Maths NBT Skills Preparation for Matrics in South Africa

Work from the inside out to simplify radical expressions, use prime factorization with perfect squares, and combine like square roots to reach simplified forms.

Identify the domain for the expression (2 − √(2 − x)) / (2x − 2) to be real by enforcing 2 − x ≥ 0 and 2x − 2 ≠ 0, i.e., x ≤ 2 and x ≠ 1.

Calculate how many tennis balls remain after donating 80 percent by using the expression T minus (M/100) times T, and simplify with a common denominator 100.

Identify domain restrictions for a trig function by examining zeros at 0 and 180 degrees and asymptotes at 90-degree intervals, concluding x = 90k for integers k.

Identify the circle's center and radius from the diagram, express the circle with (x-9)^2+(y-√17)^2=81, then compute x-intercepts by setting y=0 to get x=1 or x=17.

Learn how to calculate standard deviation by hand using the variance formula, compute the mean, deviations, squares, and the final square-root step for a four-term data set.

Rewrite the expression and form a common denominator, apply the difference of squares to factor, cancel common factors, and obtain 1/(t+1).

Apply vertical and horizontal line tests to classify relations as functions, using y = x^2 and y^2 = x to illustrate one-to-one, one-to-many, and inverse cases.

Analyze sequence types by first and second differences; this example is not arithmetic, not geometric, and not quadratic, guiding the correct choice d.

Take out the common factor 2^99 from 2^100 minus 2^99 to obtain 2^99, applying exponent rules to simplify the expression.

Construct a radius and note it is perpendicular to a tangent for triangle ADC. Apply Pythagoras to find DC = sqrt(3) R and side ratio 1 : sqrt(3) : 2.

Use equilateral triangle properties to establish 60-degree angles, apply tan 60 to obtain the gradient of line HC, and express its equation as y = sqrt(3) x + q.

This lecture analyzes solving (2x+1)/(x-1) ≤ 1, showing how assuming x>1 leads to a contradiction and concluding no solution, emphasizing careful checking to avoid assumptions.

Identify ox a B as a quarter of a unit circle with radius one, then compute the shaded area by subtracting quarter-circle areas from the unit square using pi.

Split the weird shape into a rectangle and a triangle, set the area to 100, and solve 100 = 2x + 1/2 base × height to get x = 200/9.

Apply log properties to a telescoping sum by substituting consecutive integers, canceling terms, and arriving at the result minus log 9 plus log 15.

Explore the graph of y equals minus log base 7 of x on the cartesian plane, a decreasing function that passes through (1,0) and has all real y-values.

The lecturer derives a quadratic sequence from outside in differences, identifies Tn = 1/2 n^2 + 3/2 n + 1, and computes T20 = 251 to select the correct option.

Determine the tangent to y = 2x^2 + 20 at x = -1 by computing dy/dx = 4x, giving slope -4 and point (-1,22); y = -4x + 18.

Compare the given options, apply double-angle identities to rewrite expressions as cos 2x and 2 cos^2 x − 1, and identify the correct choice by factoring.

Explore reading a box-and-whisker diagram with q1, q2, q3, and max; determine Valerie's 30th position and impact of a 2 percent credit on mean, median, interquartile range, and standard deviation.

Analyze regional graphs to identify intercepts and asymptotes, determine the correct 10x value, and confirm it from the pattern around 90.

Analyze a cubic to determine its real x-intercepts, stationary points, and inflection points; find only one real x-intercept, no real stationary points, and one inflection point.

Draws a triangle, expresses y in terms of z as y = 3z and x as 2z, then uses angle-sum to find z = 30 degrees and option b.

Determine the value of K that makes the horizontal line y=K intersect the QB graph in three distinct points between 0 and Q.

This lecture applies double-angle and pythagorean identities to simplify a trig expression, converting cos^2 x to 1 - sin^2 x and obtaining 4 sin^2 x cos^2 x.

Solve a trigonometric equation using cos^2 a and sin^2 a identities; transform to tan a, yielding tan a = -1.

Apply log laws to simplify expressions with negative exponents and move the minus between the base and the argument, including base changes like log_b(a^(-1)) and log_{1/b}(a).

Learn to work with decimals without a calculator by rewriting them as fractions and performing simple operations to identify the correct option c.

Describe how water depth in a vase changes on the graph with a constant rate, starting slow and accelerating, then becoming constant at top, and identify option C as best.

break numbers into prime bases, recognize 16 as 2^4 and 64 as 2^6, and use log properties to simplify expressions without a calculator.

Explore how to decode a three-digit number using expanded notation, identify the hundreds, tens, and units, and form and solve an equation to find the digits.

Analyze inequalities by inspecting a squared bracket and a fraction, using zero conditions and sign analysis; conclude that x must be negative (x<0), choosing option C.

Set cos alpha + sqrt3 sin alpha equal to k(sin alpha cos beta + cos alpha sin beta) and derive k=2 and beta=30 degrees.

