The question asks to identify the state where Ratnagiri mines are located. This is a factual question related to the geography of mineral resources in India.
Correct Option: C) Maharashtra
The question asks about the number of groups in the modern periodic table. This is a fundamental concept in chemistry, directly related to the organization of elements based on their properties.
Correct Option: B) The modern periodic table is structured with 18 vertical columns, each representing a group. These groups are numbered from 1 to 18.
The two roots of a quadratic equation are given as x = \( \frac{5}{3} \) and x = \( \frac{-3}{10} \). The equation can be written as:
(10x - 3) (3x - 5) = 0
(10x + 3) (3x - 5) = 0
(10x + 3) (3x + 5) = 0
(10x - 3) (3x + 5) = 0
To find the quadratic equation from its roots, we can use the property that if \(x = r_1\) and \(x = r_2\) are the roots of a quadratic equation, then the equation can be written in the factored form as \((x - r_1)(x - r_2) = 0\). We will substitute the given roots into this form and then manipulate the expressions to match one of the options.
Given the two roots of the quadratic equation:
\[ x = \frac{5}{3} \quad \text{and} \quad x = \frac{-3}{10} \]
From the first root, \(x = \frac{5}{3}\), we can write a factor:
\[ 3x = 5 \] \[ 3x - 5 = 0 \]
From the second root, \(x = \frac{-3}{10}\), we can write another factor:
\[ 10x = -3 \] \[ 10x + 3 = 0 \]
The quadratic equation is formed by multiplying these two factors and setting the product to zero:
\[ (3x - 5)(10x + 3) = 0 \]
Comparing this result with the given options, we can see that it matches option B, just with the order of factors swapped, which does not change the product.
\[ (10x + 3)(3x - 5) = 0 \]
Correct Option: B) (10x + 3) (3x - 5) = 0
This problem involves understanding directions and rotations. We need to determine the final direction of the broom after a 90-degree right turn from an initial south-west facing position.
To elaborate:
A) North-west is the correct direction because a 90° clockwise turn from South-west leads to North-west.
Arjit’s age is 3 years more than 1.5 times the age of Heera, whose age in turn is 10 years more than \( \frac{2}{3} \) the age of Deepika. If 5 times the age of Deepika is equal to 3 times the age of Arjit, what is the sum of the present ages of the trio?
This problem involves setting up a system of linear equations based on the given relationships between the ages of Arjit, Heera, and Deepika. We will define variables for each person's age, translate the word problem into mathematical equations, solve the system of equations to find individual ages, and then calculate their sum.
Define variables:
Formulate equations from the given statements:
Statement 1: Arjit's age is 3 years more than 1.5 times the age of Heera.
\[ A = 1.5H + 3 \quad \text{(Equation 1)} \]
Statement 2: Heera's age is 10 years more than \( \frac{2}{3} \) the age of Deepika.
\[ H = \frac{2}{3}D + 10 \quad \text{(Equation 2)} \]
Statement 3: 5 times the age of Deepika is equal to 3 times the age of Arjit.
\[ 5D = 3A \quad \text{(Equation 3)} \]
Solve the system of equations:
From Equation 3, we can express \(D\) in terms of \(A\):
\[ D = \frac{3}{5}A \quad \text{(Equation 4)} \]
Substitute Equation 4 into Equation 2:
\[ H = \frac{2}{3} \left( \frac{3}{5}A \right) + 10 \]
\[ H = \frac{2}{5}A + 10 \quad \text{(Equation 5)} \]
Now substitute Equation 5 into Equation 1:
\[ A = 1.5 \left( \frac{2}{5}A + 10 \right) + 3 \]
\[ A = \frac{3}{2} \left( \frac{2}{5}A + 10 \right) + 3 \]
\[ A = \frac{3}{2} \cdot \frac{2}{5}A + \frac{3}{2} \cdot 10 + 3 \]
\[ A = \frac{3}{5}A + 15 + 3 \]
\[ A = \frac{3}{5}A + 18 \]
Subtract \( \frac{3}{5}A \) from both sides:
\[ A - \frac{3}{5}A = 18 \]
\[ \frac{5A - 3A}{5} = 18 \]
\[ \frac{2A}{5} = 18 \]
\[ 2A = 18 \times 5 \]
\[ 2A = 90 \]
\[ A = 45 \]
So, Arjit's age is 45 years.
