If a rhombus has area 12 sq cm and side length 5 cm, then the length, in cm, of its longer diagonal is Options: A) \(\sqrt{37} + \sqrt{13}\) B) \(\sqrt{13} + \sqrt{12}\) C) \(\dfrac{\sqrt{37} + \sqrt{13}}{2}\) D) \(\dfrac{\sqrt{13} + \sqrt{12}}{2}\)

A. \(\sqrt{37} + \sqrt{13}\). Step 1. Let diagonals p≥q>0. Area pq/2=12 gives pq 24. Step 2. Half-diagonals and side 5 form right triangles, so p²+q²100.

If (x²+ 1/x²) = 25 and x > 0, then the value of (x⁷ + 1/x⁷) is: Options: A) 44856√3 B) 44853√3 C) 44859√3 D) 44850√3

B. 44853√3. Correct Answer: 44853√3 Given x² + 1/x² = 25 , x > 0 Using identity (a + b)² = a² + b² + 2ab I. x + 1/x: (x + 1/x)² = x² + 2 + 1/x² = 25 + 2 = 27 So, x + 1/x = 3√3 II.

Let \( x, y, \) and \( z \) be real numbers satisfying: \( 4(x^2 + y^2 + z^2) = a \), \( 4(x - y - z) = 3 + a \). Then \( a \) equals: Options: A) \( 1 \) B) \( 4 \) C) \( 3 \) D) \( 1 \dfrac{1}{3} \)

C. \( 3 \). Hit and Trial Table - MathJax Hit and Trial Table for Solving Using Options Given equations: \[ 4(x^2 + y^2 + z^2) = a \quad \text{and} \quad 4(x - y - z) = 3 + a \] Option \( a \) \( x^2 + y^2 + z^2 = \frac{a}{4} \) \( x - y - z = \frac{3 + a}{4} \) Trial Values for \( x, y, z \) Result 1 \( 1 \) \( \frac{1}{4} \) \( 1 \) \( x = 1, y = 0, z = 0 \) ❌ Too large for (1) 2 \( 4 \) \( 1 \) \( \frac{7}{4} \) \( x = 2, y = \frac{1}{8}, z = \frac{1}{8} \) ❌ Doesn't satisfy (1) 3 ✅ \( 3 \) \( \frac{3}{4} \) \( \frac{3}{2} \) Algebra confirms consistency ✅ Correct 4 \( \frac{4}{3} \) \( \f…

CAT Algebra Bible Top 10 Must Solve Algebra Questions covering 100% of Syllabus

Every real CAT question on this chapter from the past papers we have, newest year first. Passages and data sets stay together on one page.

Tap Show answer, or open the detailed solution (free login). Page 1 of 7.

Start here: the repeating CAT PYQs

Study the algebra pattern in each CAT PYQ, then practise variations. The 10 selected families contain 33 of the 33 questions tagged Algebraic identities in the reviewed set.

CAT Algebra Bible Top 10 PYQ Questions

Questions	CAT Area Covered
First 5 CAT Questions	81.82%
First 10 CAT Questions	100%
All 10 CAT Questions	100%

Importance of Algebra each year of CAT exam

CAT year	Total Algebra questions	First 6 examples	First 10 examples	All 10 examples	Outside these 10 families	Share of all Quant
2021	8	7	8	8	0	12.12%
2022	6	4	6	6	0	9.09%
2023	8	7	8	8	0	12.12%
2024	8	8	8	8	0	12.12%
2025	3	3	3	3	0	4.55%
Total	33	29	33	33	0	10.00%
Share of all 33	100%	87.88%	100%	100%	0%

Quick review for beginners and last day revision

Rule	Formula or what to do	Small example
Square of a sum	(a + b)² = a² + 2ab + b²	(x + 3)² = x² + 6x + 9
Square of a difference	(a − b)² = a² − 2ab + b²	(x − 3)² = x² − 6x + 9
Difference of squares	a² − b² = (a − b)(a + b)	x² − 9 = (x − 3)(x + 3)
Recover square sums	a² + b² = (a + b)² − 2ab	If a + b = 5 and ab = 6, square sum = 13.
Cube sum	a³ + b³ = (a + b)³ − 3ab(a + b)	Useful when a + b and ab are known.
Cube difference	a³ − b³ = (a − b)³ + 3ab(a − b)	Useful when a − b and ab are known.
Reciprocal squares	If s = x + 1/x, then x² + 1/x² = s² − 2.	Requires x ≠ 0.
Reciprocal cubes	x³ + 1/x³ = (x + 1/x)³ − 3(x + 1/x)	Higher symmetric powers can be built recursively.
Conjugate surds	(a + b√c)(a − b√c) = a² − b²c	Multiply by the conjugate to rationalise.
Zero sum of squares	For real u,v, u² + v² = 0 implies u = v = 0.	Complete squares to obtain simpler equations.

