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IOE Entrance Nepal study materials
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IOE Entrance Nepal practice questions

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Question 1Mathematics

What is the value of sin 60° + cos 30°?

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

Explanation

sin 60° = √3/2 and cos 30° = √3/2, so the sum is √3. Both angles are complementary, which is why the two values coincide.

Question 2Mathematics

The derivative of f(x) = x³ − 3x with respect to x is:

  1. 3x² − 3 Correct answer.
  2. x² − 3
  3. 3x² − 3x
  4. 3x − 3

Explanation

Apply the power rule term by term: d/dx(x³) = 3x² and d/dx(3x) = 3. Subtracting gives 3x² − 3.

Question 3Mathematics

If log₁₀ 2 = 0.3010, what is log₁₀ 8?

  1. 0.6020
  2. 2.4080
  3. 0.9030 Correct answer.
  4. 0.1505

Explanation

Since 8 = 2³, log₁₀ 8 = 3 log₁₀ 2 = 3 × 0.3010 = 0.9030. The power rule for logarithms converts the exponent into a multiplier.

Question 4Mathematics

The sum of the first 20 terms of the arithmetic sequence 3, 7, 11, 15, … is:

  1. 760
  2. 840
  3. 800
  4. 820 Correct answer.

Explanation

Here a = 3 and d = 4, so the 20th term is 3 + 19×4 = 79. Using S = (n/2)(a + l) gives 10 × (3 + 79) = 820.

Question 5Mathematics

What is the modulus of the complex number 3 − 4i?

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

Explanation

The modulus is √(3² + (−4)²) = √(9 + 16) = √25 = 5. It measures the distance of the point from the origin in the Argand plane.

Question 6Mathematics

The value of the determinant of the matrix with rows (2, 3) and (1, 4) is:

  1. −5
  2. 8
  3. 11
  4. 5 Correct answer.

Explanation

For a 2×2 matrix the determinant is ad − bc = (2)(4) − (3)(1) = 8 − 3 = 5. Order matters because the off-diagonal product is subtracted.

Question 7Mathematics

A fair die is rolled once. What is the probability of obtaining a prime number?

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

Explanation

The prime outcomes on a die are 2, 3 and 5, giving 3 favourable results out of 6. Thus the probability is 3/6 = 1/2.

Question 8Mathematics

The slope of the line passing through the points (1, 2) and (4, 11) is:

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

Explanation

Slope = (y₂ − y₁)/(x₂ − x₁) = (11 − 2)/(4 − 1) = 9/3 = 3. The rise is divided by the run between the two points.

Question 9Mathematics

What is the value of the integral of 2x with respect to x from 0 to 3?

  1. 18
  2. 9 Correct answer.
  3. 3
  4. 6

Explanation

The antiderivative of 2x is x². Evaluating from 0 to 3 gives 3² − 0² = 9, which is the area under the line y = 2x over that interval.

Question 10Mathematics

If the roots of x² − 5x + k = 0 are equal, then k equals:

  1. 25
  2. 5
  3. −25/4
  4. 25/4 Correct answer.

Explanation

Equal roots require the discriminant b² − 4ac to vanish: 25 − 4k = 0, so k = 25/4. This makes the quadratic a perfect square.

Question 11Mathematics

In how many ways can the letters of the word LEVEL be arranged?

  1. 30 Correct answer.
  2. 120
  3. 60
  4. 20

Explanation

The word has 5 letters with L repeated twice and E repeated twice. The number of distinct arrangements is 5!/(2!·2!) = 120/4 = 30.

Question 12Mathematics

The value of tan(45° + 30°) is:

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

Explanation

Using tan(A + B) = (tan A + tan B)/(1 − tan A tan B) with tan 45° = 1 and tan 30° = 1/√3 gives (1 + 1/√3)/(1 − 1/√3), which simplifies to 2 + √3.

Question 13Mathematics

The centre of the circle x² + y² − 6x + 4y − 12 = 0 is:

  1. (−6, 4)
  2. (3, −2) Correct answer.
  3. (6, −4)
  4. (−3, 2)

Explanation

For x² + y² + 2gx + 2fy + c = 0 the centre is (−g, −f). Here 2g = −6 and 2f = 4, so g = −3 and f = 2, giving centre (3, −2).

Question 14Mathematics

If a matrix A is of order 2×3 and B is of order 3×4, the order of the product AB is:

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

Explanation

Matrix multiplication is defined when the number of columns of A equals the number of rows of B, which holds here (3 = 3). The product then takes the rows of A and the columns of B, giving order 2×4.

