31.
$$99 \times 21 - \root 3 \of ? = 1968$$
(A) 1367631
(B) 111
(C) 1366731
(D) 1367
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Solution:
$$\eqalign{ & {\text{Let,}} \cr & {\text{ }}99 \times 21 - \root 3 \of x = 1968 \cr & {\text{Then,}} \cr & \Leftrightarrow 2079 - \root 3 \of x = 1968 \cr & \Leftrightarrow \root 3 \of x = 2079 - 1968 \cr & \Leftrightarrow \root 3 \of x = 111 \cr & \Leftrightarrow x = {\left( {111} \right)^3} \cr & \Leftrightarrow x = 1367631 \cr} $$
32.
If $$\sqrt {1369} + \sqrt {0.0615 + x} $$ ÃÂÃÂ ÃÂÃÂ ÃÂÃÂ = 37.25, the x is equal to ?
(A) $${10^{ - 1}}$$
(B) $${10^{ - 2}}$$
(C) $${10^{ - 3}}$$
(D) None of these
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Solution:
$$\eqalign{ & \Leftrightarrow 37 + \sqrt {0.0615 + x} = 37.25 \cr & \Leftrightarrow \sqrt {0.0615 + x} = 0.25 \cr & \Leftrightarrow 0.0615 + x = {\left( {0.25} \right)^2} = 0.0625 \cr & \Leftrightarrow x = 0.001 \cr & \Leftrightarrow x = \frac{1}{{{{10}^3}}} \cr & \Leftrightarrow x = {10^{ - 3}} \cr} $$
33.
$${1.5^2} \times \sqrt {0.0225} = ?$$
(A) 0.0375
(B) 0.3375
(C) 3.275
(D) 32.75
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Solution:
$$\eqalign{ & = {1.5^2} \times \sqrt {0.0225} \cr & = {1.5^2} \times \sqrt {\frac{{225}}{{10000}}} \cr & = 2.25 \times \frac{{15}}{{100}} \cr & = 2.25 \times 0.15 \cr & = 0.3375 \cr} $$
34.
The smallest natural number which is a perfect square and which ends in 3 identical digits lies between ?
(A) 1000 and 2000
(B) 2000 and 3000
(C) 3000 and 4000
(D) 4000 and 5000
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Solution:
The smallest such number is 1444 $$\left[ {1444 = {{\left( {38} \right)}^2}} \right]$$ It lies between 1000 and 2000.
35.
If $$\sqrt {33} = 5.745{\text{,}}$$ ÃÂÃÂ then which of the following values is approximately $$\sqrt {\frac{3}{{11}}} {\text{ ?}}$$
(A) 1
(B) 6.32
(C) 0.5223
(D) 2.035
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Solution:
$$\eqalign{ & = \sqrt {\frac{3}{{11}}} \cr & = \sqrt {\frac{{3 \times 11}}{{11 \times 11}}} \cr & = \frac{{\sqrt {33} }}{{11}} \cr & = \frac{{5.745}}{{11}} \cr & = 0.5223 \cr} $$
36.
$$\sqrt {\frac{{0.081 \times 0.484}}{{0.0064 \times 6.25}}} {\text{ }}$$ ÃÂÃÂ is equal to ?
(A) 0.9
(B) 0.99
(C) 9
(D) 99
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Solution:
Sum of decimal places in the numerator and denominator under the radical sign being the same, we remove the decimal. ∴ Given expression, $$\eqalign{ & = \sqrt {\frac{{0.081 \times 0.484}}{{0.0064 \times 6.25}}} \cr & = \sqrt {\frac{{81 \times 484}}{{64 \times 625}}} \cr & = \frac{{9 \times 22}}{{8 \times 25}} \cr & = 0.99 \cr} $$
37.
If the product of four consecutive natural numbers increased by a natural number p, is a perfect square, then the value of p is = ?
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Solution:
$$\eqalign{ & {\text{We have,}} \cr & 1 \times 2 \times 3 \times 4 = 24 \cr & {\text{And }}24 + 1 = 25\left[ {25 = {5^2}} \right] \cr & 2 \times 3 \times 4 \times 5 = 120{\text{ }} \cr & {\text{And 1}}20 + 1 = 121\left[ {121 = {{11}^2}} \right] \cr & 3 \times 4 \times 5 \times 6 = 360{\text{ }} \cr & {\text{And }}360 + 1 = 361\left[ {361 = {{19}^2}} \right] \cr & 4 \times 5 \times 6 \times 7 = 840{\text{ }} \cr & {\text{And }}840 + 1 = 841\left[ {841 = {{29}^2}} \right] \cr & \therefore p = 1 \cr} $$
38.
The square root of 123454321 is = ?
(A) 111111
(B) 12341
(C) 11111
(D) 11211
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Solution:
$$\therefore \sqrt {123454321} = 11111$$
39.
Which smallest number must be added to 710 so that the sum is a perfect cube ?
(A) 11
(B) 19
(C) 21
(D) 29
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Solution:
$$\eqalign{ & {\text{Required number to be added}} \cr & = {9^3} - 710 \cr & = 729 - 710 \cr & = 19 \cr} $$
40.
If $$\sqrt {\left( {x - 1} \right)\left( {y + 2} \right)} = 7,$$ ÃÂÃÂ ÃÂÃÂ x and y being positive whole numbers, then the values of x and y respectively are ?
(A) 8, 5
(B) 15, 12
(C) 22, 19
(D) None of these
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Solution:
$$\eqalign{ & \Leftrightarrow \sqrt {\left( {x - 1} \right)\left( {y + 2} \right)} = 7 \cr & \Leftrightarrow \left( {x - 1} \right)\left( {y + 2} \right) = {\left( 7 \right)^2} \cr & \Leftrightarrow \left( {x - 1} \right) = 7{\text{ and}}\left( {y + 2} \right) = 7 \cr & \Leftrightarrow x = 8{\text{ and }}y = 5 \cr} $$