Find the positive value of the following square root: \(\sqrt{56+\sqrt{56+\sqrt{56+\cdots}}}\) = ?
This problem involves an infinite series of nested square roots. Such problems can often be solved by setting the entire expression equal to a variable and then using the self-similar nature of the expression to form an algebraic equation.
Let the given expression be equal to \(x\):
\[ x = \sqrt{56+\sqrt{56+\sqrt{56+\cdots}}} \]Observe that the expression under the first square root, starting from the second \(\sqrt{56+\cdots}\), is identical to the original expression \(x\). So, we can substitute \(x\) back into the equation:
\[ x = \sqrt{56+x} \]To eliminate the square root, square both sides of the equation:
\[ x^2 = 56+x \]Rearrange the equation into a standard quadratic form \(ax^2+bx+c=0\):
\[ x^2 - x - 56 = 0 \]Solve the quadratic equation. We can factorize the quadratic expression. We need two numbers that multiply to -56 and add up to -1. These numbers are -8 and 7.
\[ (x-8)(x+7) = 0 \]This gives two possible values for \(x\):
\[ x-8 = 0 \implies x = 8 \]\[ x+7 = 0 \implies x = -7 \]The problem asks for the positive value of the square root. Since the square root symbol \(\sqrt{\cdot}\) conventionally denotes the principal (non-negative) square root, the result must be positive.
Therefore, we choose the positive value, \(x=8\).
B) 8 is the positive value of the given infinite square root expression.
To find the number of cricket players, we first need to determine the percentage of cricket players. Since the percentages of all other sports are given, we can subtract their sum from 100% to find the percentage of cricket players. Then, we calculate this percentage of the total number of students.
Total students = 3600
Since the remaining students are cricket players, their percentage will be:
\[ \text{Percentage of cricket players} = 100\% - \text{Percentage of other players} \] \[ \text{Percentage of cricket players} = 100\% - 93\% = 7\% \]Now, find 7% of the total number of students (3600):
\[ \text{Number of cricket players} = 7\% \text{ of } 3600 \] \[ \text{Number of cricket players} = \frac{7}{100} \times 3600 \] \[ \text{Number of cricket players} = 7 \times 36 \] \[ \text{Number of cricket players} = 252 \]Correct Option: C) 252
To find the time it takes for a sum of money to increase by a certain percentage at simple interest, we can use the simple interest formula. We'll assume an initial principal amount and then calculate the interest earned based on the given percentage increase and annual interest rate.
C) 4 years — The calculation shows that it will take 4 years for the sum of money to increase its value by 40% at a simple interest rate of 10% per annum.
The problem provides two equations involving variables \(a\) and \(b\): \(ab = 10\) and \(a^2 + b^2 = 29\). We need to find the value of \((a - b)^2\). We can use the algebraic identity for \((a - b)^2\) and substitute the given values.
Recall the algebraic identity for the square of a difference:
\[ (a - b)^2 = a^2 - 2ab + b^2 \]Rearrange the terms in the identity to group \(a^2\) and \(b^2\):
\[ (a - b)^2 = (a^2 + b^2) - 2ab \]Substitute the given values into the rearranged identity. We are given \(ab = 10\) and \(a^2 + b^2 = 29\).
\[ (a - b)^2 = (29) - 2(10) \]Perform the multiplication:
\[ (a - b)^2 = 29 - 20 \]Perform the subtraction:
\[ (a - b)^2 = 9 \]Correct Option: A) 9
To find the difference between the place values of two digits in a number, we first need to determine the place value of each specified digit. The place value of a digit depends on its position in the number. Once we have the individual place values, we subtract the smaller place value from the larger one.
Identify the number: The given number is \(128935\).
