Exercise: 3-F
Multiple Choice Questions
Q1: How many perfect squares lie between 120 and 300?
Step 1: Find the square root of the lower and upper limits.
\[
\sqrt{120} \approx 10.95 \quad \text{and} \quad \sqrt{300} \approx 17.32
\]Step 2: List the whole numbers between 10.95 and 17.32 → that is:
\[
11, 12, 13, 14, 15, 16, 17
\]Step 3: Now square each number:
\[
11^2 = 121, \quad 12^2 = 144, \quad 13^2 = 169, \quad 14^2 = 196,
\quad 15^2 = 225, \quad 16^2 = 256, \quad 17^2 = 289
\]All these values lie between 120 and 300.
Answer: c. 7 perfect squares
Q2: \(\sqrt{41-\sqrt{21+\sqrt{19-\sqrt9}}}=?\)
Step 1: Start from the innermost square root.
\[
\sqrt{9} = 3
\]Step 2: Replace in expression:
\[
\sqrt{41 – \sqrt{21 + \sqrt{19 – 3}}}
\]Step 3: Evaluate inside the next root:
\[
19 – 3 = 16 \Rightarrow \sqrt{16} = 4
\]Step 4: Now update again:
\[
\sqrt{41 – \sqrt{21 + 4}} = \sqrt{41 – \sqrt{25}} = \sqrt{41 – 5}
\]Step 5: Final steps:
\[
\sqrt{36} = 6
\]Answer: c. 6
Q3: The value of \(\sqrt{0.01}+\sqrt{0.81}+\sqrt{1.44}+\sqrt{0.0009}\) is:
Step 1: Evaluate each square root:
\[
\sqrt{0.01} = 0.1 \\
\sqrt{0.81} = 0.9 \\
\sqrt{1.44} = 1.2 \\
\sqrt{0.0009} = 0.03
\]Step 2: Add the values:
\[
0.1 + 0.9 = 1.0 \\
1.0 + 1.2 = 2.2 \\
2.2 + 0.03 = 2.23
\]Answer: d. 2.23
Q4: If \(\frac{52}{x}=\sqrt{\frac{169}{289}}\) , then the value of x is
Step 1: Simplify the square root on the right-hand side:
\[
\sqrt{\frac{169}{289}} = \frac{\sqrt{169}}{\sqrt{289}} = \frac{13}{17}
\]Step 2: Now equate:
\[
\frac{52}{x} = \frac{13}{17}
\]Step 3: Cross-multiply:
\[
52 \cdot 17 = 13 \cdot x \Rightarrow 884 = 13x
\]Step 4: Solve for \(x\):
\[
x = \frac{884}{13} = 68
\]Answer: d. 68
Q5: \(?\div\sqrt{0.25}=25\)
Step 1: Evaluate the square root:
\[
\sqrt{0.25} = 0.5
\]Step 2: Rewrite the equation:
\[
\frac{?}{0.5} = 25
\]Step 3: Multiply both sides by 0.5 to find \( ? \):
\[
? = 25 \times 0.5 = 12.5
\]Answer: a. 12.5
Q6: For what value of * the statement \(\left(\frac{\ast}{15}\right)\left(\frac{\ast}{135}\right)=1\) is true?
Step 1: Let the unknown value be \( x \), so the equation becomes:
\[
\left(\frac{x}{15}\right)\left(\frac{x}{135}\right) = 1
\]Step 2: Multiply the fractions:
\[
\frac{x^2}{15 \times 135} = 1
\]Step 3: Multiply the denominators:
\[
15 \times 135 = 2025
\]
\[
\frac{x^2}{2025} = 1
\]Step 4: Multiply both sides by 2025:
\[
x^2 = 2025
\]Step 5: Take square root:
\[
x = \sqrt{2025} = 45
\]Answer: d. 45
Q7: What percentage of the numbers from 1 to 50 have that end in the digit 1?
Step 1: List numbers from 1 to 50 that end in digit 1:
1, 9, 11, 19, 21, 29, 31, 39, 41, 49
Step 2: Count them:
There are 10 such numbers.Step 3: Total numbers from 1 to 50 = 50Step 4: Calculate percentage:
\[
\frac{10}{50} \times 100 = 20\%
\]Answer: d. 20
Q8: Which of the following cannot be the unit digit of a perfect square number?
Step 1: List all possible unit digits of perfect squares:Perfect squares end in:
\(0 → (0)^2 = 0 \)
\(1 → (1)^2 = 1, (9)^2 = 81 \)
\(4 → (2)^2 = 4, (8)^2 = 64 \)
\(5 → (5)^2 = 25 \)
\(6 → (4)^2 = 16, (6)^2 = 36 \)
\(9 → (3)^2 = 9, (7)^2 = 49\)
So, possible unit digits of perfect squares are:
0, 1, 4, 5, 6, 9❌ But 8 is not among them.Answer: c. 8
Q9: \(\frac{\sqrt{288}}{\sqrt{128}}=?\)
Step 1: Simplify the square roots:
\[
\frac{\sqrt{288}}{\sqrt{128}} = \sqrt{\frac{288}{128}} = \sqrt{\frac{9}{4}} = \frac{3}{2}
\]Step 2: The simplified value is \( \frac{3}{2} \).Answer: c. \(\frac{3}{2}\)
Q10: \(\sqrt{72}\times\sqrt{98}=?\)
Step 1: Use the property of square roots:
\[
\sqrt{72} \times \sqrt{98} = \sqrt{72 \times 98}
\]Step 2: Multiply the numbers inside the square root:
\[
72 \times 98 = 7056
\]Step 3: Take the square root of 7056:
\[
\sqrt{7056} = 84
\]Answer: d. 84
Q11: The least number by which 294 must be multiplied to make it a perfect square, is
Step 1: Find the prime factorization of 294:
\[
294 = 2 \times 3 \times 7^2
\]Step 2: To make 294 a perfect square, each prime factor should have an even power. The prime factor 2 and 3 both appear with an odd power, so we need to multiply by \( 2 \times 3 = 6 \) to make their powers even.
