If x − \( \frac{1}{x} \) = 10, then the value of x³ − \( \frac{1}{x^3} \) is:
The problem asks us to find the value of \(x^3 - \frac{1}{x^3}\) given the equation \(x - \frac{1}{x} = 10\). We can use the algebraic identity for the difference of cubes, \((a-b)^3 = a^3 - b^3 - 3ab(a-b)\), or rearrange it to \(a^3 - b^3 = (a-b)^3 + 3ab(a-b)\).
We are given the equation:
\[ x - \frac{1}{x} = 10 \]We need to find the value of \(x^3 - \frac{1}{x^3}\). Let's use the algebraic identity for the difference of cubes. The identity is:
\[ a^3 - b^3 = (a-b)(a^2 + ab + b^2) \]Alternatively, we can cube both sides of the given equation. The identity for \((a-b)^3\) is:
\[ (a-b)^3 = a^3 - b^3 - 3ab(a-b) \]Let \(a = x\) and \(b = \frac{1}{x}\). Substitute these into the identity:
\[ \left(x - \frac{1}{x}\right)^3 = x^3 - \left(\frac{1}{x}\right)^3 - 3(x)\left(\frac{1}{x}\right)\left(x - \frac{1}{x}\right) \]Simplify the expression:
\[ \left(x - \frac{1}{x}\right)^3 = x^3 - \frac{1}{x^3} - 3(1)\left(x - \frac{1}{x}\right) \]\[ \left(x - \frac{1}{x}\right)^3 = x^3 - \frac{1}{x^3} - 3\left(x - \frac{1}{x}\right) \]Now, we can rearrange the identity to solve for \(x^3 - \frac{1}{x^3}\):
\[ x^3 - \frac{1}{x^3} = \left(x - \frac{1}{x}\right)^3 + 3\left(x - \frac{1}{x}\right) \]Substitute the given value \(x - \frac{1}{x} = 10\) into this equation:
\[ x^3 - \frac{1}{x^3} = (10)^3 + 3(10) \]Calculate the values:
\[ x^3 - \frac{1}{x^3} = 1000 + 30 \]\[ x^3 - \frac{1}{x^3} = 1030 \]Correct Option: B) 1030
This is a logical reasoning problem involving arrangement based on given conditions. We need to determine the exam schedule for each person (C, D, E, F, S, T, U) on different days of the week (Monday to Sunday) and then answer the specific question about the number of people.
List the days of the week:
| Day | Person |
|---|---|
| Monday | |
| Tuesday | |
| Wednesday | |
| Thursday | |
| Friday | |
| Saturday | |
| Sunday |
Apply the first condition: "F has the exam on Friday."
| Day | Person |
|---|---|
| Monday | |
| Tuesday | |
| Wednesday | |
| Thursday | |
| Friday | F |
| Saturday | |
| Sunday |
Apply the second condition: "Only two people have their exams between T and F."
This means T can be on Tuesday (Monday, Tuesday, Wednesday, Thursday, Friday -> T _ _ F) or Saturday (Friday, Saturday, Sunday, Monday -> F _ _ T). Let's consider both cases.
Case 1: T is on Tuesday.
| Day | Person |
|---|---|
| Monday | |
| Tuesday | T |
| Wednesday | |
| Thursday | |
| Friday | F |
| Saturday | |
| Sunday |
Case 2: T is on Saturday.
| Day | Person |
|---|---|
| Monday | |
| Tuesday | |
| Wednesday | |
| Thursday | |
| Friday | F |
| Saturday | T |
| Sunday |
Apply the third condition: "Only four people have their exams between E and T."
For Case 1 (T on Tuesday): If T is on Tuesday, then E must be on Sunday (T _ _ _ _ E). This fits the condition.
| Day | Person |
|---|---|
| Monday | |
| Tuesday | T |
| Wednesday | |
| Thursday | |
| Friday | F |
| Saturday | |
| Sunday | E |
For Case 2 (T on Saturday): If T is on Saturday, then four people between E and T would mean E is on Monday (E _ _ _ T). This also fits.
| Day | Person |
|---|---|
| Monday | E |
| Tuesday | |
| Wednesday | |
| Thursday | |
| Friday | F |
| Saturday | T |
| Sunday |
Apply the fourth condition: "S has the exam immediately after U."
