Explanation

For any angle θ:  

sin²θ + cos²θ = 1

Last verified on 09-05-2026

Explanation

log₁₀ 1000 = x means 10ˣ = 1000  

Since 10³ = 1000, x = 3

Last verified on 09-06-2026

Explanation

Rationalize: 1/(2+i) × (2-i)/(2-i) = (2 - i)/(4 + 1) = (2 - i)/5 = 2/5 - (1/5)i

So imaginary part = -1/5

Last verified on 09-06-2026

Explanation

For real roots, discriminant D ≥ 0  

Given: x² – 2kx + k² – 1 = 0  

Here a=1, b=-2k, c=k² - 1

D = b² − 4ac = (-2k)² − 4(1)(k² − 1)  

D = 4k² − 4k² + 4 = 4

D = 4 > 0 for every k. 

Since D is always positive, roots are real and unequal for all real k.

Last verified on 09-06-2026

Explanation

Factor the first expression:  

a³ + b³ = (a + b)(a² − ab + b²) → sum of cubes formula

Second expression: a² − ab + b²

So LCM = product of common + non-common factors  

= (a + b)(a² − ab + b²)  

= a³ + b³

Last verified on 09-06-2026

Explanation

log₃x = 4 means 3⁴ = x  

3⁴ = 3 × 3 × 3 × 3 = 81

Last verified on 09-06-2026

Explanation

Polynomial = variables with whole number powers 0,1,2,3... only. No 1/x, no variable in denominator, no √x.

x² − 3x + √2 → √2 is just a constant, powers are 2,1,0.

So x² − 3x + √2 is the correct choice.

Last verified on 09-06-2026

Explanation

x² - 5x + 6 = 0

Factor it: Find 2 numbers that multiply to 6 and add to -5 → -2 and -3

x² - 2x - 3x + 6 = 0  

x(x - 2) - 3(x - 2) = 0  

(x - 2)(x - 3) = 0

So x - 2 = 0 → x = 2  

Or x - 3 = 0 → x = 3

Last verified on 09-05-2026

Explanation

Let the 5 consecutive odd integers be:  

x, x+2, x+4, x+6, x+8

Sum = x + (x+2) + (x+4) + (x+6) + (x+8) = 625  

5x + 20 = 625  

5x = 605  

x = 121

So the integers are: 121, 123, 125, 127, 129

Largest = 129

Last verified on 09-06-2026

Explanation

Given sinθ = 3/5, 1st quadrant

Use identity: sin²θ + cos²θ = 1

(3/5)² + cos²θ = 1  

9/25 + cos²θ = 1  

cos²θ = 1 - 9/25 = 16/25  

cosθ = ±4/5

In 1st quadrant both sin and cos are positive.  

So cosθ = 4/5.

Last verified on 04-05-2026