In a practice puzzle at a coaching centre, an ordered integer pair \((x,y)\) is called valid if it satisfies \(3x + 4|y| = 33\). How many integer ordered pairs \((x,y)\) are possible?

An ordered integer pair \((x,y)\) is considered valid if it satisfies \((|x|-3)(|y|+4)=12\). How many integer pairs \((x,y)\) satisfy this equation?

An ordered pair \((x,y)\) is called valid only when it satisfies both conditions: \(x+|y|=8\) and \(|x|+y=6\). How many ordered pairs \((x,y)\) are possible?

At a college fest stall, two types of coupons are counted as \(x\) and \(y\). The total value condition is \(2x+5y=103\), where both \(x\) and \(y\) are positive integers. How many ordered pairs \((x,y)\) satisfy this equation?

Three performance indices are denoted by \(a\), \(b\), and \(c\). It is known that \(\max(a,b,c)+\min(a,b,c)=13\) and \(\text{Median}(a,b,c)-\text{Mean}(a,b,c)=2\). Find the median of \(a\), \(b\), and \(c\).

A system of three linear equations in variables \(x\), \(y\), and \(z\) is given as \(a_1x+b_1y+c_1z=d_1\), \(a_2x+b_2y+c_2z=d_2\), and \(a_3x+b_3y+c_3z=d_3\).

Which of the following statements, if true, would imply that the above system of equations does not have a unique solution?

i. \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\ne \frac{d_1}{d_2}\) ii. \(\frac{a_1}{a_2}=\frac{a_2}{a_3}\) and \(\frac{b_1}{b_2}=\frac{b_2}{b_3}\) iii. \(a_1,a_2,a_3\) are integers; \(b_1,b_2,b_3\) are rational numbers; \(c_1,c_2,c_3\) are irrational numbers

\(b\) is defined as \((3+2\sqrt{2})^{(x^2-3)}+(3-2\sqrt{2})^{(x^2-3)}\), where \(x\) is a real number. Which of the following can be the value of \(b\)?

A shopkeeper at a weekly market usually sells \(100\) notebooks in a week at a selling price of Rs. \(150\) each. For every \(4%\) rise in the selling price, he sells \(3\) fewer notebooks in that week. If the new selling price of each notebook is Rs. \(x\), which of the following expressions represents the number of notebooks sold in that week?

In a Pune auto-parts workshop, one machine makes only nuts at the rate of \(100\) nuts per minute and must be cleaned for \(5\) minutes after every \(1000\) nuts are produced.

Another machine makes only bolts at the rate of \(75\) bolts per minute and must be cleaned for \(10\) minutes after every \(1500\) bolts are produced.

If both machines start working at the same time, what is the minimum time required to produce \(9000\) pairs of nuts and bolts?

Aarav, Bhavesh, Chirag, and Dev together bought a small college project kit for Rs. \(60\). Aarav paid one-half of the total amount paid by the other three friends. Bhavesh paid one-third of the total amount paid by the other three friends. Chirag paid one-fourth of the total amount paid by the other three friends. How much did Dev pay?

A cash-counting kiosk contains one-rupee, two-rupee, and five-rupee coins. The total number of coins is \(300\), and their total value is Rs. \(960\). If the numbers of one-rupee coins and two-rupee coins are interchanged, the total value decreases by Rs. \(40\). Find the total number of five-rupee coins.

An airline has a fixed free luggage allowance per passenger and charges excess luggage at a fixed rate per kg. Two passengers, Rajat and Priya, have \(60\) kg of luggage between them and are charged Rs. \(1200\) and Rs. \(2400\) respectively for excess luggage. Had the entire luggage belonged to one of them, the excess luggage charge would have been Rs. \(5400\). What is the weight of Priya’s luggage?

During a school fair, a father allows his son to buy chocolates, biscuits, and apples. The son may buy as many chocolates as he wishes, but he must buy biscuits equal to twice the number of chocolates, and the number of apples must be more than the total number of chocolates and biscuits together.

Each chocolate costs Rs. \(1\). Each apple costs twice as much as a chocolate, and \(4\) biscuits together cost as much as \(1\) apple.

If the numbers of chocolates, biscuits, and apples bought are all integers, which of the following can be the total amount spent?

A rectangular courtyard is completely covered with identical square tiles. The tiles along the boundary are painted white, while all the tiles strictly inside the boundary are painted red. The number of white tiles is equal to the number of red tiles. Which of the following can be the number of tiles along one edge of the courtyard?

A student visiting a Diwali card stall liked four types of cards priced at Rs. \(2.00\), Rs. \(3.50\), Rs. \(4.50\), and Rs. \(5.00\) each. He wanted to buy \(30\) cards in total, so he bought \(5\) cards each of two types and \(10\) cards each of the other two types. He paid the exact amount using only Rs. \(10\) notes. How many Rs. \(10\) notes did he give?

In a Sivakasi packaging unit, each trainee is assigned not more than \(200\) matchsticks in one session to pack into boxes. If a trainee reduces the number of matchsticks in each box by \(25\), he can fill \(3\) more boxes using the same total number of matchsticks assigned to him. Which of the following can be the number of matchsticks assigned to each trainee?