Assignment of class 12 from chapters 2,5and 6

1. CLASS XII WORKSHEET (CHAPTERS 2,5,6)
1. If xy = ex-y , show that
𝑑𝑦
𝑑𝑥
=
𝑙𝑜𝑔 𝑥
{ 𝑙𝑜𝑔 (𝑥𝑒)}2
2. Sand ispouringfroma pipe @ 12 cm3
/sec.The fallingsandformsacone on the ground insuch a way thatthe
heightof cone is1/6 of radiusof base.How fastis the heightof cone increasingwhenthe heightis4cm?
3. Findthe equationof the tangenttothe curve y = √3𝑥 − 2 whichisparallel toline 4x - 2y + 5 = 0
4. Prove that: tan-11+tan-12+tan-13= 𝜋
5. The radiusof a spherical diamondismeasuredas7 cm withan error of 0.04cm . findthe approximate errorin
calculatingitsvolume.
6. Prove that : tan-1(
𝑐𝑜𝑠 𝑥
1+𝑠𝑖𝑛 𝑥
) =
𝜋
4
-
𝜋
2
, x ∈ (−
𝜋
2
,
𝜋
2
) .
7. A closedcylinderhasvolume 2156 cm3
. What will be the radiusof itsbase so that itsT.S.A is minimum?
8. Prove that the surface areaof a solidcuboid,of square base andgivenvolume,isminimumwhenitisacube.
9. At whatpointsonthe curve x2
+ y2
-2x -4y +1 = 0, the tangentsare parallel toy – axis
10. An openbox witha square base isto be made out of a givenquantityof cardboardof area c2
square units.Show
that the maximumvolume of the box is
𝑐3
6√3
cubicunits.
11. For what value of k, the following function is continuous at x = 0 : f(x) = {
1−𝑐𝑜𝑠 4𝑥
8 𝑥2 , 𝑥 ≠ 0
𝑘 , 𝑥 = 0
12. If x = a(cos t + t sin t), y = b(sin t – t cos t), Prove that
𝑑2
𝑦
𝑑𝑥2 =
𝑏 𝑠𝑒𝑐3
𝑡
𝑎2 𝑡
.
13. Find
𝑑𝑦
𝑑𝑥
, if yx + xy + xx = ab
14. Prove that : tan-1[
√ 𝟏+𝐱 𝟐
√ 𝟏+𝐱 𝟐
+√ 𝟏−𝐱 𝟐
−√ 𝟏−𝐱 𝟐
] = π
4
+
1
2
cos−1
x2
15. Solve forx:sin -1(1 – x) – 2 sin -1 x =
π
2
16. If x = a sin 2t (1 + cos 2t) and y = b cos 2t (1 – cos2t), show that (
𝑑𝑦
𝑑𝑥
)at t =
𝜋
4
=
𝑏
𝑎
.
17. Showthat the heightof cylinderof maximumvolume thatcanbe inscribedina sphere of radiusR is2R/√3 .
18. Findthe intervalsinwhichthe functionf givenby f(x) = sinx + cos x,0≤ 𝑥 ≤ 2𝜋 isstrictlyIncreasingor
decreasing.
19. Findthe approximate value of (26)1/3
20. Findall pointsonthe curve y = 4x3
– 2x5
at whichthe tangentspassesthroughthe origin.
21. A windowisinthe formof a rectangle surmountedbyasemicircularopening.Total perimeterof windowis10m.
Findthe dimensionsof the windowtoadmitmaximumlightthroughwhole opening.
22. Evaluate : tan {
1
2
cos−1 √5
3
} .
23. Prove that the greatest integer function defined by f(x) = [x], 0<x<3 , is not differentiable at
x = 1 and x = 2.