If the sum of a number and its reciprocal is 7, then which of the options below can be such a number?
\(\frac{7+3\sqrt{5}}{2}\)
\(\frac{-7+3\sqrt{5}}{2}\)
\(\frac{3+7\sqrt{5}}{2}\)
\(\frac{3-7\sqrt{5}}{2}\)
The problem asks us to find a number such that the sum of the number and its reciprocal is 7. We can set up a quadratic equation based on this condition and solve for the number. Then, we compare the solution with the given options.
Let the number be \(x\). Its reciprocal is \(\frac{1}{x}\). According to the problem statement, the sum of the number and its reciprocal is 7.
\[ x + \frac{1}{x} = 7 \]Multiply the entire equation by \(x\) to eliminate the fraction:
\[ x \cdot x + x \cdot \frac{1}{x} = 7 \cdot x \]\[ x^2 + 1 = 7x \]Move all terms to one side to get a standard quadratic equation \(ax^2 + bx + c = 0\):
\[ x^2 - 7x + 1 = 0 \]The quadratic formula for \(ax^2 + bx + c = 0\) is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]In our equation, \(a=1\), \(b=-7\), and \(c=1\). Substitute these values into the formula:
\[ x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(1)}}{2(1)} \]\[ x = \frac{7 \pm \sqrt{49 - 4}}{2} \]\[ x = \frac{7 \pm \sqrt{45}}{2} \]Simplify \(\sqrt{45}\). We know that \(45 = 9 \times 5\), so \(\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}\).
\[ x = \frac{7 \pm 3\sqrt{5}}{2} \]The two possible values for \(x\) are:
\[ x_1 = \frac{7 + 3\sqrt{5}}{2} \]\[ x_2 = \frac{7 - 3\sqrt{5}}{2} \]We need to check which of these solutions matches one of the given options.
A) \(\frac{7+3\sqrt{5}}{2}\)
This matches one of the solutions we found for \(x\).
The question asks to identify the religious group or community for whom Nowruz is popularly known as the new year. This requires knowledge of cultural and religious festivals.
D) Parsi — Nowruz is the traditional new year for the Parsi community, who follow Zoroastrianism. It is a significant festival for them, marking the spring equinox and renewal.
If a and b are two co-prime numbers, then their LCM is:
The problem asks for the Least Common Multiple (LCM) of two co-prime numbers, 'a' and 'b'. We need to understand the definition of co-prime numbers and how it affects their LCM.
B) a \(\times\) b — As derived from the property of LCM and GCD, for co-prime numbers, their LCM is the product of the numbers.
The question asks about the period when the amount of ozone in the atmosphere began to drop sharply. This refers to the discovery and subsequent observation of the ozone hole, primarily caused by human-made chemicals.
Correct Option: A) In the 1980s. The sharp drop in atmospheric ozone, particularly the discovery of the ozone hole, was first widely observed and documented during the 1980s.
This problem involves determining the relative positions of towns based on given directional information. We will visualize the positions on a coordinate plane or by drawing a diagram to deduce the final relative position.
B) South-west — As determined in the step-by-step analysis, Town D is to the west and south of Town M, which means its position is south-west with respect to M.
This is a syllogism question. We need to analyze the given statements and determine which conclusions logically follow. We can use Venn diagrams or logical deduction to evaluate each conclusion.
C) Only conclusion I follows.
An atom of an element has the electronic configuration 2, 8, 8. To which group does it belong?
To determine the group of an element in the periodic table based on its electronic configuration, we need to identify the number of valence electrons. The number of valence electrons often corresponds to the group number for main group elements (s-block and p-block).
Correct Option: D) 18th group
The question describes a fundamental economic principle related to production, specifically how output changes when one input is varied while others are kept constant. This concept is central to the theory of production in microeconomics.
B) The law of variable proportions correctly describes the principle where increasing one factor while keeping others fixed eventually leads to a decline in the marginal product of that factor. This law is also known as the Law of Diminishing Returns.
