UP Board 10th Maths Model Paper 2027 — Full Paper With Solutions

This model paper follows the official UPMSP pattern for Class 10th Mathematics 2027. Total: 100 marks in 3 hours.

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Paper Structure

Section	Type	Marks
Section A	MCQ — 20 questions (1 mark each)	20
Section B	Short Answer — 10 questions (4 marks each)	40
Section C	Long Answer — 5 questions (8 marks each)	40

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SECTION A — MCQ (1 Mark Each)

Q1. HCF of 12 and 18 is:

(a) 2 (b) 3 (c) 6 (d) 36

Answer: (c) 6

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Q2. The discriminant of 2x² − 4x + 3 = 0 is:

(a) 8 (b) −8 (c) 16 (d) −16

Answer: (b) −8
(b² − 4ac = 16 − 24 = −8; since D < 0, no real roots)

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Q3. The 10th term of AP 2, 5, 8, 11,... is:

(a) 28 (b) 29 (c) 30 (d) 32

Answer: (b) 29
(a = 2, d = 3; a₁₀ = 2 + 9×3 = 29)

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Q4. If tan θ = 4/3, then sin θ is:

(a) 4/5 (b) 3/5 (c) 3/4 (d) 4/3

Answer: (a) 4/5
(In right triangle: opposite = 4, adjacent = 3, hypotenuse = 5; sin θ = 4/5)

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Q5. The volume of a sphere of radius 3 cm is:

(a) 36π cm³ (b) 27π cm³ (c) 24π cm³ (d) 9π cm³

Answer: (a) 36π cm³
(V = (4/3)πr³ = (4/3)π×27 = 36π)

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Q6. Probability of getting a head when a coin is tossed is:

(a) 0 (b) 1 (c) 1/2 (d) 1/4

Answer: (c) 1/2

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Q7. The midpoint of segment joining (2, 4) and (6, 8) is:

(a) (4, 6) (b) (3, 5) (c) (8, 12) (d) (4, 4)

Answer: (a) (4, 6)
(Midpoint = ((2+6)/2, (4+8)/2) = (4, 6))

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Q8. Which of the following is irrational?

(a) √4 (b) √9 (c) √2 (d) √16

Answer: (c) √2

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Q9. The distance between points (0, 0) and (3, 4) is:

(a) 5 (b) 7 (c) 3 (d) 4

Answer: (a) 5
(d = √(9 + 16) = √25 = 5)

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Q10. Area of a circle with diameter 14 cm is:

(a) 44 cm² (b) 154 cm² (c) 616 cm² (d) 308 cm²

Answer: (b) 154 cm²
(r = 7; A = πr² = (22/7)×49 = 154 cm²)

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SECTION B — Short Answer (4 Marks Each)

Q11. Solve by elimination method: 2x + 3y = 12 and 3x − 2y = 5.

Solution:
Multiply eq1 by 2: 4x + 6y = 24 ...(i)
Multiply eq2 by 3: 9x − 6y = 15 ...(ii)
Adding: 13x = 39 → x = 3
Substituting: 6 + 3y = 12 → y = 2

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Q12. Find the sum of first 20 terms of AP: 3, 7, 11, 15,...

Solution:
a = 3, d = 4, n = 20
Sₙ = n/2[2a + (n−1)d] = 20/2[6 + 76] = 10 × 82 = 820

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Q13. Prove that √3 is irrational.

Solution:
Assume √3 is rational. Then √3 = p/q where p, q are integers with HCF = 1.
Squaring: 3 = p²/q² → p² = 3q²
So 3 divides p² → 3 divides p (since 3 is prime)
Let p = 3m: 9m² = 3q² → q² = 3m²
So 3 divides q² → 3 divides q
But then HCF(p,q) ≥ 3 — contradiction!
Therefore √3 is irrational ∎

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Q14. A ladder 10 m long reaches a window 8 m above the ground. Find the distance of the foot of the ladder from the base of the wall.

Solution:
Using Pythagoras theorem:
(foot)² + 8² = 10²
(foot)² = 100 − 64 = 36
Foot of ladder = 6 m from wall

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Q15. Find the mean of the following distribution:

Class	0–10	10–20	20–30	30–40	40–50
Freq	4	8	12	6	5

Solution:
Midpoints: 5, 15, 25, 35, 45
Σfx = 20 + 120 + 300 + 210 + 225 = 875
Σf = 35
Mean = 875/35 = 25

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SECTION C — Long Answer (8 Marks Each)

Q16. Draw the graph of y = x² − 2x − 3 and find the zeroes from the graph.

Solution:
Prepare table of values:

x	−2	−1	0	1	2	3	4
y	5	0	−3	−4	−3	0	5

Key observations from graph:

• Parabola opens upward (coefficient of x² is positive)
• Graph cuts x-axis at x = −1 and x = 3
• Zeroes: x = −1 and x = 3

Verification: x² − 2x − 3 = (x−3)(x+1) = 0 → x = 3 or x = −1 ✓

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Q17. Prove: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (Pythagoras Theorem).

Solution:
Given: △ABC right-angled at B
To Prove: AC² = AB² + BC²
Construction: Draw BD ⊥ AC

Proof:
In △ABD and △ABC:
∠A = ∠A (common); ∠ADB = ∠ABC = 90°
∴ △ABD ~ △ABC (AA similarity)
→ AD/AB = AB/AC
→ AB² = AD × AC ...(i)

In △BDC and △ABC:
∠C = ∠C (common); ∠BDC = ∠ABC = 90°
∴ △BDC ~ △ABC
→ DC/BC = BC/AC
→ BC² = DC × AC ...(ii)

Adding (i) and (ii):
AB² + BC² = AC(AD + DC) = AC × AC = AC² ∎

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Practice Problems (Self Test)

1. Factorise: x³ − 6x² + 11x − 6
2. Find the quadratic equation whose roots are 5 and −3.
3. A die is thrown. Find probability of getting a number greater than 4.
4. Find the area of triangle with vertices (1,2), (3,4), (5,0).
5. How many terms of AP 5, 7, 9... must be taken to give sum 320?

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Chapter-Wise Weightage

Chapter	Marks
Real Numbers and Polynomials	12
Linear and Quadratic Equations	16
Arithmetic Progressions	10
Trigonometry	12
Coordinate Geometry	8
Mensuration (Areas and Volumes)	14
Statistics and Probability	10
Triangles and Circles	12
Constructions	6

Tip: UP Board 10th Maths Section C carries 40 marks — half the paper. Long answers require complete steps. Even if the final answer is wrong, showing correct method earns partial marks. Never leave a long answer blank.