Find the value of \(\frac{(4x^{2}-9y^{2})^{3}+(9y^{2}-49z^{2})^{3}+(49z^{2}-4x^{2})^{3}}{(2x-3y)^{3}+(3y-7z)^{3}+(7z-2x)^{3}}\) is
4xyz (2x−3y)(3y−7z)(7z−2x)
(2x+3y)(3y+7z)(7z+2x)
4xyz (2x+3y)(3y+7z)(7z+2x)
(2x−3y)(3y−7z)(7z−2x)
The problem involves simplifying a complex algebraic expression. We will use the algebraic identity: If \(a+b+c=0\), then \(a^3+b^3+c^3=3abc\). We will apply this identity to both the numerator and the denominator separately.
The problem requires us to find which two mathematical signs, when interchanged in the given equation, make the equation correct. We will test each option by interchanging the specified signs and then evaluating the expression using the BODMAS/PEMDAS rule (Brackets, Orders/Exponents, Division, Multiplication, Addition, Subtraction).
The given equation is: \({[(44+36)−(4×4)]÷(1−5)}×2=8\)
Let's evaluate the left-hand side (LHS) of the original equation first to see if it's already correct:
Substitute these values back into the equation:
\[ {[(80)−(16)]÷(-4)}×2 \]\[ {[64]÷(-4)}×2 \]\[ {-16}×2 \]\[ -32 \]The original equation evaluates to \(-32\), which is not equal to \(8\). So, we need to interchange signs.Let's test each option by interchanging the signs and then evaluating the expression.
The equation becomes: \({[(44×36)−(4+4)]÷(1−5)}+2=8\)
Substitute these values:
\[ {[(1584)−(8)]÷(-4)}+2 \]\[ {[1576]÷(-4)}+2 \]\[ {-394}+2 \]\[ -392 \]This is not equal to \(8\). So, option A is incorrect.The equation becomes: \({[(44−36)+(4×4)]÷(1+5)}×2=8\)
Substitute these values:
\[ {[(8)+(16)]÷(6)}×2 \]\[ {[24]÷(6)}×2 \]\[ {4}×2 \]\[ 8 \]This is equal to \(8\). So, option B is correct.B) +, −
To find the marked price, we need to understand the relationship between the marked price, discount percentage, and the selling price. The selling price is the marked price minus the discount. Since the discount is given as a percentage of the marked price, we can set up an equation to solve for the marked price.
Let the marked price be \( \text{MP} \).
The discount is 75% of the marked price. So, the discount amount is \( 0.75 \times \text{MP} \).
The selling price (SP) is the marked price minus the discount amount:
\[ \text{SP} = \text{MP} - (\text{Discount Percentage} \times \text{MP}) \] \[ \text{SP} = \text{MP} - (0.75 \times \text{MP}) \] \[ \text{SP} = \text{MP} (1 - 0.75) \] \[ \text{SP} = 0.25 \times \text{MP} \]We are given that the shopkeeper charges Rs 1,874, which is the selling price.
\[ \text{SP} = \text{Rs } 1,874 \]Now, we can set up the equation:
\[ 1,874 = 0.25 \times \text{MP} \]To find the marked price (\( \text{MP} \)), divide the selling price by 0.25:
\[ \text{MP} = \frac{1,874}{0.25} \] \[ \text{MP} = 1,874 \times 4 \] \[ \text{MP} = 7,496 \]Therefore, the marked price is Rs 7,496.
Correct Option: A) Rs 7,496
The question asks for the specific name of a consumer awareness program launched by the Department of Consumer Affairs in 2022. This requires knowledge of recent government initiatives related to consumer protection.
Correct Option: B) Consumer Empowerment Week
The Department of Consumer Affairs observed "Consumer Empowerment Week" from March 14th to 20th, 2022. This initiative was part of the Azadi Ka Amrit Mahotsav celebrations and focused on educating consumers about their rights and responsibilities, promoting fair trade practices, and strengthening the consumer protection framework in India.
We need to find a two-digit number based on two given conditions. We can set up a system of two linear equations using variables for the digits and solve for them. Alternatively, we can test the given options against the conditions.
Represent the two-digit number:
Let the digit in the tens' place be \(x\) and the digit in the ones' place be \(y\). The value of the two-digit number can be expressed as \(10x + y\).
Formulate the first condition into an equation:
"The sum of the digits in a two-digit number is 9."
\[ x + y = 9 \quad \text{(Equation 1)} \]Formulate the second condition into an equation:
"If the value of the number is 6 more than 5 times the digit in the ones’ place."
The value of the number is \(10x + y\).
5 times the digit in the ones' place is \(5y\).
