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SEE Optional Mathematics Nepal study materials
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SEE Optional Mathematics Nepal practice questions

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Question 1Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

For the function f(x) = 2x - 3, what is f^(-1)(5)?

  1. 4 Correct answer.
  2. 1
  3. 7
  4. 13

Explanation

Solve f(x) = 5, so 2x - 3 = 5 and x = 4. The inverse function sends the output 5 back to the input 4.

Question 2Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

A polynomial p(x) = x^3 - 4x^2 + 2x - 5 is divided by x - 2. What is the remainder?

  1. -1
  2. 1
  3. 9
  4. -9 Correct answer.

Explanation

By the remainder theorem, the remainder is p(2). Now p(2) = 8 - 16 + 4 - 5 = -9.

Question 3Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

In an arithmetic sequence with first term 3 and common difference 4, what is the 7th term?

  1. 31
  2. 24
  3. 25
  4. 27 Correct answer.

Explanation

For an arithmetic sequence, a_n = a + (n - 1)d. Thus a_7 = 3 + (7 - 1)(4) = 3 + 24 = 27.

Question 4Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

For what value of k does the quadratic equation x^2 - 6x + k = 0 have equal roots?

  1. 36
  2. 6
  3. -9
  4. 9 Correct answer.

Explanation

Equal roots occur when the discriminant b^2 - 4ac is zero. Here 36 - 4k = 0, so k = 9.

Question 5Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

If f(x) = x + 2 and g(x) = x^2, what is (g o f)(1)?

  1. 3
  2. 11
  3. 5
  4. 9 Correct answer.

Explanation

First find f(1) = 1 + 2 = 3. Then apply g, giving g(3) = 3^2 = 9.

Question 6Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

What is the sum of the first 5 terms of the geometric sequence 2, 6, 18, ...?

  1. 80
  2. 242 Correct answer.
  3. 728
  4. 486

Explanation

The first term is 2 and the common ratio is 3. Using S_n = a(r^n - 1)/(r - 1), S_5 = 2(3^5 - 1)/(3 - 1) = 242.

Question 7Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

A feasible region of a linear programming problem has vertices (0,0), (0,4), (3,2), and (5,0). What is the maximum value of P = 2x + 3y on this region?

  1. 14 Correct answer.
  2. 12
  3. 15
  4. 10

Explanation

Evaluate P at each vertex: 0, 12, 12, and 10. The greatest value is 12, attained at (0,4) and (3,2).

Question 8Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

If x - 3 is a factor of x^3 - 2x^2 + kx - 6, what is k?

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

Explanation

By the factor theorem, p(3) = 0. Substituting gives 27 - 18 + 3k - 6 = 0, so 3 + 3k = 0 and k = -1.

Question 9Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

The sum of the first n terms of an arithmetic sequence is S_n = 3n^2 + 5n. What is its 10th term?

  1. 65
  2. 62 Correct answer.
  3. 60
  4. 58

Explanation

The 10th term is S_10 - S_9. Now S_10 = 350 and S_9 = 288, so a_10 = 62. Equivalently, a_n = 6n + 2.

Question 10Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

If alpha and beta are the roots of 2x^2 - 5x + 1 = 0, what is 1/alpha + 1/beta?

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

Explanation

Use 1/alpha + 1/beta = (alpha + beta)/(alpha beta). The sum of roots is 5/2 and the product is 1/2, so the value is (5/2)/(1/2) = 5.

Question 11Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

The function f is defined by f(x) = 1/(x - 3). What is the domain of f?

  1. All real numbers greater than 3
  2. All real numbers except -3
  3. All real numbers less than 3
  4. All real numbers except 3 Correct answer.

Explanation

The denominator x - 3 must not be zero, so x = 3 is excluded from the domain. There is no square root or other restriction, so every other real number is allowed.

Question 12Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

What is the degree of the polynomial p(x) = (x^2 - 1)(x^3 + 2x)?

  1. 6
  2. 5 Correct answer.
  3. 2
  4. 3

Explanation

For a product of polynomials, the degrees add: deg((x^2 - 1)(x^3 + 2x)) = 2 + 3 = 5. Lower-degree terms do not change the highest power.

