IPMAT Algebra 2027 — Linear and Quadratic Equations, Progressions and 40 Practice Problems

April 21, 2026 · IPM Gurukul · Blog

Last Updated: April 2026

IPM 2027 | ALGEBRA COMPLETE NOTES

Linear Equations, Quadratic Equations, AP/GP/HP, Functions, Inequalities and Logarithms — with 40 practice problems at IPMAT difficulty level

📘 Overview
Algebra is one of the most heavily tested topics in IPMAT (IIM Indore) and JIPMAT. You can expect 8–12 questions on algebra across both the Quantitative Ability sections. This comprehensive guide covers every sub-topic tested — linear and simultaneous equations, quadratic equations with Vieta’s formulas, AP/GP/HP progressions, AM-GM-HM inequalities, functions, modulus inequalities, and logarithms. Work through all 40 practice problems for complete exam readiness.

📊 IPMAT vs JIPMAT Algebra: Comparison

Parameter	IPMAT Indore	JIPMAT
Algebra Questions	8–12 out of 100	6–10 out of 100
Difficulty Level	Moderate to High	Easy to Moderate
Question Type	MCQ + SA (Short Answer)	MCQ only
Most Tested Topics	Quadratics, Functions, AP/GP	Linear Equations, AP/GP
SA (Numerical Answer)	Yes (Section 3)	No

1. Linear Equations

1.1 Linear Equation in One Variable

A linear equation in one variable is of the form ax + b = 0, where a ≠ 0. The unique solution is x = −b/a.

Example: Solve 3x − 7 = 2x + 5.
3x − 2x = 5 + 7 → x = 12

1.2 Simultaneous Linear Equations (Two Variables)

A system of two equations: a₁x + b₁y = c₁ and a₂x + b₂y = c₂.

Three methods to solve:

Substitution: Express one variable in terms of the other and substitute.
Elimination: Multiply equations to make coefficients equal, then add/subtract.
Cross-multiplication: x = (b₁c₂ − b₂c₁)/(a₁b₂ − a₂b₁), y = (a₂c₁ − a₁c₂)/(a₁b₂ − a₂b₁)

Nature of Solutions:

Unique solution: a₁/a₂ ≠ b₁/b₂ (lines intersect)
No solution: a₁/a₂ = b₁/b₂ ≠ c₁/c₂ (lines are parallel)
Infinite solutions: a₁/a₂ = b₁/b₂ = c₁/c₂ (lines coincide)

Worked Example 1 (IPMAT Style):

If 2x + 3y = 12 and 4x − y = 5, find x + y.

From equation 2: y = 4x − 5. Substitute into equation 1: 2x + 3(4x − 5) = 12 → 2x + 12x − 15 = 12 → 14x = 27 → x = 27/14. Then y = 4(27/14) − 5 = 108/14 − 70/14 = 38/14 = 19/7. So x + y = 27/14 + 38/14 = 65/14.

2. Quadratic Equations

2.1 Standard Form and Discriminant

A quadratic equation is ax² + bx + c = 0 (a ≠ 0). The roots are: x = [−b ± √(b² − 4ac)] / 2a.

Discriminant D = b² − 4ac determines the nature of roots:

D > 0: Two distinct real roots
D = 0: Two equal (repeated) real roots
D < 0: Two complex conjugate roots (no real roots)

2.2 Vieta’s Formulas (Sum and Product of Roots)

If α and β are roots of ax² + bx + c = 0, then:

α + β = −b/a (Sum of roots)
αβ = c/a (Product of roots)
|α − β| = √D / |a|

Worked Example 2:

If α and β are roots of 3x² − 10x + 3 = 0, find α² + β².

α + β = 10/3, αβ = 3/3 = 1. α² + β² = (α+β)² − 2αβ = (10/3)² − 2(1) = 100/9 − 2 = 82/9.

2.3 Formation of Quadratic with Given Roots

If roots are α and β: x² − (α+β)x + αβ = 0

Worked Example 3:

Form a quadratic with roots 3 and −5. Sum = −2, Product = −15. Equation: x² + 2x − 15 = 0.

