Engineering & Architecture Admissions

JEE Main

Q. What is the value of (1 + i)^2?

A. 2i
B. 2
C. 0
D. 1 + 2i

Solution

(1 + i)^2 = 1^2 + 2*1*i + i^2 = 1 + 2i - 1 = 2i.

Correct Answer: A — 2i

Q. What is the value of 1 astronomical unit (AU) in meters?

A. 1.496 × 10^11 m
B. 1.496 × 10^9 m
C. 1.496 × 10^12 m
D. 1.496 × 10^10 m

Solution

1 astronomical unit (AU) is approximately equal to 1.496 × 10^11 meters, which is the average distance from the Earth to the Sun.

Correct Answer: A — 1.496 × 10^11 m

Q. What is the value of 1 Joule in terms of base SI units?

A. 1 kg·m²/s²
B. 1 kg·m/s²
C. 1 m²/s²
D. 1 kg·s²/m

Solution

1 Joule is defined as 1 kg·m²/s², which is the product of mass, distance squared, and time squared.

Correct Answer: A — 1 kg·m²/s²

Q. What is the value of 1 light year in meters?

A. 9.46 × 10^15 m
B. 3.00 × 10^8 m
C. 1.08 × 10^11 m
D. 6.37 × 10^6 m

Solution

1 light year is the distance light travels in one year, which is approximately 9.46 × 10^15 meters.

Correct Answer: A — 9.46 × 10^15 m

Q. What is the value of 2sin(θ)cos(θ)?

A. sin(2θ)
B. cos(2θ)
C. tan(θ)
D. sec(θ)

Solution

By the double angle identity, 2sin(θ)cos(θ) = sin(2θ).

Correct Answer: A — sin(2θ)

Q. What is the value of 5C2?

A. 10
B. 20
C. 15
D. 5

Solution

5C2 = 5! / (2!(5-2)!) = 10.

Correct Answer: A — 10

Q. What is the value of cos(0°)?

A. 0
B. 1
C. -1
D. undefined

Solution

cos(0°) = 1.

Correct Answer: B — 1

Q. What is the value of cos(60°)?

A. 0
B. 1/2
C. √3/2
D. 1

Solution

cos(60°) = 1/2.

Correct Answer: B — 1/2

Q. What is the value of cos(90°)?

A. 0
B. 1
C. -1
D. undefined

Solution

cos(90°) = 0.

Correct Answer: A — 0

Q. What is the value of cos(π/3)?

A. 0
B. 1/2
C. 1
D. √3/2

Solution

cos(π/3) = 1/2.

Correct Answer: B — 1/2

Q. What is the value of cos^(-1)(-1)?

A. 0
B. π
C. π/2
D. 2π

Solution

cos^(-1)(-1) = π, since cos(π) = -1.

Correct Answer: B — π

Q. What is the value of cos^(-1)(0)?

A. 0
B. π/2
C. π
D. 1

Solution

cos^(-1)(0) corresponds to the angle where the cosine value is 0, which is π.

Correct Answer: C — π

Q. What is the value of gravitational acceleration (g) at the surface of the Earth?

A. 9.8 m/s²
B. 10 m/s²
C. 9.81 m/s²
D. 8.9 m/s²

Solution

The standard value of gravitational acceleration at the Earth's surface is approximately 9.81 m/s².

Correct Answer: C — 9.81 m/s²

Q. What is the value of k for which the equation x^2 + kx + 9 = 0 has no real roots?

A. k < 6
B. k > 6
C. k = 6
D. k <= 6

Solution

The discriminant must be negative: k^2 - 4*1*9 < 0 => k^2 < 36 => |k| < 6, hence k > 6.

Correct Answer: B — k > 6

Q. What is the value of k for which the function f(x) = { kx + 2, x < 2; x^2 - 4, x >= 2 is continuous at x = 2?

A. 0
B. 1
C. 2
D. 3

Solution

Setting 2k + 2 = 0 gives k = 2.