Question 13Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

Which of the following sequences is geometric?

  1. 1, 4, 9, 16, ...
  2. 2, 5, 8, 11, ...
  3. 2, 4, 8, 16, ... Correct answer.
  4. 1, 1, 2, 3, ...

Explanation

A geometric sequence has a constant ratio between consecutive terms. In 2, 4, 8, 16, ..., each term is twice the previous term, so the common ratio is 2.

Question 14Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

What is the nature of the roots of the quadratic equation 3x^2 - 2x + 1 = 0?

  1. Two distinct real roots
  2. Two non-real complex roots Correct answer.
  3. Two equal real roots
  4. One real and one non-real root

Explanation

For 3x^2 - 2x + 1, the discriminant is b^2 - 4ac = (-2)^2 - 4(3)(1) = -8. A negative discriminant means the roots form a conjugate pair of non-real complex numbers.

Question 15Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

What is the sum of the infinite geometric series 1 + 1/3 + 1/9 + 1/27 + ... ?

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

Explanation

The first term is a = 1 and the common ratio is r = 1/3. For an infinite geometric series with |r| < 1, S = a/(1 - r) = 1/(1 - 1/3) = 3/2.

Question 16Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

If p(x) = 2x^3 - 3x^2 + 4x - 1, what is p(-1)?

  1. -4
  2. -10 Correct answer.
  3. 10
  4. 2

Explanation

Substitute x = -1 carefully: p(-1) = 2(-1)^3 - 3(-1)^2 + 4(-1) - 1 = -2 - 3 - 4 - 1 = -10. Keep the signs of odd and even powers separate to avoid a sign error.

Question 17Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

What is the maximum value of z = 3x + 2y subject to x ≥ 0, y ≥ 0, x + y ≤ 6, and 2x + y ≤ 8?

  1. 16
  2. 12
  3. 14 Correct answer.
  4. 10

Explanation

The feasible region has vertices (0,0), (0,6), (2,4), and (4,0). Evaluating z = 3x + 2y gives 0, 12, 14, and 12 respectively, so the maximum value is 14.

Question 18Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

If f(x) = x^2 + 1, what is f(f(1))?

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

Explanation

First evaluate the inner function: f(1) = 1^2 + 1 = 2. Then apply f again: f(2) = 2^2 + 1 = 5.

Question 19Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

In an arithmetic sequence, the 5th term is 17 and the 9th term is 33. What is the sum of the first 12 terms?

  1. 276 Correct answer.
  2. 288
  3. 264
  4. 252

Explanation

Use a + 4d = 17 and a + 8d = 33. Subtracting gives 4d = 16, so d = 4 and a = 1. Then S_12 = 12/2[2(1) + 11(4)] = 6(46) = 276.

Question 20Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

The roots of x^2 - 5x + 6 = 0 are α and β. What is the quadratic equation whose roots are α + 1 and β + 1?

  1. x^2 - 7x + 12 = 0 Correct answer.
  2. x^2 + 7x + 12 = 0
  3. x^2 - 7x - 12 = 0
  4. x^2 - 5x + 8 = 0

Explanation

The roots of x^2 - 5x + 6 = 0 are 2 and 3, so the new roots are 3 and 4. Their sum is 7 and their product is 12, giving x^2 - 7x + 12 = 0.

Question 21Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

If f(x) = 3x + 1 and g(x) = x - 2, what is (f ∘ g)(4)?

  1. 7 Correct answer.
  2. 5
  3. 13
  4. 11

Explanation

Evaluate the inner function first: g(4) = 4 - 2 = 2. Then apply f to that result: f(2) = 3(2) + 1 = 7. Therefore (f ∘ g)(4) = 7.

Question 22Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

How many terms are there in the arithmetic sequence 4, 7, 10, ..., 40?

  1. 12
  2. 14
  3. 15
  4. 13 Correct answer.

Explanation

The first term is a = 4 and the common difference is d = 3. Using the nth-term formula, 40 = 4 + (n - 1)3, so n - 1 = 12 and n = 13. Thus the sequence has 13 terms.