2.4 Maximum and Minimum of Quadratic

If a > 0: Minimum value = −D/4a at x = −b/2a (parabola opens upward)
If a < 0: Maximum value = −D/4a at x = −b/2a (parabola opens downward)

2.5 Polynomials: Factor and Remainder Theorems

Remainder Theorem: When polynomial f(x) is divided by (x − a), the remainder is f(a).

Factor Theorem: (x − a) is a factor of f(x) if and only if f(a) = 0.

Worked Example 4:

Find the remainder when f(x) = x³ − 4x² + 5x − 2 is divided by (x − 2).
f(2) = 8 − 16 + 10 − 2 = 0. Since remainder = 0, (x−2) is actually a factor.

3. Arithmetic Progressions (AP)

An AP is a sequence where consecutive terms have a constant difference d (common difference).

General term (nth term): aₙ = a + (n−1)d
Sum of n terms: Sₙ = n/2 × [2a + (n−1)d] = n/2 × [first term + last term]
Arithmetic Mean: AM of a and b = (a+b)/2
Property: If a, b, c are in AP then 2b = a + c

Worked Example 5:

The 4th and 8th terms of an AP are 11 and 23 respectively. Find the sum of first 15 terms.
a + 3d = 11, a + 7d = 23. Subtracting: 4d = 12, d = 3. Then a = 2.
S₁₅ = 15/2 × [2(2) + 14(3)] = 15/2 × [4 + 42] = 15/2 × 46 = 345.

4. Geometric Progressions (GP)

General term: aₙ = ar^(n−1)
Sum of n terms: Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1
Sum to infinity (|r| < 1): S∞ = a/(1−r)
Geometric Mean: GM of a and b = √(ab)
Property: If a, b, c are in GP then b² = ac

Worked Example 6:

Sum to infinity of GP is 27 and sum of first 3 terms is 19. Find the common ratio.
S∞ = a/(1−r) = 27 → a = 27(1−r). S₃ = a(1−r³)/(1−r) = 19 → 27(1−r³) = 19 but also a(1−r³)/(1−r) = 27(1−r)(1−r³)/(1−r) = 27(1−r³) = 19. So 1−r³ = 19/27. r³ = 8/27, r = 2/3. Then a = 27(1−2/3) = 9.

5. Harmonic Progressions (HP) and AM-GM-HM

A sequence is in HP if the reciprocals of its terms form an AP.

Harmonic Mean: HM of a and b = 2ab/(a+b)
Relationship: For two positive numbers, AM ≥ GM ≥ HM
Key identity: GM² = AM × HM
Equality holds when both numbers are equal

Worked Example 7 (AM-GM Application):

Find the minimum value of x + 1/x for x > 0.
By AM-GM: x + 1/x ≥ 2√(x × 1/x) = 2√1 = 2. Equality when x = 1/x → x = 1. Minimum value = 2.

6. Functions

Domain: Set of all valid inputs (x values)
Range: Set of all possible outputs (y values)
One-One (Injective): f(x₁) = f(x₂) implies x₁ = x₂
Onto (Surjective): Every element of co-domain has a pre-image
Bijective: Both one-one and onto (has an inverse)
Composite Function: (fog)(x) = f(g(x))
Inverse Function: f⁻¹ exists only for bijective functions; graph of f⁻¹ is reflection of f about y = x

Worked Example 8 (Function Composition):

If f(x) = 2x + 1 and g(x) = x² − 3, find fog(2) and gof(−1).
g(2) = 4 − 3 = 1. fog(2) = f(1) = 3.
f(−1) = −1. gof(−1) = g(−1) = 1 − 3 = −2.

7. Inequalities

7.1 Quadratic Inequalities (Sign Chart Method)

To solve ax² + bx + c > 0 (or < 0): find roots α, β (α < β), then test sign in each interval.

If a > 0: expression is positive for x < α or x > β, and negative for α < x < β.

