Class 10 Maths Algebra Important Questions with Answers | NCERT & PYQs

Algebra is one of the most important units of Class 10 Mathematics. Every year, 6–8 marks of direct questions in CBSE Class 10 Board Exams are asked from Algebra. The chapter includes topics like Polynomials, Pair of Linear Equations in Two Variables, Quadratic Equations, and Arithmetic Progressions (APs).

In this blog, we provide a comprehensive list of important questions with answers, including Previous Year Questions (PYQs), so that students can prepare effectively for their exams.

Key Topics in Class 10 Algebra

Polynomials – Zeros of polynomials, relationship between zeros and coefficients, division algorithm.
Pair of Linear Equations in Two Variables – Graphical method, substitution method, elimination method, cross-multiplication method, applications in word problems.
Quadratic Equations – Solution by factorisation, completing the square, quadratic formula, nature of roots.
Arithmetic Progressions (APs) – nth term of an AP, sum of n terms, applications in daily life.

Class 10 Maths Polynomials Important Questions with Answers | CBSE & PYQs

Polynomials form the foundation of Algebra in Class 10 Mathematics. Every year, 2–3 questions (4–6 marks) are directly asked in CBSE Class 10 Board Exams from this chapter. In this blog, we provide 10 important questions with detailed solutions, including previous year questions (PYQs), to help students prepare effectively.

Important Questions – Polynomials (Class 10 Maths)

Q1. Find the zeros of the polynomial $p(x) = x^2 - 5x + 6$ . Verify the relationship between zeros and coefficients.

Solution:
Factorise: $x^2 - 5x + 6 = (x-2)(x-3)$ .
Zeros = 2, 3.
Sum of zeros = 5 = $-\frac{b}{a}$ .
Product = 6 = $\frac{c}{a}$ . Verified.

Q2. (PYQ – 2019) Find a quadratic polynomial whose zeros are 4 and -3.

Solution:
If α = 4, β = -3 → sum = 1, product = -12.
Required polynomial = $x^2 - (sum)x + (product) = x^2 - x - 12$ .

Q3. If α and β are the zeros of $2x^2 - 7x + 5$ , find the value of $α^2 + β^2$ .

Solution:
$α+β = \frac{7}{2},\ αβ=\frac{5}{2}$ .
$α^2+β^2=(α+β)^2 - 2αβ = (49/4) - 5 = 29/4.$

Q4. Divide $2x^4 - 3x^3 + 3x^2 - 2x + 2$ by $x^2 - 2$ .

Solution:
Using long division:
Quotient = $2x^2 - 3x + 7$ , Remainder = $12x - 12$ .

Q5. (PYQ – 2020) If the zeros of $ax^2 + bx + c$ are equal, show that $b^2 = 4ac$ .

Solution:
Condition for equal roots: Discriminant D = 0.
D = $b^2 - 4ac = 0 → b^2 = 4ac$ .

Q6. Find a cubic polynomial with zeros 2, -1 and -3.

Solution:
Polynomial = (x-2)(x+1)(x+3).
= $(x-2)(x^2+4x+3) = x^3+2x^2-5x-6$ .

Q7. If the sum and product of the zeros of a quadratic polynomial are √2 and 1/3 respectively, find the polynomial.

Solution:
Required polynomial = $x^2 - (√2)x + 1/3$ .
Multiply through by 3: $3x^2 - 3√2x + 1$ .

Q8. (PYQ – 2021) If the polynomial $x^4 - 6x^3 + 16x^2 - 25x + k$ is exactly divisible by $x^2 - 2x + 3$ , find k.

Solution:
Let divisor = $x^2 - 2x + 3 = 0 →$ roots = 1 ± i√2.
Substitute root into dividend → remainder = 0.
On solving, k = 15.

Q9. Show that $x^2 + x + 1$ has no real zeros.

Solution:
Discriminant $D = b^2 - 4ac = 1 - 4 = -3 < 0$ .
Hence, no real zeros.

Q10. If two zeros of the cubic polynomial $2x^3 - x^2 - 5x + 2$ are √2 and -√2, find the third zero.

