## Skill Check: IITJEE XIV: Foundation Maths: Squares and square roots

Tutorial Exercises: (Reference: An old ICSE Text, Oxford Publishers, Class VIII, Abhijit Mukherjea and Sushil Kumar Sharma)

I) Find the value of the following:

(1a) $14^{2}$ (1b) $27^{2}$ (1c) $118^{2}$ (1d) $(\frac{1}{5})^{2}$ (1e) $(\frac{3}{8})^{2}$ (1f) $(1\frac{2}{5})^{2}$ (1g) $(2\frac{4}{5})^{2}$ (1h) $(0.1)^{2}$ (1i) $(5.8)^{2}$ (1j) $(3.05)^{2}$

II) Which of the following perfect squares will have even square roots and which will have odd square roots?

(2a) 441 (2b) 2916 (2c) 3969 (2d) 21609 (2e) 389376

III) Which of the following numbers are perfect squares ?

(3a) 128 (3b) 256 (3c) 2187 (3d) 6561 (3e) 6084

IV) Find the perfect square obtained by multiplying each of the following numbers by the smallest possible number:

(4a) 882 (4b) 405 (4c) 567 (4d) 1690 (4e) 7776

V) Divide the following numbers by the smallest possible number to make each one a perfect square:

(5a) 1152 (5b) 2187 (5c) 3267 (3d) 1536 (3e) 10140

VI) Find the square roots of the following natural numbers by the prime factorization method:

(6a) 225 (6b) 1296 (6c) 1764 (6d) 2401 (6e) 2916 (6f) 5184 (6g) 7225 (6h) 10816 (6i) 14400 (6j) 13456

VII) Find the square roots of the following fractions by the prime factorisation method:

(7a) $\frac{4}{9}$ (7b) $\frac{100}{121}$ (7c) $\frac{49}{64}$ (7d) $\frac{484}{625}$ (7e) $\frac{729}{900}$ (7f) $1\frac{9}{16}$ (7g) $1\frac{11}{25}$ (7h) $1 \frac{32}{49}$ (7i) $10 \frac{9}{16}$ (7j) $8\frac{1}{36}$

VIII) Find the square roots of the following decimals by the prime factorization method:

(8a) 0.0001 (8b) 1.44 (8c) 3.24 (8d) 7.29 (8e) 10.24 (8f) 30.25 (8g) 0.0049 (8h) 0.1296 (8i) 15.21 (8j) 0.2025

IX) Find the square roots of the following numbers by the division method:

(9a) 4624 (9b) 11664 (9c) 16641 (9d) 44521 (9e) 426409 (9f) 550564 (9g) 840889 (9h) 917764 (9i) 1014049 (9j) 1560001

X) Find the square roots of the following decimal fractions by the division method:

(10a) 34.81 (10b) 44.89 (10c) 5.4756 (10d) 1.3225 (10e) 4.9284 (10f) 11.2896 (10g) 4.5796 (10h)36.2404 (10i) 1.304164 (10j) 4.609609

XI) Find the square roots of the following common fractions by the division method:

(11a) $\frac{121}{289}$ (11b) $1\frac{168}{361}$ (11c) $\frac{676}{729}$ (11d) $\frac{1681}{2209}$ (11e) $\frac{5041}{7921}$

XII) Find the square roots of the following numbers, correct up to 3 decimal places:

(12a) $2$ (12b) $3$ (12c) $5$ (12d) $11$ (12e) $13$ (12f) $1458$ (12g) $27196$ (12h) 14502 (12i) $1828$ (12j) $146923$ (12k) $1.6$ (12l) $1.09$ (12m) $2.87$ (12n) $3.99$ (12o) $21.654$ (12p) $\frac{1}{5}$ (12q) $1 \frac{3}{4}$ (12r) $\frac{7}{20}$ (12s) $\frac{7}{8}$ (12t) $1\frac{7}{16}$

XIII) Find the smallest number that needs to be subtracted from 66671 in order to get a perfect square.

XIV) Find the smallest number that needs to be subtracted from 1051149 in order to get a perfect square.

XV) Find the smallest number that needs to be added to 1485155 in order to get a perfect square.

XVI) Find the smallest and greatest 5-digit numbers that are perfect squares.

XVII) The area of a square is 2.815684 square cm. Find its length.

XVIII) A farmer has 21126 seedlings, which he intends to plant in an equal number of rows and columns. What is the maximum number of rows of seedlings that can be planted? How many seedlings will be left unplanted?

XIX) Chairs need to be laid out in an equal number of rows and columns. If there are 1817 chairs in the stadium, find at least how many more chairs need to be brought in.

Regards,

Nalin Pithwa

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