7.2 Modulus Inequalities

|x| < a (a > 0): −a < x < a
|x| > a (a > 0): x < −a or x > a
|x − k| < r: k − r < x < k + r (interval around k)

8. Logarithms

Definition: log_b(a) = c ↔ b^c = a (b > 0, b ≠ 1, a > 0)
Product rule: log(mn) = log m + log n
Quotient rule: log(m/n) = log m − log n
Power rule: log(mⁿ) = n log m
Change of base: log_b(a) = log(a)/log(b) = ln(a)/ln(b)
log_b(b) = 1 and log_b(1) = 0
Log inequality: If b > 1, log_b(x) > log_b(y) ↔ x > y. If 0 < b < 1, inequality reverses.

9. Practice Problems — 40 Questions

Set A: Linear Equations (Q1–Q8)

Find the value of x if 5x − 3(2x − 4) = 18. [Ans: 3]
If 3x + 5y = 25 and 5x − 3y = 1, find x − y. [Ans: 1]
For the system 2x + ky = 4 and 4x + 8y = 12 to have no solution, find k. [Ans: k = 4]
The sum of two numbers is 25 and their difference is 7. Find the larger number. [Ans: 16]
If x/3 + y/4 = 5 and x/2 − y/3 = 2, find xy. [Ans: 48]
A man bought some articles for ₹1200. Had he bought 2 more for the same sum, each would cost ₹10 less. How many articles did he buy? [Ans: 14]
If (a+b):(b+c):(c+a) = 6:7:8 and a+b+c = 14, find c. [Ans: 6]
Solve: 2x + 3 < 11 and 3x − 1 > 5. [Ans: 2 < x < 4]

Set B: Quadratic Equations (Q9–Q16)

Find the roots of 2x² − 7x + 3 = 0. [Ans: 3, 1/2]
If roots of x² + px + q = 0 are twice the roots of x² + 3x + 2 = 0, find p and q. [Ans: p = 6, q = 8]
For what value of k does x² − 5x + k = 0 have equal roots? [Ans: k = 25/4]
If α, β are roots of 3x² − 6x + 2 = 0, find α/β + β/α. [Ans: 10/3]
The product of two consecutive even integers is 168. Find them. [Ans: 12 and 14]
Find the range of values of k for which x² − 2kx + 9 < 0 has real solutions. [Ans: k < −3 or k > 3]
If one root of 2x² + kx − 6 = 0 is 2, find the other root and k. [Ans: k = −1, other root = −3/2]
By using factor theorem, find whether (x−1) is a factor of x⁴ − x³ − x² + x. [Ans: Yes, f(1)=0]

Set C: AP, GP, HP (Q17–Q26)

Find the 15th term of the AP: 3, 7, 11, 15 … [Ans: 59]
How many terms of the AP 5, 12, 19, … are needed to give a sum of 671? [Ans: 11 terms]
Insert 4 arithmetic means between 3 and 23. [Ans: 7, 11, 15, 19]
In a GP, the 3rd term is 9 and 6th term is 243. Find the 1st term and common ratio. [Ans: a=1, r=3]
Find the sum of the infinite GP: 4, 2, 1, 1/2, … [Ans: 8]
If AM of two numbers is 13 and their GM is 12, find the numbers. [Ans: 18 and 8]
The HM of two numbers is 4. Their AM is A and GM is G. If 2A + G² = 27, find A. [Ans: A = 5]
Three numbers are in AP and their sum is 27. If 1, 3, 9 are added respectively, the result is in GP. Find the numbers. [Ans: 3, 9, 15]
In an AP, sum of first n terms is 3n² + 5n. Find the 20th term. [Ans: 123]
A ball is dropped from height 100m and rebounds to 2/3 of previous height. Find total distance travelled. [Ans: 500m]

Set D: Functions, Logarithms, Inequalities (Q27–Q40)

