Math II
Connect real and complex numbers, polynomials, quadratics, proof, circles, trigonometry, probability, and modeling.
- Problem types
- 786
- Practice variants
- 3,144
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Each problem type has four distinct practice variants. Open a preview to move among all four.
Apply a dilation factor to 1D, 2D, and 3D measures
Apply scale factors to length, area, and volume using k, k^2, and k^3 relationships.
The exponent belongs to the dimension of the measure, not to the kind of figure surrounding it. We’ll classify lengths and perimeters as one-dimensional, areas and surface areas as two-dimensional, …
Preview problemUse a scale factor to find a new length, area, or volume
Apply scale factors to length, area, and volume using k, k^2, and k^3 relationships.
The scale describes the model as an image of the real object, so direction matters before any arithmetic. We’ll identify the requested measure as a length, apply the model-to-real factor …
Preview problemFind the original measure from a scale factor and the image measure
Apply scale factors to length, area, and volume using k, k^2, and k^3 relationships.
Because the image measure is already known, the forward dilation must be undone rather than applied again. We’ll write image length as the scale factor times original length, substitute according …
Preview problemDecide similarity from all corresponding dimension ratios
Apply scale factors to length, area, and volume using k, k^2, and k^3 relationships.
Similarity requires one multiplier across every corresponding linear dimension. We’ll pair shorter side with shorter side and longer with longer, keep both ratios in the same A-to-B direction, compare the …
Preview problemInterpret dimensional measure multipliers in context
Apply scale factors to length, area, and volume using k, k^2, and k^3 relationships.
Doubling an edge does not make every measure merely double, because surface area and volume combine different numbers of length directions. We’ll translate the edge change into a linear factor, …
Preview problemClassify three lengths with the strict triangle inequality
Use triangle angle-side relationships and triangle inequality in mathematical and real-world problems.
For positive lengths, sorting reveals the one triangle-closure comparison that can fail. We’ll identify the longest segment, compare it with the sum of the other two, insist on a strict …
Preview problemFind the possible range for a missing triangle side
Use triangle angle-side relationships and triangle inequality in mathematical and real-world problems.
A third side is constrained from both directions: it must be long enough to bridge the difference and short enough for the other sides to meet. We’ll build the lower …
Preview problemOrder triangle sides from angle measures
Use triangle angle-side relationships and triangle inequality in mathematical and real-world problems.
Angle order transfers to side order only after each angle is paired with the side directly across from it. We’ll rank the angles, map each vertex to the side that …
Preview problemOrder triangle angles from side lengths
Use triangle angle-side relationships and triangle inequality in mathematical and real-world problems.
Side order transfers to angle order only through opposite pairs. We’ll rank the three side lengths, map each side to the vertex it does not touch, use longer-side-opposite-larger-angle, and then …
Preview problemIdentify separate largest-angle and longest-side fields
Use triangle angle-side relationships and triangle inequality in mathematical and real-world problems.
The prompt asks for two different objects, so we should keep the angle field separate from the side field. We’ll establish which angle is uniquely largest, locate the side directly …
Preview problemMake a contextual triangle-existence decision
Use triangle angle-side relationships and triangle inequality in mathematical and real-world problems.
A contextual route still has to pass the same strict closure test as any triangle. We’ll sort the proposed lengths, compare the longest with the combined shorter lengths, measure any …
Preview problemUse side-angle relationships to compare routes or supports
Use triangle angle-side relationships and triangle inequality in mathematical and real-world problems.
The matched side pairs hold two parts of each brace fixed, leaving the included angle to control the endpoint separation. We’ll verify that hinge setup, identify the angle between the …
Preview problemSolve algebraic triangle inequality constraints
Use triangle angle-side relationships and triangle inequality in mathematical and real-world problems.
A variable side must satisfy every pair-sum condition at once, so one inequality cannot settle the interval. We’ll write all three strict constraints, solve them separately, discard only genuinely redundant …
Preview problemClassify a triangle from its side lengths as acute, right, or obtuse
Use triangle angle-side relationships and triangle inequality in mathematical and real-world problems.
Triangle closure and angle classification use different comparisons. We’ll first verify that the two shorter sides can meet, then assign the longest side to the comparison role, compare its square …
Preview problemUse the hinge theorem informally to compare third sides or included angles in two triangles
Use triangle angle-side relationships and triangle inequality in mathematical and real-world problems.
The two congruent adjacent-side pairs make this a hinge comparison rather than a full congruence claim. We’ll confirm that the given angles are included between those sides, order the openings, …
Preview problemIntersect a third-side interval with an extra constraint
Use triangle angle-side relationships and triangle inequality in mathematical and real-world problems.
The geometric interval should be established before the extra discrete condition is applied. We’ll derive strict lower and upper bounds from the difference and sum of the fixed sides, intersect …
Preview problemWrite the equation of a circle from its center and radius
Derive the equation of a circle from center/radius and complete the square to identify center/radius.
Circle standard form records squared horizontal and vertical distances from the center. We’ll place each center coordinate inside its matching subtraction, simplify any zero shifts, square the radius on the …
Preview problemRead separate center and radius fields from circle standard form
Derive the equation of a circle from center/radius and complete the square to identify center/radius.
