Board Paper of Class 12-Science 2015 Maths Delhi(SET 2) - Solutions
General Instructions :
(i) All questions are compulsory.
(ii) Please check that this Question Paper contains 26 Questions.
(iii) Marks for each question are indicated against it.
(iv) Questions 1 to 6 in Section-A are Very Short Answer Type Questions carrying one mark each.
(v) Questions 7 to 19 in Section-B are Long Answer I Type Questions carrying 4 marks each.
(vi) Questions 20 to 26 in Section-C are Long Answer II Type Questions carrying 6 marks each.
(vii) Please write down the serial number of the Question before attempting it.
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- Question 1
If a line makes angles 90°, 60° and θ with
x,
y and z-axis respectively, where θ is acute, then find θ.
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- Question 2
Write the element
a23 of a 3 ✕ 3 matrix A = (
aij) whose elements
aij are given by
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- Question 3
Find the differential equation representing the family of curves
+ B, where A and B are arbitrary constants.
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- Question 4
Find the integrating factor of the differential equation
.
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- Question 5
If
then find the projection of
.
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- Question 6
Find λ, if the vectors
are coplanar.
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- Question 7
A bag A contains 4 black and 6 red balls and bag B contains 7 black and 3 red balls. A die is thrown. If 1 or 2 appears on it, then bag A is chosen, otherwise bag B, If two balls are drawn at random (without replacement) from the selected bag, find the probability of one of them being red and another black.
OR
An unbiased coin is tossed 4 times. Find the mean and variance of the number of heads obtained.
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- Question 9
Find the distance between the point (−1, −5, −10) and the point of intersection of the line
and the plane
x −
y +
z = 5.
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- Question 10
If sin [cot
−1 (
x+1)] = cos(tan
−1x), then find
x.
OR
If (tan
−1x)
2 + (cot
−1x)
2 =
, then find
x.
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- Question 11
If
then find
.
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- Question 12
If x = a cos θ + b sin θ, y = a sin θ − b cos θ, show that
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- Question 13
The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?
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- Question 15
Three schools A, B and C organized a mela for collecting funds for helping the rehabilitation of flood victims. They sold hand made fans, mats and plates from recycled material at a cost of Rs 25, Rs 100 and Rs 50 each. The number of articles sold are given below:
School |
|
|
|
Article |
A |
B |
C |
Hand-fans |
40 |
25 |
35 |
Mats |
50 |
40 |
50 |
Plates |
20 |
30 |
40 |
Find the funds collected by each school separately by selling the above articles. Also find the total funds collected for the purpose.
Write one value generated by the above situation.
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- Question 16
If
find
and hence find a matrix X such that
OR
If
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- Question 17
If
, using properties of determinants find the value of
f(2
x) −
f(
x).
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- Question 18
Find :
OR
Integrate the following w.r.t.
x
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- Question 20
Solve the differential equation :
OR
Find the particular solution of the differential equation
given that
y = 1, when
x = 0.
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- Question 21
If lines
intersect, then find the value of k and hence find the equation of the plane containing these lines.
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- Question 22
If A and B are two independent events such that
then find P(A) and P(B).
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- Question 23
Find the local maxima and local minima, of the function f(
x) = sin
x − cos
x, 0 <
x < 2π.
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- Question 24
Find graphically, the maximum value of z = 2
x + 5
y, subject to constraints given below :
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- Question 25
Let N denote the set of all natural numbers and R be the relation on N × N defined by (a, b) R (c, d) if ad (b + c) = bc (a + d). Show that R is an equivalence relation.
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- Question 26
Using integration find the area of the triangle formed by positive x-axis and tangent and normal of the circle
OR
Evaluate as a limit of a sum.
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