Common Mistakes 3U

7/23/2019 Common Mistakes 3U

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When skipping a question, leave a massive note and don't forget!

Double check that I have answered all parts of the question. Might be one mark at the end.

USE RADIANS AND DEGREES CORRECTLY

ALWAYS READ QUESTION AGAIN TO SEE IF IT'S BEEN ANSWERED!!

ALWAYS SIMPLIFY TO THE MOST EXTENT, IF TWO FRACTIONS, COMBINE INTO ONE,DON'T SKIP WHEN SUBSTITUTING VALUES

Remember to look at previous parts of question (i.e. part a). It might seem unrelated, but it might indeed be related to

part b.

Can the method of part 1 be repeated to figure out next part?

Trig must be in radians when differentiating, otherwise convert.

Don't skip steps. Rmb that if y = log(xa), then y/a = log(x), not y a = log(x)

When solving for x, and the x is inside absolute value or log, REMEMBER to double check as it might not lie in

domain.

Indices and Logs

When multiplying a variable, consider both positive and negative cases.

When obtaining solution of inequalities from graphs, always confirm algebraically

Inequalities

Integrating cos2x, forgetting to take half to the front

Remember half angle of cos2x involves cos4x, don't write cos2x again by accident.

When sketching label intercepts first (easier to do), then take middle value as max/min

When finding solutions, be sure to check domain, if x is between 0 and 2pi, make suitable changes, e.g. if it's x+

2pi/3, then x+2pi/3 would be between 2pi/3 and 8pi/3. Don't use general solutions unless it's necessary.

For sinx=cosx or cos2x=cosx, etc. Firstly, try to solve by trig identities before resorting to general solutions which is

very cumbersome.

When using auxiliary angle method, and =tan-1(strange number), then always provide approximation of that tan

inverse value. I.e. =tan-1(3/4)53o8'. Then proceed with the problem using that approximation. Don't use

tan-1(3/4) in place of the approximation.

For auxiliary angle method, double check whether it's TRIG FUNCTION(+) or (-). Hence making sure resulting

range and operations are applied correctly.

Remember cos(-/2 + A) = sin ( - A) = sin(A)

Trig function

Remember that auxiliary angle method can be used. Don't mix up cos double angle expansion with sin double angle expansion when using auxiliary.

General Solution

Whenever defining the base of triangles from O, define them in terms of htan(90-), where is the angle given.

Since h/cot() = htan(90-)

When finding angle subtended from the base of the cliff in the horizontal triangle, draw a top view with north

parallel lines. Centre angle will be from the sum of the alternate angles on parallel lines.

If two or three adjacent triangles, can cos rule be done in the larger triangle, then one of the smaller ones using the

same angle? Can equate and lead to interesting results.

3D trig

When substituting 5 years for n, and n is in months, substitute 60, not 5.Loan Repayments

Coordinate Geometry

Extension 1 Maths

mmon Mistakes Page 1


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For parametric of a circle x and y intercepts at 4, parametric is (4sin, 4cos), 4 not 2. There is no need to square

root it again as 4 is already the square root of 16

Division of intervals, make sure that if external, make both signs in top and bottom negative OR subst negative m or

n.

Division of intervals. Double check the direction. I.e. CP means (x1, y1) is C not P.

Take note of potential congruent triangles

Don't underestimate similar triangles. Take note, especially when their flipped, stacked on a not so obviousvertices.

Geometry

Congruent triangles and arc properties especially!

Don't forget to use exterior angle of triangles, and exterior angles of cyclic quad rules.

If there is an external point in the diagram to the circle, constructing tangents from that external point to the circle

may help. Especially if problems involve centres, or lengths (tangent secant rule).

Product of intercepts is (AX)(BX)=(CX)(DX), no matter what.

If tangent, try windsurfer. Don't just look at it, use pencil and try to label some angles.

Quote which triangle, when using exterior angle equal to sum of opposite interior angles.

When finding relationship between sides, i.e. 1/a = 1/b - 1/c, try sine rule. Anything squared or square root, try cosrule. Aim to make (side-angle) relationships the easiest by searching for the simplest angles.

Note converse to the angle in a semi-circle theorem!! The midpoint of the hypotenuse of a right angled triangle is

equidistant from all vertexes.

Circle Geometry

Substitute k value from calculator memory to double check. Don't risk straight calculation from working.