Six students, Madhu, Anshul, Avani, Aarav, Priti and Shikha, gave their presentations on different days in the month of March. The presentations were from 11th March to 16th March.
Anshul presented on 14th March and Shikha presented immediately before Anshul. Only one person presented between Shikha and Priti. Shikha presented after Priti but before Avani. Avani presented immediately before Madhu.
Who presented on 15th March?
This is a logical reasoning problem involving sequencing. We need to deduce the presentation schedule of six students based on the given clues and then identify who presented on March 15th. We will create a timeline for the presentation days and fill in the names as we deduce them.
The presentations were from March 11th to March 16th. Let's list the dates:
Clue 1: Anshul presented on 14th March.
| Date | Presenter |
|---|---|
| March 11th | |
| March 12th | |
| March 13th | |
| March 14th | Anshul |
| March 15th | |
| March 16th |
Clue 2: Shikha presented immediately before Anshul.
Since Anshul presented on March 14th, Shikha must have presented on March 13th.
| Date | Presenter |
|---|---|
| March 11th | |
| March 12th | |
| March 13th | Shikha |
| March 14th | Anshul |
| March 15th | |
| March 16th |
Clue 3: Only one person presented between Shikha and Priti.
Shikha presented on March 13th. If one person presented between Shikha and Priti, Priti could be on March 11th (with March 12th in between) or March 15th (with March 14th in between). We need to check other clues.
Clue 4: Shikha presented after Priti but before Avani.
This clue clarifies the position of Priti. Since Shikha (March 13th) presented *after* Priti, Priti must have presented before March 13th. Combining this with Clue 3 (one person between Shikha and Priti), Priti must have presented on March 11th (with March 12th being the day between Priti and Shikha).
| Date | Presenter |
|---|---|
| March 11th | Priti |
| March 12th | |
| March 13th | Shikha |
| March 14th | Anshul |
| March 15th | |
| March 16th |
Clue 5: Shikha presented before Avani.
This means Avani presented after March 13th. We already have Anshul on March 14th. So Avani must be on March 15th or March 16th.
Clue 6: Avani presented immediately before Madhu.
This means Avani and Madhu are consecutive. The only remaining consecutive slots are March 15th and March 16th. Therefore, Avani presented on March 15th and Madhu on March 16th.
| Date | Presenter |
|---|---|
| March 11th | Priti |
| March 12th | |
| March 13th | Shikha |
| March 14th | Anshul |
| March 15th | Avani |
| March 16th | Madhu |
The only remaining student is Aarav. The only remaining date is March 12th. So, Aarav presented on March 12th.
| Date | Presenter |
|---|---|
| March 11th | Priti |
| March 12th | Aarav |
| March 13th | Shikha |
| March 14th | Anshul |
| March 15th | Avani |
| March 16th | Madhu |
The question asks who presented on 15th March. From our final schedule, Avani presented on March 15th.
Correct Option: D) Avani presented on 15th March.
This problem involves algebraic identities. We are given the sum and product of two numbers and asked to find the sum of their cubes. We can use the algebraic identity for the sum of cubes, which relates \((a+b)\) and \(ab\) to \((a^3+b^3)\).
Let the two numbers be \(a\) and \(b\).
According to the problem statement:
We need to find the sum of their cubes, which is \(a^3 + b^3\).
Recall the algebraic identity for the sum of cubes:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]Alternatively, a more convenient form for this problem is:
\[ a^3 + b^3 = (a + b)^3 - 3ab(a + b) \]Substitute the given values of \((a+b)\) and \(ab\) into the identity:
\[ a^3 + b^3 = (25)^3 - 3(136)(25) \]Calculate the terms:
Now, substitute these calculated values back into the equation:
\[ a^3 + b^3 = 15625 - 10200 \] \[ a^3 + b^3 = 5425 \]Correct Option: D) 5425