Question 23Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

What is the discriminant of the quadratic equation 2x^2 - 7x + 3 = 0?

  1. -25
  2. 5
  3. 73
  4. 25 Correct answer.

Explanation

For ax^2 + bx + c = 0, the discriminant is b^2 - 4ac. Here a = 2, b = -7, and c = 3, so the discriminant is (-7)^2 - 4(2)(3) = 49 - 24 = 25.

Question 24Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

If f(x) = (2x - 1)/3, what is f^(-1)(x)?

  1. (2x + 1)/3
  2. (x + 3)/2
  3. (3x - 1)/2
  4. (3x + 1)/2 Correct answer.

Explanation

Let y = (2x - 1)/3. Swapping variables and solving gives x = (2y - 1)/3, so 3x = 2y - 1 and y = (3x + 1)/2. Hence f^(-1)(x) = (3x + 1)/2.

Question 25Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

When p(x) = 2x^3 - 5x^2 + ax - 7 is divided by x + 1, the remainder is -13. What is the value of a?

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

Explanation

By the remainder theorem, the remainder on division by x + 1 is p(-1). Compute p(-1) = -2 - 5 - a - 7 = -14 - a. Setting -14 - a = -13 gives a = -1.

Question 26Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

In a geometric sequence, the 2nd term is 6 and the 5th term is 48. What is the sum of the first 6 terms?

  1. 192
  2. 126
  3. 96
  4. 189 Correct answer.

Explanation

Let the first term be a and the common ratio be r. Then ar = 6 and ar^4 = 48, so r^3 = 8 and r = 2. This gives a = 3, and S6 = 3(2^6 - 1)/(2 - 1) = 3(63) = 189.

Question 27Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

What is the quadratic equation whose roots are 2 + √3 and 2 - √3?

  1. x^2 - 4x + 1 = 0 Correct answer.
  2. x^2 - 4x - 1 = 0
  3. x^2 + 4x - 1 = 0
  4. x^2 + 4x + 1 = 0

Explanation

The sum of the roots is (2 + √3) + (2 - √3) = 4, and the product is (2 + √3)(2 - √3) = 1. A quadratic with these roots is x^2 - (sum)x + product = 0, which gives x^2 - 4x + 1 = 0.

Question 28Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

A feasible region has vertices (1,2), (3,1), (4,4), and (0,5). What is the minimum value of z = 4x + 3y on this region?

  1. 15
  2. 28
  3. 5
  4. 10 Correct answer.

Explanation

For a linear objective function, the minimum occurs at a vertex of a bounded feasible region. The values are 4(1) + 3(2) = 10, 4(3) + 3(1) = 15, 4(4) + 3(4) = 28, and 4(0) + 3(5) = 15. Therefore the minimum value is 10.

Question 29Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

The polynomial p(x) = x^3 + ax^2 + bx + 6 has factors x - 1 and x + 2. What are the values of a and b?

  1. a = -1, b = -6
  2. a = -2, b = -5 Correct answer.
  3. a = -5, b = -2
  4. a = 2, b = -9

Explanation

By the factor theorem, p(1) = 0 and p(-2) = 0. From p(1): 1 + a + b + 6 = 0, so a + b = -7. From p(-2): -8 + 4a - 2b + 6 = 0, so 2a - b = 1. Solving these equations gives a = -2 and b = -5.

Question 30Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

A factory makes two products, A and B. Each unit of A needs 2 hours of machining and 1 hour of finishing; each unit of B needs 1 hour of machining and 2 hours of finishing. There are 100 machining hours and 80 finishing hours available. If the profit is Rs 30 per unit of A and Rs 20 per unit of B, what is the maximum profit?

  1. Rs 800
  2. Rs 1,500
  3. Rs 2,000
  4. Rs 1,600 Correct answer.

Explanation

Let x and y be the units of A and B. The constraints are 2x + y ≤ 100, x + 2y ≤ 80, and x, y ≥ 0, with profit P = 30x + 20y. The vertices are (0,0), (50,0), (0,40), and (40,20). The profits are 0, 1500, 800, and 1600, so the maximum is Rs 1,600.

