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REVISION SET- 06, , CLASS – XII, SUB – MATH, 1) If y = 3e2x + 2e3x, prove that, , d2 𝑦, dx2, , 2) If y = etan-1x, show that (x2 + 1), , d2 𝑦, dx2, , -5, , 𝑑𝑦, 𝑑𝑥, , + 6y = 0., , + (2x – 1), , 𝑑𝑦, 𝑑𝑥, , = 0., f(x) = 4x2 – 12x + 9 in [0,3], , 3) Verify Roll’s theorem for the following functions., 4) Find c of Rolle’s theorem where., , TIME – 2 Hrs., F.M. – 100, , f(x) = 2x3 + x2 – 4x and - √2 < c < √2 ., 1, , 5) It is given that for the function f(x) = x3 + bx2 + ax + 5 on *1,3+, Rolle’s theorem holds with c = 2+ √3 ., Find the values of a and b., 6) Find c such that f’(c) =, , f(b) – f(a), b–a, , 11, , 13, , , where f(x) = x3 – 3x – 1, a = - 7 , b = 7 ., , 7) Find a point on the curve y = x3 – 3x, where the tangent is parallel to the chord joining points (1,-2) and (2,2)., 8) Find the rate of change of the area of a circle with respect to its radius r when,, i) r = 5 cm, , ii) r = 4 cm, , iii) r = 3 cm., , 9) A stone is dropped into a quiet lake and waves move in circles at a speed of 4 cm per second. At the instant,, When the radius of the circular waves is 10 cm, how fast is the enclosed area increasing ?, 10) The area of an expanding rectangle is increasing at the rate of 48cm2 per sec. The length of the rectangle is, always equal to the square of the breadth. At what rate is the length increasing at the instant when the, , breadth is 4.5 cm ?, 11) The total revenue received from the sale of x units of a product is given by R(x) = 3x2 + 36x + 5., Find the marginal revenue when x = 5, whereby marginal revenue we mean the rate of change of total revenue, with respect to the number of items sold at an instant., * Find the intervals in which the following functions are increasing or decreasing :, 12) f(x) = 2x3 – 15x2 + 36x + 1, , 13) f(x) = x4 – 8x3 + 22x2 – 24x + 21, , 4 sin θ, , 𝜋, , 14) Prove that y = + 2+cos θ – θ is an increasing function of θ in 0, 2 ., 15) Prove that the tangents to the curve y = x2 – 5x + 6 at the points (2,0) and (3,0) are at right angles., 16) Find the points on the curve, , x2, 9, , y2, , + 16 = 1 at which the tangents are, i) parallel to x – axis, , 17) Find the equations of all lines having slope 2 and being tangents to the curve y +, , 2, 𝑥−3, , ii) parallel to y – axis, , = 0., , 18) Find the approximate value of f(2.01), where f(x) = 4x2 + 5x + 2., 19) Find all the points of local maxima and minima and the corresponding maximum and minimum values of the, following functions, if any, , f(x) = x3 – 6x2 + 9x + 15., , 20) Find the two positive numbers whose sum is 15 and the sum of whose squares is minimum.