Final_Assessment_Framework_Model_Question_Paper_Mathematics_HSSC-I

Addition, subtraction, and multiplication.

Vectors that lie in the same plane.

A consistent system has at least one solution, while an inconsistent system has no solutions.

î . (ĵ × k̂) = ĵ . (k̂ × î) = k̂ . (î × ĵ) = 1.

Gaussian elimination method.

The remainder when a number raised to a large exponent is divided by another number.

It can be applied to puzzles, domino effects, and economic forecasting.

In terms of components of the vectors.

Sine, cosine, and tangent functions.

The graph of cosθ is also periodic, oscillating between -1 and 1, with a period of 2π, and is an even function.

Factor Theorem.

It is a scalar value obtained from three vectors, representing the volume of the parallelepiped formed by the vectors.

Combinations are selections of objects where the order does not matter.

It relates the angles and sides of triangles, forming the basis for trigonometric identities.

Commutative Law, Associative Law, Null vector as identity, and Additive inverse.

A sequence that combines both arithmetic and geometric properties.

The maximum value is a + b and the minimum value is a - b.

𝒛 = 𝒂 + 𝒊𝒃 or (𝒂, 𝒃) where a and b are real numbers and 𝒊 = √−1.

By using cofactors and properties of determinants.

It is restricted to the set of natural numbers.

For optimal navigation paths in transportation and logistics, graphing 3D motion, and in engineering and computer graphics.

It can be calculated using the scalar triple product.

In graphic design, data encryption, seismic analysis, cryptography, transformation of geometric shapes, and social network analysis.

A sequence in which each term is obtained by adding a constant to the previous term.

To prove statements, identities, divisibility of numbers, and summation formulas.

By moving an object along a given vector.

Two complex numbers are equal if their real parts and imaginary parts are equal.

It is the component of one vector along the direction of another vector.

It gives a vector that is perpendicular to the plane containing the two vectors.

Permutations are arrangements of objects where the order matters.

By using the dot product.

𝒃 is the imaginary part of 𝒛.

𝒂 is the real part of 𝒛.

It is a method used to determine the total number of outcomes in a situation where there are multiple choices.

The complex conjugate is defined as 𝒛 = 𝒂 − 𝒊𝒃.

𝒊̂, 𝒋̂, and 𝒌̂.

A product that results in a scalar value, representing the product of their magnitudes and the cosine of the angle between them.

Their dot product equals zero.

The graph of tanθ is periodic with a period of π, has vertical asymptotes, and oscillates between -∞ and +∞.

It provides a formula for expanding expressions of the form (a + b)^n where n is a positive integer.

The scalar triple product of the vectors must be zero.

∑n = n(n + 1)/2.

∑n² = n(n + 1)(2n + 1)/6.

As sums or differences of sines and cosines.

By identifying the pattern in the sequence.

A position in an appropriate mathematical form.

The graph of sinθ is periodic, oscillating between -1 and 1, with a period of 2π.

They can be used to calculate distance, elevation, direction of structures, navigation, and mapping.

Polynomial regression, signal processing, and coding theory.

Using Pascal's triangle.

They can be used in cryptography, estimating lottery odds, calculating DNA sequences, or selecting songs for occasions.

Arithmetic and geometric sequences.

The domain is all real numbers, while the range depends on the specific function (e.g., sine and cosine range from -1 to 1).

A sequence in which each term is obtained by multiplying the previous term by a constant.

Using row operations.

In terms of components using determinant form.

Appropriate algebraic techniques.

Trigonometric ratios of allied angles, double angle, half angle, and triple angle identities.

Examples include simple interest on loans, investment planning, and traffic modeling.

A system used to define points in three-dimensional space using coordinates.

Matrix inversion method and Cramer's Rule.

By applying the theorem to estimate binomial expressions.

Even functions are symmetric about the y-axis (e.g., cosine), while odd functions are symmetric about the origin (e.g., sine).

Basic operations such as addition, subtraction, multiplication, and division.

By calculating the square root of the sum of the squares of its components.

∑n³ = [n(n + 1)/2]².

A method to prove statements for all natural numbers by showing a base case and an inductive step.