Maths NBT Skills Preparation for Matrics in South Africa

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.

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.

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).

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.

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.

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.

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.

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.

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).

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.

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.

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.

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.

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.

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.

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.

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 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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.