This means U and S must be on consecutive days (U, S).
For Case 1 (T on Tuesday, E on Sunday): The remaining empty slots are Monday, Wednesday, Thursday, Saturday. The only consecutive slots are Wednesday and Thursday. So, U is on Wednesday and S is on Thursday.
| Day | Person |
|---|---|
| Monday | |
| Tuesday | T |
| Wednesday | U |
| Thursday | S |
| Friday | F |
| Saturday | |
| Sunday | E |
For Case 2 (E on Monday, T on Saturday): The remaining empty slots are Tuesday, Wednesday, Thursday, Sunday. There are no consecutive slots available for U and S. Thus, Case 2 is invalid.
Finalize the arrangement using Case 1:
| Day | Person |
|---|---|
| Monday | |
| Tuesday | T |
| Wednesday | U |
| Thursday | S |
| Friday | F |
| Saturday | |
| Sunday | E |
Apply the fifth condition: "D does not have the exam on Monday."
The remaining empty slots are Monday and Saturday. The remaining people are C and D. Since D cannot be on Monday, D must be on Saturday. This leaves C for Monday.
| Day | Person |
|---|---|
| Monday | C |
| Tuesday | T |
| Wednesday | U |
| Thursday | S |
| Friday | F |
| Saturday | D |
| Sunday | E |
Answer the question: "How many people have their exams before D and after C?"
People before D (Saturday) are C, T, U, S, F. People after C (Monday) are T, U, S, F, D, E.
We need people who are both BEFORE D AND AFTER C. These are T, U, S, F.
There are 4 people: T, U, S, F.
A) Four. As determined in the step-by-step analysis, T, U, S, and F have their exams after C and before D.
The problem asks us to identify a pair of letter groups that follows the same pattern as the given examples: PIL : TMP and NAP : RET. We need to find the positional difference between corresponding letters in each pair to determine the underlying rule.
So, the pattern for PIL : TMP is \((+4, +4, +4)\).
So, the pattern for NAP : RET is also \((+4, +4, +4)\).
B) SWT : WAX
All letters follow the \((+4, +4, +4)\) pattern. Therefore, option B is the correct answer.
The problem asks us to calculate the total savings of a cook over a specific period. First, we need to determine her monthly savings by subtracting her monthly expenditure from her monthly income. Then, we will calculate the total savings for one-fourth of a year.
The cook's monthly income is given as ₹8,400.
\[ \text{Monthly Income} = \text{₹}8,400 \]She spends 60% of her income each month.
\[ \text{Monthly Expenditure} = 60\% \text{ of } \text{₹}8,400 \]\[ \text{Monthly Expenditure} = \frac{60}{100} \times 8,400 \]\[ \text{Monthly Expenditure} = 60 \times 84 \]\[ \text{Monthly Expenditure} = \text{₹}5,040 \]Monthly savings is the difference between monthly income and monthly expenditure.
\[ \text{Monthly Savings} = \text{Monthly Income} - \text{Monthly Expenditure} \]\[ \text{Monthly Savings} = \text{₹}8,400 - \text{₹}5,040 \]\[ \text{Monthly Savings} = \text{₹}3,360 \]We need to find the savings for one-fourth of a year. There are 12 months in a year.
\[ \text{Period} = \frac{1}{4} \text{ of a year} \]\[ \text{Period in months} = \frac{1}{4} \times 12 \text{ months} \]\[ \text{Period in months} = 3 \text{ months} \]Multiply the monthly savings by the number of months in the period.
\[ \text{Total Savings} = \text{Monthly Savings} \times \text{Number of Months} \]\[ \text{Total Savings} = \text{₹}3,360 \times 3 \]\[ \text{Total Savings} = \text{₹}10,080 \]Correct Option: A) 10,080
This problem involves identifying a pattern in letter and number relationships. We need to decode the rule applied to transform the first pair (KF 17 to HB 6) and the second pair (KH 12 to HD 1), and then apply the same rule to the third term (LP -6) to find its related term.
