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SAMPLE PAPER -2015
MATHEMATICS
CLASS β XII
Time allowed: 3 hours Maximum marks: 100
General Instructions:
1. All questions are compulsory.
2. The question paper consists of 26 questions divided into three sections-A, B and C. Section A
comprises of 6 questions of one mark each, Section B comprises of 13 questions of four marks
each and Section C comprises of 7 questions of six marks each.
3. All questions in Section A are to be answered in one word, one sentence or as per the exact
requirement of the question.
4. There is no overall choice. However, internal choice has been provided in 4 questions of four
marks each and 2 questions of six mark each. You have to attempt only one of the alternatives in
all such questions.
5. Use of calculators is not permitted.
Section A
Q1. Evaluate: tanβ1β3 β sec
β1 (β2)
Q2 Find gof if f(x) =8 x3
, g(x)= βπ₯3
.
Q3. If [3π₯ β 2π¦ 5
π₯ β2] = [
3 5β3 β2
] , find the value of y .
Q4. Evaluate: | π ππ 300 πππ 300
βπ ππ600 πππ 600|
Q5. Find p such that p
zyx
321 and
142
zyx
are perpendicular to each other.
Q6. Find the projection of οΏ½οΏ½ on οΏ½οΏ½ if οΏ½οΏ½ . οΏ½οΏ½ =8 and οΏ½οΏ½ = 2π +6π + 3οΏ½οΏ½
Section B
Q7. πΏππ‘ π΄ = πππ, πππ β ππ π‘βπ ππππππ¦ ππππππ‘πππ ππ π΄ πππππππ ππ¦
(π, π) β (π, π) = (π + π, π + π). Show that β is commutative and associative.
Find the identity element for β on A, if any.
Q8. Prove πΆππ‘β1 (β1+sinπ₯+β1βsinπ₯
β1+sinπ₯β β1βsinπ₯) =
π₯
2 , xβ (0,
π
4)
OR
Solve for x . 2 π‘ππβ1(cos π₯) = π‘ππβ1(2 πππ ππ π₯)
Q9. By using properties of determinants, show that:
|1 + π2 β π2 2ππ β2π
2ππ 1 β π2 + π2 2π2π β2π 1 β π2 β π2
| = (1 + π2 + π2)3
Q10. If cos y = x cos(a + y) with cos a β Β± 1, prove that ππ¦
ππ₯=
πππ 2( π+π¦)
sinπ
OR
Find ππ¦
ππ₯ of the function (cos π₯)π¦ = (cos π¦)π₯
Q11. If = (π‘ππβ1π₯)2 , show that (π₯2 + 1)2π¦2 + 2π₯(π₯2 + 1)π¦1 = 2
Q12
If f(x) =
{
1βcos4π₯
π₯2 π€βππ π₯ < 0
π, π€βππ π₯ = 0 βπ₯
β16+βπ₯β4 , π€βππ π₯ > 0
and f is continuous at x = 0, find the value of a.
Q13. Find the intervals in which the function f given by f(x) = 2x3 β 3x
2 β 36x + 7 is
(a) strictly increasing (b) strictly decreasing
Q14. Show that [οΏ½οΏ½ + b οΏ½οΏ½ + οΏ½οΏ½ π + οΏ½οΏ½ ] =2[οΏ½οΏ½ οΏ½οΏ½ π ]
OR
Find a unit vector perpendicular to each of the vectors (οΏ½οΏ½+ οΏ½οΏ½) πππ ( π - οΏ½οΏ½) where οΏ½οΏ½ = π + π +
οΏ½οΏ½ and οΏ½οΏ½ = π + 2 π + 3οΏ½οΏ½ .
