Quant Boosters  Hemant Malhotra  Set 12

Q26) The sum of the roots of the quadratic equation ax^2 + bx+ c = 0 is equal to the sum of the squares of their reciprocals. If a, b and c are real numbers, and a ≠ 0, then bc^2 , ca^2 and ab^2 are in
(a) G.P
(b) A.P.
(c) H.P
(d) None of these

Let the roots x and y
x + y = b/a
xy = c/a
Now x + y = (1/x^2 + 1/y^2) [ Given]
Solve this
x + y = (x^2 + y^2)/(x^2 y^2)
x+y = (x + y)^2  2xy/(xy)^2
Now equate their values
b/a = (b^2/a^2)  2(c/a)/(c/a)^2
2ca^2 = ab^2 + bc^2
So bc^2, ca^2 and ab^2 are in AP

Q27) Aman and eight of his friends took a test of 100 marks. Each of them got a different integer score and the average of their scores was 86. The score of Aman was 90 and it was more than that of exactly three of his friends. What could have been the maximum possible absolute difference between the scores of two of his friends?
(a) 83
(b) 73
(c) 54
(d) 44

We need to maximize the difference between the highest and lowest scores
So the top eight scores must have had the maximum possible values i.e.100, 99, 98, 97, 96, 90(Aman), 89 and 88.
The value of lowest score= 86 x 9  (100+ 99+ 98+ 97+ 96+ 90+ 89+ 88)= 17.
required difference = 100  17 = 83

Q28) The cost of 20 oranges and 1 kg of apple is Rs 60 while their selling price is Rs 72. The cost of 10 apples and 1 kg of orange is Rs 50 while their selling price is Rs 60. Given that the profit percent on the sale of two fruits are different, then the sum of the selling price of 5 oranges and 3 apples and the costprice of 6 kg oranges and 5 kg apples (is)
(a) can not be determined
(b) Rs 318
(c) Rs 375
(d) Rs 384
(e) none

CP of 1kg apple and 20 Oranges= 60
CP of 10 kg apple and 1kg Orange = 50
so a + 20b = 60
10a + b = 50
SP of 1kg Apple and 20 Orange = 72
SP of 10 Apple and 1kg orange = 60
x + 20y = 72
10x + y = 60
Now you can find a,b,x,y and your ans  OA = 318

Q29) Mini and Vinay are quiz masters preparing for a quiz. In 'x' minutes, Mini makes 'y' questions more than Vinay. If it were possible to reduce the time needed by each to make a question by 2 mins, then in 'x' minutes Mini would make '2y' questions more than Vinay. How many questions does Mini make in 'x' minutes?
a) 1/4[ 2(x+y)  ( 2 x^2 + 4 y^2 )^1/2 ]
b) 1/4[ 2(xy)  ( 2 x^2 + 4 y^2 )^1/2 ]
c) Either option 1 or 2
d) 1/4[ 2(xy)  ( 2 x^2  4 y^2 )^1/2 ]

Let 1 question b done by mini in m minutes and vinay in v minutes
x/m = y + x/v
x/m2 = 2y + x/v2
x(vm) = 2y(m2)(v2)
x(vm) = yvm
vm = 2(vm  2m 2v + 4)
vm = 2vm  4m  4v + 8
4m + 4v  8 = vm
4m  8 / (m4) = v
4 + 8/m4 = v and m>6
m = 5 , v=12
m = 6 , v = 8
x = 60y/7 and x = 24y
x/m = 12y/7 and 4y
x/v = 5y/7 and 3y
only A satisfies for mini .Exam Conditions approach : You can take random values of x and y & can try solve

Q30) The vertices of a triangle are (10,10), (0,0), (10,10). The number of integral points on and inside the triangle?
OA : 121

@hemant_malhotra is 19 the answer?

@hemant_malhotra root(66)/4

@hemant_malhotra Yes, n will lie between 9 and 9, inclusive of both.

@hemant_malhotra Option B

@hemant_malhotra 256 solutions.