The pattern is consistent: the first letter moves 3 steps backward, the second letter moves 4 steps backward, and the number decreases by 11.
C) IL −17 — As derived from the established pattern, the first letter moves 3 steps backward (L to I), the second letter moves 4 steps backward (P to L), and the number decreases by 11 (-6 to -17).
The question asks to identify the region in India known for Shola forests. This requires knowledge of the geographical distribution of specific forest types in India.
Correct Option: C) Western Ghats
The problem presents an analogy involving letter-clusters. We need to find the pattern or relationship between the letters in the given pair (first part of the analogy) and then apply the same pattern to find the missing letter-clusters (# and %) in the second part.
D) # = JPR, % = CKO
The pattern observed is that the first letter increases by 2 positions, the second letter increases by 1 position, and the third letter increases by 2 positions in the English alphabet.
The question asks to identify the Indian state that initiated scientific coal mining in 2025 after a decade-long ban on unscientific mining practices. This is a factual question related to recent developments in India's mining sector.
Correct Option: B) Meghalaya is the Indian state that has a history of a decade-long ban on unscientific coal mining (specifically 'rat-hole mining') and has been working towards initiating scientific coal mining practices. The ban was imposed by the National Green Tribunal (NGT) in 2014. The mention of 2025 indicates a future or ongoing plan to resume mining under regulated, scientific methods after this ban.
This is a direction and distance problem. We need to track Sudeep's movements on a 2D plane, typically using a coordinate system or by drawing a diagram, to find his final position relative to his starting point. Then, we calculate the shortest distance and the direction required to return to the starting point.
C) 7 km; West. Based on our calculations, Sudeep is 7 km to the East of Point A. To return to Point A, he must drive 7 km towards the West.
A group of numbers and symbols is coded using letter codes as per the codes given below and the conditions that follow. Study the given codes and conditions and answer the question that follows.
Note: If none of the conditions apply, then codes for the respective number/symbol are to be followed directly as given in the table.
| Number/Symbol | 2 | @ | 9 | 5 | $ | & | 3 | % | # | 7 | + | 4 | 8 | 6 |
| Code | T | F | A | J | L | E | W | Q | D | P | R | B | U | S |
Conditions:
If the first element is a symbol and the last a number, the codes for these two (the first and the last elements) are to be interchanged.
If the first element is an odd number and the last an even number, the first and last elements are to be coded as ©.
If both second and third elements are perfect squares, the third element is to be coded as the code for the second element.
If none of the above conditions are met, then the elements are to be simply coded as per the codes given in the question.
What will be the code for the following group?
@ 2 4 7 +
The problem requires us to code a given group of numbers and symbols based on a provided coding table and a set of conditions. We need to check each condition in order and apply the first one that fits. If no conditions apply, we use the direct codes from the table.
Identify the given group: The group to be coded is @ 2 4 7 +.
List the codes from the table:
@ : F2 : T4 : B7 : P+ : RCheck Condition 1: "If the first element is a symbol and the last a number, the codes for these two (the first and the last elements) are to be interchanged."
@ (a symbol).+ (a symbol, not a number).Check Condition 2: "If the first element is an odd number and the last an even number, the first and last elements are to be coded as ©."
@ (not an odd number).Check Condition 3: "If both second and third elements are perfect squares, the third element is to be coded as the code for the second element."
2 (not a perfect square).4 (a perfect square, \(2^2\)).Apply default coding: Since none of the conditions apply, we code each element directly as per the given table.
@ codes to F2 codes to T4 codes to B7 codes to P+ codes to RCombining these, the code for @ 2 4 7 + is FTBPR.
D) FTBPR