Exercise

Animals in an experiment are to be kept on a strict diet. Each animal is to receive among other things $20 \mathrm{g}$ of protein and $6 \mathrm{g}$ of fat. The laboratory technicians purchased two food mixes, $A$ and $B$. Mix $A$ has $10\%$ protein and $6\%$ fat. Mix $B$ has $20\%$ protein and $2\%$ fat. How many grams of each mix should be used?

Anuj had some chocolates and he divided them into two lots $A \text{and} B$. He sold the first lot at the rate of $₹2$ for $3$ chocolates and the second lot at the rate of $₹1$per chocolate, and got a total of $₹400$. If he had sold the first lot at a rate of$₹1$ per chocolate and the second lot at the rate of $₹4$ for $5$ chocolates, his total collection would have been $₹ 460$. Find the total number of chocolates he had.

Two linear equations in variables $x$ and $y$ are given below.

$a_{1}x+b_{1}y+c=0$

$a_{2}x+b_{2}y+c=0$

Which of the following pieces of information is independently sufficient to determine if a solution exists or not for this pair of linear equations?

I. $\frac{a_1}{b_1}=\frac{a_2}{b_2}=1$ II. $\frac{a_1}{a_2}=\frac{b_1}{b_2}$ III. $\frac{a_1}{a_2}=\frac{a_1}{b_1}\ne 1$ IV. $\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$

Some concrete water towers have been built to supply water to the localities nearby. They are usually mounted with a cylindrical tank. A water tower for a locality is 40 m high.

The water tower cast a shadow of 25 m. At the same time, a tree near it casts a shadow of 5 m. What is the height of the tree?

Two poles of heights $3 \mathrm{m}$ and $2 \mathrm{m}$ are erected upright on the ground and ropes are stretched from the top of each to the foot of the other.

The height above the ground at which the ropes cross each other is?

Which of the following are definitely similar to each other?

The break-even point for a new flavour ice cream is the number of units sold at which the total cost price of the ice cream is the same as the total selling price. The parlour launches another new flavour whose fixed cost is Rs$10000,$ variable cost is Rs $40$ and the selling price of each unit Rs $60.$

Find the number of units at the break-even point of the other new flavour. Show your work. 

A company has a locker in which valuable documents are kept. The passcode is a four digit number of the form $xyyx.$ The Chief Executive Officer (CEO) and the Vice President (VP) of the company have each been given one clue. On solving BOTH clues, the passcode that opens the locker can be found.

CEO's clue: When twice the ones digit is subtracted from the tens digit, the result is $1.$

VP's clue: Three more than the tens digit is thrice the ones digit.

Find the passcode that opens the locker. Show your work.