Section - Mathematics
A hollow spherical shell is made of a metal of density 40 g/cm3. Its internal and external radii are 1 cm and 4 cm, respectively. What is the mass (in kg) of the shell? (Take π = \( \frac{22}{7} \))
To find the mass of the hollow spherical shell, we need to calculate its volume and then multiply it by the given density. The volume of a hollow sphere is the difference between the volume of the external sphere and the internal sphere.
The formula for the volume of a sphere is \(V = \frac{4}{3}\pi r^3\).
\[ V_{\text{external}} = \frac{4}{3} \pi r_2^3 = \frac{4}{3} \times \frac{22}{7} \times (4 \text{ cm})^3 \]\[ V_{\text{external}} = \frac{4}{3} \times \frac{22}{7} \times 64 \text{ cm}^3 = \frac{5632}{21} \text{ cm}^3 \]The volume of the metal is the difference between the external and internal volumes.
\[ V_{\text{shell}} = V_{\text{external}} - V_{\text{internal}} = \frac{5632}{21} \text{ cm}^3 - \frac{88}{21} \text{ cm}^3 \]\[ V_{\text{shell}} = \frac{5632 - 88}{21} \text{ cm}^3 = \frac{5544}{21} \text{ cm}^3 \]\[ V_{\text{shell}} = 264 \text{ cm}^3 \]Mass = Density \(\times\) Volume
\[ \text{Mass} = \rho \times V_{\text{shell}} = 40 \text{ g/cm}^3 \times 264 \text{ cm}^3 \]\[ \text{Mass} = 10560 \text{ g} \]Since 1 kg = 1000 g,
\[ \text{Mass (in kg)} = \frac{10560}{1000} \text{ kg} = 10.56 \text{ kg} \]D) 10.56 kg is the calculated mass of the shell.
An amount of ₹936 is divided among three persons in the ratio of 2 : 4 : 3. The difference between the largest and the smallest shares (in ₹) in the distribution is:
To solve this problem, we first need to determine the individual shares of the three persons based on the given ratio. We will find the total number of ratio parts, divide the total amount by this sum to find the value of one ratio part, and then multiply by each person's ratio to get their share. Finally, we will calculate the difference between the largest and smallest shares.
Identify the total amount and the ratio:
Total amount to be divided = ₹936
Ratio among three persons = \(2 : 4 : 3\)
Calculate the sum of the ratio parts:
Sum of ratio parts \( = 2 + 4 + 3 = 9 \)
Determine the value of one ratio part:
Value of one ratio part \[= \frac{\text{Total amount}}{\text{Sum of ratio parts}}\]
\[ = \frac{936}{9} = 104 \]So, one ratio part is equal to ₹104.
Calculate each person's share:
Identify the largest and smallest shares:
Largest share \( = \text{₹}416 \)
Smallest share \( = \text{₹}208 \)
Calculate the difference between the largest and smallest shares:
Difference \( = \text{Largest share} - \text{Smallest share} \)
\[ = 416 - 208 = \text{₹}208 \]D) 208
The difference between the largest share (₹416) and the smallest share (₹208) is ₹208.
By how much is 65% of 75 greater than \( \frac{3}{5} \) of 15?
To solve this problem, we need to calculate two values: first, 65% of 75, and second, \( \frac{3}{5} \) of 15. Then, we will find the difference between these two values to determine how much the first is greater than the second.
Calculate 65% of 75:
To find a percentage of a number, convert the percentage to a decimal or a fraction and multiply it by the number.
\[ 65\% \text{ of } 75 = \frac{65}{100} \times 75 \]\[ = \frac{13 \times 5}{20 \times 5} \times 75 \]\[ = \frac{13}{20} \times 75 \]\[ = 13 \times \frac{75}{20} \]\[ = 13 \times \frac{15}{4} \]\[ = \frac{195}{4} \]\[ = 48.75 \]Calculate \( \frac{3}{5} \) of 15:
To find a fraction of a number, multiply the fraction by the number.
\[ \frac{3}{5} \text{ of } 15 = \frac{3}{5} \times 15 \]\[ = 3 \times \frac{15}{5} \]\[ = 3 \times 3 \]\[ = 9 \]Find the difference:
Subtract the second value from the first value to find out how much greater the first value is.
\[ \text{Difference} = (\text{65% of 75}) - (\frac{3}{5} \text{ of 15}) \]\[ = 48.75 - 9 \]\[ = 39.75 \]C) 39.75 is the correct answer, as calculated by finding 65% of 75 (which is 48.75) and subtracting \( \frac{3}{5} \) of 15 (which is 9) from it, resulting in 39.75.
