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A Level Maths questions with answers
y=af(x) - ** VERIFIED ANSWERS **✔✔stretch graph vertically by scale factor a y=f(x)+a - ** VERIFIED ANSWERS **✔✔translation in the y-axis of a y=f(ax) - ** VERIFIED ANSWERS **✔✔stretch graph horizontally by scale factor 1/a y=f(x+a) - ** VERIFIED ANSWERS **✔✔translation in the x-axis of -a y=-f(x) - ** VERIFIED ANSWERS **✔✔reflect over x-axis y=f(-x) - ** VERIFIED ANSWERS **✔✔reflection in y axis y=(ax+b) - ** VERIFIED ANSWERS **✔✔move -b horizontally, the horizontal stretch of 1/a Given a centre (a, b) and radius r, the equation of a circle is - ** VERIFIED ANSWERS **✔✔(x-a)^2+(y-b)^2=r^ Given x^2+3x+y^2-4x+10=0. How do you find the centre and radius of the circle - ** VERIFIED ANSWERS **✔✔Complete the square of the x terms, complete the square of the y terms The angle between the radius and the tangent is - ** VERIFIED ANSWERS **✔✔ 90 degrees Midpoint between A and B - ** VERIFIED ANSWERS **✔✔(average of x, average of ys)
Gradient of line between A and B - ** VERIFIED ANSWERS **✔✔Difference between y's over difference between xs Distance between A and B - ** VERIFIED ANSWERS **✔✔Use Pythagaros Equation of line between A and B - ** VERIFIED ANSWERS **✔✔m(x-x_a)=(y-y_b) kinematic equation without displacement - ** VERIFIED ANSWERS **✔✔v=u + at kinematic equation without final velocity - ** VERIFIED ANSWERS **✔✔s= ut+at^ kinematic equation without acceleration - ** VERIFIED ANSWERS **✔✔s=(u+v)/t kinematic equation without time - ** VERIFIED ANSWERS **✔✔v^2=u^2+2as The area under speed time graph is - ** VERIFIED ANSWERS **✔✔the displacement The gradient of a speed time graph is - ** VERIFIED ANSWERS **✔✔acceleration differentiate x^n - ** VERIFIED ANSWERS **✔✔nx^(n-1) Differentiate x - ** VERIFIED ANSWERS **✔✔ 1 Differentiate 3 - ** VERIFIED ANSWERS **✔✔ 0 To find the gradient of a tangent a point P on the curve - ** VERIFIED ANSWERS **✔✔Sub the x coordinate of into the gradient function
The vertical acceleration is - ** VERIFIED ANSWERS **✔✔-9. You can often find the horizontal displacement by - ** VERIFIED ANSWERS **✔✔Working with the vertical displacement The highest point of the particle is when - ** VERIFIED ANSWERS **✔✔the vertical vecocity is 0 The speed of the particle is - ** VERIFIED ANSWERS **✔✔the combination of the horizontal and vertical velocity To work out the speed of the particle you use - ** VERIFIED ANSWERS **✔✔Pythagaros To work out the angle of the speed at a certain time you use - ** VERIFIED ANSWERS **✔✔Elementary Trig P(A ∩ B)= - ** VERIFIED ANSWERS **✔✔P(A)P(B|A)=P(B)P(A|B) P(A∪B)= - ** VERIFIED ANSWERS **✔✔P(A)+P(B)-P(A∩B) Independence means in terms of the intersection - ** VERIFIED ANSWERS **✔✔P(A∩B)=P(A)P(B) Independence means in terms of a given event - ** VERIFIED ANSWERS **✔✔P(A| B)=P(A) P(A')= - ** VERIFIED ANSWERS **✔✔1-P(A) P(A'∩B) or similar requires - ** VERIFIED ANSWERS **✔✔A venn diagram
Arc length is - ** VERIFIED ANSWERS **✔✔rθ Sector area is - ** VERIFIED ANSWERS **✔✔1/2 θr^ Sin rule is - ** VERIFIED ANSWERS **✔✔a/sinA=b/sinB Cosine rule is - ** VERIFIED ANSWERS **✔✔a²=b²+c²-2bc cosA Segment area is - ** VERIFIED ANSWERS **✔✔Sector - triangle Triangle area os - ** VERIFIED ANSWERS **✔✔1/2 abSinC A fish or a ball or a book etc. is represented by - ** VERIFIED ANSWERS **✔✔a dot A fish or a ball or a book etc. is modelled by - ** VERIFIED ANSWERS **✔✔a particle We use particles in models because - ** VERIFIED ANSWERS **✔✔all of the weight is concentrated at one point Inextensible means - ** VERIFIED ANSWERS **✔✔Acceleration is constant throughout the model Newtons first law is - ** VERIFIED ANSWERS **✔✔A particle at rest or moving at constant speed will remain at rest or at constant speed unless a different force is applies Newtons third law is - ** VERIFIED ANSWERS **✔✔When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body
