Question 12 Marks
By what least number should the given number be multiplied to get a perfect square number? In case, find the number whose square is the new number.
7623
AnswerResolving 7623 into prime factors:
$7623$
$=3 \times 3 \times 7 \times 11 \times 11$
$=3^2 \times 7 \times 11^2$
Thus, to get a perfect square, the given number should be multiplied by 7 .
New number $=\left(3^2 \times 7^2 \times 11^2\right)$
$=(3 \times 7 \times 11)^2$
$=(231)^2$
Hence, the number whose square is the new number is 231 .
View full question & answer→Question 22 Marks
Using the prime factorisation method, find the following numbers are perfect squares:
$441$
AnswerA perfect square can always be expressed as a product of equal factors.
Resolving into prime factors:
$441$
$=19 \times 9$
$=7 \times 7 \times 3 \times 3$
$=7 \times 3 \times 7 \times 3$
$=21 \times 21$
$=(21)^2$
Thus, 441 is a perfect square.
View full question & answer→Question 32 Marks
By what least number should the given number be multiplied to get a perfect square number? In case, find the number whose square is the new number.
2925
AnswerResolving 2925 into prime factors:
$2925$
$=3 \times 3 \times 5 \times 5 \times 13$
$=3^2 \times 5^2 \times 13$
Thus, to get a perfect square, the given number should be multiplied by 13 .
New number $=\left(3^2 \times 5^2 \times 13^2\right)$
$=(3 \times 5 \times 13)^2$
$=(195)^2$
Hence, the number whose square is the new number is 195.
View full question & answer→Question 42 Marks
By what least number should the given number be multiplied to get a perfect square number? In case, find the number whose square is the new number.
$3380$
Answer
Resolving 3380 into prime factors:
$3380$
$=2 \times 2 \times 5 \times 13 \times 13$
$=2^2 \times 5 \times 13^2$
Thus, to get a perfect square, the given number should be multiplied by 5 .
New number $=\left(2^2 \times 5^2 \times 13^2\right)$
$=(2 \times 5 \times 13)^2$
$=(130)^2$
Hence, the new number is the square of 130.
View full question & answer→Question 52 Marks
Find the largest number of 2 digits which is a perfect square.
AnswerThe first three digit number (100) is a perfect square. Its square root is 10.
The number before 10 is 9.
Square of $9=(9)^2=81$
Thus, the largest 2 digit number that is a perfect square is 81.
View full question & answer→Question 62 Marks
Using the formula $(a-b)^2=\left(a^2-2 a b+b^2\right)$, evaluate:
$(891)^2$
Answer$(891)^2$
$=(900-9)^2$
$=(900)^2-2 \times 900 \times 9+(9)^2$
$=810000-16200+81$
$=793881$
View full question & answer→Question 72 Marks
Show that the following numbers is a perfect square. In case, find the number whose square is the given number:
$2601$
AnswerA perfect square is a product of two perfectly equal numbers.
Resolving into prime factors:
$2601$
$=9 \times 289$
$=3 \times 3 \times 17 \times 17$
$=3 \times 17 \times 3 \times 17$
$=51 \times 51$
$=(51)^2$
Thus, 2601 is the perfect square of 51.
View full question & answer→Question 82 Marks
Write a pythagorean triplet whose smallest member is:
$20$
AnswerFor every natural number $m>1,\left(2 m, m^2-1, m^2+1\right)$ is a Pythagorean triplet.
Putting $2 m=20$,
We get $m =10$
Thus, we get the triplet $(20,99,10)$.
View full question & answer→Question 92 Marks
Evaluate:
$\frac{\sqrt{1183}}{\sqrt{2023}}$
Answer$\frac{\sqrt{1183}}{\sqrt{2023}}$
$=\sqrt{\frac{1183}{2023}}$
$=\sqrt{\frac{1183\div7}{2023\div7}}$
$=\frac{\sqrt{169}}{\sqrt{289}}$
$=\frac{\sqrt{13\times13}}{\sqrt{17\times17}}$
$=\frac{13}{17}$
View full question & answer→Question 102 Marks
Write a pythagorean triplet whose smallest member is:
$14$
Answer
For every natural number $m>1,\left(2 m, m^2-1, m^2+1\right)$ is a Pythagorean triplet.
