The problem requires us to find the value of 'k' in a given equation involving decimal numbers. The approach is to first simplify both sides of the equation by performing the additions and subtractions, and then isolate 'k' to find its value.
The left side is \(1.345 + 13.45 + 0.1354 + 0.0596\).
Align the decimal points and add the numbers:
\[\begin{array}{c} \phantom{0}1.3450 \ 13.4500 \ \phantom{0}0.1354 \ \underline{+\phantom{0}0.0596} \ 14.9900 \end{array}\]So, the left side simplifies to \(14.99\).
The right side is \(11.453 - 0.543 + k\).
First, perform the subtraction:
\[\begin{array}{c} 11.453 \ \underline{-\phantom{0}0.543} \ \phantom{0}10.910 \end{array}\]So, the right side becomes \(10.91 + k\).
Now, equate the simplified left and right sides:
\[ 14.99 = 10.91 + k \]To find \(k\), subtract \(10.91\) from both sides of the equation:
\[ k = 14.99 - 10.91 \]\[\begin{array}{c} 14.99 \ \underline{-\phantom{0}10.91} \ \phantom{00}4.08 \end{array}\]Thus, \(k = 4.08\).
A) 4.08 is the correct value for \(k\).
All 20 students of a class are sitting in a straight row facing east. Ram is 8th from the right end and Sita is 16th from the left end. How many students are sitting between Ram and Sita?
To find the number of students between Ram and Sita, we first need to determine their positions from a common reference point, typically the left end. We are given Ram's position from the right end and Sita's position from the left end. We will convert Ram's position to be from the left end and then calculate the number of students between them.
The formula to convert a position from the right end to the left end (or vice versa) in a row is:
\[ \text{Position from Left End} = \text{Total Students} - \text{Position from Right End} + 1 \]Using this formula for Ram:
\[ \text{Ram's position from Left End} = 20 - 8 + 1 = 13 \]So, Ram is 13th from the left end.
Sita is at the 16th position from the left end.
Ram is at the 13th position from the left end.
Since Sita is further to the right (16th) than Ram (13th) from the left end, the number of students between them can be found by subtracting their positions and then subtracting 1 (because we are looking for students *between* them, not including them).
\[ \text{Students between Ram and Sita} = \text{Sita's position from Left} - \text{Ram's position from Left} - 1 \] \[ \text{Students between Ram and Sita} = 16 - 13 - 1 = 3 - 1 = 2 \]Therefore, there are 2 students sitting between Ram and Sita.
B) 2 is the correct answer. There are 2 students sitting between Ram and Sita.
This problem involves calculating the weight of a new person when an old person is replaced, and the average weight of the group changes. We can solve this by understanding how the total weight of the group changes due to the replacement and the resulting change in average weight.
Since the average weight of 4 persons increases by 3 kg, the total weight of the group increases by:
\[ \text{Total increase in weight} = \text{Number of persons} \times \text{Increase in average weight} \]\[ \text{Total increase in weight} = 4 \times 3 \text{ kg} = 12 \text{ kg} \]When a new person replaces an old person, the total weight of the group changes by the difference between the new person's weight and the old person's weight. Since the total weight increased, the new person must be heavier than the person replaced.
\[ \text{Weight of new person} = \text{Weight of replaced person} + \text{Total increase in weight} \]\[ \text{Weight of new person} = 53 \text{ kg} + 12 \text{ kg} \]\[ \text{Weight of new person} = 65 \text{ kg} \]B) 65 kg is the weight of the new person, as calculated by adding the total increase in weight to the weight of the person who was replaced.
When a transversal line intersects two parallel lines, the sum of the two exterior angles on the same side of the transversal is always \(180^\circ\). We are given the ratio of these two angles. We can use this property to find the individual angle measures and then calculate their difference.
Let the two parallel lines be \(L_1\) and \(L_2\), and the transversal line be \(T\).
Let the two exterior angles on the same side of the transversal be \(\angle A\) and \(\angle B\).
According to the property of parallel lines intersected by a transversal, the sum of consecutive exterior angles (or exterior angles on the same side of the transversal) is \(180^\circ\).
So, \(\angle A + \angle B = 180^\circ\).
We are given that the ratio of these two exterior angles is \(7:5\). Let the angles be \(7x\) and \(5x\).
So, \(7x + 5x = 180^\circ\).
Combine the terms:
\(12x = 180^\circ\).
Solve for \(x\):
\[ x = \frac{180^\circ}{12} = 15^\circ \]Now, find the measure of each angle:
Larger angle \(= 7x = 7 \times 15^\circ = 105^\circ\).
Smaller angle \(= 5x = 5 \times 15^\circ = 75^\circ\).
The question asks for how much more the larger angle is than the smaller one. This means we need to find the difference between the two angles.