10 Core CAT PYQ Questions

Study these first.

CAT PYQ 1: Conjugate surds and difference of squares

Understand: Check the allowed range of x first, then combine the logs and solve.

CAT 2021 Slot 1 · Quant

If \(5 - \log_{10}\sqrt{1+x} + 4\log_{10}\sqrt{1-x} = \log_{10}\frac{1}{\sqrt{1-x^2}}\), then 100x equals

CAT PYQ 2: Symmetric sums products and reciprocal powers

Understand: Surd roots with whole-number coefficients come in pairs, so use their sum and product to link b, c, m and n to a.

CAT 2022 Slot 3 · Quant

If \((3 + 2\sqrt{2})\) is a root of the equation \(ax^{2} + bx + c = 0\) and \((4 + 2\sqrt{3})\) is a root of the equation \(ay^{2} + my + n = 0\) where \(a, b, c, m\) and \(n\) are integers, then the value of \(\left(\dfrac{b}{m} + \dfrac{c - 2b}{n}\right)\) is

CAT PYQ 3: Square sum product identities in diagonal geometry

Understand: Use the area and the side to get the product and the sum of squares of the half-diagonals, then build (p + q)² and (p − q)².

CAT 2021 Slot 2 · Quant

If a rhombus has area 12 sq cm and side length 5 cm, then the length, in cm, of its longer diagonal is

CAT PYQ 4: Complete squares and use sign or equality conditions

Understand: Move everything to one side and make squares. A sum of squares is zero only when each square is zero.

CAT 2023 Slot 1 · Quant

If \(x\) and \(y\) are real numbers such that \(x^2+(x-2 y-1)^2=-4 y(x+y)\), then the value \(x-2 y\) is

CAT PYQ 5: Group common expressions and factor product constraints

Understand: Add the two equations to get a square in (x − y), solve for it, then use the condition x > y.

CAT 2021 Slot 2 · Quant

Consider the pair of equations: \(x^2 - xy - x = 22\) and \(y^2 - xy + y = 34\). If \(x > y\), then \(x - y\) equals

CAT PYQ 6: Difference or sum formulas for sequence totals

Understand: Subtract the total up to n − 1 from the total up to n to get each term.

CAT 2021 Slot 2 · Quant

For a sequence of real numbers \(x_1, x_2, ...x_n\), If \(x_1 - x_2 + x_3 - .... + (-1)^{n+1} x_n = n^2 + 2n\) for all natural numbers n, then the sum \(x_{49} + x_{50}\) equals

CAT PYQ 7: Add and subtract values at opposite polynomial roots

Understand: Put r and −r in the equation, then add and subtract the two results to isolate r² and c.

CAT 2022 Slot 2 · Quant

Let r and c be real numbers. If r and -r are roots of 5x^3 + cx^2 − 10x + 9 = 0, then c equals

CAT PYQ 8: Common base exponent identity for all parameter values

Understand: Bring both sides to the same base. If it holds for all values, pick a value that forces the exponent to be zero.

CAT 2022 Slot 3 · Quant

If \(\left(\sqrt{\dfrac{7}{5}}\right)^{3x - y} = \dfrac{875}{2401}\) and \(\left(\dfrac{4a}{b}\right)^{6x - y} = \left(\dfrac{2a}{b}\right)^{y - 6x}\), for all non-zero real values of \(a\) and \(b\), then the value of \(x + y\) is

CAT PYQ 9: Square and factor an absolute value inequality

Understand: Square both sides to get (m² − 1)(n² − 1) < 0, then count the whole-number pairs.

CAT 2021 Slot 3 · Quant

The number of distinct pairs of integers (m,n), satisfying \(|1 + mn| < |m + n| < 5\) is:

CAT PYQ 10: Expand a nested GP into simpler series

Understand: Split each bracket into a simple geometric series, sum each one and combine.