Find Heera's and Deepika's ages:
Using Equation 5 to find Heera's age:
\[ H = \frac{2}{5}(45) + 10 \]
\[ H = 2 \times 9 + 10 \]
\[ H = 18 + 10 \]
\[ H = 28 \]
So, Heera's age is 28 years.
Using Equation 4 to find Deepika's age:
\[ D = \frac{3}{5}(45) \]
\[ D = 3 \times 9 \]
\[ D = 27 \]
So, Deepika's age is 27 years.
Calculate the sum of their ages:
\[ \text{Sum} = A + H + D \]
\[ \text{Sum} = 45 + 28 + 27 \]
\[ \text{Sum} = 100 \]
The sum of their present ages is 100 years.
Correct Option: C) 100 years
Select the option that is related to the third term in the same way as the second term is related to the first term.
Bird : Nest ∷ Lion : _______?
The question asks us to find the relationship between the first pair of words and apply the same relationship to the second pair. The relationship is typically one of association, category, or function.
Correct Option: B) Den is the correct answer because a den is the dwelling place for a lion, just as a nest is for a bird.
This is a blood relation problem. We need to trace the relationships backward from Kajol's perspective to understand who "Your mother's father's own son" refers to, and then determine Preethi's relationship to Kajol based on that.
Let's break down the statement: "Your mother’s father’s own son is my husband."
"Your mother's father": For Kajol, her mother's father is her maternal grandfather.
"Your mother's father's own son": The maternal grandfather's own son would be Kajol's mother's brother. This person is Kajol's maternal uncle.
So, the statement simplifies to: "Kajol's maternal uncle is my husband."
Preethi says this. Therefore, Preethi's husband is Kajol's maternal uncle.
If Preethi's husband is Kajol's maternal uncle, then Preethi herself is the wife of Kajol's maternal uncle.
The wife of one's maternal uncle is one's maternal aunt (or simply aunt).
Thus, Preethi is Kajol's aunt.
Correct Option: D) Aunt
Rs. 10,000, invested at 30% rate of interest per annum, but compounded every four months, will give an amount of Rs. ____ in one year.
This problem involves calculating compound interest. Since the interest is compounded every four months, we need to adjust the annual interest rate and the number of compounding periods per year accordingly. The formula for compound amount will then be applied.
Since interest is compounded every four months, in one year (12 months), there will be:
\[ n = \frac{12 \text{ months}}{4 \text{ months/period}} = 3 \text{ periods} \]The annual rate is 30%. For each 4-month period, the rate will be:
\[ r = \frac{30\%}{3} = 10\% \text{ per period} \]\[ r = \frac{10}{100} = 0.10 \]The formula for the amount (\(A\)) after compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]However, we have already adjusted \(r\) to be the rate per period, so the formula becomes:
\[ A = P (1 + r)^n \]Substitute the values:
\[ A = 10000 (1 + 0.10)^3 \]\[ A = 10000 (1.1)^3 \]D) 13,310 — The calculated amount after one year, compounded every four months, is Rs. 13,310.
To determine if a number is divisible by 4, we only need to check the divisibility of the number formed by its last two digits by 4. If the number formed by the last two digits is divisible by 4, then the entire number is divisible by 4. Otherwise, it is not.
Based on these checks, only 76166 is not divisible by 4.
B) 76166 — The number formed by the last two digits, 66, is not divisible by 4 (\(66 \div 4 = 16\) with a remainder of 2). Therefore, 76166 is not divisible by 4.
This problem involves a collision between two bodies where they coalesce (stick together) after impact. In such a scenario, the principle of conservation of linear momentum applies. The total momentum of the system before the collision is equal to the total momentum of the system after the collision.
The total momentum before collision (\(P_{\text{before}}\)) is equal to the total momentum after collision (\(P_{\text{after}}\)).
\[ P_{\text{before}} = m_1 v_1 + m_2 v_2 \]\[ P_{\text{after}} = (m_1 + m_2) V \]where \(V\) is the velocity of the coalesced mass after the collision.Since the bodies coalesce, their masses add up.
\[ M_{\text{total}} = m_1 + m_2 = m + 3m = 4m \]Correct Option: C) 1 km/h