Set – I

Q.1. Find the derivative of f(x) = x² + 2x using the first principle.

Ans 1.

One of the significant concepts in calculus is differentiation. It is used to determine the change of a function if the value of the function input changes. The practical use of derivatives of a function is to find the rate of change at a particular point. This concept is commonly applied in physics, engineering, economics and computer science.

The first principle of differentiation is the most basic method to find the derivative of a

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Q.2. Evaluate ∫ (1 + cos 2x + log x) dx.

Ans 2.

Integration is the opposite of differentiation. Differentiation is finding the rate of change of a function, and integration is finding the total accumulated value of a function. In other words, when we integrate a function we are seeking to find all possible original functions that have a

Q.3. Find ∂²f/∂x² and ∂²f/∂y² for f(x, y) = e^(xy) + x² + y².

Ans 3.

Partial differentiation is a technique to apply if a function is dependent on more than one independent variable. Ordinary differentiation only has one unknown, and it is easy. However, in partial differentiation the function has two or more than two variables, and we differentiate with respect to one variable at a time with other variables constant.

This concept is naturally occurring in real-life problems where there are several inputs that determine the output. For instance, the temperature within a room is dependent upon its horizontal and vertical location. When studying how temperature varies along one direction,

Set – II

Q.4. Given r₁ = 3i + j – 3k,  r₂ = i + 2j + 3k,  r₃ = 2i – j – 3k, find the magnitude of (i) r₃  (ii) r₁ + r₂ + r₃  (iii) 2r₁ – 3r₂ + 5r₃.

Ans 4.

Vectors are mathematical objects that have two properties: magnitude and direction. The magnitude of a vector is the length or size of the vector. Direction is the direction of the vector. This is what distinguishes a vector from a regular number (a scalar) that has only magnitude. Forces, velocities, displacements and accelerations are all represented by vectors.

The vector is written in 3-D space with three components. Each of these components is along

Q.5.  Find the value of  (tan² 60 – 2 tan² 45 + sec² 30) / (2 sin² 45 · sin 90 + cos² 60 · cos² 30)  (All angles in degrees)

Ans 5.

Trigonometry is a subject that deals with the angles and sides of triangles. It is used in a variety of fields such as engineering, navigation, architecture and physics. Trigonometry is all about six functions: sine, cosine, tangents, secants, cotangents, and cosecants. These functions are related to an angle in a right triangle and the ratio of two of its sides.

Q.6.  If x + iy = √((a + ib) / (c + id)), prove that (x² + y²)² = (a² + b²) / (c² + d²)

Ans 6.

Complex numbers can be thought of as an extension of the real number system. Some equations have no solution that can be represented on a number line, although a real number can represent any point on a number line. The problem is solved by complex numbers by introducing an imaginary number, the square root of negative one. A complex number is