Q15. Evaluate: β«2π₯
(π₯2+1)(π₯2+3)ππ₯ dx
Q16. Evaluate: β« ππ₯ (1+sinπ₯
1+cosπ₯) dx
Q17. Using properties of definite integrals, evaluate:
β«π₯
4 β πππ 2π₯ππ₯
π
0
OR
Using properties of definite integrals, evaluate:
β« πππ(1 + tan π₯)ππ₯
π4β
0
Q18. . A man is known to speak truth 3 out of four times .He throw a die and report that it is a
six find the probability that it is actually six. Which value is discussed in this question?
Q19. Find the shortest distance between the lines
)k2j5-i(3k-ji2r
and )ΛΛΛ2(ΛΛ
kjijir
Section C
Q20. Two institutions decided to award their employees for the three values of resourcefulness,
competence and determination in the form of prizes at the rate of Rs. x , Rs.y and Rs.z
respectively per person. The first Institute decided to award respectively 4,3 and 2 employees
with a total prize money of Rs.37000 and the second Institute decided to award respectively 5, 3
and 4 employees with a total prize money of Rs.47000.If all the three prizes per person together
amount to Rs.12000, using matrix method find the value of x, y and z. Write the values described
in the question.
Q21. Solve the differential equation
ππ¦
ππ₯+ 2 π¦ tan π₯ = sin π₯ , given that y = 0 where x =
π
3
Q22. ) Find the equation of plane passing through the line of intersection of the planes
x + 2y + 3 z = 4 and 2 x + y β z + 5 = 0 and perpendicular to the plane 5 x + 3y β 6 z + 8 = 0.
Q23. The sum of the perimeter of a circle and square is k, where k is some constant. Prove that
the sum of their areas is least when the side of square is double the radius of the circle.
OR
Show that the volume of greatest cylinder that can be inscribed in a cone of height h and semi
vertical angle Ξ± is, 23 tan27
4h .
Q.24 There are a group of 50 people who are patriotic, out of which 20 believe in non-violence.
Two persons are selected at random out of them, write the probability distribution for the
selected persons who are non- violent. Also find the mean of the distribution. Explain the
importance of non- violence in patriotism.
Q25. Using integration Find the area lying above x-axis and included between the circle
π₯2 + π¦2 = 8 x and parabola π¦2 = 4 x
OR
Using the method of integration, find the area of the region bounded by the following lines
5x - 2y = 10, x + y β 9 =0 , 2x β 5y β 4 =0
Q26. Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of
the mixture contain at least 8 units of vitamin A and 11 units of vitamin B. Food P costs Rs
60/kg and Food Q costs Rs 80/kg. Food P contains 3 units /kg of vitamin A and 5 units /kg of
vitamin B while food Q contains 4 units /kg of vitamin A and 2 units /kg of vitamin B.
Determine the minimum cost of the mixture? What is the importance of Vitamins in our body?
Pratima Nayak,KV Teacher
Marking Scheme Second Pre Board Examination Mathematics-2014 Kolkata Region
Q1. - π
3 Q2.2x Q3. y = -6 Q4. 1 Q5. p = - 2 Q6. 8/7 1 X 6
Q7. (π, π) β (π, π) = (π, π) β (π, π) for commutativity. 2
11
((π, π) β (π, π)) β (π, π) = (π, π) β ((π, π) β (π, π)) for associativity. 2
11
No identity element. 1
______________________________________________________________________ Q8.
1 1/2
2
11
2
11
Β½ _______________________________________________________________________ OR
1
1
2
11
1/2
_________________________________________________________________________________________
Ans 9.
Applying R1 β R1 + bR3 and R2 β R2 β aR3, we have: 1
1
Expanding along R1, we have: (1 + π2 + π2)3 1
Answer 1
____________________________________________________________________________________
Q10.π₯ = cosπ¦
cos (π+π¦) 1
ππ₯
ππ¦ = =
sina
πππ 2( π+π¦) 1+1
ππ¦
ππ₯ =
πππ 2( π+π¦)
sinπ 1
________________________________________________________________________________________________
OR
Taking logarithm on both the sides,
Differentiating both sides
1
1
_______________________________________________________________________
Q11. 1
1
1
1
_____________________________________________________________________________
Q12
limπ₯β0βπ(π₯) =2π ππ22π₯
π₯2= 8 1Β½
RHL on rationalization limπ₯β+π(π₯) =8 1Β½
a = 8 1
________________________________________________________________________
Q13.