Question 15Mathematics

The mean of five observations is 12. If one observation equal to 20 is removed, the mean of the remaining observations is:

  1. 11
  2. 10 Correct answer.
  3. 12
  4. 8

Explanation

The total of all five observations is 5 × 12 = 60. Removing 20 leaves 40 spread over four observations, so the new mean is 40/4 = 10.

Question 16Mathematics

The general solution of the differential equation dy/dx = 2xy is:

  1. y = x² + C
  2. y = C e^(2x)
  3. y = C e^(x²) Correct answer.
  4. y = C x²

Explanation

Separating variables gives dy/y = 2x dx. Integrating both sides yields ln y = x² + C, so y = C e^(x²) after exponentiating.

Question 17Mathematics

If the vectors a = 2i + 3j and b = 4i − j are given, their dot product a·b is:

  1. 11
  2. 8
  3. −3
  4. 5 Correct answer.

Explanation

The dot product multiplies corresponding components and adds: (2)(4) + (3)(−1) = 8 − 3 = 5. Only matching i and j components combine.

Question 18Mathematics

The coefficient of x³ in the expansion of (1 + x)⁶ is:

  1. 6
  2. 20 Correct answer.
  3. 15
  4. 120

Explanation

By the binomial theorem the coefficient of x³ is C(6,3) = 6!/(3!·3!) = 20. This counts the ways of choosing three x factors from six.

Question 19Mathematics

The function f(x) = x³ − 3x has a local maximum at x equal to:

  1. 0
  2. 1
  3. 3
  4. −1 Correct answer.

Explanation

Setting f′(x) = 3x² − 3 = 0 gives x = ±1. The second derivative 6x is negative at x = −1, confirming a local maximum there.

Question 20Mathematics

The value of the limit of (sin 3x)/x as x approaches 0 is:

  1. 1/3
  2. 3 Correct answer.
  3. 0
  4. 1

Explanation

Using the standard limit sin θ/θ → 1 as θ → 0, write (sin 3x)/x = 3·(sin 3x)/(3x), which tends to 3 × 1 = 3.

Question 21Mathematics

If z = 1 + i, then z⁴ equals:

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

Explanation

Write z in polar form: 1 + i = √2 (cos 45° + i sin 45°). By De Moivre's theorem z⁴ = (√2)⁴ (cos 180° + i sin 180°) = 4(−1) = −4.

Question 22Mathematics

The sum of the infinite geometric series 1 + 1/3 + 1/9 + 1/27 + … is:

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

Explanation

Here a = 1 and r = 1/3, and since |r| < 1 the sum is a/(1 − r) = 1/(1 − 1/3) = 3/2. Convergence is guaranteed because the ratio lies between −1 and 1.

Question 23Mathematics

A committee of 3 is chosen from 5 men and 4 women. The probability that the committee contains exactly 2 women is:

  1. 5/21
  2. 10/21 Correct answer.
  3. 1/2
  4. 4/21

Explanation

Total committees are C(9,3) = 84. Favourable ones choose 2 women from 4 and 1 man from 5: C(4,2)·C(5,1) = 6 × 5 = 30. Thus the probability is 30/84 = 10/21.

Question 24Mathematics

The equation of the tangent to the circle x² + y² = 25 at the point (3, 4) is:

  1. 3x − 4y = 25
  2. 4x + 3y = 25
  3. x + y = 7
  4. 3x + 4y = 25 Correct answer.

Explanation

For x² + y² = r² the tangent at (x₁, y₁) is xx₁ + yy₁ = r². Substituting (3, 4) and r² = 25 gives 3x + 4y = 25.

Question 25Mathematics

The maximum value of 3 sin x + 4 cos x as x varies is:

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

Explanation

An expression a sin x + b cos x has maximum √(a² + b²). Here that is √(9 + 16) = 5, attained when the phase aligns with the coefficients.

Question 26Mathematics

If A = {1, 2, 3, 4} and B = {3, 4, 5}, what is the number of elements in the symmetric difference A Δ B?

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

Explanation

The symmetric difference A Δ B contains elements that lie in exactly one of the two sets. Removing the common elements 3 and 4 leaves {1, 2} from A and {5} from B, giving three elements.

Question 27Mathematics

The argument (principal value) of the complex number −1 + i√3 is:

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

Explanation

The point (−1, √3) lies in the second quadrant, so the argument is π minus arctan(√3/1) = π − π/3. This gives 2π/3 as the principal argument.

