The diagram shows the points P (0, 2) and Q (-4, 0). The point M is the midpoint of PQ. The line MN is perpendicular to PQ and meets the y axis at N.
(i) Show that the gradient of PQ is 12.
(ii) Find the coordinates of M.
(iii) Find the equation of the line MN.
(vi) The point R lies in the second quadrant, and PNQR is a rhombus. Find the coordinates of R.
Find the equation of the tangent to the curve y = exat the point where x = 1 .
A particle moves on a straight line so that its velocity, mv/s , at any time t seconds is given by v=(t-1)4+t2 , t ≥ 0 Find the initial velocity and show that the particle never stops. Find the initial acceleration of the particle. Find the least value of the velocity. Sketch the velocity-time graph. Find the distance travelled by the particle in the first 2 seconds.
In the diagram, the shaded region is bounded by y=loge(x-2) the x axis and the line x= 7. Find the exact value of the area of the shaded region.
A cone is inscribed in a sphere of radius a, centred at O. The height of the cone is x and the radius of the base is r, as shown in the diagram. (i) Show that the volume, V,of the cone is given by V=13π(2ax2-x3) (ii) Find the value of x for which the volume of the cone is a maximum. 3 You must give reasons why your value of x gives the maximum volume.
In the diagram, O is the centre of a circle of radius 6 cm and PQ is a chord of length 4 cm. ABCD is a rectangle constructed in the minor segment cut off by chord PQ. OM is drawn perpendicular to PQ so that M is the mid-point of both chord PQ and side AB. N is the mid-point of side CD. Let ∠CON=θ , θ in radians. Show that OM = 2 cm Show that 0<θ<1 Show that the area, a , of rectangle ABCD is given by a=43sinθ3cos θ-1 Show that dadθ=4323cos2θ-cos θ-3 Find the maximum area of the rectangle.
Use Simpson's Rule with three function values to approximate ∫0118x2-3x+1dx
O is the centre of the circle containing the arc AB. P is the midpoint of OA and Q is the midpoint of OB. ∠AOB = 120 ° OQ = OP = 5 cm (i) Find the exact length of the arc AB. (ii) Find the shaded area PQBA in exact form. Find, justifying your answer, any point(s) of inflexion on the curve y=1x3-x2+1
The diagram shows the points A(−2,2) , B(4,6) , O(0,0) and C for the parallelogram OABC. The equation of line AB is 2x − 3y +10 = 0 Do NOT prove this (i) Show that the length of AB is 213 (ii) Calculate the perpendicular distance from O to the line AB. (iii) Calculate the area of the parallelogram OABC.
Find the obtuse angle θ correct to the nearest minute
Without using calculus, sketch the curve y = ex – 2
By letting u = x13 , or otherwise, solve x23+ x13 − 6 = 0 Give your answer as an exact value.The graph below shows the function f''x Copy the diagram into your writing books and draw the primitive graph on the same axes.
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