Questions with hints

1. SAMPLE PAPER -2015
MATHEMATICS
CLASS – 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: tan–1
√3 – sec–1
(–2)
Q2 Find gof if f(x) =8 x3
, g(x)= √ 𝑥
3
.
Q3. If [
3𝑥 − 2𝑦 5
𝑥 −2
] = [
3 5
−3 −2
] , find the value of y .
Q4. Evaluate: | 𝑠𝑖𝑛 300
𝑐𝑜𝑠300
−𝑠𝑖𝑛600
𝑐𝑜𝑠600|
Q5. Find p such that
p
zyx
321
 and
142
zyx


are perpendicular to each other.
Q6. Find the projection of 𝑎⃗ on 𝑏⃗⃗ if 𝑎⃗ . 𝑏⃗⃗ =8 and 𝑏⃗⃗ = 2𝑖̂ +6𝑗̂ + 3𝑘̂

2. Section B
Q7. 𝐿𝑒𝑡 𝐴 = 𝑁𝑋𝑁, 𝑎𝑛𝑑 ∗ 𝑏𝑒 𝑡ℎ𝑒 𝑏𝑖𝑛𝑎𝑟𝑦 𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑛 𝐴 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑏𝑦
(𝑎, 𝑏) ∗ (𝑐, 𝑑) = (𝑎 + 𝑐, 𝑏 + 𝑑). Show that ∗ is commutative and associative.
Find the identity element for ∗ on A, if any.
Q8. Prove 𝐶𝑜𝑡−1
(
√1+sin 𝑥+√1−sin 𝑥
√1+sin 𝑥− √1−sin 𝑥
) =
𝑥
2
, x∈ (0,
𝜋
4
)
OR
Solve for x . 2 𝑡𝑎𝑛−1(cos 𝑥) = 𝑡𝑎𝑛−1(2 𝑐𝑜𝑠𝑒𝑐 𝑥)
Q9. By using properties of determinants, show that:
|
1 + 𝑎2
− 𝑏2
2𝑎𝑏 −2𝑏
2𝑎𝑏 1 − 𝑎2
+ 𝑏2
2𝑎
2𝑏 −2𝑎 1 − 𝑎2
− 𝑏2
| = (1 + 𝑎2
+ 𝑏2)3
Q10. If cos y = x cos(a + y) with cos a ≠ ± 1, prove that
𝑑𝑦
𝑑𝑥
=
𝑐𝑜𝑠2( 𝑎+𝑦)
sin 𝑎
OR
Find
𝑑𝑦
𝑑𝑥
of the function (cos 𝑥) 𝑦
= (cos 𝑦) 𝑥
Q11. If = (𝑡𝑎𝑛−1
𝑥)2
, show that (𝑥2
+ 1)2
𝑦2 + 2𝑥(𝑥2
+ 1)𝑦1 = 2
Q12
If f(x) =
{
1−cos4𝑥
𝑥2
𝑤ℎ𝑒𝑛 𝑥 < 0
𝑎, 𝑤ℎ𝑒𝑛 𝑥 = 0
√ 𝑥
√16+√ 𝑥−4
, 𝑤ℎ𝑒𝑛 𝑥 > 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
− 3x2
− 36x + 7 is
(a) strictly increasing (b) strictly decreasing
Q14. Show that [𝑎⃗ + b⃗⃗⃗⃗ 𝑏⃗⃗ + 𝑐⃗⃗⃗ 𝑐⃗ + 𝑎⃗ ] =2[𝑎⃗ 𝑏⃗⃗ 𝑐⃗ ]
OR
Find a unit vector perpendicular to each of the vectors (𝑎⃗+ 𝑏⃗⃗) 𝑎𝑛𝑑 ( 𝑎⃗⃗⃗⃗- 𝑏⃗⃗) where 𝑎⃗ = 𝑖̂ + 𝑗̂ +
𝑘̂ and 𝑏⃗⃗ = 𝑖̂ + 2 𝑗̂ + 3𝑘̂ .

3. Q15. Evaluate: ∫
2𝑥
(𝑥2+1)(𝑥2+3)
𝑑𝑥 dx
Q16. Evaluate: ∫ 𝑒 𝑥
(
1+sin 𝑥
1+cos 𝑥
) dx
Q17. Using properties of definite integrals, evaluate:
∫
𝑥
4 − 𝑐𝑜𝑠2 𝑥
𝑑𝑥
𝜋
0
OR
Using properties of definite integrals, evaluate:
∫ 𝑙𝑜𝑔(1 + tan 𝑥)𝑑𝑥
𝜋
4⁄
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
)kˆ2jˆ5-iˆ(3kˆ-jˆiˆ2r
and)ˆˆˆ2(ˆˆ