When eln(a) x t, this does NOT equal at; t is independent of the ln function.

Rate of growth is units/time. I.e. 210 elephants/year. Or percentage, 10%/year

When drawing the curve, MUST determine asymptote, and draw it in!!

Growth and Decay

When adding or subtracting root equations given to find the exact value, DOUBLE CHECK the operation performed,

and double check substitution.

Double check when using remainder theorem to find variables in polynomial.

Polynomial

Move everything on one side and treat this as a function.

Newton's method

Drawing f'(x), maximums and minimums are of f(x) are x-intercepts

When f''(x)


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Check whether it's a GP? If it is, apply formula for total sum and see whether it's the answer.

Treat questions deliberately and slowly, considering and even writing down all cases and double check logic.

Probability

REMEMBER that for inverse trig functions, y-axis is angle in radians and x-axis is real numbers.

If domain restriction on dy/dx, i.e. dy/dx not for x=1, -1. Then must test separately to prove certain

property involving those excluded points. I.e. constant function.

Note domain first. Remember to take into account exclusions when writing domain.

Hard subst values to make sure all domains are taken care of.

Double check resulting quadrant for answer. tan-1(2) + tan-1(3) = tan-1(-1), but note that LHS is in 2nd quadrant,

therefore, RHS = -/4 + k = 3/4 where k=1 [since 2nd Q], not -/4

When specifying domain, also remember domain restriction due to square root or log.

Since sin and tan inverse are odd, the negative can be brought out from within to make simplification easier.

Double check if answer is in domain of inverse trig, tan-1[tan(3/4)] = -/4

When making triangle, tan is opposite over adjacent, not the other way around.

When there's an absolute value within the inverse trig function, both cases must be considered. Any derivatives to

be obtained must be found for both cases.

If inside the inverse trig function, the domain is discontinuous, i.e. /x, D:{x: x>-, x


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Square root it and use the conditions to determine whether it's positive or negative.

Don't keep solution as simply v2 = , if they want it as a v(x) relationship.

i.e. v = f(x), v2/2 = f(x)2/2, d/dx both sides, LHS is acceleration.

If given v(x), and need to obtain a(x), square, divide two then differentiate in terms of x

Remember when doing rockets, that they have the initial velocity of something. Don't forget to include

this as conditions to find the constant of integration. Don't simply subst x=R, v=0 accidentally.

Remember when finding escape velocity of rocket, V>11200km/h, not equal to.

When v=sqrt(e-2x), remember that you can remove the square root along with the 2 in the power of e.

When square rooting v2, take the + or - that satisfies the given condition.

If given initial horizontal velocity, don't accidentally subst this into displacement x, rather subst into v/x

dot.

Crest to trough = 2m doesn't mean amplitude is 2m, but only 1 m.

Remember that v2 = n2(a2-x2), don't mix up the a and n, when asking for amplitude, don't give n2 by

accident. It can be confused with x=asin(nt+) where a is at the front.

Check quadrants, the equation may be set up such that a is negative. In which case, values of t in the 3rd

quadrant will yield negative answers which instead, must be positive.

More often or not, angles in radians, remember to change mode on calculator before

Remember with cos, there are two solutions, and 2-

Remember that the period is from high tide to low then to high again. Don't just count half a period

by accident.

When doing tide questions, always check whether I am indeed answering the question, did I set t=0 as a

certain time, if so, add it on.

When differentiating x=cos23t, dx/dt=2cos3t x-3sin3t = -6sin3tcos3t, don't forget when chain

differentiating the sin3t to bring out the 3.

Don't forget to divide or times the

If speed moving at 2ms-1, and substituting into v(x). Remember to consider both 2 and -2.

If the question says: Particle starts at its extreme positive position, or at origin, then that's a buzzword for

writing down x(t) and setting up appropriately.

State v2 > or = to 0. Therefore no solutions. Doesn't pass through zero.

Will particle pass through origin? Subst in x=0, must show all working: 0-0-32=-32.

Find amplitude of motion by finding extremes of motion.

Simple Harmonic Motion

If converse is required, then do the previous parts of the question provide this converse?

Look for key words, 'at least', 'no more than', which would suggest converse.

Probability

If 6th term, subst in r=5 as index number, since index has r=0 as well.

Binomial

Mi t k P 4

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Common Mistakes 3U