Question 28Mathematics

A geometric progression has first term 5 and common ratio 2. What is the sum of its first 6 terms?

  1. 310
  2. 155
  3. 315 Correct answer.
  4. 630

Explanation

The sum of n terms is a(rⁿ − 1)/(r − 1). Here 5(2⁶ − 1)/(2 − 1) = 5 × 63 = 315.

Question 29Mathematics

If the matrix A = [[3, 1], [2, 4]] has determinant 10, what is the determinant of the matrix 2A?

  1. 40 Correct answer.
  2. 80
  3. 10
  4. 20

Explanation

For an n × n matrix, det(kA) = kⁿ det(A). With n = 2 and k = 2, the determinant becomes 2² × 10 = 40.

Question 30Mathematics

The line 3x − 4y + 7 = 0 is reflected in the x-axis. What is the equation of the image line?

  1. 3x + 4y + 7 = 0 Correct answer.
  2. −3x + 4y + 7 = 0
  3. 3x + 4y − 7 = 0
  4. 3x − 4y − 7 = 0

Explanation

Reflection in the x-axis replaces y with −y. Substituting gives 3x − 4(−y) + 7 = 0, that is, 3x + 4y + 7 = 0.

Question 31Mathematics

What is the equation of the directrix of the parabola y² = 12x?

  1. x = −3 Correct answer.
  2. x = −12
  3. y = −3
  4. x = 3

Explanation

For y² = 4ax, here 4a = 12 so a = 3. The directrix of a right-opening parabola is x = −a, giving x = −3.

Question 32Mathematics

If sin A = 3/5 and A lies in the second quadrant, what is the value of cos A?

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

Explanation

In the second quadrant cosine is negative, and cos²A = 1 − 9/25 = 16/25. Hence cos A = −4/5.

Question 33Mathematics

What is the derivative of x² sin x with respect to x?

  1. 2x sin x − x² cos x
  2. x² cos x + 2x sin x Correct answer.
  3. x² sin x + 2x cos x
  4. 2x cos x

Explanation

The product rule gives d(x²)/dx · sin x + x² · d(sin x)/dx. This equals 2x sin x + x² cos x.

Question 34Mathematics

A bag contains 4 red and 6 blue balls. Two balls are drawn one after another without replacement. What is the probability that both are red?

  1. 2/15 Correct answer.
  2. 4/25
  3. 1/5
  4. 8/45

Explanation

The probability of a red ball first is 4/10, and then 3/9 for the second draw. Their product is (4/10)(3/9) = 12/90 = 2/15.

Question 35Mathematics

The variance of the data set 2, 4, 6, 8, 10 is:

  1. 8 Correct answer.
  2. 6
  3. 4
  4. 10

Explanation

The mean is 6. The squared deviations are 16, 4, 0, 4, 16, whose sum is 40. Dividing by 5 gives variance 8.

Question 36Mathematics

The value of the integral of x·eˣ with respect to x from 0 to 1 is:

  1. e + 1
  2. 1 Correct answer.
  3. e
  4. e − 1

Explanation

Integrating by parts, ∫x eˣ dx = x eˣ − eˣ. Evaluating from 0 to 1 gives (e − e) − (0 − 1) = 1.

Question 37Mathematics

The solution of the differential equation dy/dx = (x + y)/x, with y = 1 when x = 1, is:

  1. y = x ln x + x Correct answer.
  2. y = ln x + 1
  3. y = x ln x
  4. y = x² + 1

Explanation

Rewriting gives dy/dx − y/x = 1, a linear equation with integrating factor 1/x. This yields y/x = ln x + C, and the condition y(1) = 1 gives C = 1, so y = x ln x + x.

Question 38Mathematics

If the vectors a = i + 2j + 2k and b = 2i − j + 2k are given, what is the angle between them?

  1. π/6
  2. arccos(4/9) Correct answer.
  3. π/4
  4. π/2

Explanation

The dot product is 2 − 2 + 4 = 4, and each vector has magnitude 3. Using a · b = |a||b| cos θ gives cos θ = 4/9. Therefore θ = arccos(4/9), approximately 63.61°.

Question 39Mathematics

The function f(x) = x⁴ − 4x³ has an inflection point at x equal to:

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

Explanation

The second derivative is f″(x) = 12x² − 24x = 12x(x − 2). It changes sign at x = 2, where the concavity switches, so this is the inflection point.

Question 40Mathematics

What is the coefficient of x⁵ in the expansion of (2 + x)⁷?

  1. 21
  2. 168
  3. 42
  4. 84 Correct answer.