Question 31Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

A function is defined by f(x) = 2x + 5. What is the value of f^{-1}(7)?

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

Explanation

To find f^{-1}(7), solve f(x) = 7: 2x + 5 = 7, so 2x = 2 and x = 1. Thus the input that produces output 7 is 1.

Question 32Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

The nth term of an arithmetic sequence is given by a_n = 4n - 3. What is the common difference of the sequence?

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

Explanation

For an arithmetic sequence written as a_n = dn + (a_1 - d), the coefficient of n is the common difference d. Here the coefficient is 4, and indeed a_2 - a_1 = 5 - 1 = 4.

Question 33Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

For the quadratic equation 2x^2 - 8x + 6 = 0, what is the sum of its roots?

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

Explanation

For a quadratic ax^2 + bx + c = 0, the sum of roots is -b/a. Here a = 2 and b = -8, so the sum is -(-8)/2 = 4.

Question 34Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

The function f(x) = x^2 - 5x + 4 has zeros at which pair of values?

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

Explanation

Factor the quadratic as x^2 - 5x + 4 = (x - 1)(x - 4). The zeros are the x-values that make these factors zero, namely 1 and 4.

Question 35Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

An arithmetic series has first term 5 and common difference 3. What is the sum of its first 10 terms?

  1. 185 Correct answer.
  2. 170
  3. 155
  4. 200

Explanation

Use S_n = n/2[2a + (n - 1)d] with n = 10, a = 5, and d = 3. This gives S_10 = 5[10 + 27] = 185.

Question 36Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

A feasible region has vertices (0, 0), (5, 0), (3, 4), and (0, 6). What is the maximum value of P = 3x + 2y on this region?

  1. 12
  2. 15
  3. 17 Correct answer.
  4. 20

Explanation

Evaluate P = 3x + 2y at every vertex: 0 at (0, 0), 15 at (5, 0), 17 at (3, 4), and 12 at (0, 6). The largest of these values is 17.

Question 37Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

A geometric sequence has first term 3 and common ratio -2. What is its 4th term?

  1. 48
  2. 24
  3. -24 Correct answer.
  4. -16

Explanation

For a geometric sequence, a_n = a r^{n-1}. With a = 3, r = -2, and n = 4, a_4 = 3(-2)^3 = 3(-8) = -24.

Question 38Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

A geometric series has first term 8 and sum to infinity 12. What is its common ratio?

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

Explanation

For an infinite geometric series with |r| < 1, S_infinity = a/(1 - r). Substituting 12 = 8/(1 - r) gives 1 - r = 2/3, so r = 1/3.

Question 39Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

The roots of 2x^2 - 7x + 3 = 0 are α and β. What is the value of 1/α + 1/β?

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

Explanation

By Vieta's formulas, α + β = 7/2 and αβ = 3/2. Then 1/α + 1/β = (α + β)/(αβ) = (7/2)/(3/2) = 7/3.

Question 40Algebra (functions, polynomials, sequences and series, quadratic equations, linear programming)

The sum of the first n terms of an arithmetic sequence is S_n = n^2 + 3n. What is the 10th term of the sequence?

  1. 22 Correct answer.
  2. 18
  3. 24
  4. 20

Explanation

Since a_n = S_n - S_{n-1}, substitute the formula: a_n = (n^2 + 3n) - [(n - 1)^2 + 3(n - 1)] = 2n + 2. For n = 10, a_10 = 22.

Question 41Trigonometry (identities, equations, heights and distances)

If sin θ = 5/13 and θ is an acute angle, what is the value of cos θ?

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

Explanation

Use the Pythagorean identity sin²θ + cos²θ = 1. For an acute angle cos θ is positive, so cos θ = √(1 − 25/169) = √(144/169) = 12/13.

Question 42Trigonometry (identities, equations, heights and distances)

For an angle A where the expressions are defined, the value of sin(90° − A) sec A is:

  1. cot A
  2. tan A
  3. 1 Correct answer.
  4. 0

Explanation

Since sin(90° − A) = cos A, the product becomes cos A · sec A. Because sec A = 1/cos A, the result is 1.

