Class XII MATHS SAMPLE PAPER

1. MATHEMATICS SAMPLE PAPER CLASS-XII M.M. 80
SECTION-A
1. If A = (
4 6
7 5
) , thenwhat isA. ( Adj A)?
2. Showthat the function f(x) =x3
6x + 12x -18 isincreasinginR.
3. Write two pointsatwhich 𝑓( 𝑥) =
1
𝑥−[ 𝑥]
isnot continuous.
4. If f(x) = x2
+ 2 and g(x) =
x
x+1
, find gof (5).
5. P(A)=6/11, P(B) = 5/11 and P(𝐴 ∪ 𝐵) = 7/11 𝑓𝑖𝑛𝑑 𝑃(𝐵/𝐴)
6. Evaluate:∫ 𝑠𝑖𝑛7 𝑥
𝜋
−𝜋 𝑑𝑥
7. Evaluate:∫
𝑥+𝑐𝑜𝑠 6𝑥
3𝑥2+ 𝑠𝑖𝑛 6𝑥
dx
8. Write the principal value of tan-1
(√3) – cot-1
( -√3 ).
9. Find| 𝑥⃗| , if for a unitvector 𝑎⃗ ,( 𝑥⃗ − 𝑎⃗ ). ( 𝑥⃗ + 𝑎⃗ ) = 15
10. Findthe degree inthe differential equation
𝑑3 𝑦
𝑑𝑥3
+y2
+ 𝑒
𝑑𝑦
𝑑𝑥 = 0
11. Write Direction ratios of
𝑥−2
2
=
2𝑦−5
−3
= 𝑧 − 1 .
12. If A is a square matrix of order 3 such that | 𝐴𝑑𝑗 𝐴| = 225, find | 𝐴′|.
13. Write the distance between the parallelplanes2x –y + 3z = 4 and 2x – y + 3z = 18.
14. Form the differential equation of family of straight lines passing through origin.
15. A= {1,2,3} and B={4,5,6,7} and f={(1,4),(2,5),(3,6)} be a function from A to B. state whether f is one one or onto.
16. If two vectors 𝑎⃗ 𝑎𝑛𝑑 𝑏⃗⃗are :| 𝑎⃗| = 2, | 𝑏⃗⃗| = 3 and 𝑎⃗. 𝑏⃗⃗ = 4, find | 𝑎⃗ − 𝑏⃗⃗|.
17. In the matrix eqn.[
1 2
3 4
] [
4 3
2 1
] = [
8 5
20 13
]apply𝑅2 → 𝑅2 − 𝑅1
18. Showthat the functionf(x)=x3
+x2
+x+1donothave maximaor minima.
19. Evaluate:∫ 𝑙𝑜𝑔(
3+5𝑐𝑜𝑠 𝑥
3+5 𝑠𝑖𝑛 𝑥
)
𝜋
2
0 dx
20. Find the volume of the paralelopiped whose adjacent sides are represented by 𝑎⃗ , 𝑏⃗⃗ and 𝑐⃗ where
𝑎⃗= 3𝑖̂ − 2𝑗̂ + 5𝑘̂ , 𝑏⃗⃗ = 2𝑖̂ + 2𝑗̂ − 𝑘̂ , 𝑐⃗ = -4𝑖̂ + 3𝑗̂ + 2𝑘̂ .
SECTION-B
21. If E and F are independentevents, prove thatE’ and F’ are alsoindependent.
22. Verifythat y-cosy = x isa solutionof the differentialequation (ysiny+cosy+x)𝑦/=y.
23. Solve: sin -1(1 – x) – 2 sin -1 x =
π
2
24. If the radiusof a sphere ismeasuredas9m withan errorof 0.03m,find the approximate errorincalculatingthe
surface area.
25. Find the angle between the lines:
3+𝑥
3
=
1−𝑦
−5
=
𝑧+2
4
𝑎𝑛𝑑
𝑥+1
1
=
𝑦−4
1
=
5−3𝑧
−6

2. 26. If 𝑎⃗ , 𝑏⃗⃗ 𝑎𝑛𝑑 𝑐⃗ are three unitvectorssuchthat | 𝑎⃗| = 5, | 𝑏⃗⃗| = 12 and | 𝑐⃗| = 13 , and 𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗= 0⃗⃗ , findthe value of
𝑎⃗ . 𝑏⃗⃗ + 𝑏⃗⃗ . 𝑐⃗ + 𝑐⃗. 𝑎⃗ .
SECTION-C
27. Prove that the relationRinthe setA = {5, 6, 7, 8, 9} givenbyR = {(a,b) :| 𝑎 − 𝑏| isdivisible by2},isan equivalence
relation.Findall elementsrelatedtothe element6.
28. If x = a sin 2t(1 + cos 2t), y = b cos 2t( 1 – cos 2t) Show that (
𝑑𝑦
𝑑𝑥
)
𝑡=
𝜋
4
=
𝑏
𝑎
29. Evaluate:∫
𝑐𝑜𝑠𝑥 𝑑𝑥
(𝑠𝑖𝑛2 𝑥+1)(𝑠𝑖𝑛2 𝑥+4)
30.Solve the differential equation : (tan-1
y – x) dy = (1 + y2
) dx .
31. From a lotof 10 bulbs,whichinclude 3defectives,asample of 2 bulbsis drawn at random.Findthe probability
distributionof the numberof defective bulbs.
32. A dealerwishestopurchase ano.Of fansand sewingmachines.He hasonlyRs.5760 to investandhasspace for at
most20 items.A fancost himRs. 360 and a sewingmachine Rs.240. He expectstosell afanat a profitof Rs. 22
and a sewingmachine foraprofitof Rs. 18. Assumingthathe can sell all the itemsthathe buys,howshouldhe
investhismoneytomaximizehisProfit?
SECTION-D
33. Prove that: |
(𝑏 + 𝑐)2
𝑎𝑏 𝑐𝑎
𝑎𝑏 (𝑐 + 𝑎)2
𝑏𝑐
𝑐𝑎 𝑏𝑐 (𝑎 + 𝑏)2
| = 2abc(a + b + c)3
34. A tankwithrectangularbase and rectangularsides,openatthe topis to be constructedsothat its depthin2 m
and volume is8 m3
.If buildingof tankcostsRs.70 persq. meterforthe base and
Rs. 45 persq. Meterfor sides,whatisthe cost of leastexpensive tank?
35.Findthe area of the regionincludedbetweenthe curve 4y = 3x2
and line 2y = 3x + 12
36. Findthe coordinatesof the point,where the line
𝑥−2
3
=
𝑦+1
4
=
𝑧−2
2
intersectsthe plane x – y + z – 5 = 0. Also,find
the angle betweenthe lineandthe plane.