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![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.](https://image.slidesharecdn.com/classxiiworksheetchapters256-190501073135/75/Class-xii-worksheet-chapters-2-5-6-1-2048.jpg)
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- 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.