To find the least number that should be multiplied to a given number to make it a perfect square, we need to perform prime factorization of the given number. A number is a perfect square if all the exponents in its prime factorization are even. We then identify the prime factors with odd exponents and multiply the number by the smallest set of prime factors that will make all exponents even.
We start by dividing 63368 by the smallest prime numbers.
\[ \begin{array}{l} 63368 \div 2 = 31684 \\ 31684 \div 2 = 15842 \\ 15842 \div 2 = 7921 \end{array} \]Now we need to find the prime factors of 7921. We can test prime numbers. For example, \(70^2 = 4900\), \(80^2 = 6400\), \(90^2 = 8100\). So, the square root should be between 80 and 90. Numbers ending in 1 could be squares of numbers ending in 1 or 9. Let's try 89.
\[ 89 \times 89 = 7921 \]So, 7921 is \(89^2\).
Combining the factors, we get:
\[ 63368 = 2 \times 2 \times 2 \times 89 \times 89 \]This can be written in exponential form as:
\[ 63368 = 2^3 \times 89^2 \]For a number to be a perfect square, all the exponents in its prime factorization must be even. In the factorization \(2^3 \times 89^2\), the exponent of 89 is 2 (which is even), but the exponent of 2 is 3 (which is odd).
To make the exponent of 2 even, we need to multiply \(2^3\) by \(2^1\) (or simply 2). This would change \(2^3\) to \(2^4\), which has an even exponent.
Therefore, the least number to be multiplied is 2.
If we multiply 63368 by 2:
\[ 63368 \times 2 = (2^3 \times 89^2) \times 2 = 2^4 \times 89^2 \]Now, all exponents are even (4 and 2), so the resulting number is a perfect square.
\[ 2^4 \times 89^2 = (2^2 \times 89)^2 = (4 \times 89)^2 = (356)^2 = 126736 \]Correct Option: D) 2
The height of a right-angled triangular prism is 21 cm and the ratio of the sides of its base is 8:15:17. If the total area of the three lateral surfaces is 840 cm2, then what is the volume of the prism in cubic centimetres?
The problem asks for the volume of a right-angled triangular prism. We are given its height, the ratio of the sides of its base, and the total area of its three lateral surfaces. We will first use the lateral surface area and height to find the perimeter of the base. Then, using the ratio of the sides, we can determine the actual lengths of the sides of the right-angled triangular base. Finally, we will calculate the area of the base and multiply it by the height to find the volume.
Identify the properties of the base triangle: The ratio of the sides of the base is 8:15:17. Since \(8^2 + 15^2 = 64 + 225 = 289 = 17^2\), this confirms that the base is a right-angled triangle, with the sides 8x and 15x as the perpendicular sides and 17x as the hypotenuse.
Use the lateral surface area to find the perimeter of the base: The total area of the three lateral surfaces of a prism is given by the formula: \(\text{Lateral Surface Area} = \text{Perimeter of Base} \times \text{Height}\)
Given: \(\text{Lateral Surface Area} = 840 \text{ cm}^2\)
Given: \(\text{Height (h)} = 21 \text{ cm}\)
Let the sides of the base be \(8x\), \(15x\), and \(17x\). The perimeter of the base (\(P\)) is:
\[ P = 8x + 15x + 17x = 40x \]Substitute the values into the lateral surface area formula:
\[ 840 = (40x) \times 21 \]\[ 840 = 840x \]\[ x = \frac{840}{840} = 1 \]Determine the actual dimensions of the base triangle: Since \(x=1\), the sides of the right-angled triangular base are:
Calculate the area of the base: For a right-angled triangle, the area is half the product of its perpendicular sides.
\[ \text{Area of Base (A)} = \frac{1}{2} \times \text{base} \times \text{height} \]\[ A = \frac{1}{2} \times 8 \times 15 \]\[ A = 4 \times 15 = 60 \text{ cm}^2 \]Calculate the volume of the prism: The volume of a prism is given by the formula:
\[ \text{Volume} = \text{Area of Base} \times \text{Height} \]\[ \text{Volume} = 60 \text{ cm}^2 \times 21 \text{ cm} \]\[ \text{Volume} = 1260 \text{ cm}^3 \]Correct Option: D) 1260
The volume of the prism is 1260 cubic centimetres.
The question asks to identify the capital city of the Pandya dynasty. This requires knowledge of ancient Indian history, specifically the major dynasties of South India and their respective capitals.
Correct Option: D) Madurai is historically recognized as the capital of the Pandya dynasty. It was a significant center for trade, culture, and learning under Pandya rule, especially famous for the Meenakshi Amman Temple.
1.008 =?
