Latest Examination Paper 2024 - Set 4

The sum of the roots of a quadratic equation in the form ax² + bx + c = 0 is given by -b/a.

The product of the roots of a quadratic equation in the form ax² + bx + c = 0 is given by c/a.

In a quadratic equation, if the discriminant (b² - 4ac) is equal to zero, the roots are real and equal.

The probability of an event happening and the probability of that event not happening always add up to 1.

The LCM (Least Common Multiple) is the smallest number that is a multiple of all the given numbers.

The midpoint of a line segment with endpoints (x₁, y₁) and (x₂, y₂) is calculated as ((x₁ + x₂)/2, (y₁ + y₂)/2).

The distance between two points (x₁, y₁) and (x₂, y₂) is calculated as √((x₂ - x₁)² + (y₂ - y₁)²).

In an arithmetic progression (AP), the nth term (aₙ) is calculated as aₙ = a + (n - 1)d, where 'a' is the first term and 'd' is the common difference.

If sec θ - tan θ = m, then sec θ + tan θ = 1/m. This is a useful trigonometric identity that can be derived by manipulating the given equation.

The probability of an event is the ratio of the number of favorable outcomes to the total number of possible outcomes. In this case, we need to find the probability of getting a prime number from the remaining numbers after removing even numbers.

The formula represents the variance of a dataset, a measure of how spread out the data is from the mean. It involves the sum of squared deviations from the mean, weighted by the frequencies of each data point.

If the zeroes of a polynomial are twice the zeroes of another polynomial, then the relationship between the coefficients of the two polynomials can be determined. In this case, we need to find the value of 'p' in the first polynomial.

The distance between two points on a coordinate plane is calculated using the distance formula. We can find the x-coordinate of the other end of the diameter by substituting the known values into the formula.

If cos(α + β) = 0, then (α + β) must be an odd multiple of π/2. Therefore, cos (α+ β)/2 can be calculated using the trigonometric identities.

The ratio of the surface area of a sphere to the surface area of two hemispheres taken together is 1:1. This is because removing a plane from the sphere simply creates two equal halves.

The middlemost observation in a data set arranged in ascending order is called the median. This value is the central value in the data and helps to determine the average or typical point.

Two straight lines are parallel only if they have the same slope. The tangents at the endpoints of a diameter of a circle are perpendicular to the radius, therefore they are perpendicular to each other and not parallel.

A quadratic polynomial, when graphed, can have at most two x-intercepts. If a polynomial touches the x-axis at only one point, it must have a degree higher than 2.

A system of linear equations is a set of equations with the same variables. Solving it means finding the values of the variables that satisfy all equations simultaneously.

The probability of an event is the number of favorable outcomes divided by the total number of possible outcomes. In this case, we need to consider that one black card is lost, reducing the total number of cards in the deck.

The trigonometric functions work with angles and sides of a right triangle. Evaluating a trigonometric expression means substituting given angles and using the definitions of the trigonometric functions.

Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional. We can prove similarity using Angle-Angle-Angle or Side-Angle-Side similarity criteria.

An irrational number cannot be expressed as a simple fraction. We can prove a number is irrational by assuming it is rational and then showing a contradiction.

A composite number has more than two factors, including 1 and itself. We can determine if a number is composite by checking if it has any factors besides 1 and itself.

The difference between the outer and inner radii of a hollow right circular cylinder is 1 cm. The volume of the metal used in making the cylinder is 176 cm³. We need to find the outer and inner radii of the cylinder.

An arc of a circle of radius 21 cm subtends an angle of 60° at the center. We need to find the length of the arc and the area of the minor segment of the circle made by the corresponding chord.

The sum of the first and eighth terms of an arithmetic progression (AP) is 32, and their product is 60. We need to find the first term, common difference, and the sum of the first 20 terms of the AP.

In an AP of 40 terms, the sum of the first 9 terms is 153, and the sum of the last 6 terms is 687. We need to find the first term, common difference, and the sum of all the terms of the AP.

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.

In a triangle, if three perpendiculars are drawn from points on a side to the opposite side, and those perpendiculars are marked as x, y, and z, then the sum of the reciprocals of these perpendiculars is equal to the reciprocal of the side they are drawn to.

A pole 6m high is fixed on the top of a tower. The angle of elevation of the top of the pole observed from a point P on the ground is 60°, and the angle of depression of point P from the top of the tower is 45°. We need to find the height of the tower and the distance of point P from the foot of the tower.

A rectangular floor can be completely tiled with 200 square tiles. If the side length of each tile is increased by 1 unit, it would take only 128 tiles to cover the floor. We need to form a quadratic equation and solve it to find the length of the original tile.