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NEB Class 12 Mathematics study materials
NEB Class 12 Mathematics study resources

NEB Class 12 Mathematics practice questions

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Question 1Algebra

For the complex number z = 3 − 4i, what is the modulus |z|?

  1. 7
  2. √7
  3. 5 Correct answer.
  4. 25

Explanation

The modulus of a complex number a + bi is √(a² + b²). Here a = 3 and b = −4, so |z| = √(9 + 16) = √25 = 5.

Question 2Algebra

How many different arrangements can be made using all the letters of the word NEPAL, if all letters are distinct?

  1. 60
  2. 120 Correct answer.
  3. 720
  4. 24

Explanation

The word NEPAL has 5 distinct letters, so the number of arrangements is 5! = 5 × 4 × 3 × 2 × 1. This equals 120.

Question 3Algebra

Matrix A has order 2 × 3 and matrix B has order 3 × 4. What is the order of the product AB?

  1. 2 × 3
  2. 3 × 3
  3. 2 × 4 Correct answer.
  4. 4 × 2

Explanation

For AB to be defined, the number of columns of A must equal the number of rows of B, which holds here. The product then has the number of rows of A and the number of columns of B, so the order is 2 × 4.

Question 4Algebra

In the expansion of (x + 2)^8, what is the coefficient of x^5?

  1. 448 Correct answer.
  2. 56
  3. 896
  4. 224

Explanation

The general term is C(8,r) x^(8−r) 2^r. For x^5 we need 8 − r = 5, so r = 3. The coefficient is C(8,3)·2^3 = 56·8 = 448.

Question 5Algebra

What is the principal argument of the complex number z = −1 + i?

  1. π/4
  2. −π/4
  3. 3π/4 Correct answer.
  4. 5π/4

Explanation

The point (−1, 1) lies in the second quadrant, so its principal argument is between π/2 and π. The reference angle is π/4, giving arg z = π − π/4 = 3π/4.

Question 6Algebra

A committee of 3 people is to be selected from 5 men and 4 women. In how many ways can the committee contain exactly 2 women?

  1. 30 Correct answer.
  2. 60
  3. 20
  4. 40

Explanation

Choose 2 women from 4 in C(4,2) = 6 ways, and 1 man from 5 in C(5,1) = 5 ways. By the multiplication principle, the total is 6 × 5 = 30.

Question 7Algebra

If A is a 3 × 3 matrix with det(A) = 5, what is det(2A)?

  1. 10
  2. 40 Correct answer.
  3. 80
  4. 5

Explanation

Multiplying a 3 × 3 matrix by 2 multiplies each of its three rows by 2, so the determinant is multiplied by 2^3 = 8. Therefore det(2A) = 8 · det(A) = 8 · 5 = 40.

Question 8Algebra

A linear programming problem has feasible corner points (0,0), (4,0), (2,3), and (0,2). The objective Z = 5x + 4y is to be maximized. What is the maximum value of Z?

  1. 20
  2. 0
  3. 8
  4. 22 Correct answer.

Explanation

Evaluate Z at every corner: (0,0) gives 0, (4,0) gives 20, (2,3) gives 10 + 12 = 22, and (0,2) gives 8. The largest value is 22.

Question 9Algebra

The system of equations 2x + 3y = 5 and 4x + ky = 10 has infinitely many solutions when k equals:

  1. 6 Correct answer.
  2. 12
  3. 3
  4. −6

Explanation

For infinitely many solutions the two equations must be dependent, so the coefficient determinant must vanish: 2k − 12 = 0, giving k = 6. With k = 6 the second equation is exactly twice the first, so the system is consistent.

Question 10Algebra

What is the term independent of x in the expansion of (x + 1/x)^10?

  1. 120
  2. 210
  3. 252 Correct answer.
  4. 45

Explanation

The general term is C(10,r) x^(10−r)(1/x)^r = C(10,r) x^(10−2r). For the term independent of x, set 10 − 2r = 0, so r = 5. Its coefficient is C(10,5) = 252.

Question 11Algebra

For the complex number z = 2 + 3i, what is the value of z times its conjugate z̄?