If f(x) = (x+1)/(x−1) for x ≠ 1, find f(f(f(3))). [Ans: 2]
The domain of f(x) = √(4 − x²) is: [Ans: −2 ≤ x ≤ 2]
If f(x) = 3x − 2, find f⁻¹(7). [Ans: 3]
Solve: log₂(x−1) + log₂(x−3) = 3. [Ans: x = 5]
Find the value of log₃(81) − log₃(27) + log₃(9). [Ans: 3]
Solve: |2x − 3| ≤ 7. [Ans: −2 ≤ x ≤ 5]
Solve: x² − 5x + 6 < 0. [Ans: 2 < x < 3]
If log(x + y) = log x + log y, find x in terms of y. [Ans: x = y/(y−1)]
For how many integers n is n² + 4n < 0? [Ans: 3 integers: n = −3, −2, −1]
If f(x) = |x − 2| + |x + 3|, find the minimum value. [Ans: 5]
If g(x+1) = x² − 4x + 3, find g(5). [Ans: g(5) = (4)²−4(4)+3 = 3]
How many natural numbers n satisfy: (1/2)^n < 1/100? [Ans: n ≥ 7, infinitely many]
log₅(125) + log₅(1/5) = ? [Ans: 2]
(IPMAT SA Type) If 2^x = 3^y = 6^z, express 1/z in terms of 1/x and 1/y. [Ans: 1/z = 1/x + 1/y]

🎯 Exam Strategy for IPMAT Algebra

Vieta’s formulas are gold: Never expand and solve when you can use sum/product of roots directly. Saves 1–2 minutes per question.
AP: memorize sum formula by heart. Most AP questions test S_n in disguise. If the 10th term and sum of n terms are given together, set up two equations and solve.
For GP infinite sum: Verify |r| < 1 before applying formula. Forgetting this condition costs marks.
Short Answer (SA) section (IPMAT Indore): Log questions often appear here. Master change-of-base and the identity 2^x = 3^y = k type problems.
Inequalities: Quadratic inequalities — always draw a number line and mark roots. Sign chart method never fails.
Functions: Domain questions are easy marks. Remember: for √f(x), need f(x) ≥ 0; for 1/f(x), need f(x) ≠ 0; for log f(x), need f(x) > 0.

🧠 Quick Memory Tricks

Vieta’s shortcut: “Sum goes with b, Product goes with c” — α+β = −b/a (b with sign flip), αβ = c/a (c direct).
AM-GM rhyme: “Always AM ≥ GM ≥ HM, equality when they’re equal in trim.”
AP sum trick: “Sₙ = (n/2) × (first + last)” — same as average × count.
GP infinite sum mnemonic: “First over (1 minus ratio)” — works only when |r| < 1.
Log base flip: log_a(b) = 1/log_b(a) — extremely useful for IPMAT SA questions.
Modulus split rule: |x| < a means −a < x < a (between); |x| > a means outside (two separate regions).

Frequently Asked Questions

How many algebra questions come in IPMAT Indore?

IPMAT Indore typically has 8–12 algebra questions across the Quantitative Ability (MCQ) and Short Answer sections. The Short Answer section (Section III) is exclusively for QA and often tests logarithms, quadratic equations, and progressions in numerical answer format.

What is the difference between IPMAT and JIPMAT algebra difficulty?

IPMAT Indore is generally harder — it includes a Short Answer (numerical entry) section where you must compute exact values without options. JIPMAT algebra is more straightforward MCQ format. For IPMAT, focus on conceptual depth; for JIPMAT, focus on speed and accuracy.

Is Vieta’s formula important for IPMAT?

Absolutely. Vieta’s formulas (α+β = −b/a and αβ = c/a) are among the most tested algebra concepts in IPMAT. Questions often ask for α² + β², α³ + β³, or 1/α + 1/β — all solvable directly using Vieta’s without finding the actual roots.

What is the AM-GM inequality and why is it tested?

AM-GM inequality states that for positive numbers, the Arithmetic Mean is always ≥ Geometric Mean, which is ≥ Harmonic Mean. In IPMAT, it appears in optimization questions — finding minimum/maximum of expressions like x + 1/x or x²y where a constraint is given. Equality holds when all numbers are equal.

How should I prepare algebra for IPMAT in 30 days?

Week 1: Linear and simultaneous equations + quadratic equations with discriminant and Vieta’s. Week 2: AP and GP including sum formulas and infinite GP. Week 3: Functions, logarithms, and inequalities. Week 4: Full mock practice with previous year IPMAT papers. Solve at least 200 algebra questions in timed conditions.

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