To find the odd one out, we need to identify the pattern or relationship between the letters in each option. We will assign numerical positions to each letter of the English alphabet (A=1, B=2, ..., Z=26) and then look for a consistent mathematical or sequential pattern.
Therefore, Option B is the odd one out because its pattern of letter progression is different from the others.
Correct Option: B) WYZ — This option follows a pattern of +2, +1 in the alphabetical positions of its letters (W=23, Y=25, Z=26). This is different from the +2, +2 pattern observed in the other options.
The question asks to identify the hormone responsible for regulating blood sugar levels in humans. This requires knowledge of the endocrine system and the specific functions of various hormones.
D) Insulin — Insulin is a peptide hormone produced by the beta cells of the pancreatic islets. Its primary role is to lower blood glucose levels by facilitating the uptake of glucose into cells (especially muscle and fat cells) and promoting its storage as glycogen in the liver and muscles. It is the key hormone in regulating the amount of sugar in human blood, particularly in response to high blood sugar levels.
Since one zero of the polynomial is given, we can use polynomial division (synthetic division is efficient here) to reduce the degree of the polynomial. The resulting quadratic equation can then be factored or solved using the quadratic formula to find the other zeros.
Identify the given information: The polynomial is \(P(x) = x^3 + 5x^2 - 2x - 24\), and one zero is \(x = 2\).
Use the Factor Theorem: If \(x = 2\) is a zero, then \((x - 2)\) is a factor of the polynomial.
Perform Synthetic Division: Divide \(x^3 + 5x^2 - 2x - 24\) by \((x - 2)\).
\[\begin{array}{c|cccc}\text{2} & \text{1} & \text{5} & \text{-2} & \text{-24} \ & & \text{2} & \text{14} & \text{24} \ \hline & \text{1} & \text{7} & \text{12} & \text{0} \ \end{array}\]
The coefficients of the quotient are \(1, 7, 12\), and the remainder is \(0\). This means the quotient is \(x^2 + 7x + 12\).
Factor the quadratic quotient: Now we need to find the zeros of the quadratic equation \(x^2 + 7x + 12 = 0\).
We look for two numbers that multiply to \(12\) and add up to \(7\). These numbers are \(3\) and \(4\).
So, we can factor the quadratic as:
\[(x + 3)(x + 4) = 0\]
Find the other zeros: Set each factor to zero to find the zeros:
\[x + 3 = 0 \implies x = -3\]
\[x + 4 = 0 \implies x = -4\]
Conclusion: The other zeros of the polynomial are \(-3\) and \(-4\).
A) -3, -4 is the correct option as calculated by factoring the resulting quadratic equation.
If tan α = \( \frac{1}{2} \), tan β = \( \frac{1}{3} \), then find α + β.
To find the value of \(\alpha + \beta\), we can use the tangent addition formula, which states that \(\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\). We are given the values of \(\tan \alpha\) and \(\tan \beta\), so we can substitute these values into the formula.
\[ \tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \]
We are given \(\tan \alpha = \frac{1}{2}\) and \(\tan \beta = \frac{1}{3}\). Substitute these into the formula:
\[ \tan(\alpha + \beta) = \frac{\frac{1}{2} + \frac{1}{3}}{1 - \left(\frac{1}{2}\right) \left(\frac{1}{3}\right)} \]
\[ \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{3+2}{6} = \frac{5}{6} \]
\[ 1 - \left(\frac{1}{2}\right) \left(\frac{1}{3}\right) = 1 - \frac{1}{6} = \frac{6}{6} - \frac{1}{6} = \frac{6-1}{6} = \frac{5}{6} \]
\[ \tan(\alpha + \beta) = \frac{\frac{5}{6}}{\frac{5}{6}} = 1 \]
We know that \(\tan x = 1\) when \(x = 45^\circ\) (or \(\frac{\pi}{4}\) radians) in the first quadrant.
Therefore, \(\alpha + \beta = 45^\circ\).
Correct Option: B) 45°
The question asks to identify the first person to observe 'Individual Satyagraha'. This requires knowledge of the historical context and key figures involved in the Individual Satyagraha movement in India.
D) Vinoba Bhave was chosen by Mahatma Gandhi as the first satyagrahi to launch the Individual Satyagraha movement on October 17, 1940.
The question asks about biological control methods for mosquitoes. Biological control involves using natural enemies or biological processes to manage pests. We need to identify which of the given options fits this description for mosquito control.
Correct Option: C) Gambusia
What is the area of a regular hexagon of side 6 cm?
108\(\sqrt{3}\) cm²
64\(\sqrt{3}\) cm²
54\(\sqrt{3}\) cm²
72\(\sqrt{3}\) cm²
A regular hexagon can be divided into six equilateral triangles. The area of the hexagon can be found by calculating the area of one such equilateral triangle and multiplying it by six.
