NED University Mathematics Past Papers (2020-2024) Review

A comprehensive collection of 100 flashcards based on NED University Mathematics past paper transcripts (2020-2024), covering calculus, algebra, trigonometry, coordinate geometry, and vectors.

Symmetric Matrix

A square matrix $A$ such that it is equal to its transpose ($A = A^{t}$).

Skew-symmetric Matrix

A square matrix $A$ such that its transpose is equal to its negative ($A^{t} = -A$).

Idempotent Matrix

A square matrix $A$ that satisfies the condition $A^{2} = A$.

Involutory Matrix

A square matrix $A$ that satisfies the condition $A^{2} = I$, where $I$ is the identity matrix.

Identity Matrix

A diagonal matrix in which all the diagonal elements are equal to $1$; also referred to as a unit matrix.

Scalar Matrix

A diagonal matrix in which all diagonal elements are the same constant value.

Singular Matrix

A square matrix whose determinant is equal to zero ($|A| = 0$).

Periodic Matrix

A square matrix $A$ such that $A^{k+1} = A$, where $k$ is a positive integer.

Nilpotent Matrix

A square matrix $A$ for which there exists a positive integer $n$ such that $A^{n} = \text{O}$, where $\text{O}$ is the zero matrix.

Hermitian Matrix

A square matrix that is equal to its own conjugate transpose.

Determinant of Zero Row Matrix

If one row or column of any square matrix is zero, its determinant is exactly $0$.

Order of Product Matrix AB

If matrix $A$ is of order $m \times p$ and matrix $B$ is of order $p \times n$, the order of the product matrix $AB$ is $m \times n$.

Transpose of a Matrix

A new matrix formed by interchanging the rows and columns of an existing matrix, denoted as $A^{t}$ or $A'$.

Cofactor $A_{32}$

The signed minor of the element in the third row and second column of a matrix.

Adjoint of A ($Adj A$)

The transpose of the cofactor matrix of a square matrix $A$; used to find the inverse matrix via $\frac{Adj A}{|A|}$.

Empty Set

A set containing no elements, also known as a null set, which is considered a finite set.

Proper Subset

A set $A$ is a proper subset of $B$ (denoted A \tag{\text{proper subset symbol}} B) if every element of $A$ is in $B$ but $A \neq B$.

Power Set Elements

If a set has $n$ elements, the total number of elements in its power set is given by the formula $2^{n}$.

Union of Sets (A U B)

The set containing all elements that are in set $A$, in set $B$, or in both.

Cartesian Product (AxB)

The set of all ordered pairs $(a, b)$ where $a$ is from set $A$ and $b$ is from set $B$; if $A \neq B$, then $A \times B \neq B \times A$.

Difference of Sets (A - B)

The set of elements that belong to set $A$ but do not belong to set $B$.

De Morgan's Law (Union)

The complement of the union of two sets is equal to the intersection of their complements, expressed as $(A \text{ U } B)' = A' \text{ ∩ } B'$.

Domain of a Relation

The set of all first components of the ordered pairs in a relation $R$, such as $\text{Domain } R = \text{\{}1, 2, 4\text{\}}$ from $\text{\{}(1,1), (2,3), (4,5)\text{\}}$.

Irrational Numbers between 0 and 1

There are an infinite number of rational and irrational numbers possible between any two values like $0$ and $1$.

Cube Roots of Unity

The three values 1, \text{\omega}, \text{\omega}^{2} whose product is equal to $1$.

Discriminant

The part of the quadratic formula $b^{2} - 4ac$ used to determine the nature of the roots.

Imaginary Roots Condition

The condition for the roots of a quadratic equation to be complex or imaginary is when the discriminant is less than zero ($b^{2} - 4ac < 0$).

Real and Equal Roots Condition

The condition where the discriminant of a quadratic equation is zero ($b^{2} - 4ac = 0$).

Sum of Roots (lx^2 + mx + n = 0)

The sum of the roots of a quadratic equation is given by the formula $-\frac{m}{l}$.

Product of Roots (lx^2 + mx + n = 0)

The product of the roots of a quadratic equation is given by the formula $\frac{n}{l}$.

Complex Conjugate Product

For a complex number $Z = x + yi$, the product with its conjugate $Z \times \bar{Z}$ is equal to $|Z|^{2}$ or $x^{2} + y^{2}$.

Arithmetic Mean (A.M.)

The value $A$ between two numbers $a$ and $b$ calculated as $\frac{a + b}{2}$.

Geometric Mean (G.M.)

The value $G$ between two numbers $a$ and $b$ calculated as \text{\pm}\text{\sqrt{ab}}.

Harmonic Mean (H.M.)

The value $H$ between two numbers $a$ and $b$ calculated as $\frac{2ab}{a + b}$.

Mean Relationship Formula

The relationship between arithmetic, geometric, and harmonic means expressed as $G^{2} = AH$ or G = \text{\sqrt{AH}}.