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
𝑑𝑦
𝑑𝑥
+ 2 𝑦 tan 𝑥 = sin 𝑥 , given that y = 0 where x =
𝜋
3
Q22. ) Find the equation of plane passing through the line of intersection of the planes

4. x + 2y + 3 z = 4 and 2 x + y – z + 5 = 0 and perpendicular to the plane 5 x + 3y – 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 α is,  23
tan
27
4
h .
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
𝑥2
+ 𝑦2
= 8 x and parabola 𝑦2
= 4 x
OR
Using the method of integration, find the area of the region bounded by the following lines
5x - 2y = 10, x + y – 9 =0 , 2x – 5y – 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. (𝑎, 𝑏) ∗ (𝑐, 𝑑) = (𝑐, 𝑑) ∗ (𝑎, 𝑏) for commutativity.
2
1
1
((𝑎, 𝑏) ∗ (𝑐, 𝑑)) ∗ (𝑒, 𝑓) = (𝑎, 𝑏) ∗ ((𝑐, 𝑑) ∗ (𝑒, 𝑓)) for associativity.
2
1
1
No identity element. 1
______________________________________________________________________
Q8.

5. 1 1/2
2
1
1
2
1
1
½
_______________________________________________________________________
OR
1
1
2
1
1
1/2
_________________________________________________________________________________________
Ans 9.
Applying R1 → R1 + bR3 and R2 → R2 − aR3, we have: 1
1
Expanding along R1, we have: (1 + 𝑎2
+ 𝑏2)3
1
Answer 1
____________________________________________________________________________________
Q10. 𝑥 =
cos 𝑦
cos(𝑎+𝑦)
1
𝑑𝑥
𝑑𝑦
= =
sina
𝑐𝑜𝑠2( 𝑎+𝑦)
1+1
𝑑𝑦
𝑑𝑥
=
𝑐𝑜𝑠2( 𝑎+𝑦)
sin 𝑎
1
________________________________________________________________________________________________
OR
Taking logarithm on both the sides,
Differentiating both sides
1
1

6. _______________________________________________________________________
Q11. 1
1
1
1
_____________________________________________________________________________
Q12
lim 𝑥→0− 𝑓(𝑥) =
2𝑠𝑖𝑛22𝑥
𝑥2 = 8 1½
RHL on rationalization lim 𝑥→+ 𝑓(𝑥) = 8 1½
a = 8 1
________________________________________________________________________
Q13.
1
x = − 2, 3 1/2
Intervals: (- ∞,-2),( -2,3) and (3, ∞) 1
(f) is strictly increasing in (- ∞,-2) and (3, ∞) and strictly decreasing in interval ),( -2,3) 1½
___________________________________________________________________________
Q14. = {(𝑎⃗ × b⃗⃗)+(𝑎⃗ × c⃗) + (𝑏⃗⃗ × b⃗⃗) + (𝑏⃗⃗ × c⃗)} . (𝑐⃗+𝑎⃗ ) 1
= (𝑎⃗ × b⃗⃗) . 𝑐⃗ + (𝑎⃗ × c⃗). 𝑐⃗+( 𝑏⃗⃗ × c⃗). 𝑐⃗ + (𝑎⃗ × b⃗⃗) . 𝑎⃗ + (𝑎⃗ × c⃗). 𝑎⃗+( 𝑏⃗⃗ × c⃗). 𝑎⃗ 2
=[𝑎⃗ 𝑏⃗⃗ 𝑐⃗ ] + [𝑎⃗ 𝑏⃗⃗ 𝑐⃗ ]
2
1
=2[𝑎⃗ 𝑏⃗⃗ 𝑐⃗ ]
2
1
_________________________________________________________________________
OR
(𝑎⃗⃗⃗⃗⃗+ 𝑏⃗⃗) = 2𝑖̂ + 3 𝑗̂ + 4𝑘̂. ( 𝑎⃗- 𝑏⃗⃗) = 0𝑖̂ − 𝑗̂ − 2𝑘̂. 1
(𝑎⃗⃗⃗⃗⃗+ 𝑏⃗⃗)𝑋 ( 𝑎⃗⃗⃗⃗- 𝑏⃗⃗) = −2𝑖̂ + 4 𝑗̂ − 2𝑘̂.
2
1
1
|(𝑎⃗⃗⃗⃗⃗+ 𝑏⃗⃗)𝑋 ( 𝑎⃗⃗⃗⃗- 𝑏⃗⃗ | =√24 1
2
1