  1. 5
  2. 13 Correct answer.
  3. -5
  4. 13i

Explanation

For any complex number z, the product z times its conjugate equals the square of its modulus, so z z̄ = 2^2 + 3^2 = 4 + 9 = 13. The imaginary parts cancel when the conjugate is multiplied.

Question 12Algebra

How many different 3-digit numbers can be formed using the digits 1, 2, 3, and 4 if no digit is repeated?

  1. 12
  2. 4
  3. 24 Correct answer.
  4. 64

Explanation

The hundreds digit can be chosen in 4 ways, the tens digit in 3 remaining ways, and the units digit in 2 remaining ways. Thus the number of 3-digit numbers is 4 × 3 × 2 = 24.

Question 13Algebra

For the matrix A = [[1, 2], [3, 4]], what is the trace of A?

  1. 4
  2. 10
  3. 7
  4. 5 Correct answer.

Explanation

The trace of a square matrix is the sum of the entries on its main diagonal. For this matrix the diagonal entries are 1 and 4, so trace(A) = 1 + 4 = 5.

Question 14Algebra

What is the middle term in the expansion of (x + 2/x)^6?

  1. 80
  2. 160 Correct answer.
  3. 20
  4. 240

Explanation

For (x + 2/x)^6 there are seven terms, so the middle term is the fourth term. Using the general term, T4 = C(6,3) x^3 (2/x)^3 = 20 × 8 = 160.

Question 15Algebra

What is the simplified value of (1 + i)/(1 - i)?

  1. 1
  2. -i
  3. -1
  4. i Correct answer.

Explanation

Multiply numerator and denominator by the conjugate 1 + i. The denominator becomes (1 - i)(1 + i) = 2, while the numerator becomes (1 + i)^2 = 2i, so the quotient is i.

Question 16Algebra

How many diagonals does a regular octagon have?

  1. 16
  2. 28
  3. 20 Correct answer.
  4. 24

Explanation

An octagon has 8 vertices, and joining any two vertices gives C(8,2) = 28 segments. The 8 sides are not diagonals, so the number of diagonals is 28 - 8 = 20.

Question 17Algebra

If A is a 2 × 2 matrix with determinant |A| = 3, what is |adj A|?

  1. 1/3
  2. 3 Correct answer.
  3. 9
  4. 6

Explanation

For an n × n matrix A, the determinant of its adjoint satisfies |adj A| = |A|^(n-1). Here n = 2 and |A| = 3, so |adj A| = 3^(2-1) = 3.

Question 18Algebra

A linear programming problem asks to maximize Z = 3x + 2y subject to x + y ≤ 4, x ≥ 0, y ≥ 0. What is the maximum value of Z?

  1. 0
  2. 12 Correct answer.
  3. 14
  4. 8

Explanation

The feasible region has corner points (0,0), (4,0), and (0,4). At these points Z = 3x + 2y equals 0, 12, and 8 respectively, so the maximum value is 12.

Question 19Algebra

If A is a square matrix satisfying A^2 = A, what is (I + A)^3 - 7A?

  1. 0
  2. 2A
  3. I Correct answer.
  4. A

Explanation

Since A^2 = A, also A^3 = A. Expanding gives (I + A)^3 = I + 3A + 3A^2 + A^3 = I + 7A, so subtracting 7A leaves I.

Question 20Algebra

If ω is a non-real cube root of unity, what is the value of (1 + ω - ω^2)^3?

  1. 1
  2. -1
  3. 8
  4. -8 Correct answer.

Explanation

For a non-real cube root of unity, 1 + ω + ω^2 = 0, so 1 + ω = -ω^2. Then 1 + ω - ω^2 = -2ω^2, and cubing gives (-2ω^2)^3 = -8ω^6 = -8.

Question 21Algebra

For the complex number z = 1 + 2i, what is the real part of z²?

  1. 5
  2. 3
  3. -3 Correct answer.
  4. -5

Explanation

Since z² = (1 + 2i)² = 1 + 4i + 4i², and i² = −1, this simplifies to −3 + 4i. The real part is therefore −3; the imaginary part is 4.

Question 22Algebra

For what value(s) of x is the matrix M = [[x, 2], [8, x]] singular?

