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Select the graph that schematically represents BOTH y = x^{m} and y = x^{1/m} properly in the interval 0 ≤ x ≤ 1, for integer values of m, where m > 1.
Given y = x^{m} where m > 1 & y = x^{1/m} and 0 ≤ x ≤ 1
Let us consider m = 2
In that case:
y = x^{2} graph can be plotted as
Also y = x^{0.5} is represented in graph as:
∴ y = x^{m} & y = x^{1/m} is correctly represented by option 1) i.e.
y = xm ⇒ increasing slope
y = x1/m ⇒ decreasing slope
It was estimated that 52 men can complete a strip in a newly constructed highway connecting cities P and Q in 10 days. Due to an emergency, 12 men were sent to another project. How many numbers of days, more than the original estimate, will be required to complete the strip?
Concept:
M_{1 }× D_{1 }= M_{2 }× D_{2}
Where,
M_{1} = Number of men in initial condition, M_{2} = Number of men in final condition
D_{1} = Number of days required in the initial condition,
D_{2} = Number of days required in the final condition
Calculation:
Given:
M_{1 }= 52,
D_{1 }= 10, M_{2 }= 52  12 = 40,
D_{2 }= ?
52 × 10 = 40 × D_{2}
∴ D_{2 }= 13 days
Now,
Numbers of days more than the original estimate is equal to:
Number of days required in the final condition  Number of days required in the initial condition
∴ Numbers of days more than the original estimate = D_{2} _{–} D_{1}
∴ Numbers of days more than the original estimate = 13 – 10
∴ Numbers of days more than the original estimate = 3 days
M and N had four children P, Q, R and S. Of them, only P and R were married. They had children X and Y respectively. If Y is a legitimate child of W, which one of the following statements is necessarily FALSE?
Using the above symbols we get the following family tree:
From the tree diagram, we can see that R and W are the parents of Y as Y is a legitimate child of W.
Therefore it is possible that R is the father of Y and W is the wife of R. Hence the statements in options 2 and 3 are not necessarily false.
Also, from the tree diagram, we can conclude that M and N are the grandparents of Y and hence the statement in option 1 is not necessarily false.
As Y is a legitimate child of W, W is the spouse of R and not P.
Hence the statement 'W is the wife of P' is necessarily false.
There are five levels {P, Q, R, S, T} in a linear supply chain before a product reaches customers, as shown in the figure.
At each of the five levels, the price of the product is increased by 25%. If the product is produced at level P at the cost of Rs. 120 per unit, what is the price paid (in rupees) by the customers?
Concept:
Price paid by customer at end = initial cost × (increased percentage)^{n}
Where,
n = no. of levels
Calculation:
Given:
Cost of product at first level P = 120 Rupees.
Also,
It is increased by 25% in each level.
Hence,
Initial cost = 120 Rupees.,
Increased percentage = 25%,
n = 5
Now,
Price paid by customer at end = 120 × 1.25^{5} Rupees
∴ Price paid by customer at end = 366.21 Rupees.
Given that a and b are integers and a + a^{2} b^{3} is odd, which one of the following statements is correct?
Given (a + a^{2}b^{3}) is odd
⇒ a (1 + ab^{3}) is odd
As the multiplication of two numbers is odd only when those two numbers are odd.
That means ‘a’ is odd and (1 + ab^{3}) is also odd
⇒ ab^{3} is even
As ‘a’ is an odd number, b^{3} should be even number to get ab^{3} as even.
b^{3 }is even only when b is even.
Therefore, ‘a’ is odd and ‘b’ is even.
The bar graph shows the data of the students who appeared and passed in an examination for four school P, Q, R and S. The average of success rates (in percentage) of these four schools is _____.
Concept:
Average success rate of one school = Total no of passed students/Total no. of students who appeared
The average success rates of these four schools = Sum of Average success rate of all four schools/Total no. of schools
Therefore, The average success rates of these four schools will be
Calculation:
The average success rates of these four schools will be
= 59%
∴ The average success rates of these four schools = 59.0%
A person divided an amount of Rs. 100,000 into two parts and invested in two different schemes. In one he got 10% profit and in the other he got 12%. If the profit percentages are interchanged with these investments he would have got Rs.120 less. Find the ratio between his investments in the two schemes.
Let Rs. 100,000 is divided into two parts x and y respectively.
Part I: Scheme – 1 (10% profit)
After profit in Scheme – 1, he will get 1.10x rupees
Scheme – 2 (12% profit)
After profit in Scheme – 2, he will get 1.12y rupees
Total amount that he will get from Scheme 1 and Scheme 2 after applying profits of 10% and 12% respectively = 1.10x + 1.12y …1)
Part II: When profit percentages are interchanged
Scheme – 1: (12% profit), Scheme – 2: (10% profit)
Total amount after applying these profit percentages = 1.12x + 1.10y …2)
Given that,
1.10x + 1.12y – 1.12x – 1.10y = 120
0.02y – 0.02x = 120
2y – 2x = 12000
y – x = 6000 …3)
y + x = 100000 …4)
Solving 3) and 4)
2y = 106000
Y = 53000, x = 47000
Ratio between the investments = x/y = 47/53
Consider the following three statements:
(i) Some roses are red.
(ii) All red flowers fade quickly.
(iii) Some roses fade quickly.
Q. Which of the following statements can be logically inferred from the above statements?
Consider the following Venn diagram:
If all red flowers fade quickly and some roses are red, then these roses being red flowers will also fade quickly.
Hence, it can be inferred that some rose fade quickly if the statements (i) and (ii) are true.
Whereas if any of the statements (i) and (ii) are false, nothing can be inferred about statement (iii).
A rectangle becomes a square when its length and breadth are reduced by 10 m and 5 m, respectively. During this process, the rectangle loses 650 m^{2} of area. What is the area of the original rectangle in square meters?
Let the length and breadth of the rectangle is L and B respectively.
The area of the rectangle = LB
It becomes square when its length and breadth are reduced by 10 m and 5 m, respectively.
Now, the length and breadth of the square become (L – 10) and (B – 5) respectively.
As in a square the length and breadth are the same, L – 10 = B – 5
⇒ L = B + 5
The area of the square = (L – 10) (B – 5)
Given that, the rectangle loses 650 m^{2} of area
⇒ LB – 650 = (L – 10) (B – 5)
⇒ LB – 650 = LB + 50 – 10B – 5L
⇒ 10B + 5L = 700
⇒ 10B + 5(B + 5) = 700
⇒ B = 45 m
⇒ L = B + 5 = 50 m
Area of the original rectangle = LB = 50 × 45 = 2250 m^{2}
The value of the expression
Concept:
Logarithmic Properties:
Calculation:
Using
= 1
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