DCA6107 FUNDAMENTALS OF MATHEMATICS

Description

SESSION	JAN-FEB 2026
PROGRAM	MASTER OF COMPUTER APPLICATIONS (MCA)
SEMESTER	I
COURSE CODE & NAME	DCA6107 FUNDAMENTALS OF MATHEMATICS

 

 

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 function. Does not rely on any short cut rules

 

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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 derivative of the function we are integrating. Any constant is null upon differentiation so appears as an add-on.

Integration can be done in a number of different ways, depending on the function. Simple functions have the standard results applied directly. Trigonometric functions are used with well known

 

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, the other direction is kept the sam

 

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 one of three coordinate directions, known as the x-axis, y-axis and z-axis. Specific names are given to the unit vectors along these

 

 

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.

Standard Angles in Trigonometry are those values of angles where the values of all the 6 trigonometric functions are known and fixed. This is

 

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 expressed as a sum of a real and an imaginary part.

The modulus of a complex number is a number that represents the distance of the complex number from