A fruit seller has a total of 187 fruits consisting of apples, mangoes and oranges. The number of apples and mangoes are in the ratio 5 : 2. After she sells 75 apples, 26 mangoes and half of the oranges, the ratio of number of unsold apples to number of unsold oranges becomes 3 : 2. The total number of unsold fruits is

If \(a + b\sqrt{n}\) is the positive square root of \(29 - 12\sqrt{5}\), where \(a\) and \(b\) are integers and \(n\) is a natural number, then the maximum possible value of \(a + b + n\) is \(\underline{\ \ \ \ }\).

The sum of all real values of \(k\) for which \(\left(\frac{1}{8}\right)^k \times \left(\frac{1}{32768}\right)^{\frac{1}{3}} = \frac{1}{8} \times \left(\frac{1}{32768}\right)^{\frac{1}{k}}\) is

A. \( \frac{2}{3} \)    B. \( -\frac{2}{3} \)    C. \( \frac{4}{3} \)    D. \( -\frac{4}{3} \)

In the \(XY\)-plane, the area, in sq. units, of the region defined by the inequalities \[ y \ge x + 4 \quad \text{and} \quad -4 \le x^2 + y^2 + 4(x - y) \le 0 \] is

A. \(4\pi\)          B. \(2\pi\)          C. \(\pi\)          D. \(3\pi\)

Renu would take 15 days working 4 hours per day to complete a certain task whereas Seema would take 8 days working 5 hours per day to complete the same task. They decide to work together to complete this task. Seema agrees to work for double the number of hours per day as Renu, while Renu agrees to work for double the number of days as Seema. If Renu works 2 hours per day, then the number of days Seema will work, is

In September, the incomes of Kamal, Amal and Vimal are in the ratio 8 : 6 : 5. They rent a house together, and Kamal pays 15%, Amal pays 12% and Vimal pays 18% of their respective incomes to cover the total house rent in that month. In October, the house rent remains unchanged while their incomes increase by 10%, 12% and 15%, respectively. In October, the percentage of their total income that will be paid as house rent, is nearest to

Consider two sets \(A = \{2, 3, 5, 7, 11, 13\}\) and \(B = \{1, 8, 27\}\). Let \(f\) be a function from \(A\) to \(B\) such that for every element \(b \in B\), there is at least one element \(a \in A\) such that \(f(a) = b\). Then, the total number of such functions \(f\) is

A. \(540\)          B. \(537\)          C. \(665\)          D. \(667\)

Let \(x\), \(y\), and \(z\) be real numbers satisfying \[ 4(x^2 + y^2 + z^2) = a,\quad 4(x - y - 2) = 3 + a \] Then \(a\) is equal to

A. \(3\)          B. \(1\frac{1}{3}\)          C. \(1\)          D. \(4\)

The sum of all four-digit numbers that can be formed with the distinct non-zero digits a,b,c, and d, with each digit appearing exactly once in every number, is 153310+n, where n is a single digit natural number. Then, the value of (a+b+c+d+n) is

When \(10^{100}\) is divided by \(7\), the remainder is

A. \(3\)          B. \(6\)          C. \(1\)          D. \(4\)