1\( \frac{2}{25} \)
1\( \frac{1}{125} \)
1\( \frac{2}{125} \)
1\( \frac{3}{25} \)
To convert a decimal number into a mixed fraction, we first separate the whole number part from the fractional part. Then, we convert the decimal fractional part into a common fraction by writing it over a power of 10, simplifying it, and finally combining it with the whole number part.
The given number is \(1.008\).
The whole number part is \(1\).
The decimal part is \(0.008\).
The decimal part \(0.008\) can be written as a fraction by placing the digits after the decimal point over a power of 10. Since there are three digits after the decimal point, the denominator will be \(10^3 = 1000\).
\[ 0.008 = \frac{8}{1000} \]Now, simplify the fraction \(\frac{8}{1000}\) by dividing both the numerator and the denominator by their greatest common divisor (GCD). Both 8 and 1000 are divisible by 8.
So, the simplified fraction is \(\frac{1}{125}\).
Now, combine the whole number part (1) with the simplified fractional part (\(\frac{1}{125}\)) to form a mixed fraction.
\[ 1 + \frac{1}{125} = 1\frac{1}{125} \]B) \(1\frac{1}{125}\) is the correct conversion of \(1.008\) into a mixed fraction.
A toy purchased for ₹1,125 was sold at a loss of 16%. The selling price of the toy was
To find the selling price of the toy, we first need to calculate the amount of loss incurred. The loss is given as a percentage of the cost price. Once the loss amount is determined, we subtract it from the cost price to get the selling price.
The loss is 16% of the Cost Price.
\[ \text{Loss Amount} = \frac{\text{Loss Percentage}}{100} \times \text{Cost Price} \]\[ \text{Loss Amount} = \frac{16}{100} \times 1125 \]\[ \text{Loss Amount} = 0.16 \times 1125 \]\[ \text{Loss Amount} = 180 \]The Selling Price is the Cost Price minus the Loss Amount.
\[ \text{Selling Price} = \text{Cost Price} - \text{Loss Amount} \]\[ \text{Selling Price} = 1125 - 180 \]\[ \text{Selling Price} = 945 \]Thus, the selling price of the toy was ₹945.
Correct Option: B) 945
This question asks us to evaluate the strength of two arguments related to North Korea manufacturing nuclear bombs. In critical reasoning, a strong argument should be logically sound, relevant, and provide a compelling reason for or against the statement. We need to analyze each argument independently to determine its strength.
D) Neither Argument 1 nor Argument 2 is strong
Argument 1, while presenting a common justification for nuclear weapons, is not universally accepted as a strong reason given the severe risks and international opposition associated with nuclear proliferation. National security can be achieved through various means, and nuclear weapons often complicate it. Argument 2 correctly identifies a consequence (disturbing regional balance) but doesn't present a strong, universally compelling reason against the manufacturing of bombs from a policy perspective, as the statement asks for a 'should' or 'should not' decision. Both arguments are debatable and lack the definitive strength required in such critical reasoning questions.
The question asks to identify the recipient of the Padma Vibhushan award in 2018 for contributions to arts and music. This requires knowledge of prominent awardees in that specific year and category.
In 2018, the Government of India announced the Padma Awards. Among the recipients of the Padma Vibhushan, Ilaiyaraaja, the renowned music composer, was awarded for his exceptional and distinguished service in the field of Art (Music).
D) Ilaiyaraaja — Ilaiyaraaja, a legendary Indian music composer, singer, and conductor, was indeed awarded the Padma Vibhushan in 2018 for his outstanding contributions to arts and music.
To solve this problem, we need to perform successive subtractions of decimal numbers. It's crucial to align the decimal points correctly before subtracting to avoid errors. We will subtract each number one by one from left to right.
Write 396 as 396.0 to align decimal points:
\[ \begin{array}{r} 396.0 \\ - 39.6 \\ \hline 356.4 \end{array} \]Write 356.4 as 356.40 to align decimal points:
\[ \begin{array}{r} 356.40 \\ - 3.96 \\ \hline 352.44 \end{array} \]Write 352.44 as 352.440 to align decimal points:
\[ \begin{array}{r} 352.440 \\ - 0.396 \\ \hline 352.044 \end{array} \]Thus, the final result is \(352.044\).
Correct Option: B) 352.044
The question asks to identify the recipient of the Padma Shri award in 2018 for contributions to education and literature, specifically from Maharashtra. This requires knowledge of the Padma Awards recipients for the year 2018 and their respective fields and states.
Correct Option: C) Pantavane Gangadhar Vithobaji
The question asks to identify the "odd man out" from a given list of animals. This typically means finding the item that does not share a common characteristic with the others. We need to analyze the characteristics of each animal provided in the options.
C) Ostrich — The Ostrich is a bird, whereas the Tiger, Cheetah, and Elephant are all mammals. This fundamental biological classification makes the Ostrich the odd one out.