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MATHS 2 Marks Questions

A Square and A Cube · Gujarat Board (GSEB) STD 8 (GSEB - English Medium). 12 questions.

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2 Marks Questions

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12 questions · self-marked practice — reveal the answer and mark yourself.

Question 12 Marks
How many square numbers are there between 1 and 100? How many are between 101 and 200? Using the table of squares you filled earlier, enter the values below, tabulating the number of squares in each block of 100. What is the largest square less than 1000?
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Answer

The largest square less than 1000 is $31^2=961$.
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Question 22 Marks
Find how many numbers lie between two consecutive perfect squares. Do you notice a pattern?
Answer
There are exactly ' $2 n$ ' numbers between $n^2$ and $(n+1)^2$. For example: Between $3^2(n)$ and $4^2(n +1)$ there are $2 \times 3=6(2 n)$ numbers.
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Question 32 Marks
Consider the following numbers and their squares.
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If a number contains 3 zeros at the end, how many zeros will its square have at the end?
Answer
The number of zeros at the end of the square of a number is always double the number of zeros at the end of the original number. Therefore, if a number contains 3 zeros at the end, then its square will have 6 zeros.
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Question 42 Marks
If a number ends in 0, 1, 4, 5, 6, or 9, is it always a square?
Answer
We cannot determine if a number is a square just by looking at the digit in the units place. But the unit digit can tell us when a number is not a square. If a number ends with 2, 3, 7, or 8, then we can say that it is not a square. For example, 26 ends in 6 but is not a perfect square.
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Question 52 Marks
What number will you multiply by 1323 to make it a cube number?
Answer
Prime factorization of $1323=\underline{3 \times 3 \times 3} \times 7 \times 7$
Here, there is no triplet of 7.
So, 1323 is not a perfect cube. To make it a cube number, we multiply it by 7 .
$1323 \times 7=\underline{3 \times 3 \times 3} \times \underline{7 \times 7 \times 7}=9261$ which is a perfect square.
Hence, the number by which 1323 needs to be multiplied to make it a cube number is 7 .
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Question 62 Marks
The next two taxicab numbers after 1729 are 4104 and 13832. Find the two ways in which each of these can be expressed as the sum of two positive cubes.
Answer
4104 :
$\begin{array}{l}2^3+16^3=8+4096=4104 \\9^3+15^3=729+3375=4104\end{array}$

13832 :
$\begin{array}{l}2^3+24^3=8+13824=13832 \\18^3+20^3=5832+8000=13832\end{array}$
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Question 72 Marks
Can a cube end with exactly two zeroes (00)? Explain.
Answer
No. A cube cannot end with exactly two zeros because zeros in a cube occur in multiples of three. If a number ends in one zero, its cube ends in three zeros.
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Question 82 Marks
In the following pattern, fill in the missing numbers :
$\begin{array}{l}1^2 \times 2^2 \times 2^2=3^2 \\2^2 \times 3^2 \times 6^2=7^2 \\3^2 \times 4^2 \times 12^2=13^2 \\4^2 \times 5^2 \times 20^2=(\ )^2 \\9^2 \times 10^2 \times(\ )^2=(\ )^2\end{array}$
Answer
$\begin{array}{l}1^2 \times 2^2 \times 2^2=3^2 \\ 2^2 \times 3^2 \times 6^2=7^2 \\ 3^2 \times 4^2 \times 12^2=13^2 \\ 4^2 \times 5^2 \times 20^2=(\underline{21})^2 \\ 9^2 \times 10^2 \times(\underline{90})^2=(\underline{91})^2\end{array}$
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Question 92 Marks
How many numbers lie between the squares of the following numbers?
(i) 16 and 17 (ii) 99 and 100
Answer
There are $2 n$ numbers between $n^2$ and $(n+1)^2$.
(i) $16^2$ and $17^2$
Here, $n =16$ and $( n +1)=17$
Therefore, the numbers between $16^2$ and $17^2=2 n=2 \times 16=32$.

(ii) $99^2$ and $100^2$
Here, $n =99$ and $( n +1)=100$
Therefore, the numbers between the squares $99^2$ and $100^2=2 n=2 \times 99=198$.
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Question 102 Marks
Find the smallest square number that is divisible by each of the following numbers : 4, 9, and 10.
Answer
The L.C.M. of 4, 9, and 10 is 180.
So, the smallest number divisible by 4,9 , and 10 is 180 .
$\begin{array}{l}180=\underline{2 \times 2} \times \underline{3 \times 3} \times 5 \\180=2^2 \times 3^2 \times 5\end{array}$
Here, 5 has no pair.
So, 180 is not a perfect square. To make it a perfect square, we multiply it by 5.
Hence, the required smallest square number is $180 \times 5=900$.
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Question 112 Marks
Find the length of the side of a square whose area is $441\ m^2$.
Answer
Area of the square $=$ side $\times$ side $=441\ m^2$.
$\begin{array}{l}(\text {side})^2=441 \\\text {side}=\sqrt{441} \\\text {side}= \pm 21\ m\end{array}$
Since the side of a square cannot be negative, the length of the side of the square is 21 m .
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Question 122 Marks
Given $125^2=15625$, what is the value of $126^2$ ?
(i) $15625+126$
(ii) $15625+26^2$
(iii) $15625+253$
(iv) $15625+251$
(v) $15625+25^2$
Answer
$125^2=15625$
This means 15625 is the sum of the first 125 odd numbers.
$126^2=15625+127^{\text {th }}$ odd number
$127^{\text {th }}$ odd number $=(2 \times 127)-1=252-1=251$.
$\therefore 126^2=15625+251$
Therefore, (iv) $15625+251$ is the correct answer.
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