  1. ±4 Correct answer.
  2. -4 only
  3. 16
  4. 4 only

Explanation

A 2×2 matrix is singular exactly when its determinant is zero. Here det M = x·x − 2·8 = x² − 16, so x² = 16 and therefore x = ±4.

Question 23Algebra

In how many ways can 4 different books be arranged on a shelf so that two particular books are always together?

  1. 12 Correct answer.
  2. 24
  3. 48
  4. 6

Explanation

Treat the two particular books as one block, giving 3! orders of the block and the other two books. The two books inside the block can be arranged in 2! ways, so the total is 3! × 2! = 12.

Question 24Algebra

What is the sum of all binomial coefficients in the expansion of (1 + x)^n?

  1. 2^(n−1)
  2. n + 1
  3. 2^n Correct answer.
  4. 2n

Explanation

Putting x = 1 in (1 + x)^n gives (1 + 1)^n = 2^n, which is the sum of all binomial coefficients. This substitution adds every coefficient exactly once.

Question 25Algebra

If z = 1 + i√3, what is z³?

  1. -8 Correct answer.
  2. 8i
  3. -8i
  4. 8

Explanation

Write z in polar form: modulus 2 and argument 60°. Cubing gives modulus 2³ = 8 and argument 180°, so z³ = 8(cos 180° + i sin 180°) = −8.

Question 26Algebra

If A is a 3×3 matrix with det(A) = -2, what is det(A adj A)?

  1. -2
  2. 8
  3. -8 Correct answer.
  4. 4

Explanation

For any square matrix, A(adj A) = |A|I. Here |A| = −2, so A adj A = −2I, and the determinant of a 3×3 scalar multiple is (−2)³ = −8.

Question 27Algebra

A linear programming problem has constraints x + y ≤ 6, x ≤ 2, x ≥ 0, and y ≥ 0. What is the maximum value of Z = 2x + 3y?

  1. 12
  2. 18 Correct answer.
  3. 24
  4. 16

Explanation

The corner points are (0,0), (2,0), (2,4), and (0,6). Evaluating Z = 2x + 3y gives 0, 4, 16, and 18 respectively, so the maximum value is 18.

Question 28Algebra

What is the coefficient of x⁴ in the expansion of (1 + x + x²)⁵?

  1. 40
  2. 45 Correct answer.
  3. 30
  4. 50

Explanation

The coefficient of x⁴ comes from terms with j + 2k = 4 in i + j + k = 5. The possibilities (j,k) = (4,0), (2,1), and (0,2) give multinomial counts 5, 30, and 10; their sum is 45.

Question 29Algebra

How many 4-digit even numbers can be formed using the digits 0, 1, 2, 3, 4, 5 without repetition?

  1. 144
  2. 120
  3. 156 Correct answer.
  4. 180

Explanation

If the last digit is 0, the first three places can be filled in 5P3 = 60 ways. If the last digit is 2 or 4, the first digit has 4 choices, then 4 and 3 choices follow, giving 48 per last digit; total 60 + 2(48) = 156.

Question 30Algebra

From 6 boys and 4 girls, in how many ways can a committee of 5 be formed containing at least one girl?

  1. 258
  2. 252
  3. 246 Correct answer.
  4. 240

Explanation

Total committees of 5 from 10 people: C(10,5) = 252. The only ones with no girl are the all-boy committees, C(6,5) = 6, so the number containing at least one girl is 252 − 6 = 246.

Question 31Algebra

If α and β are the roots of the quadratic equation x² − 5x + 6 = 0, what is the value of α² + β²?

  1. 13 Correct answer.
  2. 12
  3. 19
  4. 25

Explanation

By Vieta's formulas, α + β = 5 and αβ = 6. The identity α² + β² = (α + β)² − 2αβ then gives 25 − 12 = 13, which also matches the direct computation 2² + 3² using the roots 2 and 3.

Question 32Algebra

In how many distinguishable ways can 6 different beads be arranged to form a necklace?

  1. 720
  2. 60 Correct answer.
  3. 120
  4. 24

Explanation

For n distinct beads on a necklace, fixing one bead removes rotational symmetry and leaves (n − 1)! orders; dividing by 2 then identifies each arrangement with its mirror image, since a necklace can be flipped over. Thus the count is 5!/2 = 120/2 = 60.