7. 1
√24
(−2𝑖̂ + 4 𝑗̂ − 2𝑘̂) ½
_________________________________________________________________________
Q15. Let x2 = t ⇒ 2x dx = dt 1/2
1
A=1/2, B=-1/2 1/2
2
_____________________________________________________________________________________________________
Q16. 1
2
1
1
½+1
_____________________________________________________________________
Q17 Use of property
∫ 𝑓((𝑥)𝑑𝑥 =
𝑎
0 ∫ 𝑓((𝑎 − 𝑥)𝑑𝑥 =
𝑎
0
, I = ∫
𝜋−𝑥
4−𝑐𝑜𝑠2 𝑥
𝑑𝑥
𝜋
0
1/2
2I = 𝜋 ∫
𝑠𝑒𝑐2 𝑥
3+4 𝑡𝑎𝑛2 𝑥
𝑑𝑥
𝜋
0
1/2
Use of property ∫ 𝑓((𝑥)𝑑𝑥 = 2
2𝑎
0 ∫ 𝑓((𝑎 − 𝑥)𝑑𝑥 𝑎𝑠 𝑓(2𝑎 − 𝑥) = 𝑓(𝑥)
𝑎
0
2I = 2𝜋/4 ∫
𝑠𝑒𝑐2 𝑥
3+4 𝑡𝑎𝑛2 𝑥
𝑑𝑥
𝜋/2
0
1
tan x = t, sec2
x dx =dt 1
& Correct result I =
𝜋2
4√3
1
_________________________________________________________________
OR 1
2
1
_________________________________________________________________________
Q18. P(T) =3/4, P(F) =1/4 1

8. 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)=
𝑃(𝑇)𝑃(
𝐸
𝑇
)
𝑃(𝑇)𝑃(
𝐸
𝑇
)+𝑃(𝐹)𝑃(
𝐸
𝐹
)
=3/8
2
1
1
Truthfulness ½
Q19.
59bb
ˆ7ˆˆ31
,ˆˆ
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 ≠ 0 so A-1
exists. X = A-1
B 1/2
Cofactors of A 2
[
−1 −1 2
−1 2 1
6 −6 3
]
Adjoint A 1/2
X = 4000 ,y = 5000, z = 3000 1½
Values ½
_______________________________________________________________
Q21. P = 2tan x, Q = sin x
I.F = 𝑠𝑒𝑐2
𝑥 1½
y 𝑠𝑒𝑐2
𝑥 = ∫ sin 𝑥 𝑠𝑒𝑐2
𝑥 𝑑𝑥 + 𝐶 1
y 𝑠𝑒𝑐2
𝑥 = sec x + C 1½
y =
1
sec 𝑥
+
𝐶
𝑠𝑒𝑐2 𝑥
= cos x + C 𝑐𝑜𝑠2
𝑥 -------------------(1) 1½
putting x =
𝜋
3
and y = 0 in eqn (1) C = -2 ½
Y= cos x - 2𝑐𝑜𝑠2
𝑥 1
__________________________________________________________________________
Q22.
Sol: The required plane is (x + 2 y + 3 z ) + k (2 x + y – 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.

9. 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 (α )
relation of h and H 1+1/2 ( figure)
1
+ ½
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
CC
P(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
38
2
245
120
1
245
87
0  2
Importance of non- violence ½
1
1

10. ______________________________________________________________
25. (1) 𝑥2
+𝑦2
= 8x
 (𝑥 − 4)2
+𝑦2
= 16 represents a circle with centre (4,0) and radius 4 units 1/2
(2) 𝑦2
=4 x represents parabola with vertex at origin and axis as x-axis. ½+ ½ ( figure)
Point of intersection of the curves are (0,0) and (4,4)
= ∫ √4𝑥
4
0
dx + ∫ √8𝑥 − 𝑥28
4
dx 1+1/2
=2∫ 𝑥
4
0
1/2
dx + ∫ √( 16 − (𝑥 − 4)28
4
dx
= 2[
2
3
𝑥
3
2⁄
]
0
4
+ [
𝑥−4
2
√16 − (𝑥 − 4)2 +
16
2
sin−1 𝑥−4
4
]
4
8
2
=
32
3
+ 4𝜋 𝑠𝑞 𝑢𝑛𝑖𝑡𝑠 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
1
1
Area of triangle ABC= area of triangle ACD + area of CDEB + area of triangle ABE
= ∫
5𝑥−10
2
4
2
𝑑𝑥 + ∫ (9 − 𝑥)𝑑𝑥 − ∫
2𝑥−4
5
7
2
7
4
𝑑𝑥
2
1
1
= ½ [ [
5𝑥2
2
− 10𝑥]
2
4
+ [9𝑥 −
𝑥2
2
]
4
7
−
1
5
[𝑥2
− 4𝑥]
2
7
2
1
1
=
21
2
𝑠𝑞 𝑢𝑛𝑖𝑡𝑠
2
1
1
-
______________________________________________________________
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 ≥ 8 … (2)
5x + 2y ≥ 11 … (3)
x, y ≥ 0 … (4)
2
1
1
Figure and shading
2
1
2
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
1
1
Marking scheme can be for any alternative method by the evaluator.
Pratima Nayak,KV Teacher