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CAT Algebra Sectional Test 4 – with Solutions

Advanced CAT Algebra practice on cubic roots, remainder theorem, log and root inequalities, greatest integer function and functional equations, with step-by-step solutions.

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Questions
22
Duration
40 min
Marks
66

Negative marking

Attempts
100
Access
With a course
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Real test conditions

Before you launch the test

This is the computer-based test format, not a quiz page: one clock, a question palette, and no way back once a section closes. Read these first.

  • It runs in full screen. The test takes over the screen when it starts, the way the real computer-based test does.
  • Leaving it is counted. Switching away is recorded and warned about; the third time, the paper is submitted for you.
  • The clock runs for 40 minutes. Set aside that long, uninterrupted. It starts when you begin, keeps running if you close the tab, and submits the paper at zero.

About this test

CAT Algebra Practice Test 4: Cubics, Log Inequalities and Functions PYQ-Style

Built for CAT aspirants in the advanced revision stage, CAT Algebra Sectional Test 4 is a solved practice set on harder Algebra ideas. It tests cubic roots, remainder theorem, logarithmic and irrational inequalities, greatest integer function, functional equations and range of functions. Aptitude360 provides a detailed solution for every question, so each new method can be studied step by step.

What this CAT Algebra Test 4 covers

  • Equations in several unknowns from word problems: prize purchases, tea stall bills and siblings' ages in AP
  • Polynomials: relations between roots and coefficients of a cubic, remainder theorem and a quadratic solved with wrongly copied coefficients
  • Inequalities with square roots, logarithms to a base less than one, modulus in a quadratic and the greatest integer function
  • Functions: self-inverse rational functions, the functional equation for products, range over positive reals and composite quadratics with four distinct roots
  • Nested surds, equal powers of different bases, AP and GP combinations and a matchstick pattern sequence
  • Applied optimisation: maximising kiosk profit, area between two modulus graphs and least value of a rational expression

Key formulas and rules for this Algebra set

  • Cubic \( ax^{3}+bx^{2}+cx+d=0 \): sum of roots \( =-\dfrac{b}{a} \), product \( =-\dfrac{d}{a} \)
  • Remainder theorem: remainder of \( P(x)\div(x-a) \) is \( P(a) \)
  • For \( 0<b<1 \): \( \log_{b}x\ge\log_{b}y \iff x\le y \), with \( x,y>0 \)
  • Greatest integer: \( [x]=n \iff n\le x<n+1 \)
  • AM-GM: \( \dfrac{a+b}{2}\ge\sqrt{ab} \) for \( a,b\ge 0 \)

Revision tips for advanced CAT Algebra

  • Flip the inequality sign whenever the log base lies between zero and one.
  • Square both sides of a root inequality only after fixing the domain and the sign of the other side.
  • Treat the greatest integer as a single variable first, then convert the integer range back to real x.
  • Sketch modulus graphs as V shapes to read areas and meeting points quickly.

Who this Algebra sectional test is for

It is meant for CAT and OMET aspirants who are comfortable with Algebra basics and want tougher, multi-concept practice before full-length mocks.

FAQs about CAT Algebra Sectional Test 4 – with Solutions

What happens to a log inequality when the base is less than 1?
When the base lies between zero and one, the log function is decreasing, so the inequality sign reverses when you remove the logs. The arguments must still be positive.
What is the remainder theorem in algebra?
The remainder theorem states that when a polynomial P(x) is divided by x minus a, the remainder is P(a). If P(a) equals zero, then x minus a is a factor of the polynomial.
How do you solve inequalities with the greatest integer function?
Replace the greatest integer of x with an integer variable and solve the resulting inequality for that integer. Then convert each valid integer n into the interval from n up to, but not including, n plus one.
What is a self-inverse function?
A self-inverse function is one whose inverse is the function itself, so applying it twice returns the original input. Its graph is symmetric about the line y equals x.

Test structure

How the paper is split, and how long each part runs. A section you have finished cannot be returned to.

SectionQuestionsTime
CAT Algebra Sectional Test - 42240 min

Marking & rules

Total marks
66 across 22 questions.
Negative marking
A wrong answer costs marks. Leaving a question unanswered costs nothing.
Attempts
100 in total. An attempt is spent the moment you begin one.

What you get when you submit

The analysis is ready the moment you submit the paper.

Score and accuracy

Marks, percentage, right, wrong and skipped, and the time you took.

Section by section

The same figures for each section, so a weak one cannot hide in an average.

The full answer key

Every question with your answer, the right one, the working, and the video where there is one.

Where you stand

Your score against the spread of everyone else who has sat this test.

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