Correct Answer: C — 2

Q. What is the value of k if f(x) = kx^2 + 2x + 1 has a minimum value of -3?

A. -1
B. -2
C. -3
D. -4

Solution

The minimum value occurs at x = -b/(2a) = -2/(2k). Setting f(-1) = -3 gives k = -2.

Correct Answer: B — -2

Q. What is the value of k if the equation x^2 + kx + 16 = 0 has no real roots?

A. -8
B. -4
C. 4
D. 8

Solution

For no real roots, the discriminant must be less than zero: k^2 - 4*1*16 < 0 => k^2 < 64 => |k| < 8.

Correct Answer: B — -4

Q. What is the value of k if the quadratic equation x^2 + kx + 16 = 0 has equal roots?

A. -8
B. -4
C. 4
D. 8

Solution

For equal roots, the discriminant must be zero: k^2 - 4*1*16 = 0, thus k^2 = 64, giving k = -8 or k = 8. The answer is -8.

Correct Answer: B — -4

Q. What is the value of k if the quadratic equation x^2 + kx + 16 = 0 has no real roots?

A. -8
B. -4
C. 4
D. 8

Solution

The discriminant must be less than zero: k^2 - 4*1*16 < 0 => k^2 < 64 => k < 8 and k > -8.

Correct Answer: B — -4

Q. What is the value of k if the quadratic equation x^2 + kx + 25 = 0 has one real root?

A. -10
B. -5
C. 5
D. 10

Solution

For one real root, the discriminant must be zero: k^2 - 4*1*25 = 0, thus k^2 = 100, giving k = -10 or k = 10.

Correct Answer: A — -10

Q. What is the value of k if the quadratic equation x^2 + kx + 9 = 0 has no real roots?

A. k < 6
B. k > 6
C. k = 6
D. k < 0

Solution

For no real roots, the discriminant must be less than zero: k^2 - 4*1*9 < 0, thus k > 6.

Correct Answer: B — k > 6

Q. What is the value of log2(8)?

A. 2
B. 3
C. 4
D. 1

Solution

log2(8) = 3 because 2^3 = 8.

Correct Answer: B — 3

Q. What is the value of log_10(1000) + log_10(0.01)?

A. 1
B. 0
C. -1
D. 2

Solution

log_10(1000) = 3 and log_10(0.01) = -2, thus 3 - 2 = 1.

Correct Answer: C — -1

Q. What is the value of log_10(1000)?

A. 1
B. 2
C. 3
D. 4

Solution

log_10(1000) = log_10(10^3) = 3.

Correct Answer: C — 3

Q. What is the value of log_2(32) - log_2(4)?

A. 1
B. 2
C. 3
D. 4

Solution

log_2(32) = 5 and log_2(4) = 2. Therefore, 5 - 2 = 3.

Correct Answer: C — 3

Q. What is the value of log_2(32) - log_2(8)?

A. 1
B. 2
C. 3
D. 4

Solution

log_2(32) = 5 and log_2(8) = 3. Therefore, 5 - 3 = 2.

Correct Answer: C — 3

Q. What is the value of log_3(27) - log_3(9)?

A. 0
B. 1
C. 2
D. 3

Solution

log_3(27) = 3 and log_3(9) = 2. Therefore, 3 - 2 = 1.

Correct Answer: B — 1

Q. What is the value of log_3(81)?

A. 2
B. 3
C. 4
D. 5

Solution

log_3(81) = log_3(3^4) = 4.

Correct Answer: C — 4

Q. What is the value of log_5(125)?

A. 2
B. 3
C. 4
D. 5

Solution

log_5(125) = log_5(5^3) = 3.

Correct Answer: B — 3

Q. What is the value of log_5(25) - log_5(5)?

A. 1
B. 0
C. -1
D. 2

Solution

log_5(25) = 2 and log_5(5) = 1. Therefore, 2 - 1 = 1.

Correct Answer: A — 1

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