FRNSW Cognitive Test cheat sheet

Syllogism validity depends only on logical form: if the premises are accepted as true, the conclusion must follow; real-world truth of a premise is irrelevant to validity.

In a valid syllogism the middle term must be distributed in at least one premise; a term is distributed when the statement says something about every member of that class (All A are B distributes A; No A are B distributes both).

Two negative premises cannot yield a valid conclusion, and if one premise is negative the conclusion must be negative; if both premises are affirmative the conclusion must be affirmative.

Contrapositive is logically equivalent to the original conditional (If P then Q = If not Q then not P); converse (If Q then P) and inverse (If not P then not Q) are not equivalent.

"Some A are B" does not imply "Some A are not B" and does not imply "All A are B"; only "No A are B" or "All A are B" plus existence claims support firm conclusions about individual members.

Passage-based questions: the correct answer must be supported by the text alone; an option that is true in general but not stated in the passage is wrong.

"Inference" questions accept only what must be true; "assumption" questions require the unstated premise the argument depends on; "strengthen/weaken" questions change the likelihood of the conclusion, not the premises.

Watch quantifier and scope words: all, some, none, only, unless, except, at least, at most. "Only A are B" means all B are A; "A unless B" means if not B then A.

Percentage change = (new - old) / old x 100; a rise of x% followed by a fall of x% leaves the value below the start, at (1 - x^2/10000) of it.

Successive percentage changes multiply: +20% then -10% gives 1.20 x 0.90 = 1.08, i.e. +8% overall, not +10%.

Reverse percentages: if a price after a 15% discount is $68, the original is 68 / 0.85 = $80; divide by the multiplier rather than adding the percentage back.

Ratios: split a total T in ratio a:b:c by computing parts = a+b+c and each share = T x (part / parts); to combine ratios a:b and b:c, scale so the shared term matches.

Rates and work: combined rate = sum of individual rates; if A finishes in a hours and B in b hours, together they take ab/(a+b) hours.

Speed = distance / time; for average speed over equal distances use the harmonic mean 2xy/(x+y), not the arithmetic mean of the two speeds.

Unit conversion in compound units: km/h to m/s multiply by 1000/3600 (divide by 3.6); area scales with the square of the length factor, volume with the cube.

Estimation check: round each number to one significant figure to bound the answer, then confirm the exact option lies within the estimated range before selecting.

In sequences, track each element separately: position (steps clockwise/counter-clockwise), rotation (degrees per step), shading/colour, size, and count of sides or dots.

Rotation rules: a figure rotated 90 degrees four times returns to the start; 45-degree steps cycle through 8 orientations; reflections reverse orientation and cannot be produced by rotation alone.

Nets: a cube net has 6 squares in one of 11 valid arrangements; opposite faces never touch in the net and are separated by exactly one square in a straight line.

Mirror images flip left-right about a vertical axis and preserve top-bottom; water images flip top-bottom about a horizontal axis and preserve left-right.

Paper folding and punching: each fold doubles the number of holes; unfold by reflecting the hole pattern across every fold line in reverse order.

Counting problems: number of cubes with exactly two painted faces on an n x n x n painted cube = 12(n-2); three painted faces = 8 corners; one painted face = 6(n-2)^2.

Odd-one-out and analogy items: identify the single attribute shared by all but one option (number of enclosed regions, symmetry lines, curved vs straight edges, open vs closed).

Matrices: read rules across rows first, then down columns; a common rule is element-wise addition or subtraction where overlapping parts cancel.

Lever balance: effort x effort arm = load x load arm about the fulcrum; a longer effort arm reduces the force needed but increases the distance moved.

Class 1 lever has the fulcrum between effort and load; class 2 has the load between (always mechanical advantage > 1); class 3 has the effort between (advantage < 1, favours speed and range).

Pulleys: a single fixed pulley changes direction only (advantage 1); a movable pulley halves the effort; with n rope segments supporting the load, ideal effort = load / n.

Gears in mesh turn in opposite directions and their speeds are inversely proportional to tooth count: rpm_A x teeth_A = rpm_B x teeth_B.

Wheel and axle or belt drive: larger wheel turns slower and with greater torque; a belt without a twist keeps both pulleys turning the same direction, a crossed belt reverses direction.

Pressure = force / area; the same force on a smaller area produces greater pressure, which is why a sharp edge or narrow point cuts or penetrates more easily.

Hydraulics: pressure is transmitted equally in a confined fluid (Pascal's principle), so force_out = force_in x (area_out / area_in).

Floating and sinking: an object floats when its density is less than the fluid's; buoyant force equals the weight of fluid displaced (Archimedes).

Friction opposes relative motion, increases with rougher surfaces and greater normal force, and is reduced by wheels, rollers, or lubrication; energy is not destroyed but converted to heat.

