perfect cubes.

  • (a)128
  • (b)343
  • (c)729
  • (d)1331
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(a) 128

(b) 343

(c) 729

(d) 1331

Q93Multiple choice

Using distributive law, find the squares of

  • (a)101
  • (b)72
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(a) 101

(b) 72

Q94Short answer

Can a right triangle with sides 6cm, 10cm and 8cm be formed? Give reason.

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Yes, because 62 + 82 = 102

Q95Short answer

Write the Pythagorean triplet whose one of the numbers is 4.

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(3, 4, 5)

Q96Multiple choice

Using prime factorisation, find the square roots of

  • (a)11025
  • (b)4761
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(a) 11025

(b) 4761

Q97Multiple choice

Using prime factorisation, find the cube roots of

  • (a)512
  • (b)2197
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(a) 512

(b) 2197

Q98Short answer

Is 176 a perfect square? If not, find the smallest number by which it should be multiplied to get a perfect square.

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No, 11

Q99Short answer

Is 9720 a perfect cube? If not, find the smallest number by which it should be divided to get a perfect cube.

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No, 75

Q100Short answer

Write two Pythagorean triplets each having one of the numbers as 5.

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3, 4, 5 and 5, 12, 13

Q101Short answer

By what smallest number should 216 be divided so that the quotient is a perfect square. Also find the square root of the quotient.

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6; 6

Q102Short answer

By what smallest number should 3600 be multiplied so that the quotient is a perfect cube. Also find the cube root of the quotient.

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60; 60

Q103Multiple choice

Find the square root of the following by long division method.

  • (a)1369
  • (b)5625
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(a) 1369

(b) 5625

Q104Multiple choice

Find the square root of the following by long division method.

  • (a)27.04
  • (b)1.44
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(a) 27.04

(b) 1.44

Q105Short answer

What is the least number that should be subtracted from 1385 to get a perfect square? Also find the square root of the perfect square.

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16; 37

Q106Short answer

What is the least number that should be added to 6200 to make it a perfect square?

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41, 79

Q107Short answer

Find the least number of four digits that is a perfect square.

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1024

Q108Short answer

Find the greatest number of three digits that is a perfect square.

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961

Q109Short answer

Find the least square number which is exactly divisible by 3, 4, 5, 6 and 8.

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3600

Q110Short answer

Find the length of the side of a square if the length of its diagonal is 10cm.

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50 or 5 2

Q111Short answer

A decimal number is multiplied by itself. If the product is 51.84, find the number.

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7.2

Q112Short answer

Find the decimal fraction which when multiplied by itself gives 84.64.

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9.2

Q113Short answer

A farmer wants to plough his square field of side 150m. How much area will he have to plough?

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22500 m2

Q114Short answer

What will be the number of unit squares on each side of a square graph paper if the total number of unit squares is 256?

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16

Q115Short answer

If one side of a cube is 15m in length, find its volume.

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3,375

Q116Short answer

The dimensions of a rectangular field are 80m and 18m. Find the length of its diagonal.

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82m

Q117Short answer

Find the area of a square field if its perimeter is 96m.

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576 m2

Q118Short answer

Find the length of each side of a cube if its volume is 512 cm3.

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8 cm

Q119Short answer

Three numbers are in the ratio 1:2:3 and the sum of their cubes is 4500. Find the numbers.

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5, 10 and 15

Q120Short answer

How many square metres of carpet will be required for a square room of side 6.5m to be carpeted.

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42.25 m2

Q121Short answer

Find the side of a square whose area is equal to the area of a rectangle with sides 6.4m and 2.5m.

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4

Q122Short answer

Difference of two perfect cubes is 189. If the cube root of the smaller of the two numbers is 3, find the cube root of the larger number.

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6

Q123Short answer

Find the number of plants in each row if 1024 plants are arranged so that number of plants in a row is the same as the number of rows.

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32

Q124Short answer

A hall has a capacity of 2704 seats. If the number of rows is equal to the number of seats in each row, then find the number of seats in each row.

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52

Q125Short answer

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?

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104

Q126Short answer

8649 students were sitting in a lecture room in such a manner that there were as many students in the row as there were rows in the lecture room. How many students were there in each row of the lecture room?

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93

Q127Short answer

Rahul walks 12m north from his house and turns west to walk 35m to reach his friend’s house. While returning, he walks diagonally from his friend’s house to reach back to his house. What distance did he walk while returning?