CAT 2023 Slot 3 · Quant

The value of \(1+\left(1+\frac{1}{3}\right) \frac{1}{4}+\left(1+\frac{1}{3}+\frac{1}{9}\right) \frac{1}{16}+\left(1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}\right) \frac{1}{64}+\cdots\), is

CAT 2025 Slot 1 · QuantMedium

If the length of a side of a rhombus is 36 cm and the area of the rhombus is 396 sq. cm, then the absolute value of the difference between the lengths, in cm, of the diagonals of the rhombus is ____

CAT 2025 Slot 3 · QuantMedium

f(x) = (x²+ 3x)(x² + 3x + 2), then the sum of all real roots of the equation √f(x) + 1 = 9701, is

CAT 2025 Slot 3 · QuantEasy

If (x²+ 1/x²) = 25 and x > 0, then the value of (x⁷ + 1/x⁷) is:

CAT 2024 Slot 1 · Quant

Let \( x, y, \) and \( z \) be real numbers satisfying:
\( 4(x^2 + y^2 + z^2) = a \),
\( 4(x - y - z) = 3 + a \).
Then \( a \) equals:

CAT 2024 Slot 1 · Quant

If \((a+b \sqrt{n})\) is the positive square root of \((29-12 \sqrt{5})\), where \(a\) and \(b\) are integers, and \(n\) is a natural number, then the maximum possible value of \((a+b+n)\) is

Frequently asked questions

Where can I find CAT algebra questions with solutions?

Every Algebra question on this page is a real CAT question from the CAT Actual Quant Papers. Tap Show answer for the answer; the detailed step-by-step solution opens after a free login. Start with the 10 CAT PYQs at the top, because they repeat the most.

Are there CAT algebra questions with answers?

Yes. Every question has its answer one tap away, free for everyone: pick an option and press Submit, or tap Show answer.

Is there a PDF of CAT algebra questions?

Yes. Tap Download PDF at the top of this page (free, after a quick OTP login) for the CAT Algebra Bible Top 10 PYQ Questions, a free PDF of the 10 most repeated CAT algebra questions. All the CAT Algebra questions are also on this page to read online, with answers and, after a free login, detailed solutions.

What are the CAT algebra previous year questions (PYQs)?

They are the real questions of the CAT Actual Quant Papers that are about algebra: 33 questions, grouped into 10 repeating CAT PYQs, with the other questions listed after them in paper order.

Which algebra questions should I practise first for CAT?

Start with the first 6 CAT PYQs, which cover about 90% of the Algebra questions. Then do all 10, which cover all of them.

How many Algebra questions came in CAT papers?

21 Algebra questions are on this page: 3 in CAT 2025, 8 in CAT 2024, 4 in CAT 2023, 3 in CAT 2022, 3 in CAT 2021. The same real question is shown once.

How difficult are CAT Algebra questions?

Of the 3 Algebra questions that carry a difficulty rating, 1 are Easy, 2 Medium and 0 Hard; the other 18 are not rated yet.

Are the solutions free?

The question and its answer are free for everyone. The detailed Cetking solution opens after a free login with your mobile number (OTP).

How is a set shown on this page?

Each passage or data set gets one page with all its questions together (2 sets here), so you never solve half of one. Questions that stand alone are shown five to a page.

Where can I practise more Algebra?

Open all CAT papers and topics for every paper, or the Cetking Arena to practise.

Last day solving checklist

Name the unknown. Include its unit and state whether it is a count, amount, rate or parameter.
Form the relationships. Translate each given condition into an equation or restriction.
Choose the method. Use the formula, substitution, elimination, a sign chart or a bound appropriate to this concept.
Check restrictions. Check integer counts, signs, nonzero denominators, domains and all original conditions.
Verify the answer. Substitute into the original question; for an extreme, show a feasible example reaches it.

All 10 repeating CAT PYQs and how much each adds

CAT PYQ	Repeating CAT PYQ	Questions	Representative	Cumulative questions	Cumulative share
1	Conjugate surds and difference of squares	7	CAT 2021 S1 Q4	7	21.21%
2	Symmetric sums products and reciprocal powers	7	CAT 2022 S3 Q14	14	42.42%
3	Square sum product identities in diagonal geometry	5	CAT 2021 S2 Q9	19	57.58%
4	Complete squares and use sign or equality conditions	4	CAT 2023 S1 Q2	23	69.70%
5	Group common expressions and factor product constraints	4	CAT 2021 S2 Q3	27	81.82%
6	Difference or sum formulas for sequence totals	2	CAT 2021 S2 Q20	29	87.88%
7	Add and subtract values at opposite polynomial roots	1	CAT 2022 S2 Q9	30	90.91%
8	Common base exponent identity for all parameter values	1	CAT 2022 S3 Q11	31	93.94%
9	Square and factor an absolute value inequality	1	CAT 2021 S3 Q18	32	96.97%
10	Expand a nested GP into simpler series	1	CAT 2023 S3 Q20	33	100%

Based on CAT Actual Quant Papers.

CAT Algebra Bible Top 10 Must Solve Algebra Questions covering 100% of Syllabus

Start here: the repeating CAT PYQs

10 Core CAT PYQ Questions

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Frequently asked questions