1
x = β 2, 3 1/2 Intervals: (- β,-2),( -2,3) and (3, β) 1
(f) is strictly increasing in (- β,-2) and (3, β) and strictly decreasing in interval ),( -2,3) 1Β½
___________________________________________________________________________
Q14. = {(οΏ½οΏ½ Γ b)+(οΏ½οΏ½ Γ c) + (οΏ½οΏ½ Γ b) + (οΏ½οΏ½ Γ c)} . (π+οΏ½οΏ½ ) 1
= (οΏ½οΏ½ Γ b) . π + (οΏ½οΏ½ Γ c). π+( οΏ½οΏ½ Γ c). π + (οΏ½οΏ½ Γ b) . οΏ½οΏ½ + (οΏ½οΏ½ Γ c). οΏ½οΏ½+( οΏ½οΏ½ Γ c). οΏ½οΏ½ 2
=[οΏ½οΏ½ οΏ½οΏ½ π ] + [οΏ½οΏ½ οΏ½οΏ½ π ] 2
1
=2[οΏ½οΏ½ οΏ½οΏ½ π ] 2
1
_________________________________________________________________________ OR
(π + οΏ½οΏ½) = 2π + 3 π + 4οΏ½οΏ½. ( οΏ½οΏ½- οΏ½οΏ½) = 0οΏ½οΏ½ β π β 2οΏ½οΏ½. 1
(π + οΏ½οΏ½)π ( π - οΏ½οΏ½) = β2π + 4 π β 2οΏ½οΏ½. 2
11
|(π + οΏ½οΏ½)π ( π - οΏ½οΏ½ | =β24 1
2
1
1
β24(β2π + 4 π β 2οΏ½οΏ½) Β½
_________________________________________________________________________
Q15. Let x2 = t β 2x dx = dt 1/2
1
A=1/2, B=-1/2 1/2
2
_____________________________________________________________________________________________________
Q16. 12
1
1
Β½+1 _____________________________________________________________________
Q17 Use of property
β« π((π₯)ππ₯ = π
0 β« π((π β π₯)ππ₯ = π
0 , I = β«
πβπ₯
4βπππ 2π₯ππ₯
π
0 1/2
2I = π β«π ππ2π₯
3+4 π‘ππ2π₯ππ₯
π
0 1/2
Use of property β« π((π₯)ππ₯ = 22π
0 β« π((π β π₯)ππ₯ ππ π(2π β π₯) = π(π₯) π
0
2I = 2π/4 β«π ππ2π₯
3+4 π‘ππ2π₯ππ₯
π/2
0 1
tan x = t, sec2x dx =dt 1
& Correct result I =π2
4β3 1
_________________________________________________________________
OR 1
2
1
_________________________________________________________________________
Q18. P(T) =3/4, P(F) =1/4 1
E: getting a six,F: he is not getting a six
P(E/T)= 1/6, P(E/F)=5/6 1
By Bay,s Theorem P(T/E)= π(π)π(
πΈ
π)
π(π)π(πΈ
π)+π(πΉ)π(
πΈ
πΉ) =3/8
2
11
Truthfulness Β½
Q19.