Rewrite in standard form to locate the y-intercept Q and x-intercept Q/3, then subtract the small triangle area from the big triangle area (base 3Q, height Q) to get 4Q^2/3.

Learn to solve nested radical problems by squaring both sides and using an infinite sequence of square roots to deduce a from 64 minus a equals eight.

apply exponent rules for multiplying like terms: add exponents to get 2^2014, as shown in question 051.

Diagnose a falling quadrilateral by identifying perpendicular intersections, classify as square, rhombus, or kite, and then select option B.

The lecture demonstrates using the remainder factor theorem to determine what to add to a polynomial so it becomes a factor, by descending powers and substituting k to obtain zero.

Solve quadratics by applying the quadratic formula to x^2 - x - 1 = 0, yielding x = (1 ± sqrt(5))/2. Sum the roots to get 1.

Practice solving a doubling of an investment with monthly-compounded interest at 12% per year using the compound interest formula and logarithms.

Apply the derivative, substitute fractions such as one over eight and one over sixty four, and apply the inverse by swapping x and y to reach the final result.

Rewrite complex fraction using B plus and A minus with 1/A and 1/B, multiply by AB, flip B minus A to A minus B, and show result is zero.

Compare expressions involving square roots using two approaches. Simplify terms to rank values and identify the second smallest, selecting option C.

Count the number of two-digit differences from 99 down to 62 using sigma notation, demonstrating a sequence approach and yielding 38 as the answer.

Multiply the counts of each letter in the word victory, using 1×2×3×4×3×2×1 to obtain 144 and identify option a as correct.

Explore limits as n grows toward infinity, showing that any term divided by infinity approaches zero, and use factoring by the highest exponent to approximate a third option.

Apply exponent rules to rewrite and combine bases, convert thousand to ten cubed, and simplify by canceling like terms to identify the correct option.

The lecturer demonstrates using the difference of squares to multiply 2188 and 2186 by computing (2188-2186) and (2188+2186), yielding 2 and 4374 respectively.

Determine where the gradient times the regional graph is negative: positive gradient below the x axis or negative gradient above, including the between a and negative eight cases.

Learn to recover the original function from a second derivative by applying the power rule and successive integration, yielding x^3/3 + (3/2) x^2 + Cx + D.

Analyze sine and cosine graphs moved down by one unit to identify x-intercepts, and derive the general solution x = 180°k for integers k.

Rewrite y = 8 - 10 sin x cos x as y = -5 sin 2x + 8 using the double-angle identity, then identify the maximum value as 13.

Solve the intersection of y=6x+3 and y=tx with y=-9 to get x=-2 and point (-2,-9); gradients between 4.5 and 6 reveal the required line.

Apply the double-angle rules to rewrite the expression as cos 4x, showing option D matches cos 4x and eliminating the other options.

Identify that triangles APD and ADC have equal areas because they lie between two parallel sides and share the same base, illustrating the area equality theorem.

determine the axis of symmetry and turning point for y = -x^2 - 2x + k by substituting x = -2 to find a and k.

Differentiate a rational function, set the numerator to zero to locate turning points, then solve x^2+4x-4=0 with the quadratic formula and simplify sqrt(32) to 4 sqrt(2).

Simplify the expression t(x) = x^3/(2x) - 4x/(2x) to obtain t(x) = (1/2)x^2 - 2, highlighting its quadratic form and selecting the correct option.

From the log graph, determine the domain: log is undefined for negative numbers and zero, so -x > 0 and x < 0, making option b the correct choice.

Learn how cubic functions behave, recognizing at least one stationary point and zero to two stationary points, with at most two inflection points in their graphs.

Investigate whether a transformed sequence derived from a geometric progression forms an arithmetic progression by comparing differences and applying logarithm rules, concluding it is arithmetic.

Solve a quadratic by factoring the difference of squares: rewrite x^2 - a as (x + sqrt(a))(x - sqrt(a)) and set to zero to obtain x = ± sqrt(a).

Solve inequalities involving x-2 squared and x+1, identify zeros at x=2 and x=-1, and determine where the expression is less than zero.

Analyze a cyclic quadrilateral angle problem to identify which option is false, concluding that the 90-a statement is not true based on angle sums.

Determine angle C from the area formula with a=17, b=20, area=85, yielding sin C=1/2. Explore the ambiguous case, showing that C can be 30 degrees or 150 degrees.

Solves a nested radical equation by isolating the square root, removing it, then cube-root both sides to eliminate the remaining root, revealing x, with x constrained to be positive.

Locate the circle center at (-4, 4√3) and set the radius to 8 from a regular hexagon with side 8. Apply Pythagoras to get sqrt(48)=4√3, then write the equation (x+4)^2+(y-4√3)^2=64.

Calculate the shaded segment area by subtracting the hexagon from the circle, using six congruent 60-degree triangles with radius eight to obtain 16 sqrt 3.

This lecture uses parallel lines and angle chasing in a circle to identify a diameter and apply exterior angle equals interior opposite angle in a cyclic quadrilateral, concluding option c.

