The sum of a number, its half, its \( \frac{1}{3} \) and 27, is 71. Find the number.
The problem can be solved by setting up a linear equation based on the given information. We will represent the unknown number with a variable, then translate the word problem into an algebraic equation, and finally solve for the variable.
B) 24 is the correct answer. When 24, its half (12), its one-third (8), and 27 are added together, the sum is \(24 + 12 + 8 + 27 = 71\).
Consider the below statements and identify the correct answer.
Statement-I: Litmus solution is a purple dye, which is extracted from lichen, a plant belonging to the division Thallophyta.
Statement-II: There are many natural materials like red cabbage leaves, turmeric, coloured petals of some flowers, which indicate the presence of acid or base in a solution.
Statement-II is true, and Statement-I is false.
Both the statements are true.
Statement-I is true, and Statement-II is false
Both Statement-I and Statement-II are false
The question asks us to evaluate two statements related to indicators used in chemistry, specifically for identifying acids and bases. We need to determine the correctness of each statement individually and then choose the option that accurately reflects their truth values.
Correct Option: B) Both the statements are true.
To find the rate of interest, we will use the formula for simple interest. First, we need to convert the time given in years and months into a single unit (years). Then, we can substitute the given values into the simple interest formula and solve for the rate.
There are 12 months in a year. So, 4 months can be converted to years by dividing by 12.
\[ 4 \text{ months} = \frac{4}{12} \text{ years} = \frac{1}{3} \text{ years} \]Total time in years:
\[ T = 3 + \frac{1}{3} = \frac{9+1}{3} = \frac{10}{3} \text{ years} \]The formula for simple interest is:
\[ SI = \frac{P \times R \times T}{100} \]Where:
We need to find \(R\). Rearranging the formula to solve for \(R\):
\[ R = \frac{SI \times 100}{P \times T} \]Now, substitute the known values:
\[ R = \frac{840 \times 100}{3600 \times \frac{10}{3}} \]Simplify the denominator:
\[ 3600 \times \frac{10}{3} = 1200 \times 10 = 12000 \]Now, calculate \(R\):
\[ R = \frac{840 \times 100}{12000} \]\[ R = \frac{84000}{12000} \]\[ R = \frac{84}{12} \]\[ R = 7 \]So, the rate of interest is 7% per annum.
Correct Option: D) 7 %
To find the ratio in which two types of sugar should be mixed, we first need to determine the cost price (CP) of the mixture. We are given the selling price (SP) of the mixture and the profit percentage. Once the CP of the mixture is known, we can use the rule of alligation to find the required ratio.
A) 4 : 3 is the correct ratio obtained by calculating the cost price of the mixture and then applying the rule of alligation.
The question asks to identify the name of an indigenous herbal medicine developed by the National Innovation Foundation - India for treating worms in livestock, as an alternative to chemical methods. This requires knowledge of specific initiatives and products related to traditional medicine and livestock health in India.
B) Wormivet is the correct answer. The National Innovation Foundation - India (NIF) has indeed developed an indigenous herbal medicine named 'Wormivet' as an alternative to chemical methods for treating endoparasites (worms) in livestock. This product is based on traditional knowledge and has undergone scientific validation.
The question asks to identify the Indian state where Tadoba National Park is located. This requires knowledge of the geographical distribution of major national parks and wildlife sanctuaries in India.
B) Maharashtra — Tadoba National Park, also known as Tadoba Andhari Tiger Reserve, is indeed located in the Chandrapur district of Maharashtra, India.
If x = −1 and x = −2 are the solutions of the quadratic equation 3x2 + px + q = 0, then what is the value of (3p−2q)?
The problem asks us to find the value of an expression involving coefficients of a quadratic equation, given its roots. We can use Vieta's formulas, which relate the roots of a polynomial to its coefficients. Alternatively, we can substitute the given roots into the equation to form a system of linear equations for the coefficients \(p\) and \(q\).