Solution:
Product of roots = $-\frac{d}{a} = -2/2 = -1$ .
So, (√2)(-√2)(third zero) = -1 → -2(third zero) = -1 → third zero = 1/2.

Tips for Students

Always memorize the relationship between coefficients and zeros:
For $ax^2+bx+c$ , sum of zeros = -b/a, product = c/a.
Practice polynomial division carefully; remainders are common exam questions.
For higher-degree polynomials, factorisation and symmetry of zeros are key.
PYQs often repeat concepts with new values — practice at least 20 similar problems.

Class 10 Maths Important Questions with Answers – Pair of Linear Equations in Two Variables | CBSE & PYQs

The topic Pair of Linear Equations in Two Variables is a crucial part of Class 10 Mathematics. In the CBSE Class 10 Board Exams, it carries around 6–8 marks and often includes application-based word problems.

Here we provide 10 important questions with step-by-step solutions, including previous year questions (PYQs), to help you prepare effectively.

Important Questions – Pair of Linear Equations in Two Variables

Q1. Solve the following pair of equations by elimination method:

$2x + 3y = 11,\quad 2x - 4y = -24$ .

Solution:
Subtracting equations: $(2x+3y) - (2x-4y) = 11 - (-24)$ .
→ $7y = 35$ .
→ $y = 5$ .
Substitute in $2x + 15 = 11$ .
→ $2x = -4,\ x = -2$ .
Answer: (-2, 5).

Q2. Solve by substitution:

$x + y = 10,\quad x - y = 4$ .

Solution:
From (1): $x = 10 - y$ .
Substitute: $10 - y - y = 4 → 10 - 2y = 4 → y = 3$ .
So, $x = 7$ .
Answer: (7, 3).

Q3. (PYQ – 2019) A boat goes 16 km downstream in 2 hours and returns in 4 hours. Find the speed of the boat in still water and the speed of the stream.

Solution:
Let boat speed = x km/h, stream speed = y km/h.
Downstream: $16/(x+y) = 2 → x+y=8$ .
Upstream: $16/(x-y) = 4 → x-y=4$ .
Solving: $x=6,\ y=2$ .
Boat speed = 6 km/h, Stream = 2 km/h.

Q4. Solve graphically:

$x + y = 5,\quad x - y = 1$ .

Solution:
First eqn: (0,5), (5,0).
Second eqn: (1,0), (2,1).
Plot and intersect → (3,2).
Answer: (3,2).

Q5. (PYQ – 2020) Find two numbers whose sum is 27 and difference is 9 using linear equations.

Solution:
Let numbers = x, y.
x+y=27 (i), x-y=9 (ii).
Add (i)+(ii): 2x=36 → x=18.
Then y=9.
Answer: 18 and 9.

Q6. Solve using cross-multiplication method:

$3x + 2y = 11,\quad 2x - 3y = -4$ .

Solution:
Equation form:
$\frac{x}{(-11)(-3)-(2)(-4)} = \frac{y}{(3)(-4)-(11)(2)} = \frac{1}{(3)(-3)-(2)(2)}$ .
Simplify → $x=1,\ y=4$ .
Answer: (1,4).

Q7. (PYQ – 2022) The sum of a two-digit number and the number obtained by reversing its digits is 66. If the digits differ by 2, find the number.

Solution:
Let digits = x, y. Number = 10x+y.
Reverse = 10y+x.
Equation 1: (10x+y)+(10y+x)=66 → 11(x+y)=66 → x+y=6.
Equation 2: x-y=2.
Solving: x=4, y=2 → Number=42.
Answer: 42.

Q8. Solve:

$x/2 + y/3 = 5,\quad x/3 - y/2 = -2$ .

Solution:
Multiply LHS by LCMs:
$3x+2y=30,\ 2x-3y=-12$ .
Multiply first by 3 → 9x+6y=90.
Multiply second by 2 → 4x-6y=-24.
Add → 13x=66 → x=66/13.
Substitute → y=72/13.
Answer: (66/13, 72/13).

Q9. (PYQ – 2021) Solve:

$7x - 15y = 2,\quad x + 2y = 3$ .