\(\text{Difference} = \text{Larger angle} - \text{Smaller angle}\)
\(\text{Difference} = 105^\circ - 75^\circ = 30^\circ\).
This problem involves understanding directions and distances. We can visualize Rahman's movement as a series of steps on a 2D plane, starting from a fixed point. By tracking his movements (north, right turn, right turn), we can determine his final position relative to the starting point.
Let's visualize the path:
Comparing the final position C \((15, 0)\) with the starting point F \((0, 0)\):
Therefore, Rahman is 15 km to the East of his starting point F.
D) 15 km towards east. This is correct because after moving 20 km North and then 20 km South, his North-South displacement is zero. His only net displacement is 15 km towards the East.
This question asks why ionic compounds have high melting and boiling points. The key to understanding this lies in the nature of the forces holding the ions together in an ionic lattice.
Correct Option: A) a large amount of energy is required to break the strong inter-ionic attraction. This option correctly identifies both the strength of the inter-ionic attraction (strong) and the amount of energy required to overcome it (large), which directly explains the high melting and boiling points of ionic compounds.
To find the resistance of the circuit, we can use the relationship between power, voltage, and resistance. The power consumed by an electric circuit is given by the formula \(P = V^2 / R\), where \(P\) is power, \(V\) is voltage, and \(R\) is resistance. We are given the power and voltage, so we can rearrange the formula to solve for resistance.
The relationship between power, voltage, and resistance is given by:
\[ P = \frac{V^2}{R} \]To find \(R\), we can rearrange the formula as:
\[ R = \frac{V^2}{P} \]Substitute \(V = 270 \text{ V}\) and \(P = 540 \text{ W}\) into the rearranged formula:
\[ R = \frac{(270 \text{ V})^2}{540 \text{ W}} \]Thus, the resistance of the circuit is 135 ohms.
B) 135 ohm — This is the correct resistance calculated using the formula \(R = V^2 / P\).
This problem involves calculating the total profit based on investment ratios, a charity deduction, and one partner's share of the profit. We will first determine the profit share of A from the remaining profit after charity, then find the total profit after charity, and finally calculate the total profit earned before charity.
Since A and B invest for the same time, their profit-sharing ratio is directly proportional to their investment ratio.
\[ \text{Ratio of investments of A and B} = 7 : 8 \] \[ \text{Ratio of profits of A and B} = 7 : 8 \]Total share units = \(7 + 8 = 15\) units.
Let the total profit after charity be \(P_{\text{after charity}}\). A's share is \(\frac{7}{15}\) of this amount.
\[ \text{A's share} = \frac{7}{15} \times P_{\text{after charity}} \]We are given that A's share is Rs 12,600.
\[ 12,600 = \frac{7}{15} \times P_{\text{after charity}} \]7% of the total profit goes to charity. This means the profit after charity represents \(100\% - 7\% = 93\%\) of the total profit.
Let the total profit earned be \(P_{\text{total}}\).
\[ 0.93 \times P_{\text{total}} = P_{\text{after charity}} \] \[ 0.93 \times P_{\text{total}} = 27,000 \] \[ P_{\text{total}} = \frac{27,000}{0.93} \] \[ P_{\text{total}} \approx 29,032.25806 \]Rounding to 2 decimal places, the total profit earned is Rs 29,032.26.
Correct Option: D) 29,032.26
To find the average score of the class as a whole, we need to calculate the total marks obtained by all students and then divide it by the total number of students. This involves calculating the total marks for each group of students and summing them up.
Total marks for a group = Number of students \(\times\) Average marks
\[ \text{Total marks (Group 1)} = 18 \times 90 = 1620 \]Average score = Total marks (Class) / Total number of students
\[ \text{Average score (Class)} = \frac{2580}{30} = 86 \]Correct Option: C) 86
The problem asks for the smallest number such that when it is tripled, the result is perfectly divisible by 5, 15, and 35. This means the tripled number must be a multiple of 5, 15, and 35. To find the smallest such number, we need to find the Least Common Multiple (LCM) of 5, 15, and 35. Once we have the LCM, we can find the original number by dividing the LCM by 3.
First, we find the prime factorization of each number:
To find the LCM, we take the highest power of all prime factors present in these numbers:
\[ \text{LCM}(5, 15, 35) = 3^1 \times 5^1 \times 7^1 = 3 \times 5 \times 7 = 105 \]The smallest number that is divisible by 5, 15, and 35 is their LCM, which is 105. According to the problem, this number (105) is the result of tripling the unknown smallest number. Let the unknown smallest number be \(x\).
So, \(3x = 105\).
Divide 105 by 3 to find \(x\):
\[ x = \frac{105}{3} = 35 \]If the smallest number is 35, then when it is tripled, it becomes \(3 \times 35 = 105\). Now, check if 105 is divisible by 5, 15, and 35:
All conditions are satisfied.
Correct Option: C) 35