Identify the given information:
Method 1: Using Vieta's formulas
For a quadratic equation \(ax^2 + bx + c = 0\), if \(\alpha\) and \(\beta\) are the roots, then:
In our case, \(a = 3\), \(b = p\), \(c = q\), \(\alpha = -1\), and \(\beta = -2\).
Sum of roots:
\[ (-1) + (-2) = -\frac{p}{3} \]\[ -3 = -\frac{p}{3} \]\[ p = 9 \]Product of roots:
\[ (-1) \times (-2) = \frac{q}{3} \]\[ 2 = \frac{q}{3} \]\[ q = 6 \]Method 2: Substituting the roots into the equation
Since \(x = -1\) is a solution, it must satisfy the equation:
\[ 3(-1)^2 + p(-1) + q = 0 \]\[ 3(1) - p + q = 0 \]\[ 3 - p + q = 0 \quad \text{(Equation 1)} \]\[ p - q = 3 \]Since \(x = -2\) is a solution, it must satisfy the equation:
\[ 3(-2)^2 + p(-2) + q = 0 \]\[ 3(4) - 2p + q = 0 \]\[ 12 - 2p + q = 0 \quad \text{(Equation 2)} \]\[ 2p - q = 12 \]Now we have a system of two linear equations:
Subtract Equation 1 from Equation 2:
\[ (2p - q) - (p - q) = 12 - 3 \]\[ 2p - q - p + q = 9 \]\[ p = 9 \]Substitute \(p = 9\) into Equation 1:
\[ 9 - q = 3 \]\[ q = 9 - 3 \]\[ q = 6 \]Calculate the value of \((3p - 2q)\):
Using the values \(p = 9\) and \(q = 6\):
\[ 3p - 2q = 3(9) - 2(6) \]\[ = 27 - 12 \]\[ = 15 \]B) The calculated value of \((3p - 2q)\) is 15.
To find the total current drawn from the battery, we first need to calculate the equivalent resistance of the two resistors connected in parallel. Once the equivalent resistance is found, we can use Ohm's Law to determine the total current.
Calculate the equivalent resistance (\(R_{eq}\)) for resistors in parallel:
When two resistors \(R_1\) and \(R_2\) are connected in parallel, their equivalent resistance is given by the formula:
Given \(R_A = 20 \ \Omega\) and \(R_B = 30 \ \Omega\).
\[ \frac{1}{R_{eq}} = \frac{1}{20 \ \Omega} + \frac{1}{30 \ \Omega} \] \[ \frac{1}{R_{eq}} = \frac{3}{60 \ \Omega} + \frac{2}{60 \ \Omega} \] \[ \frac{1}{R_{eq}} = \frac{5}{60 \ \Omega} \] \[ R_{eq} = \frac{60}{5} \ \Omega \] \[ R_{eq} = 12 \ \Omega \]Apply Ohm's Law to find the total current (\(I\)):
Ohm's Law states that \(V = IR\), where \(V\) is the voltage, \(I\) is the current, and \(R\) is the resistance. In this case, \(V\) is the battery voltage and \(R\) is the equivalent resistance.
Given \(V = 3 \ \text{V}\) and \(R_{eq} = 12 \ \Omega\).
\[ I = \frac{V}{R_{eq}} \] \[ I = \frac{3 \ \text{V}}{12 \ \Omega} \] \[ I = 0.25 \ \text{A} \]D) 0.25 A is the current through the battery, calculated using the equivalent resistance of the parallel combination and Ohm's Law.
The problem presents a number series with a missing term. We need to identify the pattern in the given numbers to find the missing value. We will analyze the relationship between consecutive terms, starting from the end of the series where all terms are known.
So, the pattern is division by 5, then division by 4, then division by 3. This means the number before 180 was divided by 2 to get 180, and the number before that (the missing term) was divided by 1 to get that number.
This pattern of dividing by 1, 2, 3, 4, 5 works perfectly.
Correct Option: D) 360
The problem asks us to modify a given number based on rules for odd and even digits and then find the difference between the largest and smallest digits in the newly formed number. We will process each digit individually according to the given rules.
Correct Option: B) 8