Solution:
From second: x=3-2y.
Substitute: 7(3-2y)-15y=2 → 21-14y-15y=2.
→ -29y=-19 → y=19/29.
Then x=3-(38/29)=49/29.
Answer: (49/29, 19/29).

Q10. Word Problem (Application)

A farmer has 100 animals (cows and hens). If the total number of legs is 290, find the number of cows and hens.

Solution:
Let cows=x, hens=y.
x+y=100.
4x+2y=290 → 2x+y=145.
Subtract: (2x+y)-(x+y)=145-100 → x=45.
Then y=55.
Answer: 45 cows, 55 hens.

Tips for Students

Practice all four methods: substitution, elimination, cross-multiplication, and graphical.
Word problems are common in boards — practice boats-streams, age, digits, and mixture problems.
Always check solutions by substituting back into original equations.

Class 10 Maths Important Questions with Answers – Quadratic Equations | CBSE & PYQs

Quadratic Equations are one of the most important topics in Class 10 Mathematics. In the CBSE Board Exam, this chapter carries 6–8 marks, usually through problem-solving and application-based word problems.

Here, we provide 10 important questions with step-by-step solutions, including Previous Year Questions (PYQs), to help students score high.

Important Questions – Quadratic Equations (Class 10 Maths)

Q1. Solve $2x^2 - 7x + 3 = 0$ by factorisation.

Solution:
$2x^2 - 7x + 3 = 2x^2 - 6x - x + 3$ .
= $2x(x-3) - 1(x-3) = (2x-1)(x-3)$ .
So, $x=\frac{1}{2}, 3$ .

Q2. (PYQ – 2021) Find the roots of $x^2 - 2x - 8 = 0$ .

Solution:
D = $b^2 - 4ac = (-2)^2 - 4(1)(-8) = 36$ .
Roots = $\frac{2 ± 6}{2} = 4, -2$ .

Q3. Solve $x^2 - 4x + 5 = 0$ . Comment on the nature of roots.

Solution:
D = (-4)² - 4(1)(5) = 16-20 = -4 < 0.
So, roots are imaginary: $x=\frac{4±i2}{2}=2±i$ .

Q4. (PYQ – 2020) Find the discriminant of $3x^2 - 2x + 1 = 0$ . Hence, find the nature of its roots.

Solution:
D = (-2)² - 4(3)(1) = 4 - 12 = -8.
Since D<0, roots are imaginary and unequal.

Q5. Solve $x^2 - 5x + 6 = 0$ using the quadratic formula.

Solution:
D = (-5)² - 4(1)(6) = 25-24=1.
Roots = $\frac{5 ± 1}{2} = 3, 2$ .

Q6. Word Problem (PYQ – 2019) The product of two consecutive positive integers is 182. Find the integers.

Solution:
Let numbers = x, x+1.
Equation: x(x+1)=182 → x²+x-182=0.
Factorise: (x+14)(x-13)=0 → x=13.
Numbers = 13, 14.

Q7. A train travels 120 km at a uniform speed. If the speed had been 10 km/h more, it would have taken 36 minutes less. Find the speed of the train.

Solution:
Let speed = x km/h. Time = 120/x.
New speed = x+10 → time = 120/(x+10).
Difference = 36/60 = 3/5 h.
Equation: 120/x - 120/(x+10) = 3/5.
On solving → x=50.
Answer: 50 km/h.

Q8. (PYQ – 2022) Find the roots of $2x^2 - 3x + 1 = 0$ by completing the square.

Solution:
Divide by 2: $x^2 - 3/2x + 1/2=0$ .
→ $x^2 - (3/2)x = -1/2$ .
Add (3/4)²=9/16 both sides.
→ $(x-3/4)^2 = 1/16$ .
→ $x=3/4 ± 1/4 → 1, 1/2$ .

Q9. Find the roots of $9x^2 - 6x + 1 = 0$ .

Solution:
D = (-6)² - 4(9)(1)=36-36=0.
So, roots are equal: $x=\frac{6}{18}=1/3$ .

