Section - Mathematics
A can lay railway track between two given stations in 20 days and B can do the same job in 12 days. With the help of C, they did the job in 2 days only. Then, C alone can do the job in
4\( \frac{8}{11} \) days
11 \( \frac{8}{11} \) days
9 \( \frac{8}{11} \) days
2 \( \frac{8}{11} \) days
This problem involves work and time. We will calculate the individual work rates of A and B, and the combined work rate of A, B, and C. Then, we will use these rates to find the work rate of C alone and subsequently the time C takes to complete the job.
Calculate the work rate of A:
A can do the job in 20 days. So, A's one day's work is:
Calculate the work rate of B:
B can do the job in 12 days. So, B's one day's work is:
Calculate the combined work rate of A, B, and C:
With the help of C, they did the job in 2 days. So, (A + B + C)'s one day's work is:
Calculate the combined work rate of A and B:
Add A's and B's one day's work:
To add these fractions, find a common denominator, which is 60.
\[ \text{(A + B)'s one day's work} = \frac{3}{60} + \frac{5}{60} = \frac{8}{60} = \frac{2}{15} \]Calculate the work rate of C alone:
C's one day's work is the difference between (A + B + C)'s one day's work and (A + B)'s one day's work:
To subtract these fractions, find a common denominator, which is 30.
\[ \text{C's one day's work} = \frac{15}{30} - \frac{4}{30} = \frac{11}{30} \]Calculate the time C takes to do the job alone:
If C's one day's work is \( \frac{11}{30} \), then C can do the job alone in \( \frac{30}{11} \) days.
Convert this improper fraction to a mixed number:
\[ \frac{30}{11} = 2 \frac{8}{11} \text{ days} \]The value of \(\sqrt{324}\) + \(\sqrt{0.0324}\) − \(\sqrt{8.41}\) is _____.
To solve this problem, we need to calculate the square root of each term individually and then perform the addition and subtraction as indicated. We will find the square roots of integers and decimals by identifying perfect squares or by using standard square root calculation methods.
We know that \(10^2 = 100\), \(20^2 = 400\). Since \(324\) ends with \(4\), its square root must end with \(2\) or \(8\). Let's try \(18\).
\[ 18 \times 18 = 324 \]So, \(\sqrt{324} = 18\).We can write \(0.0324\) as \(\frac{324}{10000}\).
\[ \sqrt{0.0324} = \sqrt{\frac{324}{10000}} = \frac{\sqrt{324}}{\sqrt{10000}} \]We already found \(\sqrt{324} = 18\).We can write \(8.41\) as \(\frac{841}{100}\).
\[ \sqrt{8.41} = \sqrt{\frac{841}{100}} = \frac{\sqrt{841}}{\sqrt{100}} \]To find \(\sqrt{841}\): We know \(20^2 = 400\) and \(30^2 = 900\). Since \(841\) ends with \(1\), its square root must end with \(1\) or \(9\). Let's try \(29\).\[ 29 \times 29 = 841 \]So, \(\sqrt{841} = 29\).Substitute the calculated values back into the original expression:
\[ \sqrt{324} + \sqrt{0.0324} - \sqrt{8.41} = 18 + 0.18 - 2.9 \]First, add \(18\) and \(0.18\):The value of the expression is \(15.28\).
Correct Option: D) 15.28
The problem describes a "buy one get one 30% off" deal for two products of the same value. We need to calculate the total amount paid by the customer. This involves finding the price of the first product and the discounted price of the second product, then summing them up.
Correct Option: A) ₹595 is the total amount paid by the customer after applying the "buy one get one 30% off" deal.
This problem involves finding the Least Common Multiple (LCM) of two numbers given their ratio and their Highest Common Factor (HCF). We will use the fundamental relationship between two numbers, their HCF, and their LCM.
Let the two numbers be \(A\) and \(B\).
We are given that the ratio of the two numbers is \(5:6\). So, we can represent the numbers as \(A = 5x\) and \(B = 6x\) for some common factor \(x\).
We are also given that their HCF (Highest Common Factor) is 7.
The HCF of two numbers represented as \(ax\) and \(bx\) (where \(a\) and \(b\) are coprime) is \(x\). In our case, \(a=5\) and \(b=6\), which are coprime. Therefore, the common factor \(x\) is equal to the HCF.
So, \(x = 7\).
Now, we can find the actual numbers:
\[ A = 5x = 5 \times 7 = 35 \]\[ B = 6x = 6 \times 7 = 42 \]We need to find the LCM (Least Common Multiple) of these two numbers, 35 and 42.
We know the relationship between two numbers, their HCF, and their LCM:
\[ \text{Product of two numbers} = \text{HCF} \times \text{LCM} \]Substitute the values:
\[ 35 \times 42 = 7 \times \text{LCM} \]Now, solve for LCM:
\[ \text{LCM} = \frac{35 \times 42}{7} \]Simplify the expression:
\[ \text{LCM} = 5 \times 42 \]\[ \text{LCM} = 210 \]B) 210 is the correct LCM of the two numbers.
To find the difference between the shares of Y and Z, we first need to determine the total ratio parts. Then, we can calculate the profit share for each individual based on their investment ratio and the total profit. Finally, subtract Z's share from Y's share.
The investment ratio of X, Y, and Z is 58 : 72 : 20.
\[ \text{Total ratio parts} = 58 + 72 + 20 = 150 \]The total profit earned is ₹3,750.
\[ \text{Profit per ratio part} = \frac{\text{Total Profit}}{\text{Total ratio parts}} = \frac{3750}{150} = 25 \]Share of Y:
\[ \text{Y's Share} = \text{Y's ratio} \times \text{Profit per ratio part} = 72 \times 25 = 1800 \]Share of Z:
\[ \text{Z's Share} = \text{Z's ratio} \times \text{Profit per ratio part} = 20 \times 25 = 500 \]A) ₹1,300 is the correct difference between the shares of Y and Z.
