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If $\log_4m + \log_4n = \log_2(m + n)$ where m and n are positive real numbers, then which of the following must be true?
Text Explanation:
$\log_4mn=\log_2(m+n)$
$\sqrt{\ mn}=(m+n)$
Squarring on both sides
$m^2+n^2+mn\ =\ 0$
Since m, n are positive real numbers, no value of m and n satisfy the above equations.
At any point of time, let x be the smaller of the two angles made by the hour hand with the minute hand on an analogue clock (in degrees). During the time interval from 2:30 p.m. to 3:00 p.m., what is the minimum possible value of x?
The difference between the hour and minute hand of a clock is given by $\left|30H-5.5m\right|$. Here H is the current hour and m represents the number of completed minutes in the current hour.
In the given time frame of 2: 30 to 3: 00 pm.
At 2 : 30 pm the angle = $\left|30\cdot2-5.5 \cdot30 \right| = 105$ degrees
At 3: 00 pm the angle = $\left|30\cdot3-5.5 \cdot0 \right| = 90$ degrees
The function of $\left|30\cdot H-5.5 \cdot m\right| =$ constantly increases as the value of m increases from 31, 32................ 59.
Because of the modulus function, the net value of the function remains positive
Between 2: 30 to 2: 59 the angle is constantly increasing. The minimum value is 2: 30 which is equal to 105 degrees which is greater than the 90 degrees when the time is 3: 00.
Hence 90 degrees is the minimum angle.
One third of the buses from City A to City B stop at City C, while the rest go non-stop to City B. One third of the passengers, in the buses stopping at City C, continue to City B, while the rest alight at City C. All the buses have equal capacity and always start full from City A. What proportion of the passengers going to City B from City A travel by a bus stopping at City C?
Let us assume there are three buses, each carrying 30 passengers.
Now it is given that one-third of the buses from City A to City B stop at City C, while the rest go non-stop to City B. This means that one bus stops at C while two buses go directly to B. This means that $2\times 30=60$ passengers directly reach to B.
For the buses that stops at C, One-third of the passengers continue to City B, that is 10 passengers in each bus continue to city B, while the rest alight at City C. Since there is only one bus which stops at city C, this means that $1\times 20=20$ passengers alight at C, while $1\times 10=10$ passengers travel from city C to city B.
We are asked what proportion of the passengers going to City B from City A travel by a bus stopping at City C. So, in total, we can see the total number of passengers who are travelling to City B is 70, while the number of passengers travelling by the buses that stop at City C is 10.
So, the proportion of the passengers going to City B from City A who travel by a bus stopping at City C = $\dfrac{10}{70}=\dfrac{1}{7}$
Rajesh, a courier delivery agent, starts at point A and makes a delivery each at points B, C and D, in that order. He travels in a straight line between any two consecutive points. The following are known: (i) AB and CD intersect at a right angle at E, and (ii) BC, CE and ED are respectively 1.3 km, 0.5 km and 2.5 km long. If AD is parallel to BC, then what is the total distance (in km) that Rajesh covers in travelling from A to D?
Based on the information, the following figure can be obtained.
Given, CE=0.5, BC = 1.3 and ED=2.5
Triangle CEB is a right-angled triangle => EB = 1.2
$\triangle$ ECB is similar to $\triangle$ EDA
EB/EC = AE/ED => AE = 6
Hence total distance travelled = AB + BC + CD = 7.2 + 1.3 + 3 = 11.5km
Let $f(x) = \frac{x^2 + 1}{x^2 - 1}$ if $x ≠ 1, -1,$ and 1 if x = 1, -1. Let $g(x) = \frac{x + 1}{x - 1}$ if $x ≠ 1,$ and 3 if x = 1.What is the minimum possible values of $\frac{f(x)}{g(x)}$ ?
$\frac{f\left(x\right)}{g\left(x\right)}=\frac{\left(x^2+1\right)}{x^2-1}\cdot\frac{\left(x-1\right)}{x+1}=\frac{\left(x^2+1\right)}{\left(x+1\right)^2}$
This function is definitely greater than 0
$let\ y=\frac{\left(x^2+1\right)}{\left(x+1\right)^2}$
=> $x^2\left(y-1\right)+2yx+\left(y-1\right)=0$ which is quadratic in x
Disctiminant should be greater than 0
$4y^2-4\left(y-1\right)^2\ge0$
=> y>=1/2
When x =1, f(x)/g(x) = 1/3
Hence either the value should be greater than 1/2 or should be equal to 1/3
Rahul has just made a $3 \times 3$ magic square, in which, the sum of the cells along any row, column or diagonal, is the same number N. The entries in the cells are given as expressions in x, y, and Z. Find N?
Sum of 3rd row = sum of 2nd column
=> 2x+4y = y+2z-1
=> 2x+3y-2z= -1 ------- (A)
Sum of diagonals are also equal
=> 3x+4y+z-1 = y+z+2x+y+z
=> x+2y-z=1 -----(B)
Solving A and B we get y= 3
Putting it in A, we get x-z = -5 ----- (C)
Sum of 1st row = sum of 2nd column
5x+5y+z = 3x+4y+2z
=> 2x +y - z =0
Since y=3, 2x-z = -3 ------ (D)
Solving C and D we get x=2 and z=7
Hence N = 36