no real roots means - ** VERIFIED ANSWERS **✔✔b^2-4ac< When a particle is in limiting equilibrium - ** VERIFIED ANSWERS **✔✔Friction=μR If a particle is moving friction acts - ** VERIFIED ANSWERS **✔✔in the direction opposite to movement If a particle is static friction acts - ** VERIFIED ANSWERS **✔✔in the direction opposite to the potential movement P=f(x)/(x-a)(x-b) - ** VERIFIED ANSWERS **✔✔ P=f(x)/(x-a)(x-b)^2 - ** VERIFIED ANSWERS **✔✔P=A/(x-a)+ B/(x-b)+C/(x-b)^ P=f(x)/g(x), if f(x) is of the same of higher order then - ** VERIFIED ANSWERS **✔✔Use algebraic division first Log(xy)= - ** VERIFIED ANSWERS **✔✔Log(x)+log(y) Log(x/y)= - ** VERIFIED ANSWERS **✔✔log(x) - log(y) log(x^n)= - ** VERIFIED ANSWERS **✔✔n log x log1= - ** VERIFIED ANSWERS **✔✔ 0 log_a(a^x)= - ** VERIFIED ANSWERS **✔✔x a^(log_a(y)) - ** VERIFIED ANSWERS **✔✔y
The definition of e is - ** VERIFIED ANSWERS **✔✔if y=a^x=dy/dx, then a=e The inverse of e^x is - ** VERIFIED ANSWERS **✔✔ln(x) if f(x) is a polynomial and f(a)=0 then - ** VERIFIED ANSWERS **✔✔(x-a) is a factor of f(x) Given that f(x) is a polynomial, and then when f(x) is divided by (x-a) there is a remainder B, this implies - ** VERIFIED ANSWERS **✔✔f(a)=b To find a stationary point - ** VERIFIED ANSWERS **✔✔the first derivative = A stationary point is a maximum if - ** VERIFIED ANSWERS **✔✔the second derivative < A stationary point is a minimum if - ** VERIFIED ANSWERS **✔✔the second derivative > A function is increasing if - ** VERIFIED ANSWERS **✔✔the first derivative ≥ 0 A function is decreasing if - ** VERIFIED ANSWERS **✔✔the first derivative ≤ 0 A function is strictly increasing if - ** VERIFIED ANSWERS **✔✔the first derivative > A point of inflection is - ** VERIFIED ANSWERS **✔✔the second derivative =0 and a higher derivative exists Concave is when - ** VERIFIED ANSWERS **✔✔the second derivative <
A unit vector has modulus - ** VERIFIED ANSWERS **✔✔ 1 The unit vector in the direction of vector a is - ** VERIFIED ANSWERS **✔✔a (1/| a|) d/dt(sint) - ** VERIFIED ANSWERS **✔✔cost d/dt(cost) - ** VERIFIED ANSWERS **✔✔-sint d/dt(tant) - ** VERIFIED ANSWERS **✔✔sec^2t d/dt(cott) - ** VERIFIED ANSWERS **✔✔-csc^2t d/dt(sect) - ** VERIFIED ANSWERS **✔✔secttant d/dt(cosect) - ** VERIFIED ANSWERS **✔✔-cosectcott d/dx(ln(f(x)) - ** VERIFIED ANSWERS **✔✔f'(x)/f(x) d/dx(d^f(x)) - ** VERIFIED ANSWERS **✔✔f'(x)e^f(x) d/dx(a^x) - ** VERIFIED ANSWERS **✔✔a^xlna d/dx(sin(fx) - ** VERIFIED ANSWERS **✔✔ Given two connected particles. How many F=ma equations should be formed - ** VERIFIED ANSWERS **✔✔ 2
What does an inelastic string imply - ** VERIFIED ANSWERS **✔✔That tension is constant throughout the model, therefore acceleration is the same for both particles Correlation describes - ** VERIFIED ANSWERS **✔✔the nature of the linear relationship between two variables The regression line of p on t is written in the form - ** VERIFIED ANSWERS **✔✔p=a+bt The coefficient b tells you - ** VERIFIED ANSWERS **✔✔the change in p for every unit of t The coefficient a tell you - ** VERIFIED ANSWERS **✔✔the value of p, when t= When describing the values a