Q10. Word Problem The sum of a number and its reciprocal is 29/10. Find the number.

Solution:
Let number = x.
Equation: $x+1/x=29/10$ .
Multiply by 10x: $10x^2+10=29x$ .
→ 10x²-29x+10=0.
Factorise: (5x-2)(2x-5)=0 → x=2/5 or 5/2.

Tips for Students

Memorize the quadratic formula:

x = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}

The discriminant (D) decides the nature of roots:
D>0 → real, unequal roots.
D=0 → real, equal roots.
D<0 → imaginary roots.
Practice word problems carefully: speed-time, age, number problems appear frequently.

Class 10 Maths Important Questions with Answers – Arithmetic Progressions (AP) | CBSE & PYQs

The chapter Arithmetic Progressions (APs) is a scoring part of Class 10 Mathematics. Every year, 6–8 marks are asked in the CBSE Board Exam from APs. The questions are usually simple but require careful application of formulas.

In this blog, we provide 10 important questions with detailed solutions, including Previous Year Questions (PYQs).

Key Formulas of Arithmetic Progressions

nth term of AP: $a_n = a + (n-1)d$
Sum of n terms: $S_n = \frac{n}{2}[2a + (n-1)d]$

Where a = first term, d = common difference, n = number of terms.

Important Questions – Arithmetic Progressions

Q1. Find the 20th term of the AP: 3, 7, 11, 15, …

Solution:
a=3, d=4, n=20.
$a_n = a + (n-1)d = 3 + 19×4 = 79.$
Answer: 79.

Q2. (PYQ – 2020) Which term of the AP 5, 11, 17, … is 81?

Solution:
a=5, d=6, $a_n=81$ .
81=5+(n-1)6 → 76=6(n-1) → n-1=76/6=38/3.
Not integer → 81 is not a term.

Q3. Find the sum of first 30 terms of the AP: 10, 15, 20, …

Solution:
a=10, d=5, n=30.
$S_n=\frac{n}{2}[2a+(n-1)d]$ .
=15[20+145]=15×165=2475.
Answer: 2475.

Q4. (PYQ – 2019) If the nth term of an AP is (7-3n), find its 11th term and 21st term.

Solution:
$a_n=7-3n$ .
11th term: 7-33=-26.
21st term: 7-63=-56.

Q5. The 7th term of an AP is 13 and the 13th term is 7 more than twice the 7th term. Find the AP.

Solution:
a+6d=13 (i).
a+12d=2×13+7=33 (ii).
Subtract: 6d=20 → d=10/3.
From (i): a=-7.
AP = -7, -7+10/3, -7+20/3, …

Q6. (PYQ – 2021) The sum of first 7 terms of an AP is 161, and its nth term is 29. Find n.

Solution:
$S_7=\frac{7}{2}[2a+6d]=161$ → 2a+6d=46 → a+3d=23. (i)
nth term: a+(n-1)d=29. (ii)
From (i): a=23-3d. Substitute in (ii): 23-3d+(n-1)d=29.
(n-4)d=6. Solve → (n=10,d=1.5) or (n=7,d=2).
Valid → n=10.

Q7. Find the sum of first 40 positive integers divisible by 9.

Solution:
AP: 9,18,27,… a=9, d=9, n=40.
$S_{40}=\frac{40}{2}[2×9+(39×9)]$ .
=20[18+351]=20×369=7380.
Answer: 7380.

Q8. (PYQ – 2018) Find the sum of first 22 terms of the AP: 3, 7, 11, …

Solution:
a=3, d=4, n=22.
$S_n=\frac{n}{2}[2a+(n-1)d]$ .
=11[6+84]=11×90=990.
Answer: 990.

Q9. If the sum of first n terms of an AP is $S_n = 5n^2+3n$ , find its 10th term.

Solution:
nth term = $S_n - S_{n-1}$ .
T₁₀= S₁₀ - S₉ = (5×100+30)-(5×81+27).
=530-432=98.
Answer: 98.

Q10. Word Problem A sum of ₹1000 is to be given in 10 monthly instalments, each instalment being ₹10 more than the preceding one. Find the first and the last instalments.