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37 m

Q128Short answer

A 5.5m long ladder is leaned against a wall. The ladder reaches the wall to a height of 4.4m. Find the distance between the wall and the foot of the ladder.

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3.3 m

Q129Long answer

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?

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900

Q130Short answer

Find three numbers in the ratio 2:3:5, the sum of whose squares is 608.

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8, 12, 20

Q131Short answer

Find the smallest square number divisible by each one of the numbers 8, 9 and 10.

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3600

Q132Short answer

The area of a square plot is 101 m2. Find the length of one side of the plot.

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10 m

Q133Short answer

Find the square root of 324 by the method of repeated subtraction.

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18

Q134Short answer

Three numbers are in the ratio 2:3:4. The sum of their cubes is 0.334125. Find the numbers.

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0.3, 0.45, 0.6

Q135Short answer

Evaluate : 3 27 + 3 0.008 + 3 0.064 2 2 2

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3.6

Q136Short answer

5 + ( ) 2 1

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50,653

Q137Short answer

6 + ( 8 ) 2

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85, 184

Q138Short answer

A perfect square number has four digits, none of which is zero. The digits from left to right have values that are: even, even, odd, even. Find the number.

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8836

Q139Short answer

Put three different numbers in the circles so that when you add the numbers at the end of each line you always get a perfect square.

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6, 19, 30

Q140Short answer

The perimeters of two squares are 40 and 96 metres respectively. Find the perimeter of another square equal in area to the sum of the first two squares.

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104

Q141Short answer

A three digit perfect square is such that if it is viewed upside down, the number seen is also a perfect square. What is the number? (Hint: The digits 1, 0 and 8 stay the same when viewed upside down, whereas 9 becomes 6 and 6 becomes 9.)

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196, 961

Q142Fill in the blanks

13 and 31 is a strange pair of numbers such that their squares 169 and 961 are also mirror images of each other. Can you find two other such pairs? 1. Quick Tricks. Let me teach you Alright Geeta. a trick, Karan. Pick any 2-digit number ending in 5. Take the tens part Tens part of 45 is 4. of the number and Its successor is 5, multiply it with so 4 × 5 = 20. its successor. Now prefix this Square of 5 is 25, so number to the number obtained square of 5. is 2025. MATHEMATICS UNIT-3 Good. Check it by actual The square is correct. multiplication. Friends, you can also try this trick for 425 or 705 or any other 3-digit number ending in 5. You will definitely enjoy it. Geeta, now let me teach you one trick. This trick can help Sure you to find the cube root n of any 4, 5 or 6-digit perfect cube orally. Alright. But what do we have to Pick any 4, 5 or 6 n1 do for it? digit perfect cube. From the right put a 91125 comma after 3 digits. 91, 125 See the digit at the units place and find the units place of its cube. SQUARE-SQUARE ROOT AND CUBE-CUBE ROOT 99 MATHEMATICS Digit at units Good! This number is place is 5 and digit the digit at units place at ones place of of the cube root. Now its cube see the digits before is also 5. the comma. Ascertain which 91 number's cube is less than this number. The cube of 4 is Absolutely. This is the 64 which is less digit at the tens place than 91. of the cube root. So it means You are right. that the cube root is 45. Friends, you can also try the same for Fascinating! 13824, 2197, 50653 or any other perfect cube of 4, 5 or 6 digit. Bye for now! 2. Cross Number Puzzle Down 1. Cube of 9. 2. Missing number to make 12, ____, 37, a pythagorean triplet. 4. Smallest number by which 248 be multiplied to make the resultant a perfect cube number. 5. Square of 75. 6. Smallest square number that is divisible by each of 5 and 11 9. Without adding, find the sum of 1 + 3 + 5 + 7 + 9 + 11. 10. Smallest number which when added to 7669 makes the resultant a perfect square. Across 2. Square of 19. 3. Look at the numbers given below and find the number which cannot be a perfect square. 81, 100, 144, 25000 7. Square root of 4489 8. Smallest natural number other than 1 which is a perfect square as well as a perfect cube number. 10. Cube root of 357911. 11. Smallest number which when subtracted from 374695 makes the resultant a perfect square number. MATHEMATICS Rough Work UNIT-3 Rough Work SQUARE-SQUARE ROOT AND CUBE-CUBE ROOT 103 MATHEMATICS Rough Work

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12, 21, 102, 201 Cross Number Puzzle