Putting $2 m=14$,
We get $m=7$
Thus, we get the triplet $(14,48,50)$.
View full question & answer→Question 112 Marks
Find the square root of number by using the method of prime factorisation:
225
AnswerBy prime factorisation method:
$225=3\times3\times5\times5$
$\therefore\sqrt{225}=(3\times5)=15$
View full question & answer→Question 122 Marks
Evaluate:$\sqrt{9.8596}$
Answer$\begin{array}{c|c} &3.14 \\ \hline 3 & 9.\ \overline{85}\ \overline{96}\\& -9\ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \hline61 &\ \ \ \ 85\ \ \ \ \ \ \\ &-61\ \ \ \ \ \ \\ \hline624 &\ 2496 \\ &-2496\ \ \\ \hline &\ \times \end{array}$
$\therefore\sqrt{9.8596}=3.14$
View full question & answer→Question 132 Marks
Find the square root of number by using the method of prime factorisation:
11025
AnswerBy prime factorisation method: $11025=3\times3\times5\times5\times7\times7$$\therefore\sqrt{11025}=(3\times5\times7)=105$
View full question & answer→Question 142 Marks
Write a pythagorean triplet whose smallest member is:
$6$
AnswerFor every natural number $m>1,\left(2 m, m^2-1, m^2+1\right)$ is a Pythagorean triplet.
Putting $2 m=6$
We get $m =3$
Thus, we get the triplet $(6,8,10)$.
View full question & answer→Question 152 Marks
Evaluate:
$94 \times 106$
Answer$(100-6) \times(100+6)$
$= {\left[(100)^2-(6)^2\right] }$
$= (10000-36)$
$=9964$
View full question & answer→Question 162 Marks
Show that the following numbers is a perfect square. In case, find the number whose square is the given number:
$7056$
AnswerA perfect square is a product of two perfectly equal numbers.
Resolving into prime factors:
$7056$
$=12 \times 588$
$=12 \times 7 \times 84$
$=12 \times 7 \times 12 \times 7$
$=(12 \times 7)^2$
$=(84)^2$
Thus, 7056 is the perfect square of 84 .
View full question & answer→Question 172 Marks
Write a pythagorean triplet whose smallest member is:
$16$
AnswerFor every natural number $m>1,\left(2 m, m^2-1, m^2+1\right)$ is a Pythagorean triplet.
Putting $2 m=16$,
We get $m =8$
Thus, we get the triplet $(16,63,65)$.
View full question & answer→Question 182 Marks
Evaluate:
$(105)^2-(104)^2$
AnswerWe have,
$(n+1)^2-n^2=(n+1)+n$
Taking $n=104$ and $(n+1)=105$
We get,
$(105)^2-(104)^2=(105+104)=209$
View full question & answer→Question 192 Marks
By what least number should the given number be divided to get a perfect square number? In case, find the number whose square is the new number.
$4500$
AnswerResolving 4500 into prime factors:
$4500$
$=2 \times 2 \times 3 \times 3 \times 5 \times 5 \times 5$
$=2^2 \times 3^2 \times 5^2 \times 5$
Thus, to get a perfect square, the given number should be divided by 5 .
New number obtained $=\left(2^2 \times 3^2 \times 5^2\right)$
$=(2 \times 3 \times 5)^2$
$=(30)^2$
Hence, the new number is the square of 30.
