To determine which of the given statements are true, we need to substitute the provided values of \(x\), \(y\), and \(z\) into each expression and evaluate them. Then, we compare the calculated results with the given results in each statement.
\(x = 7\)
\(y = 4\)
\(z = 9\)
Substitute the values:
\[ 7 + 4 + 9 \]\[ 11 + 9 \]\[ 20 \]The statement says \(x + y + z = 20\), which matches our calculation. So, Statement I is TRUE.Substitute the values:
\[ 7 - 4 + 9 \]\[ 3 + 9 \]\[ 12 \]The statement says \(x - y + z = 13\), but our calculation gives \(12\). So, Statement II is FALSE.Substitute the values:
\[ 7 + 4 - 9 \]\[ 11 - 9 \]\[ 2 \]The statement says \(x + y - z = 3\), but our calculation gives \(2\). So, Statement III is FALSE.Substitute the values:
\[ -7 + 4 - 9 \]\[ -3 - 9 \]\[ -12 \]The statement says \(-x + y - z = -12\), which matches our calculation. So, Statement IV is TRUE.Statements I and IV are TRUE.
Correct Option: C) I and IV
A sum of money was invested for 2 years at a fixed interest rate. Had that money been invested at a 5% higher rate of interest, it would have gained Rs. 200 more. Find the sum.
The problem involves simple interest. We are given a scenario where a sum of money is invested for a certain period, and we need to find the principal amount. The key information is the additional interest gained if the rate were 5% higher. We can set up an equation based on the simple interest formula to solve for the principal.
Let the principal sum be \(P\).
Let the original interest rate be \(R\%\) per annum.
The time period of investment is \(T = 2\) years.
The formula for simple interest (SI) is:
\[ \text{SI} = \frac{P \times R \times T}{100} \]According to the problem, if the interest rate were 5% higher, i.e., \((R+5)\%\), the interest gained would be Rs. 200 more.
Let \(\text{SI}_1\) be the interest at rate \(R\%\) and \(\text{SI}_2\) be the interest at rate \((R+5)\%\).
\[ \text{SI}_1 = \frac{P \times R \times 2}{100} \] \[ \text{SI}_2 = \frac{P \times (R+5) \times 2}{100} \]We are given that \(\text{SI}_2 - \text{SI}_1 = 200\).
Substitute the expressions for \(\text{SI}_1\) and \(\text{SI}_2\) into the equation:
\[ \frac{P \times (R+5) \times 2}{100} - \frac{P \times R \times 2}{100} = 200 \]Factor out common terms:
\[ \frac{2P}{100} \left[ (R+5) - R \right] = 200 \]Simplify the expression inside the brackets:
\[ \frac{2P}{100} \left[ 5 \right] = 200 \]Further simplification:
\[ \frac{10P}{100} = 200 \] \[ \frac{P}{10} = 200 \]Solve for \(P\):
\[ P = 200 \times 10 \] \[ P = 2000 \]Therefore, the sum of money is Rs. 2,000.
Correct Option: D) Rs. 2,000
The question asks to identify the world's first programmer. This requires knowledge of the history of computing and key figures involved in its early development.
Correct Option: B) Ada Lovelace is widely recognized as the world's first programmer. She was the daughter of the poet Lord Byron and worked with Charles Babbage on his Analytical Engine in the mid-19th century. She wrote what is considered the first algorithm specifically designed to be processed by a machine, demonstrating how the Analytical Engine could go beyond simple calculations to perform a sequence of operations, such as calculating Bernoulli numbers.
Which country has won the maximum number of Oscar awards to date in the category of foreign films? (as of 2016)
The question asks to identify the country with the most Oscar wins in the Best Foreign Language Film category as of 2016. This requires knowledge of historical Academy Award statistics.
Correct Option: A) Italy
This problem involves comparing the purchase price (Cost Price) of a certain number of items with the selling price of a different number of items. We need to determine if there's a profit or loss percentage. The key is to find the relationship between the Cost Price (CP) and Selling Price (SP) of a single item or an equivalent number of items.
Set up the initial equation:
Given that the purchase price of 30 items is equal to the selling price of 40 items, we can write this relationship mathematically:
\[ 30 \times \text{CP} = 40 \times \text{SP} \]Find the ratio of CP to SP:
Rearrange the equation to find the ratio of CP to SP:
\[ \frac{\text{CP}}{\text{SP}} = \frac{40}{30} = \frac{4}{3} \]Determine profit or loss:
From the ratio \[\frac{\text{CP}}{\text{SP}} = \frac{4}{3}\], we can infer that \(\text{CP} = 4x\) and \(\text{SP} = 3x\) for some value \(x\). Since the Cost Price (\(4x\)) is greater than the Selling Price (\(3x\)), there is a loss.
Calculate the loss amount:
Loss \( = \text{CP} - \text{SP} = 4x - 3x = x \)
Calculate the loss percentage:
The loss percentage is calculated with respect to the Cost Price:
\[ \text{Loss Percentage} = \frac{\text{Loss}}{\text{CP}} \times 100\% \]\[ \text{Loss Percentage} = \frac{x}{4x} \times 100\% \]\[ \text{Loss Percentage} = \frac{1}{4} \times 100\% \]\[ \text{Loss Percentage} = 25\% \]Therefore, there is a 25% loss.
C) 25% loss — As calculated, the cost price of 30 items equals the selling price of 40 items, leading to a 25% loss.
What is the SI unit of pressure?
Newton per square centimetre (N/cm2)
Newton-square metre (N·m2)
Newton per square metre (N/m2)
Newton-square centimetre (N·cm2)
The question asks for the SI unit of pressure. To determine this, we need to recall the definition of pressure and the SI units for the quantities involved in its definition.
C) Newton per square metre (N/m2) is the correct SI unit of pressure, as derived from its definition of force per unit area.
The question asks for the specific term used to describe the study of maps. This requires knowledge of various scientific and artistic fields and their definitions.
Correct Option: D) Cartography is the art, science, and technology of making maps, as well as their study and use. Therefore, the study of a map is called cartography.
The question asks to identify the Indian state that has decided to implement the Bharat Net service to connect all gram panchayats through optical fiber. This requires knowledge of recent government initiatives and state-specific decisions regarding the Bharat Net project.
D) Uttar Pradesh. The Uttar Pradesh government decided to implement the Bharat Net service to connect all its gram panchayats through optical fiber, aiming to provide high-speed internet connectivity to rural areas.
A cycle wheel completes 5000 rounds in 44 km. Find the radius of wheels? (π = \( \frac{22}{7} \))
To determine how many numbers are completely divisible by 175, we need to divide each given number by 175 and check if the remainder is zero. If the remainder is zero, the number is completely divisible by 175.
The numbers completely divisible by 175 are 875, 1575, 2625, and 3325. There are 4 such numbers.
A) 4 — As determined by the step-by-step division, exactly 4 numbers (875, 1575, 2625, 3325) from the given list are completely divisible by 175.