Identify the properties of a regular hexagon: A regular hexagon has six equal sides and six equal interior angles. When its vertices are connected to its center, it forms six congruent equilateral triangles.
Determine the side length of the equilateral triangles: If the side of the regular hexagon is \(s\), then the side of each equilateral triangle formed inside it is also \(s\).
Given side of the hexagon, \(s = 6 \text{ cm}\).
Recall the formula for the area of an equilateral triangle: The area of an equilateral triangle with side length \(a\) is given by the formula:
\[ \text{Area of equilateral triangle} = \frac{\sqrt{3}}{4} a^2 \]Calculate the area of one equilateral triangle: Substitute \(a = 6 \text{ cm}\) into the formula:
\[ \text{Area of one triangle} = \frac{\sqrt{3}}{4} (6)^2 \] \[ \text{Area of one triangle} = \frac{\sqrt{3}}{4} \times 36 \] \[ \text{Area of one triangle} = 9\sqrt{3} \text{ cm}^2 \]Calculate the total area of the regular hexagon: Since there are six such equilateral triangles, multiply the area of one triangle by 6:
\[ \text{Area of hexagon} = 6 \times (\text{Area of one triangle}) \] \[ \text{Area of hexagon} = 6 \times 9\sqrt{3} \] \[ \text{Area of hexagon} = 54\sqrt{3} \text{ cm}^2 \]Correct Option: C) 54\(\sqrt{3}\) cm²
This problem involves work and time. We will use the concept that if a person or a group of people can complete a work in 'n' days, then their one day's work is \(1/n\) of the total work. We will calculate the combined work rate of P, Q, R, and S, and then subtract the work rates of P, Q, and R to find the work rate of S alone.
First, let's find the one day's work for each given combination:
P, Q, R, and S together can do the work in 20 days.
\[ (\text{P} + \text{Q} + \text{R} + \text{S})\text{'s 1 day's work} = \frac{1}{20} \]P and Q together can do the work in 50 days.
\[ (\text{P} + \text{Q})\text{'s 1 day's work} = \frac{1}{50} \]R alone can do the work in 60 days.
\[ \text{R's 1 day's work} = \frac{1}{60} \]Now, we want to find S's 1 day's work. We can do this by subtracting the combined work of (P+Q) and R from the combined work of (P+Q+R+S):
\[ \text{S's 1 day's work} = (\text{P} + \text{Q} + \text{R} + \text{S})\text{'s 1 day's work} - [(\text{P} + \text{Q})\text{'s 1 day's work} + \text{R's 1 day's work}] \]Substitute the values:
\[ \text{S's 1 day's work} = \frac{1}{20} - \left(\frac{1}{50} + \frac{1}{60}\right) \]Calculate the sum inside the parenthesis first. Find the Least Common Multiple (LCM) of 50 and 60. The LCM of 50 and 60 is 300.
\[ \frac{1}{50} + \frac{1}{60} = \frac{6}{300} + \frac{5}{300} = \frac{6+5}{300} = \frac{11}{300} \]Now substitute this back into the equation for S's 1 day's work:
\[ \text{S's 1 day's work} = \frac{1}{20} - \frac{11}{300} \]Find the LCM of 20 and 300. The LCM is 300.
\[ \text{S's 1 day's work} = \frac{15}{300} - \frac{11}{300} = \frac{15-11}{300} = \frac{4}{300} \]Simplify the fraction:
\[ \text{S's 1 day's work} = \frac{4}{300} = \frac{1}{75} \]If S's 1 day's work is \(1/75\), then S alone can do the work in 75 days.
B) 75
A) 60 — This would be the answer if R alone could do the work, or if S's work rate was \(1/60\). Our calculation shows S's work rate is \(1/75\).
C) 65 — There is no logical calculation path that leads to 65 days. It might arise from incorrect arithmetic or misinterpretation of the problem.
D) 90 — This value is not derived from the correct application of the work and time principles in this problem.
This problem involves the concept of division algorithm. When a number is divided by another number, it can be expressed in the form of dividend = divisor × quotient + remainder. We are given the remainder when a number is divided by 234. We need to find the remainder when the same number is divided by 13. We can use the property that if the original divisor is a multiple of the new divisor, then we can find the new remainder by dividing the original remainder by the new divisor.
D) 10 is the correct remainder when the number is divided by 13.
The problem provides the ratio of A's age to B's age and information about A's age after 6 years. We need to find B's current age. We can use variables to represent their current ages based on the given ratio, then use the future age information to find the value of the variable, and finally calculate B's present age.