Which of the following should be added to each of the four numbers 4, 8, 12, 22, to make them proportional?
\( \frac{4}{3} \)
\( \frac{3}{4} \)
\( \frac{8}{3} \)
\( \frac{2}{6} \)
To make four numbers proportional, the ratio of the first two numbers must be equal to the ratio of the last two numbers. We need to find a value, let's call it \(x\), that when added to each of the given numbers (4, 8, 12, 22) makes them proportional. This means \((4+x) : (8+x) = (12+x) : (22+x)\).
Let the number to be added to each of the four numbers be \(x\).
The new numbers will be \( (4+x) \), \( (8+x) \), \( (12+x) \), and \( (22+x) \).
For these numbers to be proportional, the ratio of the first two must be equal to the ratio of the last two:
\[ \frac{4+x}{8+x} = \frac{12+x}{22+x} \]Cross-multiply to solve for \(x\):
\[ (4+x)(22+x) = (12+x)(8+x) \]Expand both sides of the equation:
Left side:
\[ 4(22) + 4x + 22x + x^2 = 88 + 26x + x^2 \]Right side:
\[ 12(8) + 12x + 8x + x^2 = 96 + 20x + x^2 \]Set the expanded expressions equal to each other:
\[ 88 + 26x + x^2 = 96 + 20x + x^2 \]Subtract \(x^2\) from both sides:
\[ 88 + 26x = 96 + 20x \]Rearrange the terms to solve for \(x\). Subtract \(20x\) from both sides:
\[ 88 + 6x = 96 \]Subtract 88 from both sides:
\[ 6x = 96 - 88 \]\[ 6x = 8 \]Divide by 6 to find \(x\):
\[ x = \frac{8}{6} \]\[ x = \frac{4}{3} \]To verify, substitute \(x = \frac{4}{3}\) back into the ratios:
First ratio:
\[ \frac{4 + \frac{4}{3}}{8 + \frac{4}{3}} = \frac{\frac{12+4}{3}}{\frac{24+4}{3}} = \frac{\frac{16}{3}}{\frac{28}{3}} = \frac{16}{28} = \frac{4}{7} \]Second ratio:
\[ \frac{12 + \frac{4}{3}}{22 + \frac{4}{3}} = \frac{\frac{36+4}{3}}{\frac{66+4}{3}} = \frac{\frac{40}{3}}{\frac{70}{3}} = \frac{40}{70} = \frac{4}{7} \]Since both ratios are equal to \( \frac{4}{7} \), the value \( x = \frac{4}{3} \) is correct.
Correct Option: A) \( \frac{4}{3} \)
The density of an object of mass 300 kg and volume 150 m3 will be _______.
To find the density of an object, we use the fundamental formula that relates mass and volume. Density is defined as mass per unit volume. We will apply this formula using the given values for mass and volume and determine the correct unit.
Density (\(\rho\)) is calculated by dividing the mass (\(m\)) of an object by its volume (\(V\)).
\[ \rho = \frac{m}{V} \]The unit \(\frac{\text{kg}}{\text{m}^3}\) can also be written as kg m\(^{-3}\).
Therefore, the density of the object is 2 kg m\(^{-3}\).
Correct Option: A) 2 kg m\(^{-3}\)
The question asks to identify the odd one out from a given group of words. This is a classification type of question where we need to find a common characteristic among most items and identify the one that does not share that characteristic.
A) D (Swimmer) — Violinist, Singer, and Dancer are all professions related to performing arts or music. A Swimmer is primarily involved in a sport or athletic activity, making it the odd one out.
This problem involves substituting given symbols with mathematical operators and then evaluating the resulting expression using the order of operations (BODMAS/PEMDAS).
The given expression is \(28 \text{N} 5 \text{L} 50 \text{P} 5 \text{M} 4\).
Replacing the symbols with their corresponding operators, we get:
\[ 28 \times 5 + 50 \div 5 - 4 \]BODMAS stands for Brackets, Orders (powers/roots), Division, Multiplication, Addition, Subtraction. PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. Both rules dictate the same order of operations.
First, perform Division and Multiplication from left to right.
The expression now becomes:
\[ 140 + 10 - 4 \]The value of the expression is \(146\).
Correct Option: B) 146
To find the total volume of the rocket, we need to calculate the volume of its individual components and then subtract the volume of the internal cut-out. The rocket consists of a conical upper part and a cylindrical lower part. An internal semicircular cut is made at the base. We will calculate the volume of the cone, the volume of the cylinder, and the volume of the hemisphere (since a semicircular cut with a given radius implies a hemispherical void if it's a 3D object like a rocket base), and then combine these volumes appropriately.
Calculate the volume of the conical part:
Given:
The formula for the volume of a cone is \(V_c = \frac{1}{3} \pi r_c^2 h_c\).
\[ V_c = \frac{1}{3} \pi (3 \text{ m})^2 (3 \text{ m}) = \frac{1}{3} \pi (9 \text{ m}^2) (3 \text{ m}) = 9 \pi \text{ m}^3 \]Calculate the volume of the cylindrical part:
Given:
The formula for the volume of a cylinder is \(V_{cyl} = \pi r^2 h_{cyl}\).
\[ V_{cyl} = \pi (3 \text{ m})^2 (10 \text{ m}) = \pi (9 \text{ m}^2) (10 \text{ m}) = 90 \pi \text{ m}^3 \]Calculate the volume of the internal cut-out:
The base of the rocket is cut in a semicircular form from the inside to make room for the engine, with a radius of 3 m. This implies a hemispherical void at the base of the cylinder.
