1. CLASS XII WORKSHEET (CHAPTERS 2,5,6)
1. If xy = ex-y , show that
ππ¦
ππ₯
=
πππ π₯
{ πππ (π₯π)}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 = β3π₯ β 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(
πππ π₯
1+π ππ π₯
) =
π
4
-
π
2
, x β (β
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 β 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
π3
6β3
cubicunits.
11. For what value of k, the following function is continuous at x = 0 : f(x) = {
1βπππ 4π₯
8 π₯2 , π₯ β 0
π , π₯ = 0
12. If x = a(cos t + t sin t), y = b(sin t β t cos t), Prove that
π2
π¦
ππ₯2 =
π π ππ3
π‘
π2 π‘
.
13. Find
ππ¦
ππ₯
, if yx + xy + xx = ab
14. Prove that : tan-1[
β π+π± π
β π+π± π
+β πβπ± π
ββ πβπ± π
] = Ο
4
+
1
2
cosβ1
x2
15. Solve forx:sin -1(1 β x) β 2 sin -1 x =
Ο
2
16. If x = a sin 2t (1 + cos 2t) and y = b cos 2t (1 β cos2t), show that (
ππ¦
ππ₯
)at t =
π
4
=
π
π
.
17. Showthat the heightof cylinderof maximumvolume thatcanbe inscribedina sphere of radiusR is2R/β3 .
18. Findthe intervalsinwhichthe functionf givenby f(x) = sinx + cos x,0β€ π₯ β€ 2π isstrictlyIncreasingor
decreasing.
19. Findthe approximate value of (26)1/3
20. Findall pointsonthe curve y = 4x3
β 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
cosβ1 β5
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.