Syllogism validity depends only on logical form: if the premises are accepted as true, the conclusion must follow; real-world truth of a premise is irrelevant to validity.

In a valid syllogism the middle term must be distributed in at least one premise; a term is distributed when the statement says something about every member of that class (All A are B distributes A; No A are B distributes both).

Two negative premises cannot yield a valid conclusion, and if one premise is negative the conclusion must be negative; if both premises are affirmative the conclusion must be affirmative.

Contrapositive is logically equivalent to the original conditional (If P then Q = If not Q then not P); converse (If Q then P) and inverse (If not P then not Q) are not equivalent.

"Some A are B" does not imply "Some A are not B" and does not imply "All A are B"; only "No A are B" or "All A are B" plus existence claims support firm conclusions about individual members.

Passage-based questions: the correct answer must be supported by the text alone; an option that is true in general but not stated in the passage is wrong.

"Inference" questions accept only what must be true; "assumption" questions require the unstated premise the argument depends on; "strengthen/weaken" questions change the likelihood of the conclusion, not the premises.

Watch quantifier and scope words: all, some, none, only, unless, except, at least, at most. "Only A are B" means all B are A; "A unless B" means if not B then A.

Percentage change = (new - old) / old x 100; a rise of x% followed by a fall of x% leaves the value below the start, at (1 - x^2/10000) of it.

Successive percentage changes multiply: +20% then -10% gives 1.20 x 0.90 = 1.08, i.e. +8% overall, not +10%.

Reverse percentages: if a price after a 15% discount is $68, the original is 68 / 0.85 = $80; divide by the multiplier rather than adding the percentage back.

Ratios: split a total T in ratio a:b:c by computing parts = a+b+c and each share = T x (part / parts); to combine ratios a:b and b:c, scale so the shared term matches.

Rates and work: combined rate = sum of individual rates; if A finishes in a hours and B in b hours, together they take ab/(a+b) hours.

Speed = distance / time; for average speed over equal distances use the harmonic mean 2xy/(x+y), not the arithmetic mean of the two speeds.

Unit conversion in compound units: km/h to m/s multiply by 1000/3600 (divide by 3.6); area scales with the square of the length factor, volume with the cube.

Estimation check: round each number to one significant figure to bound the answer, then confirm the exact option lies within the estimated range before selecting.

In sequences, track each element separately: position (steps clockwise/counter-clockwise), rotation (degrees per step), shading/colour, size, and count of sides or dots.

Rotation rules: a figure rotated 90 degrees four times returns to the start; 45-degree steps cycle through 8 orientations; reflections reverse orientation and cannot be produced by rotation alone.

Nets: a cube net has 6 squares in one of 11 valid arrangements; opposite faces never touch in the net and are separated by exactly one square in a straight line.

Mirror images flip left-right about a vertical axis and preserve top-bottom; water images flip top-bottom about a horizontal axis and preserve left-right.

Paper folding and punching: each fold doubles the number of holes; unfold by reflecting the hole pattern across every fold line in reverse order.

Counting problems: number of cubes with exactly two painted faces on an n x n x n painted cube = 12(n-2); three painted faces = 8 corners; one painted face = 6(n-2)^2.

Odd-one-out and analogy items: identify the single attribute shared by all but one option (number of enclosed regions, symmetry lines, curved vs straight edges, open vs closed).

Matrices: read rules across rows first, then down columns; a common rule is element-wise addition or subtraction where overlapping parts cancel.

Lever balance: effort x effort arm = load x load arm about the fulcrum; a longer effort arm reduces the force needed but increases the distance moved.

Class 1 lever has the fulcrum between effort and load; class 2 has the load between (always mechanical advantage > 1); class 3 has the effort between (advantage < 1, favours speed and range).

Pulleys: a single fixed pulley changes direction only (advantage 1); a movable pulley halves the effort; with n rope segments supporting the load, ideal effort = load / n.

Gears in mesh turn in opposite directions and their speeds are inversely proportional to tooth count: rpm_A x teeth_A = rpm_B x teeth_B.

Wheel and axle or belt drive: larger wheel turns slower and with greater torque; a belt without a twist keeps both pulleys turning the same direction, a crossed belt reverses direction.

Pressure = force / area; the same force on a smaller area produces greater pressure, which is why a sharp edge or narrow point cuts or penetrates more easily.

Hydraulics: pressure is transmitted equally in a confined fluid (Pascal's principle), so force_out = force_in x (area_out / area_in).

Floating and sinking: an object floats when its density is less than the fluid's; buoyant force equals the weight of fluid displaced (Archimedes).

Friction opposes relative motion, increases with rougher surfaces and greater normal force, and is reduced by wheels, rollers, or lubrication; energy is not destroyed but converted to heat.