Solution:
Let first instalment=a, d=10, n=10.
Total =1000= (10/2)[2a+(10-1)10].
→1000=5[2a+90].
→200=2a+90 →2a=110 →a=55.
Last instalment= a+(n-1)d=55+90=145.
Answer: First=₹55, Last=₹145.

Tips for Students

Always write down a, d, n before solving.
Learn to use both formulas for nth term and sum.
Word problems (like instalments, consecutive terms, divisibility) are common in boards.
PYQs often repeat in a new form, so practice thoroughly.
Summary
Algebra is a high-weightage unit in Class 10 Maths Board Exams.
Practice 10–15 questions daily from each chapter.
Revise formulas and PYQs, as CBSE repeats concepts often.
With consistent practice, Algebra can become the most scoring section of Mathematics.

Class 10: Maths Whole Algebra (Important Questions & PYQs) | Toppers Hub

Class 10 Maths Algebra Important Questions with Answers | NCERT & PYQs

Key Topics in Class 10 Algebra

Class 10 Maths Polynomials Important Questions with Answers | CBSE & PYQs

Important Questions – Polynomials (Class 10 Maths)

Q1. Find the zeros of the polynomial p(x)=x2−5x+6p(x) = x^2 - 5x + 6p(x)=x2−5x+6. Verify the relationship between zeros and coefficients.

Q2. (PYQ – 2019) Find a quadratic polynomial whose zeros are 4 and -3.

Q3. If α and β are the zeros of 2x2−7x+52x^2 - 7x + 52x2−7x+5, find the value of α2+β2α^2 + β^2α2+β2.

Q4. Divide 2x4−3x3+3x2−2x+22x^4 - 3x^3 + 3x^2 - 2x + 22x4−3x3+3x2−2x+2 by x2−2x^2 - 2x2−2.

Q5. (PYQ – 2020) If the zeros of ax2+bx+cax^2 + bx + cax2+bx+c are equal, show that b2=4acb^2 = 4acb2=4ac.

Q6. Find a cubic polynomial with zeros 2, -1 and -3.

Q7. If the sum and product of the zeros of a quadratic polynomial are √2 and 1/3 respectively, find the polynomial.

Q8. (PYQ – 2021) If the polynomial x4−6x3+16x2−25x+kx^4 - 6x^3 + 16x^2 - 25x + kx4−6x3+16x2−25x+k is exactly divisible by x2−2x+3x^2 - 2x + 3x2−2x+3, find k.

Q9. Show that x2+x+1x^2 + x + 1x2+x+1 has no real zeros.

Q10. If two zeros of the cubic polynomial 2x3−x2−5x+22x^3 - x^2 - 5x + 22x3−x2−5x+2 are √2 and -√2, find the third zero.

Tips for Students

Class 10 Maths Important Questions with Answers – Pair of Linear Equations in Two Variables | CBSE & PYQs

Important Questions – Pair of Linear Equations in Two Variables

Q1. Solve the following pair of equations by elimination method:

Q2. Solve by substitution:

Q3. (PYQ – 2019) A boat goes 16 km downstream in 2 hours and returns in 4 hours. Find the speed of the boat in still water and the speed of the stream.

Q4. Solve graphically:

Q5. (PYQ – 2020) Find two numbers whose sum is 27 and difference is 9 using linear equations.

Q6. Solve using cross-multiplication method:

Q7. (PYQ – 2022) The sum of a two-digit number and the number obtained by reversing its digits is 66. If the digits differ by 2, find the number.

Q8. Solve:

Q9. (PYQ – 2021) Solve:

Q10. Word Problem (Application)

Tips for Students

Class 10 Maths Important Questions with Answers – Quadratic Equations | CBSE & PYQs

Important Questions – Quadratic Equations (Class 10 Maths)

Q1. Solve 2x2−7x+3=02x^2 - 7x + 3 = 02x2−7x+3=0 by factorisation.

Q2. (PYQ – 2021) Find the roots of x2−2x−8=0x^2 - 2x - 8 = 0x2−2x−8=0.