Question 43Trigonometry (identities, equations, heights and distances)

A vertical pole 10 m high casts a shadow on level ground when the sun's angle of elevation is 45°. What is the length of the shadow?

  1. 10√3 m
  2. 10√3/3 m
  3. 20 m
  4. 10 m Correct answer.

Explanation

In the right triangle formed by the pole, its shadow and the sun's ray, tan 45° = height / shadow length. Since tan 45° = 1, the shadow length equals the pole height, 10 m.

Question 44Trigonometry (identities, equations, heights and distances)

Simplify (1 + tan²θ)/(1 + cot²θ), where the expressions are defined.

  1. cot²θ
  2. sec²θ
  3. tan²θ Correct answer.
  4. 1

Explanation

Use 1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ. Then sec²θ / csc²θ = (1/cos²θ)/(1/sin²θ) = sin²θ/cos²θ = tan²θ.

Question 45Trigonometry (identities, equations, heights and distances)

The general solution of 2 cos²θ − 1 = 0 is:

  1. 360°n ± 45°
  2. 90°n ± 45°
  3. 180°n ± 45° Correct answer.
  4. 180°n ± 60°

Explanation

Rearrange to cos²θ = 1/2, so cos θ = ±1/√2. These values occur at 45°, 135°, 225° and 315°, and the pattern repeats every 180°, giving θ = 180°n ± 45°.

Question 46Trigonometry (identities, equations, heights and distances)

From a point on level ground, the angle of elevation of the top of a tower is 30°. After walking 20 m straight towards the tower, the angle of elevation becomes 60°. Find the height of the tower.

  1. 10 m
  2. 20 m
  3. 20√3 m
  4. 10√3 m Correct answer.

Explanation

Let h be the height and x the original horizontal distance. Then h = x tan30° = x/√3 and h = (x − 20)tan60° = √3(x − 20). Equating gives x = 3(x − 20), so x = 30 m and h = 30/√3 = 10√3 m.

Question 47Trigonometry (identities, equations, heights and distances)

For an angle θ where the expressions are defined, (1 − cos θ)/(1 + cos θ) is equal to:

  1. (sec θ − tan θ)²
  2. (csc θ − cot θ)² Correct answer.
  3. (csc θ + cot θ)²
  4. (1 + cos θ)²/sin²θ

Explanation

Multiply numerator and denominator by 1 − cos θ: (1 − cos θ)²/(1 − cos²θ) = (1 − cos θ)²/sin²θ. This is ((1 − cos θ)/sin θ)² = (csc θ − cot θ)², since csc θ − cot θ = (1 − cos θ)/sin θ.

Question 48Trigonometry (identities, equations, heights and distances)

If tan θ + cot θ = 2, what is the value of sin θ cos θ?

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

Explanation

Write tan θ + cot θ = sin θ/cos θ + cos θ/sin θ = (sin²θ + cos²θ)/(sin θ cos θ) = 1/(sin θ cos θ). Hence 1/(sin θ cos θ) = 2, so sin θ cos θ = 1/2.

Question 49Trigonometry (identities, equations, heights and distances)

From the top of a 60 m high building, a man observes two cars on opposite sides of the building. The angles of depression of the cars are 30° and 45°. Find the distance between the cars.

  1. 120 m
  2. 60(√3 − 1) m
  3. 60(√3 + 1) m Correct answer.
  4. 60√3 m

Explanation

Each car is on the opposite side, so the horizontal distances from the building are 60 cot30° = 60√3 m and 60 cot45° = 60 m. Because the cars lie on opposite sides, add the distances: 60√3 + 60 = 60(√3 + 1) m.

Question 50Trigonometry (identities, equations, heights and distances)

What is the exact value of cos 20° cos 40° cos 80°?

  1. 1/8 Correct answer.
  2. 1/2
  3. √3/8
  4. 1/4

Explanation

Use the identity cos A cos B = [cos(A + B) + cos(A − B)]/2, or multiply by 8 sin20° and repeatedly use 2 sin x cos x = sin 2x. The product reduces to sin160°/(8 sin20°) = 1/8 because sin160° = sin20°.