Question 33Algebra

If A is a 3 × 3 skew-symmetric matrix, what is the value of det(A)?

  1. 1
  2. It depends on the entries of A
  3. −1
  4. 0 Correct answer.

Explanation

A skew-symmetric matrix satisfies Aᵀ = −A, so det(A) = det(Aᵀ) = det(−A) = (−1)³ det(A) = −det(A). Therefore 2 det(A) = 0, forcing det(A) = 0 for every skew-symmetric matrix of odd order.

Question 34Algebra

What is the coefficient of x⁻² in the expansion of (2x − 1/x²)⁷?

  1. 560
  2. −560 Correct answer.
  3. −280
  4. 280

Explanation

The general term is C(7, r)(2x)^(7−r)(−x⁻²)^r = C(7, r)·2^(7−r)·(−1)^r·x^(7−3r). Setting 7 − 3r = −2 gives r = 3, so the coefficient is C(7, 3)·2⁴·(−1)³ = 35·16·(−1) = −560.

Question 35Trigonometry

Convert 150° to radian measure.

  1. 7π/6
  2. 3π/4
  3. 5π/6 Correct answer.
  4. 2π/3

Explanation

To convert degrees to radians, multiply by π/180. Thus 150° = 150π/180 = 5π/6. Equivalently, 150° is 30° short of 180°, so the radian measure is π − π/6 = 5π/6.

Question 36Trigonometry

The period of the function f(x) = cos(3x) is

  1. 6π
  2. π/3
  3. 3π
  4. 2π/3 Correct answer.

Explanation

For a function cos(kx), the period is 2π/|k| because the argument kx must increase by 2π. Here k = 3, so the period is 2π/3. The coefficient 3 compresses the usual 2π period by a factor of three.

Question 37Trigonometry

The general solution of the equation cos θ = 0 is

  1. θ = (2n+1)π/2 Correct answer.
  2. θ = nπ
  3. θ = nπ/2
  4. θ = 2nπ

Explanation

The cosine function equals zero at the odd multiples of π/2, namely π/2, 3π/2, 5π/2, and their negatives. These are exactly the values generated by (2n+1)π/2 for integer n. The expression nπ gives the zeros of sine, not cosine.

Question 38Trigonometry

The principal value of sin⁻¹(−1/2) is

  1. 5π/6
  2. −π/3
  3. −π/6 Correct answer.
  4. 7π/6

Explanation

The principal value of sin⁻¹ lies in [−π/2, π/2]. Since sin(−π/6) = −1/2 and −π/6 lies in that interval, the principal value is −π/6. Angles such as 7π/6 and 5π/6 lie outside the principal range.

Question 39Trigonometry

The general solution of 2 sin²θ − 1 = 0 is

  1. θ = 2nπ ± π/4
  2. θ = nπ ± π/3
  3. θ = nπ/2 ± π/4
  4. θ = nπ ± π/4 Correct answer.

Explanation

Rearrange to sin²θ = 1/2, so sinθ = ±1/√2. The solutions combine to θ = nπ ± π/4, where the ± sign accounts for both positive and negative values of sinθ. The term nπ supplies the period π of the squared equation.

Question 40Trigonometry

For sin 2θ ≠ 0, the expression (1 − cos 2θ)/sin 2θ simplifies to

  1. cot θ
  2. sec θ
  3. tan θ Correct answer.
  4. cos θ

Explanation

Use the double-angle identities 1 − cos 2θ = 2 sin²θ and sin 2θ = 2 sinθ cosθ. Substituting gives (2 sin²θ)/(2 sinθ cosθ) = sinθ/cosθ = tanθ. The quotient is therefore the tangent of θ, not its reciprocal.

Question 41Trigonometry

The maximum value of 3 sin θ + 4 cos θ is

  1. 7
  2. 1
  3. 25
  4. 5 Correct answer.

Explanation

For constants a and b, the maximum of a sinθ + b cosθ is √(a² + b²). Here a = 3 and b = 4, so the maximum is √(9 + 16) = 5. This value is attained when the sine and cosine terms align with the coefficient vector.