Q3. Solve x2−4x+5=0x^2 - 4x + 5 = 0x2−4x+5=0. Comment on the nature of roots.

Q4. (PYQ – 2020) Find the discriminant of 3x2−2x+1=03x^2 - 2x + 1 = 03x2−2x+1=0. Hence, find the nature of its roots.

Q5. Solve x2−5x+6=0x^2 - 5x + 6 = 0x2−5x+6=0 using the quadratic formula.

Q6. Word Problem (PYQ – 2019) The product of two consecutive positive integers is 182. Find the integers.

Q7. A train travels 120 km at a uniform speed. If the speed had been 10 km/h more, it would have taken 36 minutes less. Find the speed of the train.

Q8. (PYQ – 2022) Find the roots of 2x2−3x+1=02x^2 - 3x + 1 = 02x2−3x+1=0 by completing the square.

Q9. Find the roots of 9x2−6x+1=09x^2 - 6x + 1 = 09x2−6x+1=0.

Q10. Word Problem The sum of a number and its reciprocal is 29/10. Find the number.

Tips for Students

Class 10 Maths Important Questions with Answers – Arithmetic Progressions (AP) | CBSE & PYQs

Key Formulas of Arithmetic Progressions

Important Questions – Arithmetic Progressions

Q1. Find the 20th term of the AP: 3, 7, 11, 15, …

Q2. (PYQ – 2020) Which term of the AP 5, 11, 17, … is 81?

Q3. Find the sum of first 30 terms of the AP: 10, 15, 20, …

Q4. (PYQ – 2019) If the nth term of an AP is (7-3n), find its 11th term and 21st term.

Q5. The 7th term of an AP is 13 and the 13th term is 7 more than twice the 7th term. Find the AP.

Q6. (PYQ – 2021) The sum of first 7 terms of an AP is 161, and its nth term is 29. Find n.

Q7. Find the sum of first 40 positive integers divisible by 9.

Q8. (PYQ – 2018) Find the sum of first 22 terms of the AP: 3, 7, 11, …

Q9. If the sum of first n terms of an AP is Sn=5n2+3nS_n = 5n^2+3nSn​=5n2+3n, find its 10th term.

Q10. Word Problem A sum of ₹1000 is to be given in 10 monthly instalments, each instalment being ₹10 more than the preceding one. Find the first and the last instalments.

Tips for Students

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Q1. Find the zeros of the polynomial $p(x) = x^2 - 5x + 6$ . Verify the relationship between zeros and coefficients.

Q3. If α and β are the zeros of $2x^2 - 7x + 5$ , find the value of $α^2 + β^2$ .

Q4. Divide $2x^4 - 3x^3 + 3x^2 - 2x + 2$ by $x^2 - 2$ .

Q5. (PYQ – 2020) If the zeros of $ax^2 + bx + c$ are equal, show that $b^2 = 4ac$ .

Q8. (PYQ – 2021) If the polynomial $x^4 - 6x^3 + 16x^2 - 25x + k$ is exactly divisible by $x^2 - 2x + 3$ , find k.

Q9. Show that $x^2 + x + 1$ has no real zeros.

Q10. If two zeros of the cubic polynomial $2x^3 - x^2 - 5x + 2$ are √2 and -√2, find the third zero.

Q1. Solve $2x^2 - 7x + 3 = 0$ by factorisation.

Q2. (PYQ – 2021) Find the roots of $x^2 - 2x - 8 = 0$ .

Q3. Solve $x^2 - 4x + 5 = 0$ . Comment on the nature of roots.

Q4. (PYQ – 2020) Find the discriminant of $3x^2 - 2x + 1 = 0$ . Hence, find the nature of its roots.

Q5. Solve $x^2 - 5x + 6 = 0$ using the quadratic formula.

Q8. (PYQ – 2022) Find the roots of $2x^2 - 3x + 1 = 0$ by completing the square.

Q9. Find the roots of $9x^2 - 6x + 1 = 0$ .

Q9. If the sum of first n terms of an AP is $S_n = 5n^2+3n$ , find its 10th term.