The distance between two electric poles of length 25 metres and 36 metres is x metres. If the two angles of elevation of their respective top from the bottom of the other are complementary to each other, then the value of x (in metres) is:
This problem involves trigonometry, specifically the concept of angles of elevation and complementary angles. We will set up two right-angled triangles formed by the poles and the distance between them, use the tangent function, and apply the condition that the angles of elevation are complementary.
Let the heights of the two electric poles be \(h_1 = 25\) metres and \(h_2 = 36\) metres. Let the distance between the poles be \(x\) metres.
Consider the first pole of height \(h_1 = 25\) m. The angle of elevation of its top from the bottom of the second pole is \(\theta_1\). From the diagram, we can form a right-angled triangle where the height is \(h_1\), the base is \(x\), and the angle is \(\theta_1\).
So, we have:
\[ \tan(\theta_1) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h_1}{x} = \frac{25}{x} \quad \ldots (1) \]Now, consider the second pole of height \(h_2 = 36\) m. The angle of elevation of its top from the bottom of the first pole is \(\theta_2\). Similarly, we form another right-angled triangle where the height is \(h_2\), the base is \(x\), and the angle is \(\theta_2\).
So, we have:
\[ \tan(\theta_2) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h_2}{x} = \frac{36}{x} \quad \ldots (2) \]The problem states that the two angles of elevation are complementary. This means their sum is \(90^\circ\).
So, \(\theta_1 + \theta_2 = 90^\circ\).
From this, we can write \(\theta_2 = 90^\circ - \theta_1\).
Substitute this into equation (2):
\[ \tan(\theta_2) = \tan(90^\circ - \theta_1) \]Using the trigonometric identity \(\tan(90^\circ - A) = \cot(A)\), we get:
\[ \tan(\theta_2) = \cot(\theta_1) \]We also know that \(\cot(A) = \frac{1}{\tan(A)}\). So,
\[ \tan(\theta_2) = \frac{1}{\tan(\theta_1)} \]Now, substitute the expressions for \(\tan(\theta_1)\) and \(\tan(\theta_2)\) from equations (1) and (2) into this relationship:
\[ \frac{36}{x} = \frac{1}{\frac{25}{x}} \]\[ \frac{36}{x} = \frac{x}{25} \]Cross-multiply to solve for \(x\):
\[ x^2 = 36 \times 25 \]\[ x^2 = 900 \]Take the square root of both sides:
\[ x = \sqrt{900} \]\[ x = 30 \]Since \(x\) represents a distance, it must be positive. Therefore, the value of \(x\) is 30 metres.
B) 30
A supplier offers a trade discount of 20% on a product priced at ₹5,000. Additionally, there is a scheme discount of 10% on the already discounted price. What is the final price of the product after both discounts?
To find the final price after two successive discounts, we first calculate the price after the trade discount. Then, we apply the scheme discount to this new discounted price. It's important to note that discounts are applied sequentially, not cumulatively on the original price.
The original price of the product is ₹5,000.
The trade discount is 20%.
\[ \text{Trade Discount Amount} = 20\% \text{ of } \text{₹}5,000 \]\[ = \frac{20}{100} \times 5,000 \]\[ = \text{₹}1,000 \]\[ \text{Price after Trade Discount} = \text{Original Price} - \text{Trade Discount Amount} \]\[ = 5,000 - 1,000 \]\[ = \text{₹}4,000 \]The scheme discount is 10% on the already discounted price (₹4,000).
\[ \text{Scheme Discount Amount} = 10\% \text{ of } \text{₹}4,000 \]\[ = \frac{10}{100} \times 4,000 \]\[ = \text{₹}400 \]\[ \text{Final Price} = \text{Price after Trade Discount} - \text{Scheme Discount Amount} \]\[ = 4,000 - 400 \]\[ = \text{₹}3,600 \]D) ₹3,600 is the final price of the product after both discounts.
If the third proportional of 9 and 30 be x, then what is the value of x?
The third proportional of two numbers \(a\) and \(b\) is a number \(c\) such that \(a, b, c\) are in continued proportion. This means the ratio of the first to the second is equal to the ratio of the second to the third. We will set up this proportion and solve for the unknown third proportional.
Let the two given numbers be \(a = 9\) and \(b = 30\).
Let the third proportional be \(x\). According to the definition of third proportional, \(a, b, x\) are in continued proportion.
