Class 11 Mathematics Fix Question 2083 | CEE Nexus
NEB EXAM 2083 / 2026
Mathematics Master Question Bank
Comprehensive Collection of All Image Questions + NEB High-Weightage Extras
1. Logic, Sets, and Real Numbers
Q1.Construct the truth table for the implication p → q.
Hint: The implication is only False (F) when the antecedent (p) is True (T) and the consequent (q) is False (F). Otherwise, it is always True.
Q2.Prove that (A ∪ B)' = A' ∩ B' using the element method. Repeated
Hint: Start by letting x ∈ (A ∪ B)'. This implies x ∉ (A ∪ B), which means x ∉ A AND x ∉ B. Therefore, x ∈ A' ∩ B'.
Q3.State the law of Trichotomy for real numbers.
Hint: For any two real numbers a and b, exactly one of the following is true: a < b, a = b, or a > b.
Q4.Solve the absolute value inequality: |2x - 5| ≤ 9.
Hint: Remove the bars by setting -9 ≤ 2x - 5 ≤ 9. Add 5 to all sides, then divide by 2. Result: -2 ≤ x ≤ 7.
Q5.Define an equivalence relation with an example.
Hint: A relation is an equivalence relation if it is Reflexive, Symmetric, and Transitive. Example: "is equal to" on the set of integers.
Q6.If f(x) = 2x + 3 and g(x) = x², find the composite function (g ˆ f)(x).
Hint: (g ˆ f)(x) means g(f(x)). Substitute 2x+3 into x² to get (2x+3)² = 4x² + 12x + 9.
2. Functions and Polynomials
Q7. Define a Periodic Function and find the period of f(x) = sin(2x).
Hint: A function is periodic if f(x+T) = f(x). For sin(ax), the period is 2π/|a|. Here, Period = 2π/2 = π.
Q8. State the Remainder Theorem and Factor Theorem.
Hint: Remainder Theorem: If P(x) is divided by (x-a), the remainder is P(a). Factor Theorem: If P(a)=0, then (x-a) is a factor of P(x).
Q9. If x³ - 6x² + 11x - 6 is divided by (x - 1), find the remainder using the Remainder Theorem.
Hint: Replace x with 1 in the polynomial: (1)³ - 6(1)² + 11(1) - 6 = 1 - 6 + 11 - 6 = 0. (This also means x-1 is a factor).
Q10. Find the condition that the roots of the equation ax² + bx + c = 0 are reciprocal to each other.
Hint: Let roots be α and 1/α. Product of roots = α * (1/α) = c/a. Therefore, 1 = c/a, which gives the condition **a = c**.
Q11. Solve the cubic equation: x³ - 7x + 6 = 0 using Synthetic Division. Important
Hint: Test small integers. x=1 is a root (1-7+6=0). Use synthetic division with 1 to reduce to a quadratic: (x-1)(x²+x-6)=0, then factorize the quadratic.
3. Matrices and Determinants (Part I)
Q12. Define a Symmetric and Skew-symmetric matrix with examples.
Hint: Symmetric: Aᵀ = A. Skew-Symmetric: Aᵀ = -A (diagonal elements must be zero).
Q13. If A = [[1, 2], [3, 4]], find the Adjoint of A.
Hint: For a 2x2 matrix, swap the diagonal elements and change the signs of the off-diagonal elements. Result: [[4, -2], [-3, 1]].
Q14. Prove that |Aᵀ| = |A| for any square matrix A of order 2.
Hint: Let A = [[a, b], [c, d]]. Then |A| = ad-bc. Aᵀ = [[a, c], [b, d]]. |Aᵀ| = ad-cb. Since ad-bc = ad-cb, they are equal.
Q15. Using properties of determinants, show that: | [1, 1, 1], [a, b, c], [bc, ca, ab] | = (a-b)(b-c)(c-a). Most Repeated
Hint: Apply C2 → C2 - C1 and C3 → C3 - C1. Take (b-a) and (c-a) common from columns 2 and 3, then expand.
4. Systems of Linear Equations
Q16. Solve the following system using Cramer's Rule: 3x + 2y = 8 2x - y = 3
Hint: Calculate D = |[3, 2], [2, -1]|. Then find Dx (replace x-column with [8, 3]) and Dy (replace y-column). x = Dx/D, y = Dy/D.
Q17. Solve using the Matrix Method (Inverse Method): x + y + z = 6 2x - y + z = 3 x + 2y - z = 2
Hint: Write in form AX = B. Find A⁻¹ using (1/|A|) * adj(A). Then multiply A⁻¹B to find the values of x, y, and z.
