The problem asks us to find the next number in the given series: 5, 6, 14, 41, ?. We need to identify the pattern or rule governing the sequence.
The differences are 1, 8, 27. These numbers are perfect cubes:
This suggests that the pattern for the differences is a sequence of consecutive cubes: \(1^3, 2^3, 3^3, \dots\).
Following this pattern, the next difference should be \(4^3\).
\[ 4^3 = 4 \times 4 \times 4 = 64 \]To find the next term, we add the predicted difference to the last known term (41).
\[ \text{Next term} = 41 + 64 = 105 \]The calculated next term is 105, which matches option C.
C) 105 is the correct answer because the pattern of differences between consecutive terms is \(1^3, 2^3, 3^3, \dots\). The next difference is \(4^3 = 64\), and adding this to the last term \(41 + 64 = 105\).
The question asks about the applicability limit of Newlands' Law of Octaves. This law states that when elements are arranged in increasing order of atomic masses, the properties of every eighth element are similar to the first. We need to recall the specific element up to which this law was found to be valid.
A) Ca — Newlands' Law of Octaves was applicable only up to Calcium (Ca). After Calcium, the periodic repetition of properties was not observed.
The problem asks us to find the focal length of a spherical mirror given its radius of curvature. We will use the fundamental relationship between the radius of curvature and the focal length for spherical mirrors.
Thus, the focal length of the spherical mirror is 20 cm.
B) 20 cm. This is the correct focal length calculated using the relationship \(f = R/2\).
The sum of the interior angles of a quadrilateral is equal to:
The problem asks for the sum of the interior angles of a quadrilateral. We can solve this by using the general formula for the sum of interior angles of a polygon or by dividing the quadrilateral into triangles.
Both methods confirm that the sum of the interior angles of a quadrilateral is \(360^\circ\).
Correct Option: C) 360°
The question describes the fundamental principles of heredity, specifically how genetic material (chromosomes) is passed from parents to offspring, ensuring the maintenance of the species' chromosome number. This process involves germ cells contributing half the genetic material and their fusion restoring the full complement. We need to identify which scientific work or concept best explains this mechanism.
A) the results of the Mendel’s experiments. Mendel's laws, particularly the Law of Segregation, explain how each parent contributes one allele (on a chromosome) to the offspring, and the combination of these restores the diploid number, ensuring genetic stability.
The question asks about the gas evolved during the thermal decomposition of limestone. Limestone is primarily composed of calcium carbonate. Thermal decomposition refers to the breakdown of a compound by heat.
Correct Option: B) Carbon dioxide is the gas evolved when limestone (calcium carbonate) undergoes thermal decomposition, producing calcium oxide and carbon dioxide.
To find the selling price when there is a loss, we first calculate the amount of loss based on the given percentage and the cost price. Then, we subtract this loss amount from the cost price to determine the selling price.
Alternatively, we can directly calculate the selling price using the formula:
\[ \text{Selling Price} = \text{Cost Price} \times \left(1 - \frac{\text{Loss Percentage}}{100}\right) \]\[ \text{Selling Price} = 800 \times \left(1 - \frac{15}{100}\right) \]\[ \text{Selling Price} = 800 \times \left(\frac{100 - 15}{100}\right) \]\[ \text{Selling Price} = 800 \times \frac{85}{100} \]\[ \text{Selling Price} = 8 \times 85 \]\[ \text{Selling Price} = \text{Rs } 680 \]D) Rs 680 — The selling price is Rs 680, which is obtained by subtracting the 15% loss (Rs 120) from the cost price (Rs 800).
A sells an article to B at a gain of 16%. B sells it to C at a loss of 15%, and C sells it to D at a gain of 20%. If the difference between the profits earned by C and A is ₹248, then the loss (in ₹) incurred by B is:
We need to find the loss incurred by B. We will assume an initial cost price for article A and then calculate the cost/selling prices at each step for A, B, C, and D. We will then use the given difference in profits to form an equation and solve for the initial cost price, which will allow us to calculate B's loss.
Let the cost price of the article for A be \(\text{₹}P\).
A sells to B at a gain of 16%.
Selling price for A = Cost price for B
\[ \text{CP}_{\text{B}} = P \times \left(1 + \frac{16}{100}\right) = P \times \frac{116}{100} = 1.16P \]B sells to C at a loss of 15%.
