Q. 1: Meena scores 40% in an examination and after review, even though her score is increased by 50%, she fails by 35 marks. If her post-review score is increased by 20%, she will have 7 marks more than the passing score. The percentage score needed for passing the examination is
1. 60
2. 75
3. 80
4. 70
Let the total Maximum marks in exam = 100x
Meena’s score = 40% of 100x = 40x
Her marks after review = 40x+ 50% of 40x = 60x
Her marks after post review = 60x + 20% of 60x = 72x
As per question 72x – 7 = 60x+ 35
12x = 42
x = 42/12 = 7/2
so passing marks = 60x + 35 = 60*7/2 + 35 =245
total marks = 100x = 350
Required percentage score needed for passing the examination = 245/350 = 70%
Let the length of race = x
When the first beat the second by 11 metres and the third by 90 metres, distance traveled by 2nd and 3rd horse at that time will be x -11 and x -90 respectively.
When the second beat the third by 80 metres traveled by 2nd and 3rd horse at that time will be x and x -80 respectively.
Ratio of speed of 2nd and 3rd horse = ratio of distance traveled = (x-11)/(x-90) = x/(x-80)
(x-11)*(x-80) = x(x-90)
By solving we get x = 880
Q. 3: The number of solution to the equation |x| (6x^2 + 1) = 5x^2 is?
case 1) If x >0
X*(6x^2 +1) = 5x^2
6x^3 -5x^2 +x = 0
x(6x^2 -5x +1) = 0
x(6x^2 -3x -2x +1) =0
x(2x -1)*(3x -1) =0
x = 0, ½ or 1/3
number of solutions = 3
case 2) if x < 0
– X*(6x^2 +1) = 5x^2
6x^3 + 5x^2 +x = 0
x(6x^2 +5x +1) = 0
x(6x^2 +3x +2x +1) =0
x(2x +1)*(3x +1) =0
x = 0, -1/2 or -1/3
Thus x can take 5 different values 0,1/2, 1/3, -1/2 and -1/3
Total number of solutions = 5
Q. 4: The product of two positive numbers is 616. If the ratio of the difference of their cubes to the cube of their difference is 157:3, then the sum of the two numbers is
1. 58
2. 50
3. 95
4. 85
Let the number be a and b.
So ab = 616
Given, (a^3 –b^3)/(a-b)^3 = 157/3
(a-b)*(a^2 + b^2 +ab)/(a-b)*(a^2 + b^2 -2ab) = 157/3
Let a^2 + b^2 =k
So (k+ab)/(k-2ab) = 157/3
3k + 3ab = 157k – 314ab
154k = 117ab = 317*616
k = 317*616/154 = 1268
As we know (a+b)^2 = (a^2 + b^2) + 2ab = k + 2ab = 1268 + 2*616
(a+b)^2 =2500 = 50^2
So a+b = 50
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