NATA Maths PYQs: Most Repeated Algebra Questions with Step-by-Step Solutions

Many students preparing for NATA spend most of their time memorising Algebra formulas but struggle when similar questions appear in a different format. Topics such as quadratic equations, logarithms, matrices, determinants, AP & GP, and permutations & combinations are frequently tested, making previous-year question practice essential for improving speed and accuracy.

A smarter revision strategy is to practise the most repeated NATA Maths PYQs chapter-wise before attempting mixed questions. This collection brings together 15 frequently asked Algebra questions with detailed step-by-step solutions, helping you revise concepts, strengthen problem-solving methods and build confidence for the Mathematics section.

Most Repeated NATA Maths Algebra PYQs 

Start with chapter-wise PYQs: Solve NATA Maths Most Repeated Questions topic-wise – Quadratic Equations, Logarithms, AP & GP, Matrices and Determinants – before attempting mixed practice.

1. Sum and Product of Roots

Question: If α\alphaα and β\betaβ are the roots of 2x2−4x+1=02x^2-4x+1=02x2−4x+1=0, find 1α+1β\frac{1}{\alpha}+\frac{1}{\beta}α1+β1.

Solution:

For ax2+bx+c=0ax^2+bx+c=0ax2+bx+c=0:

• Sum of roots = −ba\frac{-b}{a}a−b

• Product of roots = ca\frac{c}{a}ac

Here:

• α+β=2\alpha\beta=2α+β=2

• αβ=12\alpha\beta=\frac12αβ=21

1α+1β=α+βαβ=21/2=4\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}=\frac{2}{1/2}=4α1+β1=αβα+β=1/22=4

Answer: 444

2. Logarithm Equation

Question: Solve log⁡2x+log⁡2(x−2)=3\log_2x+\log_2(x-2)=3log2x+log2(x−2)=3.

Solution:

log⁡2x(x−2)x(x−2)=3\log_2x(x−2)x(x-2)=3log2x(x−2)x(x−2)=3

x(x−2)=8x(x-2)=8x(x−2)=8

x2−2x−8=0x^2-2x-8=0x2−2x−8=0

(x−4)(x+2)=0(x-4)(x+2)=0(x−4)(x+2)=0

Only x=4x=4x=4 satisfies the logarithm condition.

Answer: 444

3. Arithmetic Progression

Question: The 5th term of an AP is 19, and the 11th term is 43. Find the 20th term.

Solution:

a+4d=19a+4d=19a+4d=19

a+10d=43a+10d=43a+10d=43

Subtracting:

6d=24⇒d=46d=24\Rightarrow d=46d=24⇒d=4

a=3a=3a=3

a20=3+19(4)=79a_{20}=3+19(4)=79a20=3+19(4)=79

Answer: 79

4. Infinite Geometric Progression

Question: Find the sum of:

1+13+19+127+⋯1+\frac13+\frac19+\frac1{27}+\cdots1+31+91+271+⋯

Solution:

Here a=1a=1a=1 and r=13r=\frac13r=31.

S∞=11−1/3=32S_\infty=\frac{1}{1-1/3}=\frac32S∞=1−1/31=23

Answer: 32\frac3223

5. Matrix Determinant

Question: If

A=(2413)A=\begin{pmatrix}2&4\\1&3\end{pmatrix}A=(2143)

find ∣A−1∣|A^{-1}|∣A−1∣.

Solution:

∣A∣=(2)(3)−(4)(1)=2|A|=(2)(3)-(4)(1)=2∣A∣=(2)(3)−(4)(1)=2

∣A−1∣=1∣A∣=12|A^{-1}|=\frac1{|A|}=\frac12∣A−1∣=∣A∣1=21

Answer: 12\frac1221

6. Complex Numbers

Question: Find the modulus of

z=3+4i1−2iz=\frac{3+4i}{1-2i}z=1−2i3+4i

Solution:

∣3+4i∣=5|3+4i|=5∣3+4i∣=5

∣1−2i∣=5|1-2i|=\sqrt5∣1−2i∣=5

∣z∣=55=5|z|=\frac5{\sqrt5}=\sqrt5∣z∣=55=5

Answer: 5\sqrt55

7. Binomial Theorem

Question: Find the middle term of

(x+2x)6\left(x+\frac2x\right)^6(x+x2)6

Solution:

The middle term is the 4th term.