Step 3: Multiply 294 by 6:
\[
294 \times 6 = 1764
\]Step 4: Verify if 1764 is a perfect square:
\[
\sqrt{1764} = 42 \quad (\text{which is a whole number})
\]Answer: c. 6
Q12: The least number by which 1470 must be divided to get a number which is a perfect square, is
Step 1: Find the prime factorization of 1470:
\[
1470 = 2 \times 3 \times 5 \times 7^2
\]Step 2: To make the number a perfect square, each prime factor should appear with an even power. The prime factors 2, 3, and 5 have odd powers, so we need to divide by \( 2 \times 3 \times 5 = 30 \) to make their powers even.
Step 3: Divide 1470 by 30:
\[
\frac{1470}{30} = 49
\]Step 4: Verify if 49 is a perfect square:
\[
\sqrt{49} = 7 \quad (\text{which is a whole number})
\]Answer: d. 30
Q13: What is the least number to be added to 7700 to make it a perfect square?
Step 1: Find the square root of 7700:
\[
\sqrt{7700} \approx 87.7
\]Step 2: The next whole number greater than 87.7 is 88. Calculate its square:
\[
88^2 = 7744
\]Step 3: Subtract 7700 from 7744 to find the least number to be added:
\[
7744 – 7700 = 44
\]Answer: a. 44
Q14: \(\sqrt{110.25}\times\sqrt{0.01}\div\sqrt{0.0025}-\sqrt{420.25}\) equals
Step 1: Simplify each square root:
\[
\sqrt{110.25} = 10.5, \quad \sqrt{0.01} = 0.1, \quad \sqrt{0.0025} = 0.05, \quad \sqrt{420.25} = 20.5
\]Step 2: Substitute the values into the expression:
\[
10.5 \times 0.1 \div 0.05 – 20.5
\]Step 3: Perform the operations:
\[
10.5 \times 0.1 = 1.05, \quad 1.05 \div 0.05 = 21, \quad 21 – 20.5 = 0.5
\]Answer: a. 0.50
Q15: The greatest four digit perfect square number is
Step 1: The greatest four-digit number is 9999. We need to find the largest perfect square less than or equal to 9999.
Step 2: Find the square root of 9999:
\[
\sqrt{9999} \approx 99.99
\]
The largest whole number less than or equal to 99.99 is 99.
Step 3: Square 99 to get the largest perfect square:
\[
99^2 = 9801
\]Answer: b. 9801
Q16: A gardener plants 17956 trees in such a way that there are as many rows as there are trees in a row. The number of trees in a row is
Step 1: The number of rows and the number of trees in each row are the same. Therefore, the number of trees in a row is the square root of the total number of trees:
\[
\sqrt{17956}
\]Step 2: Find the square root of 17956:
\[
\sqrt{17956} = 134
\]Answer: a. 134
Q17: \(\sqrt[3]{4\frac{12}{125}}=?\)
Step 1: Convert the mixed fraction into an improper fraction:
\[
4 \frac{12}{125} = \frac{4 \times 125 + 12}{125} = \frac{512}{125}
\]Step 2: Find the cube root of the fraction:
\[
\sqrt[3]{\frac{512}{125}} = \frac{\sqrt[3]{512}}{\sqrt[3]{125}} = \frac{8}{5}
\]Step 3: Convert the fraction to a mixed number:
\[
\frac{8}{5} = 1 \frac{3}{5}
\]Answer: b. \(1\frac{3}{5}\)
Q18: By what least number must 21600 be multiplied so as to make it a perfect cube?
Step 1: Find the prime factorization of 21600:
\[
21600 = 2^5 \times 3^3 \times 5^2
\]Step 2: For the number to be a perfect cube, each prime factor’s exponent must be a multiple of 3.
– \(2^5\) needs one more factor of 2 to make it a multiple of 3.
– \(3^3\) is already a multiple of 3.
– \(5^2\) needs one more factor of 5 to make it a multiple of 3.
Step 3: Multiply by 10 to make the exponents of all primes multiples of 3:
\[
21600 \times 10 = 216000
\]Step 4: Verify the result:
\[
216000 = 2^6 \times 3^3 \times 5^3
\]
which is a perfect cube.
Answer: a. 10
Q19: What is the smallest number by which 3600 be divided to make it a perfect cube?
Step 1: Find the prime factorization of 3600:
\[
3600 = 2^4 \times 3^2 \times 5^2
\]Step 2: For the number to be a perfect cube, each prime factor’s exponent must be a multiple of 3.
– \(2^4\) needs one more factor of 2 to make the exponent 5 (next multiple of 3).
– \(3^2\) needs one more factor of 3 to make the exponent 3.
– \(5^2\) needs one more factor of 5 to make the exponent 3.
Step 3: To make it a perfect cube, divide by the necessary factors to get each exponent to a multiple of 3:
\[
\frac{3600}{2 \times 3 \times 5} = \frac{3600}{30} = 120
\]So, divide by 30 to make 3600 a perfect cube.
Answer: d. 450
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