Given:
The formula for the volume of a hemisphere is \(V_h = \frac{2}{3} \pi r_h^3\).
\[ V_h = \frac{2}{3} \pi (3 \text{ m})^3 = \frac{2}{3} \pi (27 \text{ m}^3) = 18 \pi \text{ m}^3 \]Calculate the total volume of the rocket:
The total volume is the sum of the volume of the cone and the cylinder, minus the volume of the hemispherical cut-out.
\[ V_{\text{total}} = V_c + V_{cyl} - V_h \] \[ V_{\text{total}} = 9 \pi \text{ m}^3 + 90 \pi \text{ m}^3 - 18 \pi \text{ m}^3 \] \[ V_{\text{total}} = (9 + 90 - 18) \pi \text{ m}^3 \] \[ V_{\text{total}} = (99 - 18) \pi \text{ m}^3 \] \[ V_{\text{total}} = 81 \pi \text{ m}^3 \]Correct Option: D) 81π
The question asks to identify the Indian multinational corporation (MNC) that received the "Innovator of the Year 2017" award at the Times India Awards. This requires specific knowledge of business awards and their recipients in India for that particular year.
Correct Option: B) Mahindra & Mahindra was awarded the title of Innovator of the Year 2017 at the Times India Awards. This recognition was for their consistent focus on innovation and their diverse product portfolio.
The question asks to identify the first Indian to win an individual gold medal at the 2008 Summer Olympic Games. This requires knowledge of India's Olympic history, specifically regarding the Beijing Olympics.
Correct Option: D) Abhinav Bindra was the first Indian to win an individual gold medal at the 2008 Summer Olympic Games. He achieved this feat in the Men's 10-meter Air Rifle event, making history as India's first-ever individual Olympic gold medalist.
To find the difference between the amounts received by X and Z, we first need to determine the individual amounts each person receives. This can be done by using the given ratio and the total amount distributed. We will calculate the value of one ratio unit and then multiply it by the respective ratio parts for X and Z to find their shares. Finally, we will subtract the amount of X from the amount of Z.
Understand the given information:
Calculate the sum of the ratio parts:
Sum of ratio parts \( = 4 + 5 + 6 = 15 \)
Determine the value of one ratio unit:
If 15 ratio parts correspond to 900 rupees, then one ratio part corresponds to:
\[ \text{Value of one ratio unit} = \frac{\text{Total amount}}{\text{Sum of ratio parts}} = \frac{900}{15} = 60 \text{ rupees} \]Calculate the amount received by X:
Amount of X \( = \text{X's ratio part} \times \text{Value of one ratio unit} = 4 \times 60 = 240 \text{ rupees} \)
Calculate the amount received by Z:
Amount of Z \( = \text{Z's ratio part} \times \text{Value of one ratio unit} = 6 \times 60 = 360 \text{ rupees} \)
Calculate the difference between the amount of X and Z:
Difference \( = \text{Amount of Z} - \text{Amount of X} = 360 - 240 = 120 \text{ rupees} \)
C) 120 rupees is the difference between the amount of X and Z.
To find the maximum number of factors a two-digit positive integer less than or equal to 29 can have, we need to list all such integers and determine the number of factors for each. The number of factors for an integer \(N\) with prime factorization \(N = p_1^{a_1} \cdot p_2^{a_2} \cdot \ldots \cdot p_k^{a_k}\) is given by the product \((a_1+1)(a_2+1)\ldots(a_k+1)\).
C) 8 — The integer 24 has 8 factors (1, 2, 3, 4, 6, 8, 12, 24), which is the maximum among two-digit integers less than or equal to 29.
Find the value of \( \sin^{4}\frac{\pi}{4} + 4\sin^{4}\frac{\pi}{4}\sin^{4}\frac{\pi}{3} + \sin^{4}\frac{\pi}{6} \)
\( \frac{7}{8} \)
\( \frac{-7}{2} \)
\( \frac{5}{8} \)
\( \frac{3}{2} \)
To solve this problem, we need to evaluate the trigonometric values for the given angles and then substitute them into the expression. We will use the standard values of sine for angles \(\frac{\pi}{4}\) (45 degrees), \(\frac{\pi}{3}\) (60 degrees), and \(\frac{\pi}{6}\) (30 degrees).
The expression is \[\sin^{4}\frac{\pi}{4} + 4\sin^{4}\frac{\pi}{4}\sin^{4}\frac{\pi}{3} + \sin^{4}\frac{\pi}{6}\]
Substitute the calculated values:
\[ \frac{1}{4} + 4\left(\frac{1}{4}\right)\left(\frac{9}{16}\right) + \frac{1}{16} \]To add these fractions, find a common denominator, which is 16.
\[ \frac{4}{16} + \frac{9}{16} + \frac{1}{16} = \frac{4 + 9 + 1}{16} = \frac{14}{16} \]A) The calculated value of the expression is \( \frac{7}{8} \).