T4=(63)x3(2x)3T_4=\binom63x^3\left(\frac2x\right)^3T4=(36)x3(x2)3

=20×8=160=20\times8=160=20×8=160

Answer: 160

8. Permutation of Letters

Question: In how many ways can the letters of ARCH be arranged so that vowels are never together?

Solution:

ARCH contains only one vowel (A), so the condition is always satisfied.

Total arrangements:

4!=244!=244!=24

Answer: 24

9. Combination

Question: A committee of 4 is formed from 6 men and 4 women containing at least 2 women.

Solution:

Total:

90+24+1=11590+24+1=11590+24+1=115

Answer: 115

10. Absolute Value Inequality

Question: Solve ∣2x−3∣<5|2x-3|<5∣2x−3∣<5.

Solution:

−5<2x−3<5-5<2x-3<5−5<2x−3<5

−2<2x<8-2<2x<8−2<2x<8

−1<x<4-1<x<4−1<x<4

Answer: x∈(−1,4)x\in(-1,4)x∈(−1,4)

11. Singular Matrix

Question: Find the value of kkk if

(k236)\begin{pmatrix}k&2\\3&6\end{pmatrix}(k326)

is singular.

Solution:

A singular matrix has determinant zero.

6k−6=06k-6=06k−6=0

k=1k=1k=1

Answer: 1

12. Linear Equations

Question: For what value of ppp does the system have infinitely many solutions?

x+2y=3x+2y=3x+2y=3

3x+py=93x+py=93x+py=9

Solution:

13=2p=39\frac13=\frac2p=\frac3931=p2=93

p=6p=6p=6

Answer: 6

13. Roots in Arithmetic Progression

Question: The roots of

x3−6x2+11x−6=0x^3-6x^2+11x-6=0x3−6x2+11x−6=0

are in AP. Find the roots.

Solution:

Let the roots be:

a−d, a, a+da-d,\ a,\ a+da−d, a, a+d

From the sum of roots:

3a=6⇒a=23a=6\Rightarrow a=23a=6⇒a=2

From the product:

(2−d)(2)(2+d)=6(2-d)(2)(2+d)=6(2−d)(2)(2+d)=6

d=1d=1d=1

Roots:

1, 2, 3

14. Change of Base

Question: Evaluate:

log⁡35×log⁡59\log_35\times\log_59log35×log59

Solution:

log⁡5log⁡3×log⁡9log⁡5\frac{\log5}{\log3}\times\frac{\log9}{\log5}log3log5×log5log9

=log⁡39=2=\log_39=2=log39=2

Answer: 2

15. Sum of Squares

Question: Find

12+22+32+⋯+1021^2+2^2+3^2+\cdots+10^212+22+32+⋯+102

Solution:

Use:

Sn=n(n+1)(2n+1)6S_n=\frac{n(n+1)(2n+1)}6Sn=6n(n+1)(2n+1)

S10=10×11×216=385S_{10}=\frac{10\times11\times21}6=385S10=610×11×21=385

Answer: 385

How Can You Use NATA Maths PYQs to Score Better in Algebra?

Practising previous year questions is most effective when you follow a structured revision strategy instead of solving questions randomly. Focus on understanding the method behind each problem, strengthen frequently tested formulas and revisit repeated question patterns to improve both speed and accuracy.

• Start with chapter-wise PYQs: Solve questions topic-wise – Quadratic Equations, Logarithms, AP & GP, Matrices and Determinants – before attempting mixed practice.

• Revise formulas regularly: Keep a separate formula sheet for logarithms, sequences, determinants, permutations and combinations, and review it daily.
• Prioritise high-scoring topics: Refer to the NATA Maths Topics to identify important areas, while practising Matrix, determinant and permutation questions to improve speed and accuracy.

• Write complete working steps: Even while practising MCQs, solve the calculation step by step to minimise errors and improve conceptual clarity.

• Reattempt repeated PYQs: Rather than solving many new questions, revisit previous year Algebra questions multiple times until you can solve them confidently within the exam time limit.

Success in the NATA Mathematics section comes from consistent practice with questions that reflect the actual exam pattern. Revising these repeated Algebra PYQs helps strengthen concepts across quadratic equations, logarithms, matrices, sequences, determinants and permutations while improving calculation speed and problem-solving accuracy.