Selling price for B = Cost price for C
\[ \text{CP}_{\text{C}} = \text{CP}_{\text{B}} \times \left(1 - \frac{15}{100}\right) = 1.16P \times \frac{85}{100} = 1.16P \times 0.85 = 0.986P \]C sells to D at a gain of 20%.
Selling price for C = Cost price for D
\[ \text{CP}_{\text{D}} = \text{CP}_{\text{C}} \times \left(1 + \frac{20}{100}\right) = 0.986P \times \frac{120}{100} = 0.986P \times 1.2 = 1.1832P \]Calculate the profit earned by A:
\[ \text{Profit}_{\text{A}} = \text{SP}_{\text{A}} - \text{CP}_{\text{A}} = 1.16P - P = 0.16P \]Calculate the profit earned by C:
\[ \text{Profit}_{\text{C}} = \text{SP}_{\text{C}} - \text{CP}_{\text{C}} = 1.1832P - 0.986P = 0.1972P \]The difference between the profits earned by C and A is ₹248.
\[ \text{Profit}_{\text{C}} - \text{Profit}_{\text{A}} = 248 \]\[ 0.1972P - 0.16P = 248 \]\[ 0.0372P = 248 \]\[ P = \frac{248}{0.0372} \]\[ P = \frac{2480000}{372} \]\[ P = \frac{620000}{93} \]\[ P = \frac{200000}{31} \]\[ P = 6451.6129 \approx 6451.61 \]Let's recheck the calculation of P. If we simplify \(\frac{248}{0.0372}\), we get:\[ P = \frac{248}{0.0372} = \frac{248 \times 10000}{372} = \frac{2480000}{372} \]Dividing 248 by 372:\[ \frac{248}{372} = \frac{62 \times 4}{93 \times 4} = \frac{62}{93} \]So, \[ P = \frac{62}{93} \times 10000 = \frac{2 \times 31}{3 \times 31} \times 10000 = \frac{2}{3} \times 10000 = \frac{20000}{3} \]This gives \(P = 6666.66...\). Let's assume there might be a simpler value for P that would result in integer values for the profits. Let's re-evaluate the calculation. If \(P = \frac{20000}{3}\), then: \[\text{Profit}_{\text{A}} = 0.16 \times \frac{20000}{3} = \frac{3200}{3}\] \[\text{Profit}_{\text{C}} = 0.1972 \times \frac{20000}{3} = \frac{3944}{3}\] \[\text{Profit}_{\text{C}} - \text{Profit}_{\text{A}} = \frac{3944 - 3200}{3} = \frac{744}{3} = 248\]. This confirms \(P = \frac{20000}{3}\) is correct.Now, calculate the loss incurred by B.
Loss by B = Cost price for B - Selling price for B
\[ \text{Loss}_{\text{B}} = \text{CP}_{\text{B}} - \text{SP}_{\text{B}} \]\[ \text{Loss}_{\text{B}} = 1.16P - 0.986P = (1.16 - 0.986)P = 0.174P \]Substitute the value of \(P\):\[ \text{Loss}_{\text{B}} = 0.174 \times \frac{20000}{3} \]\[ \text{Loss}_{\text{B}} = \frac{174}{1000} \times \frac{20000}{3} \]\[ \text{Loss}_{\text{B}} = \frac{174 \times 20}{3} \]\[ \text{Loss}_{\text{B}} = 58 \times 20 \]\[ \text{Loss}_{\text{B}} = 1160 \]The loss incurred by B is ₹1,160.
B) 1,160
When four quantities \(a, b, c, d\) are in proportion, it means that the ratio of the first two is equal to the ratio of the last two. Mathematically, this is expressed as \(a:b::c:d\) or \[\frac{a}{b} = \frac{c}{d}\]. We will first substitute the given value of \(x\) into the expressions to find the numerical values of the first three terms and then use the proportion property to find \(y\).
Correct Option: B) 6
The power input to the circuit when charge of Q coulomb is flowing for t seconds for a potential difference of V volt is:
P = \(\frac{Vt}{Q}\)
P = VQt
P = \(\frac{VQ}{t}\)
P = \(\frac{Qt}{V}\)
To determine the power input to the circuit, we need to recall the fundamental definitions of electric current, potential difference, and power. Power is defined as the rate at which work is done or energy is transferred. In an electrical circuit, this relates to the potential difference and the flow of charge over time.