and b your answers must be in - ** VERIFIED ANSWERS **✔✔context Extrapolation is - ** VERIFIED ANSWERS **✔✔bad The PMCC describes - ** VERIFIED ANSWERS **✔✔the linear correlation between two variables PMCC is between - ** VERIFIED ANSWERS **✔✔1 and - Given umbrella sales=a+b( monthly rain), and PMCC is near 1 - ** VERIFIED ANSWERS **✔✔As monthly rain increase, umbrella sales increas Given long john sales=a+b( average temperature), and PMCC is near -1 - ** VERIFIED ANSWERS **✔✔As temperature decreases, sales of long johns increase
if H_1: p≠0.2, the C.R. will be in the form - ** VERIFIED ANSWERS **✔✔x≤a1, x≥a nCr - ** VERIFIED ANSWERS **✔✔n!/(r!(n-r)!) (a+b)^5 - ** VERIFIED ANSWERS **✔✔5C0(a)^5(b)^0+5C1(a)^4(b)^1+5C2(a)^3(b)^2+5C3(a)^2(b)^3+5C4(a)^ (b)^4+5C5(a)^0(b)^ The integration of x^n is - ** VERIFIED ANSWERS **✔✔(1/n+1)x^(n+1) When integrating indefinitely you must include - ** VERIFIED ANSWERS **✔✔+ C To find the area under a curve above the x axis. - ** VERIFIED ANSWERS **✔✔The biggest number goes on top of the integral sign To find the area under a curve below the x axis. - ** VERIFIED ANSWERS **✔✔The smallest number goes on top of the integral sign To find the area between two curves - ** VERIFIED ANSWERS **✔✔Take the bottom curve from the top curve and integrate sin (A+B) - ** VERIFIED ANSWERS **✔✔sinAcosB+cosAsinB cos(A+B) - ** VERIFIED ANSWERS **✔✔cosAcosB-sinAsinB sin(2x) - ** VERIFIED ANSWERS **✔✔2sinxcosx cos(2x) - ** VERIFIED ANSWERS **✔✔cos^2x-sin^2x cos2A - ** VERIFIED ANSWERS **✔✔1-2sin^2A
cos2B - ** VERIFIED ANSWERS **✔✔2cos^2B- cos^2(x) - ** VERIFIED ANSWERS **✔✔(1+cos2x)/ sin^2(x) - ** VERIFIED ANSWERS **✔✔(1-cos2x)/ A mapping - ** VERIFIED ANSWERS **✔✔transforms one set of numbers into a different set of numbers A function - ** VERIFIED ANSWERS **✔✔is a mapping where every input has one output A one to one mapping is a - ** VERIFIED ANSWERS **✔✔function A one to many mapping is - ** VERIFIED ANSWERS **✔✔a function A many to one mapping - ** VERIFIED ANSWERS **✔✔is not a function A function has an inverse only if it is - ** VERIFIED ANSWERS **✔✔one to one The domain is - ** VERIFIED ANSWERS **✔✔what goes in (x-axis)` The range is - ** VERIFIED ANSWERS **✔✔what comes out (y axis) The domain for the inverse is the same as - ** VERIFIED ANSWERS **✔✔the range of the original function The range of the inverse is the same as - ** VERIFIED ANSWERS **✔✔the domain of the original function
To find a stationary point of an implicit function - ** VERIFIED ANSWERS **✔✔Differentiate; set dy/dx=0, Sub the equation back into the original equation Integrate 1/(bx+c) - ** VERIFIED ANSWERS **✔✔(1/b) ln(bx+c) Integrate 1/(bx+c)^2 - ** VERIFIED ANSWERS **✔✔(-1/b) (bx+c)^- Integrate (f'(x))/(f(x)) - ** VERIFIED ANSWERS **✔✔ln(f(x)) if x>0 The graph y=f(|x|) is - ** VERIFIED ANSWERS **✔✔y=f(x) if x<0 The graph y=f(|x|) is - ** VERIFIED ANSWERS **✔✔reflection of y=f(-x) if x>0, the graph y=f|(x)| is - ** VERIFIED ANSWERS **✔✔y=f(x), except all parts below the x-axis are reflected to above the x-axis Given X∼N(10, 4). What is the mean? - ** VERIFIED ANSWERS **✔✔ 10 Given X∼N(10, 4). What is the standard deviation? - ** VERIFIED ANSWERS **✔✔ 2 To find P(X<a)=0.7. The area to input in the inverse normal mode is - ** VERIFIED ANSWERS **✔✔0. To find P(X>a)=0.7. The area to input in the inverse normal mode is - ** VERIFIED ANSWERS **✔✔0. P(X≤10) - ** VERIFIED ANSWERS **✔✔P(X<10) Given X∼N(μ, σ ) and P(X<4)=0.85. You normalize by - ** VERIFIED ANSWERS² **✔✔P(Z<(4-μ)/σ)=0.