Q18. Apply Gauss-Elimination method to solve: x + 2y + 3z = 14 3x + y + 2z = 11 2x + 3y + z = 11
Hint: Convert the augmented matrix [A|B] into an upper triangular matrix using row operations (R2 → R2 - 3R1, etc.), then use back-substitution.
5. Complex Numbers
Q19. If z = 1 + i√3, find its modulus and principal argument.
Q26. Prove that nPr = n! / (n-r)! from the fundamental principle of counting.
Hint: The first place can be filled in n ways, second in (n-1), ..., r-th in (n-r+1) ways. Product: n(n-1)...(n-r+1). Multiplying and dividing by (n-r)! gives the factorial formula.
Q27. If nC12 = nC8, find the value of nC17. Repeated
Hint: Use the property nCx = nCy &implies; x+y = n. Here, n = 12 + 8 = 20. Then calculate 20C17, which is equal to 20C3 = (20×19×18)/(3×2×1).
Q28. Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?
Hint: Selection: 7C3 × 4C2. Arrangement of 5 selected letters: 5!. Total = (7C3 × 4C2) × 5!.
8. Binomial Theorem
Q29. Find the 5th term in the expansion of (x - 2y)¹⁰.
Hint: Use General Term T(r+1) = nCr aⁿ⁻ʳ xʳ. Here r = 4, n = 10, a = x, and x = -2y. T5 = 10C4 (x⁶) (-2y)⁴.
Q30. Find the middle term in the expansion of (x/a + a/x)¹².
Hint: Since n=12 (even), there is one middle term: (n/2 + 1) = 7th term. T7 = 12C6 (x/a)⁶ (a/x)⁶ = 12C6.
Q31. Find the term independent of x in (√x + 1/3x²)¹⁰. Most Important
Hint: Write the general term T(r+1). Simplify the powers of x and set the final exponent of x to zero. Solve for r and calculate the coefficient.
Q32. Use the Binomial Theorem to expand (1.01)⁵ up to 4 decimal places.
Hint: Write as (1 + 0.01)⁵. Expand using 1 + nC1(0.01) + nC2(0.01)² + ... Usually, the first 3 or 4 terms are enough for the required accuracy.
9. Trigonometry
Q33. Prove that: cos(A+B) = cosA cosB - sinA sinB using a geometric method.
Hint: Use a unit circle or a rectangle diagram where the diagonal makes angles A and B with the sides. Projection of lengths on the axes will lead to the identity.
Q34. Solve the general solution for: √3 sin θ + cos θ = √2. High Weightage
Hint: Divide both sides by √( (√3)² + 1²) = 2. This gives (√3/2) sin θ + (1/2) cos θ = √2/2. Rewrite as sin(θ + 30°) = sin 45°. Apply general formula θ = nπ + (-1)ⁿα.
Q35. In any triangle ABC, prove that: a = b cos C + c cos B (Projection Law).
Hint: Draw a perpendicular from vertex A to side BC. BC is divided into two segments: c cos B and b cos C. Their sum equals side a.
Q37. Find the condition that two lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 are perpendicular.
Hint: The product of their slopes must be -1. (m₁ × m₂ = -1). This simplifies to a₁a₂ + b₁b₂ = 0.
Q38. Find the coordinates of the foot of the perpendicular from (1, 2) to the line x - 3y + 4 = 0.
Hint: Find the equation of the line perpendicular to the given line passing through (1,2). Then solve the two equations simultaneously.
Q39. Find the equation of a circle passing through the points (1, 0), (0, 1), and (1, 1).
Hint: Use the general equation x² + y² + 2gx + 2fy + c = 0. Substitute the three points to get three equations, solve for g, f, and c.
Q40. Show that the line 3x + 4y = 25 touches the circle x² + y² = 25. Repeated
Hint: A line touches a circle if the perpendicular distance from the center (0,0) to the line equals the radius (5). Distance = |3(0)+4(0)-25| / √(3²+4²) = 25/5 = 5.
11. Vectors
Q41. If &vec;a = 2i - j + 2k and &vec;b = i + 3j - k, find the unit vector perpendicular to both &vec;a and &vec;b.
Hint: First find the cross product &vec;a × &vec;b using a determinant. Then divide that result by its own magnitude to get the unit vector.
Q42. Prove by vector method that the diagonals of a rhombus bisect each other at right angles. Most Repeated
Hint: Let &vec;a and &vec;b be the vectors representing two adjacent sides. The diagonals are (&vec;a+&vec;b) and (&vec;a-&vec;b). Show their dot product is zero because |&vec;a|=|&vec;b| in a rhombus.
Q43. Find the scalar projection of &vec;a = i - 2j + k on &vec;b = 4i - 4j + 7k.
