The question asks to identify the winner of the women's singles title at the Syed Modi International Badminton Tournament in 2022. This requires knowledge of recent sports events and their results.
Correct Option: D) PV Sindhu
PV Sindhu, a two-time Olympic medalist and former world champion, won the women's singles title at the Syed Modi International Badminton Tournament in 2022. She defeated fellow Indian shuttler Malvika Bansod in the final match with a score of 21-16, 21-16.
Find the roots of \(\sqrt{2x+9}\) + x = 13.
To find the roots of the given equation, we need to isolate the square root term, square both sides to eliminate the square root, and then solve the resulting quadratic equation. Finally, it's crucial to check the solutions in the original equation, as squaring both sides can introduce extraneous roots.
Isolate the square root term:
The given equation is \(\sqrt{2x+9} + x = 13\).
Subtract \(x\) from both sides:
\[ \sqrt{2x+9} = 13 - x \]Square both sides:
Squaring both sides will eliminate the square root:
\[ (\sqrt{2x+9})^2 = (13 - x)^2 \]This simplifies to:
\[ 2x+9 = 169 - 26x + x^2 \]Rearrange into a quadratic equation:
Move all terms to one side to form a standard quadratic equation \(ax^2 + bx + c = 0\):
\[ x^2 - 26x - 2x + 169 - 9 = 0 \]Combine like terms:
\[ x^2 - 28x + 160 = 0 \]Solve the quadratic equation:
We can solve this by factoring. We need two numbers that multiply to 160 and add up to -28. These numbers are -8 and -20.
\[ (x - 8)(x - 20) = 0 \]This gives two potential roots:
\[ x - 8 = 0 \Rightarrow x = 8 \]or
\[ x - 20 = 0 \Rightarrow x = 20 \]Check for extraneous roots:
It is essential to substitute these values back into the original equation \(\sqrt{2x+9} + x = 13\) to ensure they are valid roots.
Check for \(x = 8\):
\[ \sqrt{2(8)+9} + 8 = 13 \]Simplify inside the square root:
\[ \sqrt{16+9} + 8 = 13 \]Further simplify:
\[ \sqrt{25} + 8 = 13 \]Calculate the square root:
\[ 5 + 8 = 13 \]This simplifies to:
\[ 13 = 13 \]Since this is true, \(x = 8\) is a valid root.
Check for \(x = 20\):
\[ \sqrt{2(20)+9} + 20 = 13 \]Simplify inside the square root:
\[ \sqrt{40+9} + 20 = 13 \]Further simplify:
\[ \sqrt{49} + 20 = 13 \]Calculate the square root:
\[ 7 + 20 = 13 \]This simplifies to:
\[ 27 = 13 \]Since this is false, \(x = 20\) is an extraneous root and not a valid solution to the original equation.
Therefore, the only valid root is \(x = 8\).
Correct Option: C) 8 only
If the area of an equilateral triangle is 25\(\sqrt{3}\) cm², then the length of each side of the triangle is:
To find the length of each side of an equilateral triangle given its area, we use the formula for the area of an equilateral triangle. We will set the given area equal to the formula and solve for the side length.
The area of an equilateral triangle with side length \(a\) is given by:
\[ \text{Area} = \frac{\sqrt{3}}{4} a^2 \]We are given that the area of the equilateral triangle is \(25\sqrt{3}\) cm\(^2\). So, we can write:
\[ \frac{\sqrt{3}}{4} a^2 = 25\sqrt{3} \]Divide both sides by \(\sqrt{3}\):
\[ \frac{1}{4} a^2 = 25 \]Multiply both sides by 4:
\[ a^2 = 25 \times 4 \]\[ a^2 = 100 \]Take the square root of both sides:
\[ a = \sqrt{100} \]\[ a = 10 \text{ cm} \]A) 10 cm is the correct answer. The calculation shows that the side length of the equilateral triangle is 10 cm.
To answer this question, one needs to recall the major physical divisions of India as classified by geographers. These divisions are based on distinct physiographic characteristics such as mountains, plains, plateaus, deserts, and coastal features.
B) 6 — India is broadly divided into six major physiographic regions, each with unique geological and climatic characteristics.
The question asks to identify the specific portal launched by the Ministry of Social Justice and Empowerment in October 2021, aimed at providing employment opportunities for senior citizens in India. This requires knowledge of recent government initiatives and their names.
Correct Option: C) SACRED
The SACRED portal (Senior Able Citizens for Re-Employment in Dignity) was launched by the Ministry of Social Justice and Empowerment in October 2021. Its objective is to provide a platform for senior citizens to find employment opportunities, thereby promoting their active participation in the workforce and ensuring their dignity.
The question asks to identify the Olympic Games where Neeraj Chopra won India's first-ever gold medal in athletics in 2021. This requires knowledge of major sporting events and India's achievements in them.
A) Tokyo. Neeraj Chopra won the gold medal in the men's javelin throw at the 2020 Summer Olympics, which were held in Tokyo from 23 July to 8 August 2021 due to the COVID-19 pandemic. This was India's first-ever Olympic gold medal in athletics.
The problem provides a relationship between Rani's age and her sister's age, and then gives the sister's age. We need to use this information to find Rani's age by setting up a simple algebraic equation or by direct calculation.
Let Rani's age be \(R\) years.
According to the problem, Rani's sister's age is 4 years more than Rani's age. So, we can write:
\[ \text{Sister's age} = \text{Rani's age} + 4 \]We know the sister's age is 28 years. So, substitute this into the equation:
\[ 28 = R + 4 \]To find \(R\), subtract 4 from both sides of the equation:
\[ R = 28 - 4 \]\[ R = 24 \]Rani's age is 24 years.
Correct Option: B) 24 years
The problem presents a series of number pairs and asks us to find the missing number in the last pair based on the relationship observed in the previous pairs. We need to identify the pattern that connects the first number to the second number in each given pair.
Let the first number be \(N_1 = 14\) and the second number be \(N_2 = 203\).
We can try to find a relationship involving squaring \(N_1\) or multiplying it by some factor.
Let's try \(N_1^2\): \(14^2 = 196\).
The difference between \(N_2\) and \(N_1^2\) is \(203 - 196 = 7\).
Notice that \(7\) is half of \(N_1\) (\(14/2 = 7\)).
So, the pattern for the first pair seems to be \(N_2 = N_1^2 + \frac{N_1}{2}\).
Let's verify: \(14^2 + \frac{14}{2} = 196 + 7 = 203\). This matches.
Let the first number be \(N_3 = 16\) and the second number be \(N_4 = 264\).
Using the same pattern: \(N_4 = N_3^2 + \frac{N_3}{2}\).
Let's verify: \(16^2 + \frac{16}{2} = 256 + 8 = 264\). This also matches.
Let the first number be \(N_5 = 18\) and the missing number be \(N_6\).
Using the established pattern: \(N_6 = N_5^2 + \frac{N_5}{2}\).
Substitute \(N_5 = 18\):
\[ N_6 = 18^2 + \frac{18}{2} \] \[ N_6 = 324 + 9 \] \[ N_6 = 333 \]Thus, the missing number is 333.
Correct Option: D) 333