Use the conjugate to remove the denominator: multiply 4/(√11−√7) by (√11+√7)/(√11+√7). The denominator becomes 4, the numerator simplifies to 4(√11+√7), yielding √11+√7 (option B).

Calculate the remaining pool after removing one tenth and then five percent, simplify to 9/10 x minus 5% of 9/10 x, and find that 85.5% of x remains.

Analyze the inequality f(x) = (x-5)^2 < -3 and show there is no solution since a square is always nonnegative.

Model compound interest with a timeline: invest 8800 rand at 6% per year, compounded monthly, with two ten-thousand rand withdrawals, and determine the balance after the youngest withdrawal.

Solve a sine equation for theta minus 28 degrees using a reference angle of 50 degrees and quadrants iii and iv. Obtain the solution between 0 and 360 degrees as 258 degrees.

Use the double-angle identity cos(2x)=1-2 sin^2 x to solve for sin^2 15°, substituting cos 30°=sqrt(3)/2 and isolating sin^2 15° to obtain the result.

The lecture shows how to use first and second differences to detect a non-linear sequence, extend the pattern of increasing differences, and determine the seventh term as 103.

Differentiate to get gradient y' = 3x^2 − 6x + 12, locate turning point at x = 1, y = 9, and analyze the minimum value and gradient behavior.

Identify t as a root, an x value, specifically -7. Then compute t minus 70 squared, using the one over step, and the answer is a.

Explore solving exponential equations from question 107, showing 5^x-1 = 0 gives x=0 while 2^x+4 = 0 has no solution because 2^x is never negative; thus x=0.

Explore solving a trig expression using sum and difference identities: combine cos 3x cos 2x with sin 3x sin 2x to derive a single cosine term and simplify.

Factor out x to rewrite the equation as 3^x minus one plus thirty three x squared equals zero. Exponential and parabola graphs intersect at zero, giving x=0.

Equate the square's area to the circle's area via s^2 = pi r^2, then solve for s in terms of r (or r in terms of s) using square roots.

demonstrates factoring out v squared and manipulating v squared plus v to relate to eight plus seven, then computes 64 plus 7 to reach 71 and identify option d.

Apply Pythagoras to a rectangle with a 50 unit perimeter, relate the diagonal r to width w, and minimize r by setting its derivative to zero.

Find the vertex of the downward parabola -10x^2 + 20x + 16; with x = 1, substitute to get y = 26, so the maximum value is 26 (option C).

Convert 100 meters in 9 seconds to kilometers per hour by dividing by 1000 and by 60 twice, yielding 40 km/h as the speed.

compute the total surface area of a right triangular pyramid with four equilateral faces, using side length two, by multiplying one face area by four to choose option d.

Convert the equations to standard form, compute intercepts, and determine the angle between two lines using tangent values; apply the exterior angle of a triangle to conclude a 15-degree angle.

Learn key probability ideas including mutually exclusive and complementary events, the complement rule, and the P(A or B) formula, with examples identifying false statements.

Determine the number of coding permutations for twelve on/off switches by multiplying the options, yielding two to the power of twelve.

Work through a step-by-step trig simplification using sine, cosine, and tangent, applying quadrant sign rules and cofunction identities, and canceling terms to arrive at a cosecant form.

Derive the triangle K area in V as A = 1/2 t^2 sin(angle K) and analyze the resulting sine graph to determine its maximum and minimum values.

Optimize a revenue problem for a 72 rand paintball gun sold at 80 units; price increases by 4 rand, sales drop by 2, reaching weekly income at n = 12.

Analyze a sad face parabola gradient graph to identify turning points and determine option d as the most correct, recognizing the turning point's y-value may mislead.

Compute the probability of drawing at least one pink paper clip from an envelope containing four yellow, three white, and two pink clips, without replacement, across multiple draws.

Explore whether the question illustrates independent, mutually exclusive, complementary, or dependent events, and how latent state considerations influence the problem.

Manipulate inner expression using sine and cosine, convert sine squared to 1 minus cosine squared, apply the difference of squares to simplify, and yield 1 over (1 minus cos^2 a).

Explore how powers of two multiplied by five over three form a mixed fraction pattern, and show that even exponents yield a final value of two.

Compute the probability of A or B for independent events using P(A or B) = P(A) + P(B) - P(A)P(B), applying P(A)=0.888 and P(B)=0.1 to get 0.8992.

Determine how many ways four rugby nations can occupy the top three positions, given equal chances to win. Compute 4 × 3 × 2, yielding 24.

Count arrangements of Hannah’s six letters, a palindrome idea, by dividing six factorial by two factorial for each repeated letter. Conclude the probability of spelling Hannah in order is 1/90.

Substitute two points on the graph of 4x = a sin x + b to compute a and b, use sine values and quadrants, and solve the resulting simultaneous equations.

Identify the first term and common ratio of a geometric sequence, determine convergence, and compute the sum to infinity using a = 2 and r = 3/8, yielding 16/5.