High School Mathematics Exam Quiz

The highest common factor (HCF) of two or more integers is the largest positive integer that divides each of the integers without leaving a remainder.

The degree of a linear polynomial is 1.

The system of equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 has a unique solution if the ratio of their coefficients is not equal: a1/a2 ≠ b1/b2.

The quadratic equation ax^2 + bx + c = 0, where a ≠ 0, has two equal real roots if the discriminant is equal to zero: b^2 - 4ac = 0.

The distance between two points P(-7, 7) and Q(-2, -3) is calculated using the distance formula: √((x2 - x1)^2 + (y2 - y1)^2).

An arithmetic progression (A.P.) is a sequence of numbers where the difference between any two consecutive terms is constant. The formula for the nth term of an A.P. is a + (n-1)d, where 'a' is the first term and 'd' is the common difference.

The discriminant of a quadratic equation ax^2 + bx + c = 0 is given by b^2 - 4ac. It determines the nature of the roots of the equation.

The 11th term of the A.P. -3, -1/2, 2,... is calculated by using the formula for the nth term of an A.P. a + (n - 1)d, where 'a' is the first term and 'd' is the common difference.

A system of linear equations has a unique solution if the ratio of their coefficients is not equal. This means the lines represented by the equations intersect at a single point.

A quadratic equation has no real roots when its discriminant is negative. This means the parabola doesn't touch the x-axis.

The arithmetic mean (AM) of a set of numbers is the sum of all the numbers divided by the total count of numbers.

The standard deviation measures how spread out a set of data points are from their mean.

The formula for the area of a triangle is half the product of its base and height.

The volume of a cone is calculated as one-third the product of the area of its base and its height.

The surface area of a sphere is four times the product of pi and the square of its radius.

The volume of a sphere is four-thirds the product of pi and the cube of its radius.

The Basic Proportionality Theorem states that a line drawn parallel to one side of a triangle divides the other two sides proportionally.

The coordinates of the point that divides a line segment joining two points in a given ratio can be calculated using the section formula.

The value of 𝑐𝑜𝑠𝐴 and 𝑡𝑎𝑛𝐴 can be found using the trigonometric identities. Since 𝑠𝑖𝑛𝐴 is given, we can use Pythagorean identity to find 𝑐𝑜𝑠𝐴 and then the ratio of 𝑠𝑖𝑛𝐴 and 𝑐𝑜𝑠𝐴 to get the value of 𝑡𝑎𝑛𝐴.

The lengths of tangents drawn from an external point to a circle are equal. This can be proven by considering congruent triangles formed by the tangents and the radii.

The probability of two independent events happening is the product of their individual probabilities.

The probability of an event not happening is 1 minus the probability of it happening.

The probability of any event happening is always between 0 and 1, inclusive.

A quadratic equation is an equation where the highest power of the variable is 2. In the given equation, the term with the highest power of x is x² making it a quadratic equation. Therefore, the statement is false.

The common difference of an arithmetic progression (A.P.) is the constant value added to any term to get the next term. In this sequence, the common difference is -2 since each term is 2 less than the previous one. Therefore, the statement is true.

Similar triangles have the same shape but different sizes. All equilateral triangles are similar, but not all isosceles triangles are similar. Consider isosceles triangles with different vertex angles. They won't be similar. The statement is false.

The midpoint of a line segment is the point that divides the segment into two equal parts. The midpoint formula is: [(x1 + x2)/2, (y1 + y2)/2]. Applying this to the given points A(3, -1) and B(6, 4), the midpoint is (9/2, 3/2), not (9, 3). Therefore, the statement is false.

The probability of the complement of an event E, denoted as P(E̅), is the probability that event E does not occur. It is calculated as 1 minus the probability of event E occurring: P(E̅) = 1 - P(E). Therefore, the statement is true.

The empirical relationship between the mean, median, and mode is an observed pattern in data that is approximately true for many datasets. However, it is not a strict mathematical rule and may not hold true in all cases. The statement is true, but it is important to remember that this is an empirical relationship and may not always be completely accurate.

The probability of an event not happening is 1 minus the probability of the event happening.

The probability of drawing an ace from a standard 52-card deck is 4/52, or 1/13. There are four aces in the deck.

The probability of not drawing an ace from a standard 52-card deck is 48/52, or 12/13. There are 48 non-aces in the deck.

The probability of drawing a red ball from a bag containing 3 red balls and 5 black balls is 3/8.

The probability of not drawing a red ball from a bag containing 3 red balls and 5 black balls is 5/8.

An irrational number cannot be expressed as a fraction - it's a decimal that continues forever without repeating.

A number ending in 0 is divisible by 10, which means it must be divisible by both 2 and 5.

The height of the tower is 30√3 meters. Use the tangent function (tan 30° = opposite / adjacent) and solve for the height.

For a quadratic equation 𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0, where 𝑎 ≠ 0, the discriminant (𝐷) is the value that determines the nature of the roots (solutions). It is calculated using the formula: 𝐷 = 𝑏² - 4𝑎𝑐.

The sum (𝑆) of the first 𝑛 terms of an arithmetic progression (A.P.) is calculated using the formula: 𝑆 = (𝑛/2) * [2𝑎 + (𝑛 - 1)𝑑], where 𝑎 is the first term and 𝑑 is the common difference.

All circles are similar. This means that any two circles can be made to have the same shape by scaling one of them up or down. They have the same shape but different sizes.

The point where a tangent line touches a circle is called the point of contact. It is the only common point between the tangent line and the circle.

The sum of the probabilities of all elementary events in an experiment is always equal to 1. This means that the probability of something happening or not happening is certain.

At any point on a circle, the tangent line is perpendicular to the radius drawn through that point. This means the angle formed between the tangent and the radius is 90 degrees.

The area of a sector of a circle is calculated using the formula: Area of Sector = (𝜃/360°) * πr², where 𝜃 is the angle of the sector in degrees and 𝑟 is the radius of the circle.

The curved surface area of a hemisphere is calculated using the formula: Curved Surface Area = 2πr², where 𝑟 is the radius of the hemisphere.

The curved surface area of a cylinder is calculated using the formula: Curved Surface Area = 2πrh, where 𝑟 is the radius of the base and ℎ is the height of the cylinder.

The value of sin 0° (sine of 0 degrees) is always equal to 0.