Explanation

The general term is C(7, k) · 2^(7−k) · xᵏ. For x⁵, k = 5, giving C(7, 5) · 2² = 21 × 4 = 84. The power of 2 comes from the two factors that contribute constants.

Question 41Physics

A body moves along a straight line with uniform acceleration. Which quantity-time graph is a straight line with non-zero slope?

  1. Velocity versus time Correct answer.
  2. Velocity versus displacement
  3. Acceleration versus displacement
  4. Displacement versus time

Explanation

For uniform acceleration, velocity changes at a constant rate, so v = u + at is linear in time and the v-t graph is a straight line with slope equal to the acceleration. Displacement versus time is a parabola, not a straight line.

Question 42Physics

A stone is dropped from rest and falls freely. Taking g = 10 m/s², how far does it fall in the first 3 seconds?

  1. 90 m
  2. 45 m Correct answer.
  3. 15 m
  4. 30 m

Explanation

Using s = ut + ½gt² with u = 0 gives s = ½ × 10 × 3² = 45 m. The distance grows with the square of time, so tripling the time multiplies the fall by nine relative to one second.

Question 43Physics

A force of 20 N acts on a 4 kg body initially at rest on a frictionless surface. What is the body's speed after 5 s?

  1. 20 m/s
  2. 100 m/s
  3. 25 m/s Correct answer.
  4. 5 m/s

Explanation

Newton's second law gives a = F/m = 20/4 = 5 m/s². Then v = u + at = 0 + 5 × 5 = 25 m/s.

Question 44Physics

Two bodies of masses m and 2m moving with equal kinetic energies have speeds in the ratio:

  1. 1 : √2
  2. √2 : 1 Correct answer.
  3. 1 : 2
  4. 2 : 1

Explanation

Kinetic energy is ½mv², so for equal KE, v ∝ 1/√m. The lighter body has mass ratio 1:2, so its speed ratio to the heavier is √2:1.

Question 45Physics

A bullet of mass 20 g moving at 300 m/s embeds in a 1.98 kg block at rest on a frictionless surface. The speed of the combined mass is about:

  1. 30 m/s
  2. 6 m/s
  3. 3 m/s Correct answer.
  4. 1.5 m/s

Explanation

Momentum is conserved: 0.02 × 300 = 6 kg·m/s. The combined mass is 2.00 kg, so v = 6/2 = 3 m/s. Kinetic energy is not conserved in this perfectly inelastic collision.

Question 46Physics

A wheel starting from rest reaches 20 rad/s in 4 s with uniform angular acceleration. The angular acceleration is:

  1. 80 rad/s²
  2. 4 rad/s²
  3. 5 rad/s² Correct answer.
  4. 0.2 rad/s²

Explanation

Angular acceleration is the change in angular velocity divided by time: α = (20 − 0)/4 = 5 rad/s². This parallels linear kinematics with ω replacing v.

Question 47Physics

A satellite orbits Earth in a circular orbit. If the orbital radius is increased four times, the orbital speed becomes:

  1. Unchanged
  2. One quarter of the original
  3. Twice the original
  4. Half the original Correct answer.

Explanation

For a circular orbit, gravitational force provides centripetal force: GMm/r² = mv²/r, so v = √(GM/r), meaning v ∝ 1/√r. Increasing r four times reduces v by a factor of two.

Question 48Physics

The escape velocity from Earth's surface is about 11.2 km/s. The escape velocity from a planet with the same density but twice Earth's radius would be:

  1. 5.6 km/s
  2. 11.2 km/s
  3. 44.8 km/s
  4. 22.4 km/s Correct answer.

Explanation

Escape velocity is v = √(2GM/R). With equal density, M ∝ R³, so v ∝ R. Doubling the radius therefore doubles the escape velocity to 22.4 km/s.

Question 49Physics

A simple pendulum has period T on Earth. On a planet where g is one quarter of Earth's value, its period becomes:

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

Explanation

The period of a simple pendulum is T = 2π√(L/g), so T ∝ 1/√g. Reducing g to one quarter multiplies the period by √4 = 2.

Question 50Physics

A wave has frequency 50 Hz and wavelength 4 m. Its speed and the time period are respectively:

  1. 12.5 m/s and 0.02 s
  2. 200 m/s and 0.02 s Correct answer.
  3. 50 m/s and 4 s
  4. 200 m/s and 0.2 s

Explanation

Wave speed is v = fλ = 50 × 4 = 200 m/s. The time period is T = 1/f = 1/50 = 0.02 s.