Question 42Trigonometry

In triangle ABC, the sides are a = 13, b = 14 and c = 15, where a is opposite angle A. The value of cos A is

  1. 12/13
  2. 3/5 Correct answer.
  3. 4/5
  4. 5/13

Explanation

By the cosine rule, cos A = (b² + c² − a²)/(2bc), where a is the side opposite angle A. Substituting 13, 14 and 15 gives (196 + 225 − 169)/(2·14·15) = 252/420 = 3/5. Hence angle A is acute and cos A = 3/5.

Question 43Trigonometry

The general solution of tan 2θ = cot θ is

  1. θ = (2n+1)π/6 Correct answer.
  2. θ = nπ ± π/6
  3. θ = (2n+1)π/2
  4. θ = nπ/3

Explanation

Rewrite cotθ as tan(π/2 − θ), which is valid where both functions are defined. Then tan2θ = tan(π/2 − θ) gives 2θ = nπ + π/2 − θ, so 3θ = nπ + π/2 and θ = (2n+1)π/6. The integer n records the periodicity of the tangent equation.

Question 44Trigonometry

The expression 2 tan⁻¹(1/3) is equal to

  1. tan⁻¹(1/3)
  2. tan⁻¹(3/2)
  3. tan⁻¹(3/4) Correct answer.
  4. tan⁻¹(2/3)

Explanation

Use the double-angle formula tan(2 tan⁻¹x) = 2x/(1 − x²) when |x| < 1. With x = 1/3 this becomes (2/3)/(1 − 1/9) = (2/3)/(8/9) = 3/4. Because 2 tan⁻¹(1/3) lies in (0, π/2), it equals tan⁻¹(3/4).

Question 45Trigonometry

A circular arc of radius 9 cm subtends a central angle of π/4 radian. What is the length of the arc?

  1. 9π/4 cm Correct answer.
  2. 9π/2 cm
  3. 18π cm
  4. 9π/8 cm

Explanation

For a circle, arc length is s = rθ when θ is measured in radians. Substituting r = 9 cm and θ = π/4 gives s = 9π/4 cm, so the arc length is 9π/4 cm.

Question 46Trigonometry

The exact value of sin(7π/6) is

  1. √3/2
  2. −1/2 Correct answer.
  3. −√3/2
  4. 1/2

Explanation

The angle 7π/6 lies in the third quadrant, where sine is negative. Its reference angle is π/6, and sin(π/6) = 1/2, so sin(7π/6) = −1/2.

Question 47Trigonometry

In triangle ABC, b = 8 cm, c = 6 cm and the included angle A = 30°. What is the area of the triangle?

  1. 12 cm² Correct answer.
  2. 12√3 cm²
  3. 24√3 cm²
  4. 24 cm²

Explanation

The area of a triangle with two sides and the included angle is (1/2)bc sin A. Substituting b = 8, c = 6 and A = 30° gives (1/2)(8)(6)(1/2) = 12 cm².

Question 48Trigonometry

The principal value of cos⁻¹(−1/√2) is

  1. −π/4
  2. π/4
  3. 3π/4 Correct answer.
  4. 5π/4

Explanation

The principal range of cos⁻¹ is [0, π], so the answer must lie in that interval. The angle in [0, π] whose cosine is −1/√2 is 3π/4.

Question 49Trigonometry

The general solution of tan θ = 1, where n is an integer, is

  1. θ = nπ + π/4 Correct answer.
  2. θ = nπ/2 + π/4
  3. θ = 2nπ + π/4
  4. θ = nπ − π/4

Explanation

Since tan(π/4) = 1, and tangent has period π, all solutions are given by θ = nπ + π/4. The nπ term accounts for every full period of the tangent function.

Question 50Trigonometry

For values of x for which both sides are defined, (1 + tan²x)/(1 + cot²x) simplifies to

  1. 1
  2. tan²x Correct answer.
  3. cot²x
  4. sec²x

Explanation

Use the identities 1 + tan²x = sec²x and 1 + cot²x = csc²x. Then the quotient is sec²x / csc²x = (1/cos²x)/(1/sin²x) = sin²x/cos²x = tan²x.