Common Ratio (r)

The constant factor found by dividing any term in a Geometric Progression (G.P.) by the preceding term.

Common Difference (d)

The constant value added to each term to get the next term in an Arithmetic Progression (A.P.).

Sum to Infinity (G.P.)

The sum of an infinite geometric series where $|r| < 1$, calculated as S_{\text{\infty}} = \frac{a}{1 - r}.

Collinear Points

Points that lie on the same straight line; for points $(x_1, y_1), (x_2, y_2), (x_3, y_3)$, the slope between any two pairs must be equal.

Binomial Coefficient

The numerical factor in the expansion of $(a + b)^{n}$, often denoted as $\binom{n}{r}$ or $^nC_r$.

Middle Term in (x^2 + 1)^n

The term(s) located in the center of a binomial expansion; if $n$ is even, there is one middle term, if odd, there are two.

Term Independent of x

The specific term in a binomial expansion that does not contain the variable $x$ ($x^0$).

Permutations of 'REARRANGE'

The total number of words formed by arranging all letters of a word with repetitions, using the formula $\frac{n!}{n_1!n_2!...}$.

Probability of an Event P(A)

The ratio of the number of favorable outcomes to the total number of possible outcomes in a sample space.

Sample Space for Two Dice

The set of $36$ possible outcomes when two dice are rolled simultaneously.

Probability of Same Faces

The probability of getting the same number on both dice when two are rolled, which is $\frac{6}{36} = \frac{1}{6}$.

Probability of Red King

The probability of drawing a red king from a deck of $52$ cards, which is $\frac{2}{52} = \frac{1}{26}$.

Face Cards

The Kings, Queens, and Jacks in a standard deck of $52$ cards, totaling $12$ cards.

Non-face Cards

The cards in a standard deck that are not Kings, Queens, or Jacks, totaling $40$ cards ($52 - 12$).

Limit of (x lnx) as x approaches 0

The mathematical limit \text{\lim}_{x \to 0} (x \text{\ln(x)}) = 0.

Derivative of a^x

The rate of change of an exponential function with base $a$, given by \frac{d}{dx}(a^{x}) = a^{x}\text{\ln(a)}.

Derivative of ln(sin x)

The derivative is calculated as \frac{d}{dx}(\text{\ln(sin(x))}) = \frac{1}{\text{\sin(x)}} \times \text{\cos(x)} = \text{\cot(x)}.

Gradient of a Curve

The value of the derivative $\frac{dy}{dx}$ at a specific point on the curve.

Second Derivative

The derivative of the derivative of a function, denoted as $f''(x)$ or $\frac{d^{2}y}{dx^{2}}$.

Integration of tan x

The antiderivative \text{\int} \text{\tan(x)}\text{\(dx)} = \text{\ln(sec(x))} + C.

Integration of e^x

The antiderivative \text{\int} e^{x}\text{\(dx)} = e^{x} + C.

Integration of 1/x

The antiderivative \text{\int} \frac{1}{x}\text{\(dx)} = \text{\ln(x)} + C.

Integration by Parts Formula

A method of integration based on the product rule, often summarized as \text{\int} u \text{\(dv)} = uv - \text{\int} v \text{\(du)}.

Partial Fractions

A method to decompose a complex rational fraction into a sum of simpler fractions.

Slope of a Line (m)

The ratio of the vertical change to the horizontal change between two points $(x_1, y_1)$ and $(x_2, y_2)$, calculated as $m = \frac{y_2 - y_1}{x_2 - x_1}$.

Parallel Lines Slope Condition

Two non-vertical lines are parallel if and only if their slopes are equal ($m_1 = m_2$).

Perpendicular Lines Slope Condition

Two lines are perpendicular if the product of their slopes is $-1$ ($m_1 \times m_2 = -1$).

Undefined Slope

The slope of a vertical line, which is parallel to the y-axis and perpendicular to the x-axis.

Equation of a Circle

The set of all points $(x, y)$ at a fixed distance $r$ (radius) from a center $(h, k)$, given by $(x - h)^{2} + (y - k)^{2} = r^{2}$.

Eccentricity of a Circle

The constant value denoting the deviation of a conic section from being circular; for a circle, $e = 0$.

Equation of a Parabola

The locus of points equidistant from a fixed point (focus) and a fixed line (directrix); standard forms include $y^{2} = 4ax$ or $x^{2} = 4ay$.

Focus of a Parabola

A fixed point used to define a parabola; for $y^{2} = 12x$, the focus is at $(3, 0)$, where $4a = 12$, so $a = 3$.

Latus Rectum (Parabola)

A chord passing through the focus perpendicular to the axis of the parabola; its length is given by $4a$.

Eccentricity of a Parabola

In the context of conic sections, the eccentricity of any parabola is exactly $e = 1$.

Ellipse

The locus of points where the sum of the distances to two fixed points (foci) is constant.

