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Subjective 2026

Math Set 1

LONG ANSWER (5 MARKS)
100% FixGraph
1. Solve graphically: x + 3y = 6 and 2x - 3y = 12.
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Points: (0,2), (6,0) & (0,-4), (6,0). Intersect at (6,0).
VVIH & D
2. Statue 1.6m tall. Angle of elevation of top 60° & top of pedestal 45°. Find pedestal height.
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Ans: 0.8(√3 + 1) m. (Rationalize 1.6/(√3-1)).
VVIConstruction
3. Construct triangle sides 5,6,7cm & another 7/5 times of it.
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Draw base 7cm. Make triangle. Extend base. Draw ray at acute angle, mark 7 pts. Join 5th to end, draw parallel from 7th.
VVITheorem
4. State and Prove Pythagoras Theorem.
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In a right triangle, square of hypotenuse equals sum of squares of other two sides. Proof uses similar triangles (AA similarity).
VVIMensuration
5. Toy is a cone on hemisphere (r=3.5). Total h=15.5. Find TSA.
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Cone h=12, l=12.5. TSA = πrl + 2πr². Ans: 214.5 cm².
VVIStatistics
6. Find Median: CI 0-10 to 50-60. F: 5, 8, 20, 15, 7, 5.
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N=60. Median class 20-30. Use Formula: L + [(N/2 - cf)/f] × h. Ans: 28.5.
VVIH & D
7. Angle of elevation of building from foot of tower 30°, tower from building 60°. Tower 50m. Find Building.
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Ans: 16.66 m (50/3 m).
VVITheorem
8. State and Prove Thales (BPT) Theorem.
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If a line is parallel to one side intersecting others, it divides them in same ratio. Use Area of Triangle method.
SHORT ANSWER (2 MARKS)
9. Prove that 3 + 2√5 is irrational.
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Assume rational. Rearrange to √5 = (a/b - 3)/2. RHS is rational, LHS irrational. Contradiction.
10. Find HCF and LCM of 96 and 404 by prime factorization.
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HCF = 4, LCM = 9696.
11. Find zeroes of quadratic polynomial x² - 2x - 8.
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(x-4)(x+2)=0. Zeroes are 4, -2.
12. If zeroes of x² - px + 6 are in ratio 3:2, find p.
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Ans: p = ±5.
13. Solve for x & y: 2x + 3y = 11 and 2x - 4y = -24.
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x = -2, y = 5. (Also find m for y=mx+3, m=-1).
14. Find roots of 2x² - 7x + 3 = 0 using Quadratic Formula.
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D = 49 - 24 = 25. Roots: (7±5)/4 → 3, 1/2.
15. Find 31st term of AP whose 11th term is 38 and 16th is 73.
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a=-32, d=7. t31 = a+30d = 178.
16. Which term of AP 3, 8, 13, 18... is 78?
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78 = 3 + (n-1)5. Ans: 16th term.
17. Find sum of first 1000 positive integers.
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n(n+1)/2 = 1000*1001/2 = 500500.
18. Prove √((1+sinA)/(1-sinA)) = secA + tanA.
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Multiply numerator/denominator by √(1+sinA).
19. Evaluate: 2tan²45° + cos²30° - sin²60°.
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2(1)² + (√3/2)² - (√3/2)² = 2.
20. If tan(A+B)=√3 and tan(A-B)=1/√3, find A and B.
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A+B=60, A-B=30. Ans: A=45°, B=15°.
21. Find distance between (2,3) and (4,1).
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√[(4-2)² + (1-3)²] = √[4+4] = 2√2.
22. Find ratio in which y-axis divides line joining (5,-6) and (-1,-4).
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Use Section Formula (x=0). Ans: 5:1.
23. Find area of triangle with vertices (2,3), (-1,0), (2,-4).
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1/2 |x1(y2-y3)...| Ans: 10.5 sq units.
24. Prove length of tangents from external point are equal.
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RHS Congruence rule (Radius ⊥ Tangent).
25. Two concentric circles radii 5cm, 3cm. Find length of chord of larger circle tangent to smaller.
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Pythagoras in half chord triangle (3-4-5). Total length = 2*4 = 8cm.
26. Find area of sector with radius 6cm and angle 60°.
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(θ/360)πr² = 1/6 * 22/7 * 36. Ans: 132/7 cm².
27. Metallic sphere radius 4.2cm melted to cylinder radius 6cm. Find h.
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Vol Sphere = Vol Cyl. Ans: 2.74 cm.
28. Probability of getting a King of Red color from 52 cards.
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Red Kings (Heart+Diamond) = 2. Prob = 2/52 = 1/26.
29. Find Mean of: 10, 25, 15, 12, 18.
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Sum = 80, N = 5. Mean = 80/5 = 16.
30. Find Mode of: 2, 6, 4, 5, 0, 2, 1, 3, 2, 3.
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Most frequent value is 2. Mode = 2.