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Ram purchased 3 oranges at Rs. 20 each and 5 mangoes at Rs. 38 each and then sold them by offering discounts of 25% and 20% on the fixed marked prices of oranges and mangoes, respectively. If he earned a profit of 50% on selling the oranges alone, and a total profit of 32% on selling all the fruits, then the marked price of each mango was ___

The remainder when $7^{103} + 7^{101}$ is divided by $9$ is ___

If $n$ is an integer such that $\frac{|n+6|-|n-3|}{\sqrt{100-n^3}} \geq 0$, then the number of possible values of $n$ is ___

In a company, initially the ratio of the foreign and domestic workers was 5 : 8. When some foreign and 12 domestic workers left the company, this ratio became 2 : 3. Later, when 10 more foreign workers were replaced by 10 new domestic workers, this ratio became 3 : 7. The initial number of domestic workers in the company was ___

The number of ways in which three distinct integers can be chosen from the set $\{1, 2, \ldots, 9\}$ such that their product is divisible by 4, is ___

Savitri borrowed 10000 rupees from a bank for a period of two years at a fixed interest rate of 10% per annum, compounded semi-annually, and paid back 5025 rupees at the end of first year. Then, the amount, in rupees, to be paid at the end of second year is ___

A man starts walking from point A to point B at 4 km/hr. After 30 minutes, a woman starts from point A at 6 km/hr. If the woman reaches point B 20 minutes earlier than the man, then the distance between A and B in km is ___

If $x, y$ are real numbers and equations $x^2 - 12x + 35 = 0$ and $x^2 + ax + 105 = 0$ have at least one common root, then the minimum possible value of $y^2 + 4y - 5a$ is ___

Let $ABCD$ be a cyclic quadrilateral with $AB = 15, BC = 20, CD = 24$ and $AC = 25$. Then $AD$ equals ___

Let $S$ denote an arithmetic progression whose first term is either 132 or 158, and the common difference is an even integer less than 10. If the $n^{\text{th}}$ term of $S$ is 174, then the number of possible distinct values of $n$ is ___

The Celsius scale of temperature can be converted to Fahrenheit scale using the relationship $F = 1.8C + 32$, where $F$ and $C$ are temperature recorded in degrees Fahrenheit and Celsius, respectively. Ajay noted down the temperature of a chemical reaction as $x$ degrees Fahrenheit but reported it as $x$ degrees Celsius by mistake. If this led to a recording of the temperature that was 144% higher than the actual temperature in Celsius scale, then $x$ equals ___

Let $M = \begin{bmatrix} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2 \end{bmatrix}$ where $a, b$ and $c$ are real numbers such that $a + b + c = 0$ and $abc \neq 0$. If $\det M = 0$, then the maximum possible value of $\frac{a^2+b^2}{c^2}$ is ___