To find the missing number in the series, we need to identify the pattern or relationship between consecutive terms. This often involves looking at the differences between numbers, differences of differences (second-order differences), or other mathematical operations.
B) 197 is the correct answer. The pattern involves a second-order difference series of consecutive odd numbers (3, 5, 7, 9, 11...).
This is a syllogism problem. We need to determine which conclusions logically follow from the given statements. We can use Venn diagrams or the rules of syllogism to evaluate each conclusion.
Analyze the Statements:
Combining these, we get a chain relationship: Lamps \(\subset\) Lanterns \(\subset\) Torches \(\subset\) Candles. This means all lamps are lanterns, all lanterns are torches, all torches are candles, and consequently, all lamps are also torches and all lamps are also candles.
Evaluate Conclusion I: Some candles are lamps.
Evaluate Conclusion II: All lanterns are lamps.
Evaluate Conclusion III: No lantern is a lamp.
Based on the evaluation, only Conclusion I follows.
C) Only conclusion I follows
The question asks to identify the main river to which the Nubra and Shyok rivers are tributaries. This requires knowledge of the major river systems of India, particularly those originating in the Himalayas.
D) Indus — The Nubra and Shyok rivers are indeed important tributaries of the Indus River. The Shyok River originates from the Rimo Glacier and flows through Ladakh, eventually joining the Indus River. The Nubra River, originating from the Siachen Glacier, joins the Shyok River before the latter merges with the Indus.
To find the percentage of votes received by the winning candidate, we first need to calculate the total number of votes cast. Then, we can use the formula for percentage: \[\text{Percentage} = \left( \frac{\text{Part}}{\text{Whole}} \right) \times 100\%\].
The winner received 1854 votes.
The loser received 618 votes.
Total votes = Votes received by winner + Votes received by loser
\[ \text{Total votes} = 1854 + 618 = 2472 \]Percentage of votes for winning candidate = \[\left( \frac{\text{Votes received by winner}}{\text{Total votes}} \right) \times 100\%\]
\[ \text{Percentage} = \left( \frac{1854}{2472} \right) \times 100\% \]\[ \text{Percentage} = 0.75 \times 100\% \]\[ \text{Percentage} = 75\% \]D) 75
The winning candidate received 75% of the total votes.
Find the positive difference between the roots of the equation 2x² − 7x + 5 = 0.
−\( \frac{3}{2} \)
\( \frac{5}{2} \)
−\( \frac{5}{2} \)
\( \frac{3}{2} \)
To find the positive difference between the roots of a quadratic equation of the form \(ax^2 + bx + c = 0\), we can first find the roots using the quadratic formula or by factorization, and then calculate their difference. Alternatively, we can use the properties of roots: if \(\alpha\) and \(\beta\) are the roots, then \(\alpha + \beta = -b/a\) and \(\alpha \beta = c/a\). The difference between the roots is given by \(|\alpha - \beta| = \frac{\sqrt{b^2 - 4ac}}{|a|}\).
Identify the coefficients:
The given quadratic equation is \(2x^2 - 7x + 5 = 0\).
Comparing this with the standard form \(ax^2 + bx + c = 0\), we have:
Method 1: Using the quadratic formula to find the roots.
The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
Substitute the values of \(a\), \(b\), and \(c\):
\[ x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(2)(5)}}{2(2)} \]\[ x = \frac{7 \pm \sqrt{49 - 40}}{4} \]\[ x = \frac{7 \pm \sqrt{9}}{4} \]\[ x = \frac{7 \pm 3}{4} \]The two roots are:
\[ x_1 = \frac{7 + 3}{4} = \frac{10}{4} = \frac{5}{2} \]\[ x_2 = \frac{7 - 3}{4} = \frac{4}{4} = 1 \]The positive difference between the roots is \(|x_1 - x_2|\):
\[ \left|\frac{5}{2} - 1\right| = \left|\frac{5}{2} - \frac{2}{2}\right| = \left|\frac{3}{2}\right| = \frac{3}{2} \]Alternatively, \[|1 - \frac{5}{2}| = |\frac{2}{2} - \frac{5}{2}| = |-\frac{3}{2}| = \frac{3}{2}\].
Method 2: Using the properties of roots.
For a quadratic equation \(ax^2 + bx + c = 0\), if \(\alpha\) and \(\beta\) are the roots, then:
Using the difference formula:
\[ |\alpha - \beta| = \frac{\sqrt{(-7)^2 - 4(2)(5)}}{|2|} \]\[ |\alpha - \beta| = \frac{\sqrt{49 - 40}}{2} \]\[ |\alpha - \beta| = \frac{\sqrt{9}}{2} \]\[ |\alpha - \beta| = \frac{3}{2} \]Both methods yield the same result.
D) \( \frac{3}{2} \)
The positive difference between the roots is \( \frac{3}{2} \).
All 30 students of a class are standing in a straight row facing the north. Ashok is 16th from the right end, while Aruna is 19th from the left end. How many students stand between Ashok and Aruna?
To find the number of students between Ashok and Aruna, we first need to determine their positions from the same end (either left or right). We are given Ashok's position from the right end and Aruna's position from the left end. We can convert one of these positions to be from the opposite end and then use the formula for finding the number of people between two individuals in a row.