Reading circle form requires undoing two different encodings: subtraction hides each center coordinate, while the right side stores the radius squared. We’ll match both binomials to the standard pattern, rewrite …
Preview problemDerive a circle equation from the distance formula
Derive the equation of a circle from center/radius and complete the square to identify center/radius.
A circle equation grows directly from its fixed-distance definition. We’ll express the distance from a general point to the center, set that distance equal to the radius, square both nonnegative …
Preview problemComplete the square to convert a circle equation to standard form
Derive the equation of a circle from center/radius and complete the square to identify center/radius.
Completing the square must preserve the equation while turning each variable group into a distance square. We’ll isolate the constant, group x-terms and y-terms, derive each compensation value from half …
Preview problemComplete the square and report center and radius separately
Derive the equation of a circle from center/radius and complete the square to identify center/radius.
This conversion has two stages: first create center-radius form without changing the equation, then decode its geometric parameters. We’ll complete and balance both variable groups, factor the squares, reverse the …
Preview problemClassify a circle equation by the sign of radius squared
Derive the equation of a circle from center/radius and complete the square to identify center/radius.
Not every completed-square equation describes a positive-radius real circle, so the right side needs a sign check. We’ll convert to center-radius form, identify the center and radius-squared value, classify the …
Preview problemWrite a circle equation from the endpoints of a diameter
Derive the equation of a circle from center/radius and complete the square to identify center/radius.
Diameter endpoints determine the circle in two different ways: their midpoint locates the center, while their separation determines twice the radius. We’ll average coordinates, compute the full endpoint distance, halve …
Preview problemWrite a circle equation from its center and a point on the circle
Derive the equation of a circle from center/radius and complete the square to identify center/radius.
The given point is not another center; it supplies the center-to-boundary displacement. We’ll place the known center into standard form, subtract coordinates to get the horizontal and vertical changes, use …
Preview problemClassify a point as on, inside, or outside a circle
Derive the equation of a circle from center/radius and complete the square to identify center/radius.
Point position can be decided without taking a square root. We’ll compute the point’s squared distance from the center, compare it with the circle’s radius squared, and interpret a smaller …
Preview problemGraph a circle from center, radius, and four exact anchors
Derive the equation of a circle from center/radius and complete the square to identify center/radius.
An exact circle graph starts with parameters before any sketching. We’ll decode the center signs, take the nonnegative square root for the radius, move that distance left and right while …
Preview problemFind a missing parameter in a circle equation
Derive the equation of a circle from center/radius and complete the square to identify center/radius.
The point-on-circle condition turns the missing radius into a distance calculation. We’ll substitute the boundary point into the shifted squares, evaluate the horizontal and vertical differences, combine their squares to …
Preview problemInterpret a contextual circle boundary and covered region
Derive the equation of a circle from center/radius and complete the square to identify center/radius.
The equation with equality describes only the coverage boundary, while the context asks about the whole covered region. We’ll read the source location and linear reach, distinguish radius from radius …
Preview problemCompare two circles by records and spatial position
Derive the equation of a circle from center/radius and complete the square to identify center/radius.
Two circles’ radii alone do not determine how their boundaries meet; the distance between centers completes the comparison. We’ll read both circle records, compute that distance, compare it with the …
Preview problemComplete a circle equation and classify a point consistently
Derive the equation of a circle from center/radius and complete the square to identify center/radius.
One consistent circle record should drive every part of the classification. We’ll complete and balance both variable squares, factor to reveal the center and radius squared, compute the point’s squared …
Preview problemDerive separate parabola features from focus and directrix
Derive the equation of a parabola from a focus and directrix.
A parabola’s main features come from one perpendicular focus-to-directrix construction. We’ll project the focus onto the line, take the midpoint for the vertex, use the perpendicular orientation for the axis, …
Preview problemDerive a vertical parabola equation from focus and directrix
Derive the equation of a parabola from a focus and directrix.
The equation can be derived directly from the parabola’s equal-distance definition. We’ll first locate the vertex and signed focal distance, set a general point’s focus distance equal to its perpendicular …
Preview problemDerive a horizontal parabola equation from focus and directrix
Derive the equation of a parabola from a focus and directrix.
A vertical directrix forces a horizontal parabola, so the y-expression will be squared. We’ll project the focus horizontally, use the midpoint for the vertex and direction for signed p, equate …
Preview problemWrite the standard form of a parabola from its vertex, orientation, and p-value
Derive the equation of a parabola from a focus and directrix.
Orientation decides which coordinate expression is squared, while the vertex supplies both shifts. We’ll use the vertical template indicated by the opening, preserve the sign of p, substitute the parameters …
Preview problemRead separate focus and directrix fields from a parabola equation
Derive the equation of a parabola from a focus and directrix.
Reading a parabola equation backward starts by matching its orientation and shifts to standard form. We’ll identify the vertex, solve the unsquared-coordinate coefficient as four times p, move from the …
Preview problemConvert a parabola equation to standard focus-directrix form
Derive the equation of a parabola from a focus and directrix.
Converting to focus-directrix form is an equivalent rearrangement, so both sides must be scaled together. We’ll clear the fractional coefficient, put the squared coordinate first, preserve the zero vertex shifts, …
Preview problem