Find the place value of 2: In the number \(128935\), the digit 2 is in the ten thousands place.
\[ \text{Place value of 2} = 2 \times 10000 = 20000 \]Find the place value of 3: In the number \(128935\), the digit 3 is in the tens place.
\[ \text{Place value of 3} = 3 \times 10 = 30 \]Calculate the difference: Subtract the place value of 3 from the place value of 2.
\[ \text{Difference} = \text{Place value of 2} - \text{Place value of 3} \]\[ \text{Difference} = 20000 - 30 \]\[ \text{Difference} = 19970 \]Correct Option: D) 19970
The question asks to identify the country where the Chiapas plateau is located. This requires knowledge of world geography, specifically the location of major physical features like plateaus.
Correct Option: A) Mexico is the country where the Chiapas plateau is located. It is a prominent geographical feature in the southern part of the country, known for its mountainous terrain and indigenous communities.
To find the lateral surface area of a cylinder, we use the formula \(2 \pi r h\), where \(r\) is the radius and \(h\) is the height. We will substitute the given values of radius and height into this formula and calculate the result.
Radius of the cylinder, \(r = 25 \text{ cm}\)
Height of the cylinder, \(h = 42 \text{ cm}\)
The lateral surface area of a cylinder is given by:
\[ \text{LSA} = 2 \pi r h \]Using the approximate value of \(\pi = \frac{22}{7}\):
\[ \text{LSA} = 2 \times \frac{22}{7} \times 25 \text{ cm} \times 42 \text{ cm} \]First, simplify the terms:
\[ \text{LSA} = 2 \times 22 \times 25 \times \frac{42}{7} \text{ cm}^2 \]\[ \text{LSA} = 2 \times 22 \times 25 \times 6 \text{ cm}^2 \]\[ \text{LSA} = 44 \times 25 \times 6 \text{ cm}^2 \]\[ \text{LSA} = 1100 \times 6 \text{ cm}^2 \]\[ \text{LSA} = 6600 \text{ cm}^2 \]Thus, the lateral surface area of the cylinder is \(6600 \text{ cm}^2\).
Correct Option: C) 6600 cm² is the correct lateral surface area calculated using the formula \(2 \pi r h\).
This problem involves ratios and algebraic manipulation. We are given the initial ratio of students in three classes and how this ratio changes after an equal increase in students in each class. We need to find the total number of students after the increase. We will use a common multiplier for the initial ratio to set up equations and solve for the unknown.
Represent initial number of students:
Let the initial number of students in the three classes be \(2x\), \(3x\), and \(5x\) respectively, where \(x\) is a common multiplier.
Represent number of students after increase:
If 30 students are increased in each class, the new number of students in each class will be:
Formulate equations based on the new ratio:
The new ratio is given as \(4 : 5 : 7\). We can form equations using any two pairs of classes:
Using Class 1 and Class 2:
\[ \frac{2x + 30}{3x + 30} = \frac{4}{5} \]Solve for \(x\):
Cross-multiply the equation:
Calculate the number of students in each class after the increase:
Substitute \(x = 15\) into the expressions for the increased number of students:
Let's verify the new ratio: \(60 : 75 : 105\). Dividing by 15, we get \(4 : 5 : 7\), which matches the given new ratio.
Calculate the total number of students after the increase:
Add the number of students in all three classes after the increase:
B) 240 is the total number of students after the increase.
The question asks to identify the country known as the 'Hermit Kingdom'. This is a common historical and geopolitical nickname for a specific nation known for its isolationist policies.
Correct Option: A) North Korea is widely known as the 'Hermit Kingdom' due to its long history of isolationist policies, both historically as part of Korea and in its modern form.
The question asks to identify the reform movement that occurred during Lord William Bentinck's tenure in 1829. We need to recall the major social reforms initiated by Lord William Bentinck and their respective years.
C) Prohibition of Sati — Lord William Bentinck, with the support of Indian reformers like Raja Ram Mohan Roy, played a crucial role in abolishing the practice of Sati in 1829. This was a landmark social reform aimed at ending the ritualistic burning of widows on their husband's funeral pyre.