physics-Basic Mathematics(Basic-L-2)

Which of the following is the correct option?

• tan²Θ = sec²Θ + 1

• cosec²Θ = 1-cot²Θ

• cos²Θ = 1+sin²Θ

Answer explanation

The correct option is secΘ-tanΘ = 1/(secΘ+tanΘ). This choice matches the given correct answer.

Which of the following is the incorrect option?

• sin²Θ + cos²Θ = 1

• sin²Θ = 1-cos²Θ

• cos²Θ = 1+sin²Θ

• sin²Θ - cosΘ = 1

Answer explanation

The correct option is sin^2Θ + cos^2Θ = 1, which is a fundamental trigonometric identity known as the Pythagorean identity.

Which of the following is the correct option?

• cosec²Θ = 1+cot²Θ

• cosec²Θ = 1-cot²Θ

• cot²Θ = 1+cosec²Θ

• sec²Θ = tan²Θ - 1

Answer explanation

The correct option is cosec^2Θ = 1+cot^2Θ because it is a trigonometric identity that relates the cosecant and cotangent functions.

Which of the following is the incorrect option?

Answer explanation

The incorrect option is cosC+cosD = 2cos(\frac{C+D}{2})sin(\frac{C-D}{2}) because it should be cosC+cosD = 2cos(\frac{C+D}{2})cos(\frac{C-D}{2})

Calculate (1001)^1/3 using binomial approximation

• 8.164444

• 9.128485

• 10.003333

• 11.123333

Answer explanation

Using binomial approximation, (1001)^1/3 is approximated as 1000^(1/3) + (1/3)*1000^(-2/3) = 10 + 1/3 = 10.003333. Therefore, the correct answer is 10.003333.

• 2ax - b

• 3ax + b

• 2ax + b

• ax² + b

Answer explanation

The derivative of y = ax^2 + bx + c with respect to x is 2ax + b. The derivative of ax^2 is 2ax, and the derivative of bx is b.

• 25x⁴

• 42x⁴

• 7x⁴

• 35x⁴

Answer explanation

To find dx/dy of y = 7x^5, differentiate with respect to x: dy/dx = 35x^4. Then, to find dy/dx, multiply by dx/dy = 1/(dy/dx) = 1/(35x^4) = 35x^4.

Differentiate the following wrt x, y = 4x³+3x²+x+7

• dy/dx = 12x² + 6x + 1

• dy/dx = 14x² + 8x + 1

• dy/dx = 8x² + 2x + 1

• dy/dx = 10x² + 4x + 1

Answer explanation

To differentiate y = 4x^3 + 3x^2 + x + 7 with respect to x, apply the power rule to each term. The derivative is dy/dx = 12x^2 + 6x + 1, which matches the correct choice.

• tanx

• -sinx

• cscx

• secx

Answer explanation

The derivative of cosx is -sinx. Therefore, the correct answer is -sinx.

• cosec²x

• cosecx.cotx

• cosecx.cosecx

• -cosecx .cotx

Answer explanation

The correct derivative of y = cosecx is -cosecx.cotx. The derivative of cosecx is -cosecx.cotx using the chain rule.