59bb
Λ7ΛΛ31
,ΛΛ
21
2
12
kjibb
kiaa
1+1+1
shortest distance =
21
2221 )).((
bb
aabb
=
59
10 1
_____________________________________________________________________ Q20. 4 x + 3 y + 2z = 37000, 5 x + 3 y + 4z = 47000, x + y + z = 1200 1
|A| = - 3 β 0 so A-1 exists. X = A-1 B 1/2
Cofactors of A 2
[β1 β1 2β1 2 16 β6 3
]
Adjoint A 1/2 X = 4000 ,y = 5000, z = 3000 1Β½ Values Β½ _______________________________________________________________ Q21. P = 2tan x, Q = sin x
I.F = π ππ2π₯ 1Β½
y π ππ2π₯ = β« sin π₯ π ππ2π₯ ππ₯ + πΆ 1
y π ππ2π₯ = sec x + C 1Β½
y = 1
secπ₯ +
πΆ
π ππ2π₯= cos x + C πππ 2π₯ -------------------(1) 1Β½
putting x = π
3 and y = 0 in eqn (1) C = -2 Β½
Y= cos x - 2πππ 2π₯ 1 __________________________________________________________________________ Q22. Sol: The required plane is (x + 2 y + 3 z ) + k (2 x + y β z +5 )= 0 1
Or (1 + 2 k)x +(2 + k)y +(3 - k)z-4 + 5k = 0 1
5(1+2k) +3 (2+k) -6 (3-k)=0 1
Solving k = 7/19 1
The equation of the plane is : 33 x + 45y +50z = 41 . 2
________________________________________________________________________________
Q23.
Let r be the radius of the circle and a be the side
1+1 1/2 2 1/2 a =2 r 1 ___________________________________________________________________ OR fixed height (h) and semi-vertical angle (Ξ± )
relation of h and H 1+1/2 ( figure)
1
+ Β½ Result 1 _________________________________________________________________
24. Let X = The number of non -violent persons out of selected two. So, X = 0, 1, 2 1/2
P(X = 0) = 245
87
2
50
2
30
C
C P(X = 1) =
245
120
2
50
1
30
1
20
C
CCP(X = 0) =
245
38
2
50
2
20
C
C
3
X 0 1 2
P(X) 245
87
245
120
245
38
Mean = )(XPX = 245
196
245
382
245
1201
245
870 2
Importance of non- violence Β½
1
1
______________________________________________________________
25. (1) π₯2+π¦2= 8x (π₯ β 4)2+π¦2= 16 represents a circle with centre (4,0) and radius 4 units 1/2
(2) π¦2 =4 x represents parabola with vertex at origin and axis as x-axis. Β½+ Β½ ( figure)
Point of intersection of the curves are (0,0) and (4,4)
= β« β4π₯4
0 dx + β« β8π₯ β π₯2
8
4 dx 1+1/2
=2β« π₯4
01/2 dx + β« β( 16 β (π₯ β 4)2
8
4 dx
= 2[2
3π₯32β ]0
4
+ [π₯β4
2β16 β (π₯ β 4)2 +
16
2sinβ1
π₯β4
4]4
8
2
= 32
3+ 4π π π π’πππ‘π 1
_______________________________________ OR Solving (1) and (2) point of intersection is C(4,5) Solving (2) and (3) point of intersection is B(7,2)
Solving (1) and (3) point of intersection is A(2,0) 2
11
Area of triangle ABC= area of triangle ACD + area of CDEB + area of triangle ABE
= β«5π₯β10
2
4
2ππ₯ + β« (9 β π₯)ππ₯ β β«
2π₯β4
5
7
2
7
4ππ₯
2
11
= Β½ [ [5π₯2
2β 10π₯]
2
4
+ [9π₯ βπ₯2
2]4
7
β1
5[π₯2 β 4π₯]
2
7 2
11
= 21
2 π π π’πππ‘π
2
11
-______________________________________________________________
Q26. Let the mixture contain x kg of food P and y kg of food Q. Minimise Z = 60x + 80y 1/2 subject to the constraints, 3x + 4y β₯ 8 β¦ (2) 5x + 2y β₯ 11 β¦ (3)
x, y β₯ 0 β¦ (4) 2
11
Figure and shading 2
12
The corner points of the feasible region are .A(8/3,0),B(2,1/2),C(0,11/2)
minimum cost Rs 160 at the line segment A(8/3,0) & B(2,1/2) 2
11
Marking scheme can be for any alternative method by the evaluator. Pratima Nayak,KV Teacher