View full question & answer→Question 202 Marks
Evaluate:$\sqrt{1.69}$
Answer$\begin{array}{c|c} &1.3 \\ \hline 1 & \bar{1}\ \overline{.69}\\& -1\ \ \ \ \ \ \ \ \ \\ \hline23 &\ \ 69\\ &-69\\ \hline &\ \ \ \times \end{array}$
$\therefore\sqrt{1.69}=1.3$
View full question & answer→Question 212 Marks
Evaluate:$\sqrt{75.69}$
Answer$\begin{array}{c|c} &8.7 \\ \hline 8 & \overline{75}\ \overline{.69}\\&-64 \ \ \ \ \ \ \ \ \ \\ \hline167 &\ \ \ \ \ \ \ \ \ 1169\ \ \ \ \ \\ &\ \ \ \ \ -1169\ \ \ \ \ \ \\ \hline &\ \ \ \ \ \ \ \ \ \ \times \end{array}$
$\therefore\sqrt{75.69}=8.7$
View full question & answer→Question 222 Marks
Evaluate:$\sqrt{156.25}$
Answer$\begin{array}{c|c} &12.5 \\ \hline 1 & \overline{156}\ \overline{.25}\\& -1\ \ \ \ \ \ \ \ \ \ \ \ \ \\ \hline28 &\ 56\ \ \ \ \ \\ &-56\ \ \ \ \ \ \\ \hline5 &\ \ \ \ \ \ \ \ \ 25 \\ &\ \ \ \ \ \ -25 \\ \hline &\ \ \ \ \ \ \ \ \ \ \times \end{array}$
$\therefore\sqrt{156.25}=12.5$
View full question & answer→Question 232 Marks
Using the formula $(a-b)^2=\left(a^2-2 a b+b^2\right)$, evaluate:
$(689)^2$
Answer$(689)^2$
$=(700-11)^2$
$=(700)^2-2 \times 700 \times 11+(11)^2$
$=490000-15400+121$
$=474721$
View full question & answer→Question 242 Marks
By what least number should the given number be divided to get a perfect square number? In case, find the number whose square is the new number.
$3380$
AnswerResolving 3380 into prime factors:
$3380$
$=2 \times 2 \times 5 \times 13 \times 13$
$=2^2 \times 5 \times 13^2$
Thus, to get a perfect square, the given number should be divided by 5 .
$\text { New number obtained }=\left(2^2 \times 13^2\right)$
$=(2 \times 13)^2$
$=(26)^2$
Hence, the new number is the square of 26 .
View full question & answer→Question 252 Marks
Using the prime factorisation method, find the following numbers are perfect squares:
$4225$
AnswerA perfect square can always be expressed as a product of equal factors.
Resolving into prime factors:
$4225$
$=25 \times 169$
$=5 \times 5 \times 13 \times 13$
$=5 \times 13 \times 5 \times 13$
$=65 \times 65$
$=(65)^2$
Thus, 4225 is a perfect square.
View full question & answer→Question 262 Marks
Find the least number which must be subtracted from 2509 to make it a perfect square.
AnswerFinding the square root of 2509 by division we find that 9 is left as remainder
$\begin{array}{c|c} & 50 \\ \hline 5 & \overline{25}\ \overline{09}\\& 25 \ \ \ \ \ \\ \hline100 &\ \ \ \ \ \ \ 09\\ &\ \ \ \ \ \ \ 00\\ \hline &\ \ \ \ \ \ \ 09 \end{array}$
9 must be subtracted to get the perfect square 100
Least number to be subtracted = 9
View full question & answer→Question 272 Marks
Find the square root of number by using the method of prime factorisation:
17424
AnswerBy prime factorisation method: $17424=2\times2\times2\times2\times3\times3\times11\times11$$\therefore\sqrt{17424}=(2\times2\times3\times11)=132$
View full question & answer→Question 282 Marks
Evaluate:
$78 \times 82$
Answer$=(80-2) \times(80+2)$
$= {\left[(80)^2-(2)^2\right] }$
$= (6400-4)$
$=6396$
View full question & answer→Question 292 Marks
By what least number should the given number be divided to get a perfect square number? In case, find the number whose square is the new number.
$4056$
Answer
Resolving 4056 into prime factors:
$4056$
$=2 \times 2 \times 2 \times 3 \times 13 \times 13$
$=2^2 \times 2 \times 3 \times 13^2$
Thus, to get a perfect square, the given number should be divided by 6 , which is a product of 2 and 3.
$\text { New number }$
$=(2 \times 13)^2$
$=(26)^2$
Hence, the new number is the square of 26.
View full question & answer→Question 302 Marks
Evaluate:
$\sqrt{17956}$
Answer$\begin{array}{c|c} & 134 \\ \hline 1 & \bar1\ \overline{79}\ \overline{56}\\& 1\ \ \ \ \ \ \ \ \ \ \\ \hline23 &79\\ &69\ \\ \hline264 &1056\\ &1056\\\hline &\times \end{array}$
$\sqrt{17956}=134$
View full question & answer→Question 312 Marks
Evaluate:
$\frac{\sqrt{80}}{\sqrt{405}}$
Answer$\frac{\sqrt{80}}{\sqrt{405}}$
$=\sqrt{\frac{80}{405}}$
$=\sqrt{\frac{16}{81}}$
$=\frac{\sqrt{16}}{\sqrt{81}}$
$=\frac{4}{9}$
View full question & answer→Question 322 Marks
By what least number should the given number be multiplied to get a perfect square number? In case, find the number whose square is the new number.
