This problem involves calculating time taken for different modes of transport and setting up an equation based on the given conditions. We will use the formula \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\) to find the time taken in each scenario and then form an equation based on the difference in total time.
Identify given information:
Calculate time for the first scenario:
Calculate time for the second scenario:
Formulate the equation:
According to the problem, the second scenario takes 1 hour more than the first scenario.
\[ t_2 = t_1 + 1 \] \[ \frac{16}{T} + 6 = 4 + \frac{24}{T} + 1 \]Solve the equation for \(T\):
\[ \frac{16}{T} + 6 = 5 + \frac{24}{T} \] \[ 6 - 5 = \frac{24}{T} - \frac{16}{T} \] \[ 1 = \frac{24 - 16}{T} \] \[ 1 = \frac{8}{T} \] \[ T = 8 \text{ km/hr} \]Correct Option: B) 8 km/h
Find the odd one out.
The task is to identify the odd one out from the given options based on the symbols shown in the diagram.
Correct Option: D) Option D is the odd one out because it uses an English letter instead of a Greek letter.
Consider the given statement followed by two conclusions numbered I and II. Assume the statement to be true, then consider the two conclusions together and decide which of them logically follows beyond a reasonable doubt from the information given in the statement. Statement:
Some parks are circles.
Some circles are crossroads.
Conclusions:
I. All crossroads are circles.
II. All crossroads are parks.
Both the conclusions follow
Only conclusion I follows
Only conclusion II follows
Neither I nor II follows
This is a syllogism question. We need to evaluate the given conclusions based on the provided statements. We can use Venn diagrams to visualize the relationships between the categories (Parks, Circles, Crossroads) and determine which conclusions logically follow.
Let's represent the sets as P (Parks), C (Circles), and X (Crossroads).
From "Some parks are circles," we draw two overlapping circles for P and C.
From "Some circles are crossroads," we draw a third circle for X, overlapping with C. The crucial point is that the statement does not specify any direct relationship between P and X. X can overlap with C in a way that it also overlaps with P, or it can overlap with C without overlapping with P.
The statement says "Some circles are crossroads," which implies that there are some crossroads that are circles. However, it does not say that ALL crossroads are circles. There could be crossroads that are not circles. For example, if we have 100 crossroads, and 20 of them are circles, the statement "Some circles are crossroads" is true, but "All crossroads are circles" is false. Thus, this conclusion does not logically follow.
The statements establish a relationship between Parks and Circles, and between Circles and Crossroads. There is no direct or indirect statement that links all crossroads to parks. "Some parks are circles" and "Some circles are crossroads" does not guarantee any overlap between crossroads and parks, let alone that all crossroads are parks. It's possible that the 'some circles' that are 'crossroads' are not the 'some circles' that are 'parks'. Therefore, this conclusion does not logically follow.
D) Neither I nor II follows. Both conclusions make claims that are not necessarily true based on the given statements. The statements only indicate 'some' relationships, which do not allow for 'all' conclusions or direct relationships between non-adjacent categories (Parks and Crossroads) without further information.
Choose the mirror image for the following figure when the mirror is kept at the AB line.
To find the mirror image of a figure, we need to reflect it across the specified mirror line. In this case, the mirror line is labeled as AB, positioned vertically to the right of the figure.
Upon reflection, the circle should have the diagonal line flipped horizontally, and the straight line should extend in the opposite direction.
Correct Option: C) The image matches the correct reflection, with the diagonal line and straight line in the correct positions.
Which Bollywood superstar was honoured with the 24th Crystal Award at the World Economic Forum (WEF - 2018) in Davos, Switzerland?
The question asks to identify the Bollywood superstar who received the 24th Crystal Award at the World Economic Forum (WEF) in Davos, Switzerland, in 2018. This is a factual question related to current affairs and awards. Recalling major international awards received by prominent Indian personalities is key.
C) Shahrukh Khan — He was indeed honored with the 24th Crystal Award at the World Economic Forum (WEF - 2018) in Davos for his philanthropic work.
The question asks about the impact of greenhouse gases on Earth's average atmospheric temperature. This requires understanding the basic mechanism of the greenhouse effect.