To determine which number divides 862026417, we will use divisibility rules for each option provided. We will check for divisibility by 6, 5, 7, and 11.
Let the given number be \(N = 862026417\).
Divisibility by 6:
Divisibility by 5:
Divisibility by 7:
Divisibility by 11:
Based on the divisibility tests, only 7 divides 862026417.
C) 7 is the correct answer because, as shown by the divisibility test, 862026417 is divisible by 7.
The value of \( \frac{2}{5} \) + \(\left(\frac{1}{1+\frac{5}{7}}\right)\) − \( \frac{2}{5} \) is
\( \frac{2}{7} \)
\( \frac{10}{17} \)
\( \frac{7}{12} \)
\( \frac{14}{13} \)
The problem asks us to evaluate a given expression involving fractions. We need to simplify the expression by performing the operations in the correct order, following the rules of arithmetic (BODMAS/PEMDAS).
Correct Option: C) \( \frac{7}{12} \)
If cotA + cosA = p, cotA - cosA = q, then what is the value of p² - q²?
4\(\sqrt{pq}\)
3\(\sqrt{pq}\)
2\(\sqrt{pq}\)
\(\sqrt{pq}\)
We are given two equations involving \(\cot A\) and \(\cos A\), and we need to find the value of \(p^2 - q^2\). We will use the algebraic identity \(a^2 - b^2 = (a+b)(a-b)\) and then substitute the given expressions for \(p\) and \(q\) to simplify the result. Finally, we will express the answer in terms of \(pq\).
Given the equations:
We need to find the value of \(p^2 - q^2\). Using the algebraic identity \(a^2 - b^2 = (a+b)(a-b)\), we can write:
\[ p^2 - q^2 = (p+q)(p-q) \]First, let's find \(p+q\):
\[ p+q = (\cot A + \cos A) + (\cot A - \cos A) \]\[ p+q = \cot A + \cos A + \cot A - \cos A \]\[ p+q = 2\cot A \]Next, let's find \(p-q\):
\[ p-q = (\cot A + \cos A) - (\cot A - \cos A) \]\[ p-q = \cot A + \cos A - \cot A + \cos A \]\[ p-q = 2\cos A \]Now, substitute the values of \(p+q\) and \(p-q\) into the expression for \(p^2 - q^2\):
\[ p^2 - q^2 = (2\cot A)(2\cos A) \]\[ p^2 - q^2 = 4\cot A \cos A \]We also need to find the product \(pq\):
\[ pq = (\cot A + \cos A)(\cot A - \cos A) \]\[ pq = (\cot A)^2 - (\cos A)^2 \]\[ pq = \cot^2 A - \cos^2 A \]Let's simplify \(pq\) further using the identity \(\cot A = \frac{\cos A}{\sin A}\):
\[ pq = \frac{\cos^2 A}{\sin^2 A} - \cos^2 A \]\[ pq = \cos^2 A \left(\frac{1}{\sin^2 A} - 1\right) \]\[ pq = \cos^2 A \left(\frac{1 - \sin^2 A}{\sin^2 A}\right) \]\[ pq = \cos^2 A \left(\frac{\cos^2 A}{\sin^2 A}\right) \]\[ pq = \cos^2 A \cot^2 A \]Now, we have \(p^2 - q^2 = 4\cot A \cos A\) and \(pq = \cos^2 A \cot^2 A\). We need to express \(p^2 - q^2\) in terms of \(\sqrt{pq}\).
From \(pq = \cos^2 A \cot^2 A\), we can take the square root:
\[ \sqrt{pq} = \sqrt{\cos^2 A \cot^2 A} \]\[ \sqrt{pq} = |\cos A \cot A| \]Assuming \(A\) is in a quadrant where \(\cos A \cot A\) is positive (which is usually implied in such problems unless specified otherwise), we can write:
\[ \sqrt{pq} = \cos A \cot A \]Comparing this with our expression for \(p^2 - q^2\):
\[ p^2 - q^2 = 4(\cot A \cos A) \]\[ p^2 - q^2 = 4\sqrt{pq} \]A) The derived value \(p^2 - q^2 = 4\sqrt{pq}\) matches this option.
To find the final price after successive discounts, we need to apply each discount one by one to the current price. First, calculate the price after the first discount. Then, apply the second discount to this new price.
Alternatively, we can use the formula for successive discounts. If the marked price is \(MP\) and the successive discounts are \(d_1\%\) and \(d_2\%\), the final selling price \(SP\) is given by:
\[ SP = MP \times \left(1 - \frac{d_1}{100}\right) \times \left(1 - \frac{d_2}{100}\right) \]Applying this formula:A) ₹1,710 is the final price the customer pays after two successive discounts.
To simplify the given algebraic expression, we need to distribute the terms, combine like terms, and perform the arithmetic operations carefully.
\(x(5x - 9) = x \cdot 5x - x \cdot 9 = 5x^2 - 9x\)
\(7(x^2 - 2) = 7 \cdot x^2 - 7 \cdot 2 = 7x^2 - 14\)
The expression becomes: \((5x^2 - 9x) + (7x^2 - 14) + 17\)
The simplified expression is \(12x^2 - 9x + 3\).
Correct Option: C) \(12x^2 - 9x + 3\)