Total number of students: There are 30 students in the class.
Ashok's position from the right end: Ashok is 16th from the right end.
Aruna's position from the left end: Aruna is 19th from the left end.
Find Ashok's position from the left end:
The formula to find a person's position from the opposite end is: \(\text{Position from opposite end} = \text{Total students} - \text{Position from given end} + 1\).
So, Ashok's position from the left end will be:
Determine the number of students between them:
Now we have:
Aruna's position from the left end = 19th
Ashok's position from the left end = 15th
Since Aruna's position (19th) is greater than Ashok's position (15th) from the left end, Aruna is to the right of Ashok.
The number of students between them can be calculated as: \(\text{Difference in positions} - 1\).
Number of students between Ashok and Aruna = \(19 - 15 - 1 = 4 - 1 = 3\).
C) Three
Find the value of \(\sqrt{9604}\).
To find the square root of a number like 9604, we can use a few methods. One common method is the long division method for square roots. Another approach is to estimate the square root by looking at the number's range and its last digit, then checking the options provided.
Thus, the square root of 9604 is 98.
B) 98 is correct because \(98^2 = 9604\).
To find the magnification of the image formed by a converging mirror, we need to use the mirror formula and the magnification formula. First, we'll determine the image distance using the given object distance and focal length. Then, we'll calculate the magnification using the object and image distances.
Identify the given values and sign conventions:
Apply the mirror formula to find the image distance (\(v\)):
The mirror formula is:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]Substitute the values into the mirror formula:
\[ \frac{1}{20} = \frac{1}{v} + \frac{1}{-40} \]\[ \frac{1}{20} = \frac{1}{v} - \frac{1}{40} \]Solve for \(\frac{1}{v}\):
\[ \frac{1}{v} = \frac{1}{20} + \frac{1}{40} \]\[ \frac{1}{v} = \frac{2}{40} + \frac{1}{40} \]\[ \frac{1}{v} = \frac{3}{40} \]\[ v = \frac{40}{3} \text{ cm} \]Calculate the magnification (\(m\)):
The magnification formula is:
\[ m = -\frac{v}{u} \]Substitute the values of \(v\) and \(u\) into the magnification formula:
\[ m = -\frac{\left(\frac{40}{3}\right)}{(-40)} \]\[ m = -\left(-\frac{40}{3 \times 40}\right) \]\[ m = -\left(-\frac{1}{3}\right) \]\[ m = +\frac{1}{3} \approx +0.33 \]Wait! Let's recheck the focal length sign convention for a converging mirror. For a converging mirror (concave mirror), the focal length is generally taken as negative when using the Cartesian sign convention where light travels from left to right and the pole is the origin. Let's re-evaluate with \(f = -20 \text{ cm}\).
Re-identify the given values with standard sign conventions:
Apply the mirror formula:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]Substitute the values:
\[ \frac{1}{-20} = \frac{1}{v} + \frac{1}{-40} \]\[ -\frac{1}{20} = \frac{1}{v} - \frac{1}{40} \]Solve for \(\frac{1}{v}\):
\[ \frac{1}{v} = -\frac{1}{20} + \frac{1}{40} \]\[ \frac{1}{v} = -\frac{2}{40} + \frac{1}{40} \]\[ \frac{1}{v} = -\frac{1}{40} \]\[ v = -40 \text{ cm} \]Calculate the magnification (\(m\)):
\[ m = -\frac{v}{u} \]\[ m = -\frac{(-40)}{(-40)} \]\[ m = -(1) \]\[ m = -1 \]The negative sign for magnification indicates that the image is real and inverted. Since \(|m|=1\), the image is the same size as the object. This scenario corresponds to an object placed at the center of curvature (\(C = 2f\)) of a concave mirror, where the image is also formed at \(C\), real, inverted, and same size.
B) −1 is the correct magnification. This indicates a real, inverted image of the same size as the object, formed at the center of curvature.
The question asks for the specific term used to describe a woman's first menstrual flow during puberty. This is a direct recall question from biology, specifically human reproductive health.
D) menarche — Menarche is the medical term for the first menstrual period in a female. It signifies the onset of puberty and reproductive capability.
Refer to the following letter and symbol series and answer the question.
(Left) E # R & H & @ J & M N # B & V && D # P @ W % & Q & A % & # E & (Right)
How many such letters are there in the series which are immediately preceded and immediately followed by the symbol '&'?
The problem asks us to identify the number of letters in a given series that are immediately preceded and immediately followed by the symbol '&'. We need to scan the series from left to right and look for the pattern '& Letter &' (Symbol - Letter - Symbol).
&) - Letter - Symbol (&).E # R & H & @ J & M N # B & V && D # P @ W % & Q & A % & # E &E # R & H &: Here, 'H' is preceded by '&' and followed by '&'. So, H is one such letter.@ J & M N # B & V && D # P @ W % & Q &: Here, 'Q' is preceded by '&' and followed by '&'. So, Q is another such letter.A % & # E &: Here, 'E' is preceded by '&' and followed by '&'. So, E is a third such letter.D) Three is the correct option because we found three letters (H, Q, E) that fit the given condition of being immediately preceded and immediately followed by the symbol '&'.