Correct Option: A) Increases. Greenhouse gases absorb outgoing longwave radiation from Earth and re-emit it, trapping heat and thus increasing the average temperature of the Earth's atmosphere.
If 3 sec2 x – 2 tan2 x = 6 and 0° ≤ x ≤ 90° then x =?
The problem requires finding the value of \(x\) given a trigonometric equation \(3 \sec^2 x - 2 \tan^2 x = 6\) and the range \(0^\circ \le x \le 90^\circ\). We will use the fundamental trigonometric identity \(\sec^2 x = 1 + \tan^2 x\) to convert the equation into a single trigonometric function, \(\tan x\), and then solve for \(x\).
Given the equation:
\[ 3 \sec^2 x - 2 \tan^2 x = 6 \]Use the trigonometric identity \(\sec^2 x = 1 + \tan^2 x\) to substitute \(\sec^2 x\) in the given equation:
\[ 3 (1 + \tan^2 x) - 2 \tan^2 x = 6 \]Distribute the 3:
\[ 3 + 3 \tan^2 x - 2 \tan^2 x = 6 \]Combine the \(\tan^2 x\) terms:
\[ 3 + (3 - 2) \tan^2 x = 6 \] \[ 3 + \tan^2 x = 6 \]Isolate \(\tan^2 x\):
\[ \tan^2 x = 6 - 3 \] \[ \tan^2 x = 3 \]Take the square root of both sides:
\[ \tan x = \pm \sqrt{3} \]Given the range \(0^\circ \le x \le 90^\circ\), \(x\) is in the first quadrant. In the first quadrant, all trigonometric functions are positive. Therefore, \(\tan x\) must be positive:
\[ \tan x = \sqrt{3} \]Recall the standard trigonometric values. We know that \(\tan 60^\circ = \sqrt{3}\). Therefore:
\[ x = 60^\circ \]A) \(60^\circ\) is the correct value for \(x\) that satisfies the given equation and the condition \(0^\circ \le x \le 90^\circ\).
B) \(30^\circ\): If \(x = 30^\circ\), then \(\tan 30^\circ = \frac{1}{\sqrt{3}}\) and \(\sec 30^\circ = \frac{2}{\sqrt{3}}\). Substituting these into the original equation:
\[ 3 \left(\frac{2}{\sqrt{3}}\right)^2 - 2 \left(\frac{1}{\sqrt{3}}\right)^2 = 3 \left(\frac{4}{3}\right) - 2 \left(\frac{1}{3}\right) = 4 - \frac{2}{3} = \frac{12-2}{3} = \frac{10}{3} \ne 6 \]So, \(30^\circ\) is incorrect.
C) \(45^\circ\): If \(x = 45^\circ\), then \(\tan 45^\circ = 1\) and \(\sec 45^\circ = \sqrt{2}\). Substituting these into the original equation:
\[ 3 (\sqrt{2})^2 - 2 (1)^2 = 3(2) - 2(1) = 6 - 2 = 4 \ne 6 \]So, \(45^\circ\) is incorrect.
D) \(90^\circ\): \(\tan 90^\circ\) and \(\sec 90^\circ\) are undefined. Therefore, \(x\) cannot be \(90^\circ\).
How many triangles are there in the following figure?
To determine the total number of triangles in the given figure, we need to systematically count all possible triangles, considering different sizes and combinations.
Correct Option: C) 26
In the given figure, AOB is a straight line. ∠AOC = 67° and OD is the bisector of ∠BOC. What is the value of ∠BOD in degrees?
To find the value of \( \angle BOD \), we need to understand the relationship between the angles given in the diagram. \( \angle AOB \) is a straight line, meaning it measures \( 180^\circ \). \( OD \) bisects \( \angle BOC \), so \( \angle BOD \) is half of \( \angle BOC \).
Correct Option: A) 56.5°
One picometre is equal to _______.
This question tests knowledge of SI prefixes, specifically the prefix "pico". Understanding the standard values associated with these prefixes is crucial for competitive exams.
D) \(10^{-12}\) m. As established, the prefix "pico" corresponds to a factor of \(10^{-12}\).