$3332$
Answer
Resolving 3332 into prime factors:
$3332$
$=2 \times 2 \times 7 \times 7 \times 17$
$=2^2 \times 7^2 \times 17$
Thus, to get a perfect square, the given number should be multiplied by 17 .
$\text { New number }=\left(2^2 \times 7^2 \times 17^2\right)$
$=\left(2 \times 7 \times 17^2\right)$
$=(238)^2$
Hence, the new number is the square of 238.
View full question & answer→Question 332 Marks
Evaluate:
$(38)^2-(37)^2$
Answer
We have,
$(n+1)^2-n^2=(n+1)+n$
Taking $n=37$ and $(n+1)=38$,
We get,
$(38)^2-(37)^2=(38+37)=75$
View full question & answer→Question 342 Marks
Find the value of using the column method:
$(23)^2$
AnswerGiven number $23=20+3$
Here,
$a =20$ and $b =3$
| $a ^2$ |
$2ab$
|
$b_2$
|
| $(20)^2=400$ |
$2 × 20 × 3 = 120$
|
$(3)^2=9$
|
$\therefore(23)^2=(400+120+9)=529$ View full question & answer→Question 352 Marks
Evaluate:
$\sqrt{19600}$
Answer$\begin{array}{c|c} & 140 \\ \hline 1 & \bar1\ \overline{96}\ \overline{00}\\& 1\ \ \ \ \ \ \ \ \ \ \\ \hline24 &96\\ &96\ \\ \hline280 &00\\ &00\\\hline &\times \end{array}$
$\sqrt{19600}=140$
View full question & answer→Question 362 Marks
Using the formula $(a+b)^2=\left(a^2+2 a b+b^2\right)$, evaluate:
$(310)^2$
Answer$(310)^2$
$=(300+10)^2$
$=(300)^2+2 \times 300 \times 10+(10)^2$
$=(90000+6000+100)$
$=96100$
View full question & answer→Question 372 Marks
Express 100 as the sum of 10 odd numbers.
AnswerWe know that $n ^2$ is equal to the sum of first n odd numbers.
$100=10^2$
$=$ Sum of 10 odd numbers $=(1+3+5+7+9+11+13+15+17+19)$
View full question & answer→Question 382 Marks
By what least number should the given number be multiplied to get a perfect square number? In case, find the number whose square is the new number.
$2156$
AnswerResolving 2156 into prime factors:
$2156$
$=2 \times 2 \times 7 \times 7 \times 11$
$=\left(2^2 \times 7^2 \times 11\right)$
Thus to get a perfect square, the given number should be multiplied by 11 ,
$\text { New number }=\left(2^2 \times 7^2 \times 11\right)$
$=\left(2 \times 7 \times 11^2\right)$
$=(154)^2$
Hence, the new number is the square of 154 .
View full question & answer→Question 392 Marks
Evaluate:
$\sqrt{7056}$
Answer$\begin{array}{c|c} & 84 \\ \hline 8 & \overline{70}\ \overline{56}\\& 64\ \ \ \ \ \\ \hline164 &656\\ &656\\ \hline &\times \end{array}$
$\sqrt{7056}=84$
View full question & answer→Question 402 Marks
Evaluate:
$\sqrt{14161}$
Answer$\begin{array}{c|c} & 119 \\ \hline 1 & \bar1\ \overline{41}\ \overline{61}\\& 1\ \ \ \ \ \ \ \ \ \ \\ \hline21 &41\\ &21\ \\ \hline229 &2061\\ &2061\\\hline &\times \end{array}$
$\sqrt{14161}=119$
View full question & answer→Question 412 Marks
Evaluate:$\sqrt{10.0489}$
Answer$\begin{array}{c|c} &3.17 \\ \hline 3 & \overline{10}.\ \overline{04}\ \overline{89}\\& -9\ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \hline61 &104\ \ \ \ \ \ \\ &-61\ \ \ \ \ \ \ \\ \hline627 &\ 4389 \\ &-4389\ \ \\ \hline &\ \times \end{array}$
$\therefore\sqrt{10.0489}=3.17$
View full question & answer→Question 422 Marks
Using the prime factorisation method, find the following numbers are perfect squares:
1176
AnswerA perfect square can always be expressed as a product of equal factors.
