Algebra in CAT: 1998 – 2006 (Part II)

The very fact that the Algebra portion in CAT has undergone a slight transformation over the past seven years has led to one of the oft-repeated question: “Is CAT still a Class X game?” Maybe there is some truth behind this concern, especially since the importance of Arithmetic in CAT has reduced over the years.

In continuation of my last article, let us now take a look at CAT from 2001 to 2003 to understand the changing trend.

CAT 2001:
This particular paper placed a lot of premium on Algebra. There were two questions based on simple linear inequalities; one on forming equations given a set of 4 conditions; and the other was a set of 2 equations from which a third equation had to be formed. This question gauged your ability to identify the numbers, with which the two equations needed to be multiplied so that the resultant combination led to the third equation.

There was one question on quadratic equations—sum of and product of roots linking to the coefficients of the quadratic equation.

Two questions were asked on the concept of AM greater than or equal to GM. Questions of this type were to appear in a few later CATs as well. Both the questions were quite simple. All one needed to do was take all the variables as equal.

There were three questions based on some user-defined function, with averages combined into it. No prior algebraic knowledge was needed to solve these. One just had to interpret the data and know the concept of averages.

One question was asked on Arithmetic progression, which had a combination of numbers. The question went like this: “A student added some consecutive natural numbers from 1 , but one number was added twice. He got a sum of 1000. Which number was added twice?” Questions like these had appeared in earlier CATs in different avatars.

On the whole, it was a simple paper on the Algebra front.

CAT 2002:
Probably the toughest CAT when it comes to Algebra. The sheer number of concepts that were checked and new types of questions that were introduced made the paper Algebra heavy.

There were two questions on Data Sufficiency: one on inequalities and the other on modulus. Both were easy to solve.

The toughest question in Algebra in the 2002 paper, which had earlier appeared in the Joint Entrance Examination was: “x+y+z=5 and xy+yz+xz=3. What is the largest value that x can take?” To solve this question, one needed some algebraic manipulation and application of AM > or = GM concept. I personally feel that there has hardly ever been a tougher question asked in CAT on Algebra.

One more question was based on the application of AM > or =GM concept. It appeared in the form of a packaging problem on how many Samosas to pack vis-à-vis maximization of revenues for Davji Shops.

The concept of Series was checked in three questions. The first was an AP-GP series: 2+5x+9x^2+14x^3…where |x| < 1. One needed to know the method to solve such questions. A similar question reappeared in a subsequent CAT.

In the second question, the nth term of a series was given and one asked to guess the general nature of the odd and even terms, whether +ve or –ve. All one needed to do was list the first 5 to 6 terms and notice the pattern.

The third was a repetition of a concept from the previous CAT. “A student was asked to add a first few natural numbers …missed one number… the sum came to be 575…which number did he miss?” The concept of the sum of an A.P could have been used to solve this question.

There were two questions based on identities, the first was: x^2+5y^2+z^2 =2y(2x+z). To answer this question, one needed to know how to rewrite the identity as sum of 3 squares.

The other question asked one to find out the number by which 7^6n – 6^6n is divisible. If you knew how to factorize a^n – b^n when n is even or odd, it was a sitter. Some others solved the question by putting n =1.

Functions and graphs were not really predominant in this CAT. There were only two questions on the former. In the first, a logarithmic function, one just needed to know properties of logs, or simply by putting x= a and y=b, one could have cracked the question.

The second one was tricky. Two functions were defined as L(x,y) = [x] +[y] +[xy] and R(x,y) = [2x] +[2y]. One had to find whether L(x,y) could be > or = or < than R(x,y). This was the first time that the Greatest Integer function was introduced in Algebra. One needed to know the number properties of such functions. It was a conceptually tough question for those not comfortable with Algebra. We have not seen GIF in subsequent CATs.

A simple question was asked on quadratic properties. One had to find the number of real roots for an expression (A/x) + (B/ x -1) = 1. It needed rewriting the expression and applying standard fundamental concepts of quadratics. However, the form of the expression stumped quite a few.

One question that many people answered in spite of not knowing how to solve but just by applying the Pythagorean triplets was the one which said u^m + v^m = w^m…where u,v,w,m are all integers. There were choices relating m and the max / min of u,v,w. If one were to attempt, proving the relationship it would have needed significant amount of expertise. But CAT was checking out if you could relate it to the Pythagorean triplets.

Algebraic manipulation was required for the question where pqr=1 was to be used in simplifying an expression having 3 terms given in terms of p, q and r. It needed a bit of smart work to do it the straight way. Else one could have put p=q=r=1 and arrived at the answer right away!

Overall, the number of questions and variety of concepts made this a tough Algebra section. Nevertheless it was always possible even in such a CAT to apply fundamentals to make tough questions simpler. In fact, that is exactly what CAT is expecting you to work on.

CAT 2003:
The year when the CAT paper got leaked. The retest, however, proved to be less conceptual in nature.

The graphs resurfaced, appearing in 3 questions. In each of these, the number of points of intersection of two curves needed to be found. In one question, one had to figure that out between y=2^x and y=x+1 and in the other, between y=1/x and y= logx. Those who knew how the graphs looked could guess correctly. Others were lucky enough to realize that at x=0 and x=1, in the first two graphs the y coordinate was the same. In the third one, cubic and one quadratic were given with the task of finding the common roots. One just needed to find the points of intersection (which was very easy, given the nature of the equation, the trademark of CAT questions) and check whether they were the roots, too.

There was one question which could have been solved very easily using graphs, the one on min of the function max(5-x,x+2). This concept reappeared in CAT after a gap of nine years.

There were questions on similar lines in which one was asked to find the maximum or minimum values of the square of 4 integers, whose sum was of the form 4K+1. It could also have been categorized as a ‘numbers’ question, but the knowledge that if you are close to the mean, the sum of the squares would be the least would have made this a simple question.

There was one question based on a set of three linear equations with three variables. One had to find the condition for which they had at least one solution. Just plugging the choices and checking the answers would have done the trick.

There were three questions on Arithmetic progression. In the first, one needed to know how to write the nth term in terms of “a” and “d”. The smart way of approaching was to apply the concept that the middle term is also the average of all the terms. In the second, one had to find which term would be just short of a sum of 288 when the series 1+2+3…is written. The third question was a further application—one needed to know that the first and the last terms were same as the second and the second last, and so on.

The other question on series was on AP and GP: “n questions in a paper…the number of students getting more than j mistakes was 2^(n-j)…there were 4095 mistakes in all…find the number of students. Now, 4095 is close to 4096, which is 2^12. One needed to look at the choice of numbers. Those who checked with n = 3 got the answer. An example how in some cases mathematical induction works.

There was one question on modulus, which was a little difficult for quite a few people. This particular concept appeared for the first time in CAT—min value of |x-a| + |b-x|+ |x-c|…

There was one quadratic given in terms of some unknown coefficients and one was asked to find what could be the minimum value of the square of the roots. The coefficients were such that one just had to express the sum of the squares of the roots in terms of the sum and the product of the roots. Then, the solution would have been obvious…

The last was a question based on the fact that if x is +ve x + 1/x is not less than 2. This had to be applied to a given expression.

This CAT had quite a few questions on Algebra. The paper was just a shade lesser than the previous year’s CAT, in terms of the width of concepts covered, nevertheless the new concepts that came into picture were points of intersections of curves, minimum values in Modulus expressions.

In a nutshell, the retest was a lot better!