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CBSE 12th Mathematics 2014 Unsolved Paper Delhi Board
TIME - 3HR. | QUESTIONS - 29
THE MARKS ARE MENTIONED ON EACH QUESTION __________________________________________________________________________
SECTION β A
Question number 1 to 10 carry 1 marks each:
Q.1. Let * be a binary operation, on the set of all non- zero real number, given by
π β π =ππ
π πππ π, π π πΉ β {π}.
Find the value of π,given that π β (π β π) = ππ. 1 marks
Q. 2. ππ π¬π’π§ (πππβπ π
π+ πππβππ) = π, then find the value of π. 1 πππππ
Q. 3. ππ π [π ππ π
] + [π ππ π
] = [π πππ π
] , ππππ (π β π). 1 πππππ
Q. 4. Solve the following matrix equation 1 marks
ππ¨π« π:, [π π]. [π π
βπ π] = π.
Q. 5. π°π |ππ ππ π
| = |π βππ π
| ,πππππ πππ πππππ ππ π. 1 πππππ
Q. 6. Write the antiderivative of (πβπ +π
βπ) . 1 πππππ
Q. 7. Evaluate: 1 marks
β«π π
π + ππ .
π
π
Q. 8. Find the projection of the vector οΏ½ΜοΏ½ + ππΜ + ππ Μ on the vector
ποΏ½ΜοΏ½ β ππΜ + ποΏ½ΜοΏ½ . 1πππππ
Q. 9. π°π οΏ½ββοΏ½ πππ οΏ½ββοΏ½ are two unit vectors such that οΏ½ββοΏ½ + οΏ½ββοΏ½ is also a unit vector, then find the
angle between οΏ½ββοΏ½ πππ οΏ½ββοΏ½ . 1 marks
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Q.10. write the vector equation of the plane, passing through the point (π, π, π) and
Parallel to the plane οΏ½βοΏ½ . (οΏ½ΜοΏ½ + πΜ + οΏ½ΜοΏ½ ) = π. 1 πππππ
SECTION-B
Question number 11 to 22 carry 4 marks each:
Q. 11. π³ππ π¨ = {π, π, π, β¦ . . , } and R be the relation in π Γ π defined by (π, π) πΉ (π, π ) π’π π
+π = π + π ππ¨π« (π, π), (π, π ) ππ π Γπ. Prove that R is an equivalence relation. Also obtain the equivalence class [(2,5)]. 4 marks
Q. 12. Prove that 4 marks
πππβπ (βπ + πππ π + βπ β πππ π
βπ + πππ π β βπ β πππ π) =
π
π; π π (π,
π
π) .
OR
Prove that
π πππβπ (π
π) + πππβπ (
πβπ
π) + π πππβπ (
π
π) =
π
π .
Q. 13. Using properties of determinants, prove that 4 marks
|
ππ π β π β π ππππ ππ π β π β π
π β π β π ππ ππ| = (π + π + π)π
Q.14. Differentiate πππβπ (βπβππ
π) π°π’ππ‘ respect to ππ¨π¬βπ (ππ βπ β ππ),
π°π‘ππ§ π β π. 4 marks
Q.15. If π = ππ, π©π«π¨π―π ππ‘ππ 4 marks
π ππ
π ππβ
π
π(π π
π π)π
βπ
π= π.
Q.14. Find the intervals in which the function π(π) = πππ β πππ β ππππ + π ππ 4 marks
(a) strictly increasing
(b) strictly decreasing
OR
Find the equations of the tangent and normal to the curve π = π ππππ π½ πππ π =
π πππππ½ ππ π½ = π
π.
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Q17. Evaluate: 4 marks
β«ππππ π ππππ π
πππππ πππππ π π.
OR
Evaluate: β«(π β π)βππ + ππ β ππ π π.
Q.18. Find the particular solution of the differential equation ππβπ β ππ π π +π
π π π =
π, π π’π―ππ§ ππ‘ππ π = π π°π‘ππ§ π = π, 4 marks
Q.19. Solve the following differential equation: 4 marks
(ππ β π)π π
π π+ πππ =
π
ππ β π.
Q.20. Prove that, for any three vectors οΏ½ββοΏ½ , οΏ½ββοΏ½ , οΏ½βοΏ½ . 4 marks
[ οΏ½ββοΏ½ + οΏ½ββοΏ½ , οΏ½ββοΏ½ + οΏ½βοΏ½ , π ββ + οΏ½ββοΏ½ ] = π [ οΏ½ββοΏ½ , οΏ½ββοΏ½ , οΏ½βοΏ½ ]
OR
Vectors οΏ½ββοΏ½ , οΏ½ββοΏ½ and οΏ½βοΏ½ are such that οΏ½ββοΏ½ + οΏ½ββοΏ½ + οΏ½βοΏ½ = οΏ½ββοΏ½ and |οΏ½ββοΏ½ | = π, |οΏ½ββοΏ½ | = π and |οΏ½βοΏ½ | = π
find the angle between οΏ½ββοΏ½ πππ οΏ½ββοΏ½ .
Q.21. Show that the lines: 4 marks
π + π
π=
π + π
π=
π + π
ππππ
π β π
π=
π β π
π=
π β π
π
Intersect. Also, find their point of intersection.
Q.22. Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls? Given that 4 marks
(i) the youngest is a girl.
(ii) at least one is a girl.
SECTION β C
Question number 11 to 22 carry 4 marks each:
Q.23. Two schoolβs P and Q want to award their selected students on the values of
discipline, politeness and punctuality. The school P wants to award Rs y each and Rs z each for the three respective values to its 3, 2 and 1 students with total award money of Rs 1,000. School Q wants to spend Rs 1,500 to award its 4, 1 and 3 students on the respective values (by giving the same award money for the three values as
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before). If the total amount of awards for one prize on each value is Rs 600, using matrices, find the award money for each value.
A part from the above three values, suggest one more value for awards. 6 marks
Q.24. Show that the semi-vertical angle of the cone of the maximum volume and of given
slant height is πππβπ π
βπ. 6 marks
Q. 25. Evaluate: 6 marks
β«π π
π + βππππ
π π
π π
.
Q. 26. Find the area of the region in the first quadrant enclosed by the π β ππππ,
The line π = π and the circle ππ + ππ = ππ. 6 marks
Q.27. Find the distance between the point (7,2,4) and the plane determined by the points A (2,5-3), B (-2, -3,5) and C (5,3, -3). 6 marks
OR
Find the distance of the point (-1, -5, -10) from the point of intersection of the line
οΏ½βοΏ½ = ππ Μ β π Μ + ππ Μ + π (ππ Μ + ππ Μ + ππ Μ) and the plane οΏ½βοΏ½ . (π Μ β π Μ + π Μ) = π.
Q.28. A dealer in rural area wishes to purchase a number of sewing machine. He has only Rs5,760 to invest and has space for at most 20 items for strong. An electronic sewing machine cost him Rs360 and a manually operated sewing machine Rs240. He can sell an electronic sewing machine at a profit of Rs22 and a manually operated sewing machine at a profit of Rs18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit? Make it as a LPP and solve it graphically. 6 marks
Q. 29. A card from a pack of 52 playing cards is lost. From the remaining cards of the pack, three cards are drawn at random (without replacement) and are found to be all spades. Find the probability of the lost card being a spade. 6 marks
OR
From a lot of 15 bulbs which include 5 defectives, a sample of 4 bulbs is drawn one by one with replacement. Find the probability distribution of number of defective bulbs. Hence find the mean of the distribution.
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