Resolving into prime factors:
1176
= 7 × 168
= 7 × 21 × 8
= 7 × 7 × 3 × 2 × 2 × 2
1176 cannot be expressed as a product of two equal numbers.
Thus, 1176 is not a perfect square.
View full question & answer→Question 432 Marks
Evaluate:
$\sqrt{9025}$
Answer$\begin{array}{c|c} & 95 \\ \hline 9 & \overline{90}\ \overline{25}\\& 81\ \ \ \ \ \\ \hline185 &925\\ &925\\ \hline &\times \end{array}$
$\sqrt{9025}=95$
View full question & answer→Question 442 Marks
Using the formula $(a+b)^2=\left(a^2+2 a b+b^2\right)$, evaluate:
$(508)^2$
Answer$(508)^2$
$=(500+8)^2$
$=(500)^2+2 \times 500 \times 8+(8)^2$
$=(250000+8000+64)$
$=258064$
View full question & answer→Question 452 Marks
Evaluate $\sqrt{3}$ up to two places of decimal.
Answer$\begin{array}{c|c} &1.732 \\ \hline 1 &3. \overline{00}\ \overline{00}\ \overline{00}\\& -1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \hline27 &200\ \ \ \ \ \ \ \ \ \ \ \\ &-189\ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \hline343 &\ 1100\ \ \ \ \ \ \\ &-1029\ \ \ \ \ \ \ \ \\ \hline \hline3462 &\ \ \ \ \ \ \ 7100\ \ \ \ \ \ \\ &\ \ \ \ -6924\ \ \ \ \ \ \ \ \\ \hline&\ \ \ \ \ \ \ \ \ \ \ \ 176\ \ \ \ \ \ \ \ \end{array}$
$\therefore\sqrt{3}=1732=1.73$
(Correct up to two places of decimal)
View full question & answer→Question 462 Marks
Find the square root of number by using the method of prime factorisation:
15876
AnswerBy prime factorisation method: $15876=2\times2\times3\times3\times3\times3\times7\times7$$\therefore\sqrt{15876}=(2\times3\times3\times7)=126$
View full question & answer→Question 472 Marks
By what least number should the given number be divided to get a perfect square number? In case, find the number whose square is the new number.
$1575$
AnswerResolving 1575 into prime factors:
$1575$
$=3 \times 3 \times 5 \times 5 \times 7$
$=3 \times 5 \times 7$
Thus, to get a perfect square, the given number should be divided by 7 .
New number obtained $=\left(3^2 \times 5^2\right)$
$=(3 \times 5)^2$
$=(15)^2$
Hence, the new number is the square of 15.
View full question & answer→Question 482 Marks
Show that the following numbers is a perfect square. In case, find the number whose square is the given number:
$1225$
AnswerA perfect square is a product of two perfectly equal numbers.
Resolving into prime factors:
$1225$
$=25 \times 49$
$=5 \times 5 \times 7 \times 7$
$=5 \times 7 \times 5 \times 7$
$=35 \times 35$
$=(35)^2$
Thus, 1225 is the perfect square of 35.
View full question & answer→Question 492 Marks
Evaluate:
$\sqrt{\frac{121}{256}}$
Answer$\sqrt{\frac{121}{256}}$
$=\frac{\sqrt{121}}{\sqrt{256}}$
$=\sqrt{\frac{11\times11}{16\times16}}$
$=\frac{11}{16}$
View full question & answer→Question 502 Marks
Using the prime factorisation method, find the following numbers are perfect squares:
$576$
AnswerA perfect square can always be expressed as a product of equal factors.
Resolving into prime factors:
$576$
$=64 \times 9$
$=8 \times 8 \times 3 \times 3$
$=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3$
$=24 \times 24$
$=(24)^2$
Thus, 576 is a perfect square
View full question & answer→