So, we have:
\[ 10x + y = 5y + 6 \quad \text{(Equation 2)} \]Simplify Equation 2:
Subtract \(y\) from both sides:
\[ 10x = 4y + 6 \]Solve the system of equations:
From Equation 1, we can express \(x\) in terms of \(y\):
\[ x = 9 - y \]Substitute \(x\) into the simplified Equation 2:
Substitute \(x = 9 - y\) into \(10x = 4y + 6\):
\[ 10(9 - y) = 4y + 6 \]\[ 90 - 10y = 4y + 6 \]Solve for \(y\):
Add \(10y\) to both sides:
\[ 90 = 14y + 6 \]Subtract 6 from both sides:
\[ 84 = 14y \]Divide by 14:
\[ y = \frac{84}{14} \]\[ y = 6 \]Solve for \(x\):
Substitute \(y = 6\) back into Equation 1 (\(x = 9 - y\)):
\[ x = 9 - 6 \]\[ x = 3 \]Form the number:
The digit in the tens' place is \(x = 3\), and the digit in the ones' place is \(y = 6\).
The number is \(10x + y = 10(3) + 6 = 30 + 6 = 36\).
Verify the conditions with the number 36:
D) 36. Both conditions are satisfied by the number 36.
In this question, a group of numbers/symbols is coded using letters as per the table given below and the conditions which follow. The correct combination of codes following the conditions is your answer.
| Number/Symbol | 4 | 6 | 7 | $ | # | @ | 2 |
| Codes | H | F | M | E | T | P | Z |
Conditions:
(i) If the first element is an even number and the last element is also an even number, the codes for these two (the first and the last elements) are to be interchanged.
(ii) If the second element is an even number and the fifth element is a symbol, the codes for these two (the second and the fifth elements) are to be interchanged.
(iii) If the fourth element is a symbol and the sixth element is also a symbol, both these elements (the fourth and the sixth elements) are to be coded as N.
Question:
4 6 $ 7 4 # @ 2
Z H E M F T P H
Z F E M H T P H
H F E M H T P Z
Z F E N H N P H
The problem requires us to code a given sequence of numbers and symbols based on a provided coding table and a set of conditions. We need to apply the conditions sequentially and modify the codes as specified.
First, let's write down the given sequence and its initial codes from the table:
Sequence: 4 6 $ 7 4 # @ 2
Codes: H F E M H T P Z
Now, let's apply the conditions one by one:
After applying all conditions, the final code sequence is: Z F E M H T P H
B) Z F E M H T P H is the correct combination of codes after applying all the conditions.
A shopkeeper bought 12 dozen eggs at the rate of ₹5 per egg. 12 eggs broke in transit. He sold the remaining eggs at the rate of ₹6 per egg. Find his percentage of profit.
To find the percentage profit, we need to calculate the total cost price (CP) and the total selling price (SP). The profit percentage is then calculated using the formula: \[\text{Profit Percentage} = \frac{\text{SP} - \text{CP}}{\text{CP}} \times 100\].
1 dozen = 12 eggs.
Number of eggs bought = \(12 \text{ dozen} \times 12 \text{ eggs/dozen} = 144 \text{ eggs}\).
Cost per egg = ₹5.
Total CP = \(144 \text{ eggs} \times \text{₹}5/\text{egg} = \text{₹}720\).
Number of eggs broken = 12 eggs.
Remaining eggs = \(144 - 12 = 132 \text{ eggs}\).
Selling price per egg = ₹6.
Total SP = \(132 \text{ eggs} \times \text{₹}6/\text{egg} = \text{₹}792\).
Profit = SP - CP
Profit = \(\text{₹}792 - \text{₹}720 = \text{₹}72\).
Percentage Profit = \(\frac{\text{Profit}}{\text{CP}} \times 100\)
Percentage Profit = \(\frac{72}{720} \times 100\)
Percentage Profit = \(\frac{1}{10} \times 100 = 10\%\).
Correct Option: B) 10 %
The question asks about the ministry responsible for launching India's first 'One Health Consortium'. This is a factual question related to government initiatives and ministries. Identifying the correct ministry requires knowledge of which government body typically handles scientific research and public health initiatives that integrate human, animal, and environmental health.
D) Ministry of Science and Technology — The Department of Biotechnology (DBT), which operates under the Ministry of Science and Technology, launched India's first 'One Health Consortium'. This consortium aims to undertake surveillance of important zoonotic diseases, understand the spread of antimicrobial resistance, and develop a holistic framework for 'One Health' in India.
The question asks for the year in which the Chauri Chaura incident occurred. This is a factual recall question from Indian history, specifically related to the Non-Cooperation Movement.
Correct Option: B) 1922
The problem requires us to find which interchange of signs will make the given equation correct. We need to test each option by swapping the specified signs in the original equation and then evaluate the expression using the order of operations (BODMAS/PEMDAS).
The given equation is: \(5 \times 4 \div 10 - 8 + 2 = 8\)
Let's evaluate the original equation first to see if it's already correct:
Since \(-4 \neq 8\), the original equation is incorrect.
Now let's test each option:
A) +, -
When '+' and '-' are interchanged, the equation becomes \(5 \times 4 \div 10 + 8 - 2\). Following the order of operations:
Thus, \(8 = 8\), which is correct.