1. 1
SAMPLE PAPER
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: tanβ1
β3 β secβ1
(β2)
Q2 Find gof if f(x) =8 x3
, g(x)= β π₯
3
.
Q3. If [
3π₯ β 2π¦ 5
π₯ β2
] = [
3 5
β3 β2
] , find the value of y .
Q4. Evaluate: | π ππ300
πππ 300
βπ ππ600
πππ 600|
Q5. Find p such that
p
zyx
321
ο½ο½ and
142
zyx
ο½ο½
ο
are perpendicular to each other.
Q6. Find the projection of πβ on πββ if πβ . πββ =8 and πββ = 2πΜ +6πΜ + 3πΜ
2. 2
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+sin π₯+β1βsin π₯
β1+sin π₯β β1βsin π₯
) =
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
2π
2π 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¦)
sin π
OR
Find
ππ¦
ππ₯
of the function (cos π₯) π¦
= (cos π¦) π₯
Q11. If = (π‘ππβ1
π₯)2
, show that (π₯2
+ 1)2
π¦2 + 2π₯(π₯2
+ 1)π¦1 = 2
Q12
If f(x) =
{
1βcos4π₯
π₯2
π€βππ π₯ < 0
π, π€βππ π₯ = 0
β π₯
β16+β π₯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 3x2
β 36x + 7 is
(a) strictly increasing (b) strictly decreasing
Q14. Show that [πβ + bββββ πββ + πβββ πβ + πβ ] =2[πβ πββ πβ ]
3. 3
OR
Find a unit vector perpendicular to each of the vectors (πβ+ πββ) πππ ( πββββ- πββ) where πβ = πΜ + πΜ +
πΜ and πββ = πΜ + 2 πΜ + 3πΜ .
Q15. Evaluate: β«
2π₯
(π₯2+1)(π₯2+3)
ππ₯ dx
Q16. Evaluate: β« π π₯
(
1+sin π₯
1+cos π₯
) dx
Q17. Using properties of definite integrals, evaluate:
β«
π₯
4 β πππ 2 π₯
ππ₯
π
0
OR
Using properties of definite integrals, evaluate:
β« πππ(1 + tan π₯)ππ₯
π
4β
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
)kΛ2jΛ5-iΛ(3kΛ-jΛiΛ2r
and)ΛΛΛ2(ΛΛ
ο«ο«ο«ο½
ο«οο«ο«ο½
ο
ο¬
ο²
ο²
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
4. 4
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, ο‘ο° 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
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