Labratory flask being filled with liquids, close-up

The topic is very popular among the aptitude tests & IQ tests world wide. It is basically the application of the fundamental arithmetic concepts like percentage, ratio, average etc. The mixtures under consideration will be homogeneous in nature.

TYPES OF MIXTURES:- A mixture is not necessarily made up of liquids or solids. There can be various types such as.

  • A liquid and a solid
  • Two liquids
  • Two solids
  • Profit percent on two products
  • Boys & girls in a group
  • A journey covered with different speeds
  • and many more
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similartrianglesSimilar triangles present one of the biggest tools students have to solve many geometry questions. And yet, whenever the opportunity arises, the students often fail to spot the similarity between two triangles. Sometime they spot the similarity but fail to apply the ratios correctly. Today we will start with some familiar cases in which we apply similarity of triangles and then move on to some tricky and difficult cases. We hope that after solving all these problems, you’ll not have any issue in spotting and solving similarity.

In practical scenario, we always prove that two triangles are similar by proving that their corresponding angles are equal. For example, look at the very simple figure below:

two similar triangles

If you can prove that \angle BAC = \angle QPR and \angle ABC = \angle PQR then \triangle ABC and \triangle PQR are similar. In which case the ratios of the corresponding sides are equal. The best way to write the ratio of the sides is to write both the triangles in the order of the angle which are equal (for example here \angle A = \angle P, \angle B = \angle Q and \angle C = \angle R so write ABC and PQR only) and then write the ratio by picking same corresponding points from the two written triangles. Therefore,

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theory of equationsIn CAT and other MBA entrance exams, there are many questions which fall in the domain of equations (quadratic, cubic, quartic..) and the properties of their roots. Deriving these properties will not only help us remember them but also give us the basic concepts to solve all kinds of equations and their problems. In this chapter we shall learn how the properties of roots of a general polynomial equation are derived. So let’s begin!

Let’s take a polynomial P(x) in a single variable x:

Let P(x) = a_0(x)^n + a_1(x)^{n-1} + a_2(x)^{n-2} + ... + a_{n-1}x + a_n where a_1, a_2 etc. are constants (associated with decreasing powers of x) and n, n-1, etc. are the whole number powers of the variable x. The highest power of the variable x is known as the ‘degree’ of the polynomial P(x).

For example, P(x) = x^5 - 3x + 1 is a polynomial with degree 5 with a_0 = 1, a_1 = 0, a_2 = 0, a_3 = 0, a_4 = -3, a_5 = 1

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