Major Axis of an Ellipse

The longest diameter of an ellipse, passing through its center and both foci.

Minor Axis of an Ellipse

The shortest diameter of an ellipse, perpendicular to the major axis at the center.

Eccentricity of an Ellipse

A measure of how 'stretched' an ellipse is; it is always less than one ($e < 1$).

Directrix of an Ellipse

A fixed line perpendicular to the major axis, used along with the focus to define the ellipse; the distance from center is $\frac{a}{e}$.

Hyperbola

The set of all points such that the absolute difference of the distances to two fixed points (foci) is constant.

Transverse Axis

The segment of length $2a$ that passes through the vertices and foci of a hyperbola.

Conjugate Axis

The segment of length $2b$ perpendicular to the transverse axis of a hyperbola at its center.

Asymptotes of a Hyperbola

The lines that the branches of a hyperbola approach but never touch as they extend toward infinity; for $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, they are y = \text{\pm}\frac{b}{a}x.

Eccentricity of a Hyperbola

A measure of the opening of the branches of a hyperbola; it is always greater than one ($e > 1$).

Sine Rule (Law of Sines)

The relationship in any triangle \text{\frac{a}{\sin(A)}} = \text{\frac{b}{\sin(B)}} = \text{\frac{c}{\sin(C)}}.

Cosine Rule (Law of Cosines)

For any triangle, the square of a side equals the sum of the squares of the other sides minus twice their product times the cosine of the included angle, e.g., a^2 = b^2 + c^2 - 2bc\text{\cos(A)}.

Pythagoras Theorem

A special case of the Law of Cosines for right-angled triangles (\text{\angle C} = 90^{\circ}), where $c^{2} = a^{2} + b^{2}$.

Range of y = cos x

The set of all possible output values for the cosine function, which is $[-1, 1]$ or -1 \text{\le} y \text{\le} 1.

Radical Equation

An equation in which a variable is under a radical sign, such as \text{\sqrt{x + 1}} + \text{\sqrt{x - 1}} = \text{\sqrt{x^{2} + x + 1}}.

Exponential Equation

An equation where the variable appears in the exponent, such as $12^{x} + 18 = 468$.

Dot Product (Scalar Product)

The product of the magnitudes of two vectors and the cosine of the angle between them (\text{\mathbf{A}} \text{\cdot} \text{\mathbf{B}} = |A||B|\text{\cos(\theta)}).

Cross Product (Vector Product)

A vector quantity obtained by multiplying the magnitudes of two vectors and the sine of the angle between them (\text{\mathbf{A}} \text{\times} \text{\mathbf{B}} = |A||B|\text{\sin(\theta)}\text{\mathbf{n}}).

Perpendicular Vectors Condition

Two non-zero vectors are perpendicular if and only if their scalar (dot) product is zero (\text{\mathbf{A}} \text{\cdot} \text{\mathbf{B}} = 0).

Parallel Vectors Condition

Two non-zero vectors are parallel if and only if their vector (cross) product is the zero vector (\text{\mathbf{A}} \text{\times} \text{\mathbf{B}} = 0).

Magnitude of a Vector

The length of a vector \text{\mathbf{A}} = ai + bj + ck, calculated as \text{\sqrt{a^2 + b^2 + c^2}}.

Projection of Vector A onto B

The scalar value representing the component of vector \text{\mathbf{A}} in the direction of \text{\mathbf{B}}, calculated as \frac{\text{\mathbf{A}} \text{\cdot} \text{\mathbf{B}}}{|B|}.

Scalar Area of a Triangle

The area of a triangle with vertices \text{\mathbf{A}}, \text{\mathbf{B}}, and \text{\mathbf{C}} can be found using the magnitude of half the cross product of two side vectors: \frac{1}{2}|\text{\mathbf{AB}} \text{\times} \text{\mathbf{AC}}|.

Displacement

A vector quantity representing the change in position of an object, often denoted as $S$, where velocity is its first derivative ($v = \frac{dS}{dt}$).

Velocity

The rate of change of displacement with respect to time, given by the derivative $v = \frac{dS}{dt}$.

Moment of Force

The turning effect of a force \text{\mathbf{F}} applied at a point relative to another point, calculated as the cross product of the position vector \text{\mathbf{r}} and force \text{\mathbf{F}} (\text{\mathbf{M}} = \text{\mathbf{r}} \text{\times} \text{\mathbf{F}}).

Direction Cosines

The cosines of the angles between a vector and the positive coordinate axes ($x$, $y$, and $z$).

Collinear Vectors

Vectors that lie on the same line or are parallel to each other.

Unit Vector along x-axis

The standard basis vector denoted as $i$ (or \text{\mathbf{i}}).

Unit Vector along y-axis

The standard basis vector denoted as $j$ (or \text{\mathbf{j}}).

Unit Vector along z-axis

The standard basis vector denoted as $k$ (or \text{\mathbf{k}}).