Mathematics-I - January 2024 Exam

The dot product of two vectors is a scalar quantity that represents the projection of one vector onto the other. It is calculated by multiplying the magnitudes of the vectors and the cosine of the angle between them.

A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The roots of a quadratic equation are the values of x that satisfy the equation.

The roots of a quadratic equation can be found using the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a. The discriminant (b² - 4ac) tells us the nature of the roots.

A logarithm is the exponent to which a base must be raised to produce a given number. In the equation logₐb = c, a is the base, b is the argument, and c is the logarithm.

The value of a logarithm can be rewritten using the change of base formula: logₐb = logc b /logc a, where c is any positive number other than 1.

Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, defined as i² = -1.

The conjugate of a complex number a + bi is a - bi. Multiplying a complex number by its conjugate results in a real number.

The modulus of a complex number a + bi is the distance from the origin to the point representing the complex number in the complex plane. It is calculated as |a + bi| = √a² + b².

The cross product of two vectors is a vector that is perpendicular to both of the original vectors. Its magnitude is equal to the area of the parallelogram defined by the two vectors.

The binomial theorem provides a formula for expanding expressions of the form (a + b)ⁿ, where n is a non-negative integer. It states that (a + b)ⁿ = ∑(k=0 to n) ⁿCₖ * aⁿ⁻ᵏ * bᵏ.

A derivative is used to find the instantaneous rate of change of a function. It measures the slope of the tangent line to the curve at a particular point.

The domain of a function is the set of all possible input values (x) for which the function is defined.

The limit of a function as x approaches a particular value (c) is the value that the function approaches as x gets arbitrarily close to c. It is denoted by lim_(x→c) f(x).

An odd function is a function that satisfies the property f(-x) = -f(x) for all values of x in its domain.

An even function is a function that satisfies the property f(-x) = f(x) for all values of x in its domain.

The inverse of a function is another function that 'undoes' the original function. They are denoted by f⁻¹(x).

Integration is the process of finding the area under a curve. It is the inverse operation of differentiation.

A unit vector is a vector with a magnitude of 1. It is used to represent direction.

The projection of a vector a onto a vector b is the component of a that lies in the direction of b. It is calculated as Proj_ba = ((a · b) / |b|²) * b.

Parallel vectors have the same direction. They can be expressed as multiples of each other.

Perpendicular vectors have an angle of 90 degrees between them. Their dot product is equal to zero.

A function is called a one-to-one function if each input value has a unique output value. It passes the horizontal line test.

The trigonometric functions sine, cosine, and tangent are defined in terms of the ratios of sides in a right triangle.

The inverse trigonometric functions - arcsine (sin⁻¹), arccosine (cos⁻¹), and arctangent (tan⁻¹) - are used to find the angle corresponding to a given trigonometric ratio.

The Pythagorean identity states that sin²θ + cos²θ = 1 for any angle θ. It is a fundamental relationship in trigonometry.

The angle addition formula for tangent states that tan(A + B) = (tan A + tan B) / (1 - tan A tan B). It helps calculate the tangent of the sum of two angles.

The derivative of a constant function is always zero. This means that the slope of the tangent line to a constant function is always zero.

The power rule of differentiation states that the derivative of xⁿ is nxⁿ⁻¹. This rule is fundamental for differentiating polynomial functions.