Simplify: \(\frac{\sqrt[3]{5832}}{\sqrt[4]{10000}}\) × \( \frac{5}{54} \) × 78
To simplify the given expression, we need to evaluate the cube root and the fourth root terms first. Then, we will perform the multiplication operations in a step-by-step manner.
We need to find the cube root of 5832. We can do this by prime factorization or by recognizing common cubes.
\[ 5832 = 2 \times 2916 = 2^2 \times 1458 = 2^3 \times 729 \]\[ 729 = 9^3 \]\[ \text{So, } 5832 = 2^3 \times 9^3 = (2 \times 9)^3 = 18^3 \]\[ \sqrt[3]{5832} = 18 \]We need to find the fourth root of 10000.
\[ 10000 = 10 \times 1000 = 10^2 \times 100 = 10^3 \times 10 = 10^4 \]\[ \sqrt[4]{10000} = 10 \]The original expression is:
\[ \frac{\sqrt[3]{5832}}{\sqrt[4]{10000}} \times \frac{5}{54} \times 78 \]\[ \text{Substitute the calculated roots:} \]\[ \frac{18}{10} \times \frac{5}{54} \times 78 \]First, simplify the fraction \(\frac{18}{10}\):
\[ \frac{18}{10} = \frac{9}{5} \]\[ \text{Now the expression becomes:} \]\[ \frac{9}{5} \times \frac{5}{54} \times 78 \]Cancel out the 5 in the numerator and denominator:
\[ \frac{9}{\cancel{5}} \times \frac{\cancel{5}}{54} \times 78 = \frac{9}{54} \times 78 \]\[ \text{Simplify } \frac{9}{54}: \]\[ \frac{9}{54} = \frac{1}{6} \]\[ \text{Now the expression is:} \]\[ \frac{1}{6} \times 78 \]Thus, the simplified value of the expression is 13.
Correct Option: B) 13
To solve this problem, we need to track Sundar's movements on a 2D plane. We can represent his path using coordinates or by calculating the net displacement in the East-West and North-South directions. Since all turns are 90 degrees, we can visualize the path as a series of perpendicular segments.
C) 2 km to the north
To find the final percentage change in salary, we can assume an initial salary and then apply the percentage reduction and subsequent percentage increase sequentially. Finally, we compare the final salary with the initial salary to calculate the overall percentage decrease.
C) 6.5% is the correct answer. The final salary is 6.5% lower than the initial salary.
The problem asks us to find the value of \(x\) given that 260 is the mean proportion between \(x\) and 338. We will use the definition of mean proportion to set up an equation and solve for \(x\).
Understand Mean Proportion: If \(b\) is the mean proportion between \(a\) and \(c\), then the relationship is given by:
\[ \frac{a}{b} = \frac{b}{c} \]This implies \(b^2 = ac\).
Identify the Given Values: In this problem, 260 is the mean proportion, so \(b = 260\). The other two numbers are \(x\) and 338, so we can set \(a = x\) and \(c = 338\).
Set up the Equation: Substitute the values into the mean proportion formula \(b^2 = ac\):
\[ (260)^2 = x \times 338 \]Calculate \((260)^2\):
\[ 260^2 = 260 \times 260 = 67600 \]Solve for \(x\):
\[ 67600 = 338x \]Divide both sides by 338:
\[ x = \frac{67600}{338} \]To simplify the division, notice that \(338 \times 2 = 676\). Therefore:
\[ x = \frac{676 \times 100}{338} = 2 \times 100 \] \[ x = 200 \]Correct Option: A) 200
The question asks to identify which component is NOT a part of the Central Processing Unit (CPU). To answer this, we need to understand the basic architecture of a CPU and its primary components.
Correct Option: B) Graphic Card
To find the average score of the entire class, we need to calculate the total score of all students and then divide it by the total number of students. The total score can be found by summing the total scores of the two groups of students.
The total score for a group is given by the product of the number of students and their average score.
\[ \text{Total Score}_1 = \text{Number of students}_1 \times \text{Average Score}_1 \]\[ \text{Total Score}_1 = 12 \times 60 = 720 \]Correct Option: A) 61.2
Study the given Venn diagram carefully and answer the question that follows.
The numbers in different sections indicate the number of persons. How many chefs are there who are also teachers but not lawyers?
To find the number of chefs who are also teachers but not lawyers, we need to analyze the given Venn diagram and identify the section that represents this specific group.
Correct Option: B) 9
To find the average of the first 14 whole numbers, we first need to identify what whole numbers are. Whole numbers start from 0. Then, we list the first 14 whole numbers, calculate their sum, and divide the sum by the count of numbers (which is 14).
Alternatively, for an arithmetic progression, the average is simply the average of the first and the last term. The first term is 0 and the last term is 13.\[ \text{Average} = \frac{\text{First term} + \text{Last term}}{2} = \frac{0 + 13}{2} = \frac{13}{2} = 6.5 \]
Correct Option: A) 6.5
The problem presents an analogy based on letter-clusters. We need to identify the pattern or relationship between the letters in the given pair (BNW : %) and apply the same logic to find the missing cluster (# : VEC).
A) # = XHA, % = ZKY. This option correctly identifies the missing letter clusters based on the consistent pattern of \(-2, -3, +2\) applied to the alphabetical positions of the letters.
The question asks to identify the Indian state that is landlocked, meaning it does not have a coastline. This requires knowledge of the geographical location of Indian states.
B) Rajasthan is a landlocked state in India and does not have a coastline. It is situated in the north-western part of the country.