The nth term of a geometric series is - ** VERIFIED ANSWERS **✔✔ar^(n-1) The sum to n terms of a geometric series is - ** VERIFIED ANSWERS **✔✔(a(1- r)^n)/(1-r) The infinite sum of a geometric series is - ** VERIFIED ANSWERS **✔✔a/(1-r) The necessary condition for a geometric series that converges is - ** VERIFIED ANSWERS **✔✔|r|< Given Asinx+Bcosx, R= - ** VERIFIED ANSWERS **✔✔R^2=A^2+B^ Given Asinx+Bcosx= R sin(x+θ). The first thing any reasonable human does is - ** VERIFIED ANSWERS **✔✔Expand R sin(x+θ)=Rsinxcosθ+Rcosxsinθ Given Asinx+Bcosx=Rsinxcosθ+Rcosxsinθ - ** VERIFIED ANSWERS **✔✔A=Rcosθ, B=Rsinθ Given A=Rcosθ, B=Rsinθ - ** VERIFIED ANSWERS **✔✔B/A=tanθ In a ladder/hinge question the three equations you need are - ** VERIFIED ANSWERS **✔✔up=down, left=right, clockwise=anticlockwise When working with a hinge there are - ** VERIFIED ANSWERS **✔✔two parts to the reaction force, horizontal and vertical In a horizontal moments question you need up= - ** VERIFIED ANSWERS **✔✔down In a moments question you need clockwise moments= - ** VERIFIED ANSWERS **✔✔anti clockwise moments
Given ∫f(x)dx, and a substitution u=g(x). The first step is - ** VERIFIED ANSWERS **✔✔∫f(x) (dx/du) du Given u=sin x, dx/du is found by - ** VERIFIED ANSWERS **✔✔Differentiate u, and flip. ie du/dx=-cos x, dx/du=1/-cosx Given ∫f(x)dx, and a substitution u=g(x). The second step is - ** VERIFIED ANSWERS **✔✔to replace all x terms with u terms ∫f(x)g'(x)dx= - ** VERIFIED ANSWERS **✔✔f(x)g(x)-∫f'(x)g(x)dx If your second integral is more complicated than your first integral - ** VERIFIED ANSWERS **✔✔Go back and start again, but define the f(x) and g'(x) the other way round L in Liate is - ** VERIFIED ANSWERS **✔✔logs i in Liate is - ** VERIFIED ANSWERS **✔✔indices t in Liate is - ** VERIFIED ANSWERS **✔✔trig a in liate is - ** VERIFIED ANSWERS **✔✔a^x e in liate is - ** VERIFIED ANSWERS **✔✔e^x The volume of a cube is increasing at a rate of 5cm^3. - ** VERIFIED ANSWERS **✔✔dv/dt= Population grows at a rate proportional to the population - ** VERIFIED ANSWERS **✔✔dp/dt=kp
dv/dt - ** VERIFIED ANSWERS **✔✔=(dv/dx)(dx/dt) Given dy/dx=f(x)g(y). The first step is - ** VERIFIED ANSWERS **✔✔Divide by g(y). Leaving (1/g(y)) dy/dx=f(x) Given dy/dx=f(x)g(y). The second step is - ** VERIFIED ANSWERS **✔✔Integrate both sides leaving ∫(1/g(y))dy=∫f(x) dx Given dy/dx=f(x)g(y). The third step is - ** VERIFIED ANSWERS **✔✔Carry out both integrations. Given x=t+2, y=4+t^2 - ** VERIFIED ANSWERS **✔✔rearrange both to find t=, then set equal to each other, y=4+(x-2)^ x=cos^2(t), y=sin(t) - ** VERIFIED ANSWERS **✔✔find what cost=, and sint= then use cos^2t+sin^2t=1 hence x+y^2= Given x=3+sec^2(t), y=4tan(t) - ** VERIFIED ANSWERS **✔✔Find what tant=, and sect= then use tan^2t+1=sec^2t hence (y/4)^2+1=(x-3)^ dy/dx parametrically = - ** VERIFIED ANSWERS **✔✔dy/dx (dt/dx) Parametric integration is - ** VERIFIED ANSWERS **✔✔∫y (dx/dt)dt (1+x)^n= - ** VERIFIED ANSWERS **✔✔1+nx+n(n-1)(x^2)/2!+n(n-1)(n-2) (x^3)/3!+... Given (a+bx)^n - the first step is - ** VERIFIED ANSWERS **✔✔a^n (1+(b/a)x)^n Given (1+(b/a)x)^n the range of values that an expansion is valid for is - ** VERIFIED ANSWERS **✔✔|(b/a)x|<