Hint: Use the formula: Projection = (&vec;a · &vec;b) / |&vec;b|. Dot product = (1*4) + (-2*-4) + (1*7) = 19. Magnitude of &vec;b = 9.
12. Calculus: Limits and Continuity
Q44. Evaluate the limit: lim (x→0) [ (√(1+x) - √(1-x)) / x ].
Hint: Rationalize the numerator by multiplying with the conjugate (√(1+x) + √(1-x)). After simplifying, the x in the denominator will cancel out.
Q45. If f(x) = { x²-1 for x<2, and 3 for x=2, and 2x-1 for x>2 }, check the continuity at x=2. Important
Hint: Find LHL (lim x→2⁻), RHL (lim x→2⁺), and f(2). If LHL = RHL = f(2), the function is continuous. Here 3=3=3, so it is continuous.
Hint: Since it is in 0/0 form, differentiate numerator and denominator twice. 1st deriv: (e^x - 1)/2x. 2nd deriv: e^x/2. Limit is 1/2.
Q47. State the three conditions for a function f(x) to be continuous at a point x=a.
Hint: 1. f(a) must be defined. 2. Limit f(x) as x→a must exist. 3. Limit f(x) as x→a must equal f(a).
13. Derivatives (Differentiation)
Q48. Find from the first principle the derivative of: f(x) = √(sin x). High Weightage
Hint: Use f'(x) = lim (h→0) [√(sin(x+h)) - √(sin x)] / h. Rationalize the numerator, then use the formula for sin C - sin D.
Q49. Differentiate y = x^(sin x) with respect to x.
Hint: Take natural log (ln) on both sides: ln y = sin x ln x. Differentiate using the product rule on the right side, then multiply by y to find dy/dx.
Q50. Find dy/dx if x = a(θ + sin θ) and y = a(1 - cos θ).
Hint: This is parametric differentiation. Find dx/dθ and dy/dθ separately. Then dy/dx = (dy/dθ) / (dx/dθ). Simplify using half-angle formulas.
Q51. State and prove Rolle's Theorem.
Hint: If f(x) is continuous in [a,b], differentiable in (a,b), and f(a)=f(b), then there exists at least one 'c' in (a,b) such that f'(c)=0.
14. Anti-Derivatives (Integration)
Q52. Evaluate the integral: ∫ [ dx / (x² - a²) ]
Hint: This is a standard formula. Use partial fractions: 1/(x-a)(x+a) = [1/2a] * [1/(x-a) - 1/(x+a)]. The result is (1/2a) log |(x-a)/(x+a)| + C.
Q53. Evaluate: ∫ x² e^x dx using integration by parts. Repeated
Hint: Use the ILATE rule (u = x², dv = e^x dx). You will need to apply integration by parts twice to reach the final answer.
Q54. Evaluate the definite integral: ∫₀ᵝ/² [ (sin x) / (sin x + cos x) ] dx.
Hint: Use the property ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a-x) dx. Adding the original integral and the modified one will result in 2I = ∫₀ᵝ/² 1 dx. Result is π/4.
Q55. Find the area bounded by the ellipse x²/a² + y²/b² = 1 using integration.
Hint: Solve for y = (b/a)√(a²-x²). Area = 4 * ∫₀ᵃ y dx. Use the formula for √(a²-x²) integration. Final Area = πab.
15. Statistics and Probability
Q56. Calculate the Mean Deviation from the Median for the following data: 10, 15, 18, 20, 25.
Hint: First find the Median (18). Then find the absolute differences |x - Median| for each value. Mean Deviation = Σ|x - Median| / n.
Q57. Find the Standard Deviation and Variance of the first 10 natural numbers. Frequently Asked
Hint: Use the formula for SD of first n natural numbers: σ = √[(n² - 1) / 12]. For n=10, σ = √[99 / 12] ≈ 2.87. Variance is σ².
Q58. State and prove the Addition Law of Probability for two non-mutually exclusive events.
Hint: P(A ∪ B) = P(A) + P(B) - P(A ∩ B). Prove this using a Venn diagram showing that the intersection is counted twice if not subtracted.
Q59. A bag contains 4 red, 5 black, and 6 white balls. Three balls are drawn at random. Find the probability that they are of different colors.
Hint: P = (Ways to pick 1R, 1B, 1W) / (Total ways to pick 3). P = (4C1 × 5C1 × 6C1) / 15C3.
Q60. In a single throw of two dice, what is the probability of getting a total sum of 9 or 11?
Hint: Possible outcomes for sum 9: (3,6), (4,5), (5,4), (6,3). For sum 11: (5,6), (6,5). Total favorable = 6. Total outcomes = 36. P = 6/36 = 1/6.
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