This means that the ratio of the first number to the second number is equal to the ratio of the second number to the third number. Mathematically, this can be written as:
\[ \frac{a}{b} = \frac{b}{x} \]Substitute the given values of \(a\) and \(b\) into the proportion:
\[ \frac{9}{30} = \frac{30}{x} \]To solve for \(x\), cross-multiply the terms:
\[ 9 \times x = 30 \times 30 \]Calculate the product on the right side:
\[ 9x = 900 \]Divide both sides by 9 to find the value of \(x\):
\[ x = \frac{900}{9} \]\[ x = 100 \]Thus, the value of the third proportional \(x\) is 100.
Correct Option: D) 100
Zeeshan travels 252 km at 63 km/hr, the next 150 km at 30 km/hr and the next 680 km at 68 km/hr. What is his average speed (in km/hr, rounded off to two decimal places) for the whole journey?
To find the average speed for the entire journey, we need to calculate the total distance traveled and the total time taken. Average speed is defined as the total distance divided by the total time.
Rounding \(56.94736\) to two decimal places gives \(56.95\ \text{km/hr}\).
Correct Option: D) 56.95
What is the difference between a single discount of 48% on ₹1,450 and two successive discounts of 10% and 40% on the same amount?
To find the difference, we need to calculate the final price after a single discount and the final price after two successive discounts on the given amount. Then, we find the difference between the two discount amounts.
The marked price is ₹1,450.
\[ \text{Single Discount Amount} = 48\% \text{ of } \text{₹}1,450 \]\[ = \frac{48}{100} \times 1450 \]\[ = 48 \times 14.5 \]\[ = \text{₹}696 \]First, calculate the effective single discount percentage for two successive discounts of \(d_1\) and \(d_2\). The formula for the effective discount percentage is:
\[ \text{Effective Discount} = \left(d_1 + d_2 - \frac{d_1 \times d_2}{100}\right)\% \]\[ = \left(10 + 40 - \frac{10 \times 40}{100}\right)\% \]\[ = \left(50 - \frac{400}{100}\right)\% \]\[ = (50 - 4)\% \]\[ = 46\% \]Alternatively, we can calculate the final price after each discount:Price after first discount of 10%:
\[ = 1450 - (10\% \text{ of } 1450) \]\[ = 1450 - \frac{10}{100} \times 1450 \]\[ = 1450 - 145 \]\[ = \text{₹}1305 \]Price after second discount of 40% on the reduced price:
\[ = 1305 - (40\% \text{ of } 1305) \]\[ = 1305 - \frac{40}{100} \times 1305 \]\[ = 1305 - 522 \]\[ = \text{₹}783 \]The total discount amount for successive discounts is the original price minus the final price:\[ \text{Successive Discount Amount} = 1450 - 783 \]\[ = \text{₹}667 \]Difference = Single Discount Amount - Successive Discount Amount
\[ = 696 - 667 \]\[ = \text{₹}29 \]B) ₹29 is the correct answer. The difference between the single discount of 48% (₹696) and the two successive discounts of 10% and 40% (₹667) is ₹29.
Simplify: 4((\( \frac{3}{4} \))x² − 25x + 17) − 3(x² + 8x − 13)
To simplify the given algebraic expression, we need to distribute the constants into their respective parentheses and then combine like terms. This involves careful handling of signs, especially when subtracting an entire expression.
The first part of the expression is \(4\left(\frac{3}{4}x^2 - 25x + 17\right)\).
Distributing \(4\):
\[ 4 \times \frac{3}{4}x^2 - 4 \times 25x + 4 \times 17 \] \[ 3x^2 - 100x + 68 \]The second part of the expression is \(-3(x^2 + 8x - 13)\).
Distributing \(-3\):
\[ -3 \times x^2 - 3 \times 8x - 3 \times (-13) \] \[ -3x^2 - 24x + 39 \]The full expression becomes:
\[ (3x^2 - 100x + 68) + (-3x^2 - 24x + 39) \]Remove the parentheses and combine like terms:
\[ 3x^2 - 100x + 68 - 3x^2 - 24x + 39 \]Correct Option: D) −124x + 107
Shahid and Mira have to travel from Delhi to Kanpur in their respective cars. Shahid is driving at 42 km/hr while Mira is driving at 98 km/hr. Find the time taken by Mira to reach Kanpur if Shahid takes 14 hours.
This problem involves calculating distance, speed, and time. We are given the speeds of two individuals, Shahid and Mira, and the time taken by Shahid to cover a certain distance. We need to find the time taken by Mira to cover the same distance. The key is to first calculate the total distance from Delhi to Kanpur using Shahid's speed and time, and then use this distance along with Mira's speed to find her travel time.
C) Mira takes 6 hours to reach Kanpur.