Board Paper of Class 12-Science 2015 Maths Abroad(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
, then write the cofactor of the element
a21 of its 2
nd row.
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- Question 2
Write the sum of the order and degree of the differential equation
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- Question 3
Write the solution of the differential equation
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- Question 4
Find the unit vector in the direction of the sum of the vectors
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- Question 5
Find the area of a parallelogram whose adjacent sides are represented by the vectors
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- Question 6
Find the sum of the intercepts cut off by the plane
on the coordinate axes.
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- Question 8
Three machines E
1, E
2 and E
3 in a certain factory producing electric bulbs, produce 50%, 25% and 25% respectively, of the total daily output of electric bulbs. It is known that 4% of the bulbs produced by each of machines E
1 and E
2 are defective and that 5% of those produced by machine E
3 are defective. If one bulb is picked up at random from a day's production, calculate the probability that it is defective.
OR
Two numbers are selected at random (without replacement) from positive integers 2, 3, 4, 5, 6 and 7. Let X denote the larger of the two numbers obtained. Find the mean and variance of the probability distribution of X.
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- Question 9
The two vectors
represent the two sides vectors
respectively of triangle ABC. Find the length of the median through A.
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- Question 10
Find the equation of a plane which passes through the point (3, 2, 0) and contains the line
.
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- Question 11
If 2 tan
−1 (cos θ) = tan
−1 (2 cosec θ), (θ ≠ 0), then find the value of θ.
OR
If
, then find the value of θ.
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- Question 12
If
and I is the identity matrix of order 2, then show that A
2= 4 A − 3 I. Hence find A
−1.
OR
If
, then find the values of a and b.
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- Question 13
Using properties of determinants, prove the following :
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- Question 14
Evaluate :
OR
Evaluate :
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- Question 15
Find whether the following function is differentiable at
x = 1 and
x = 2 or not :
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- Question 16
In a parliament election, a political party hired a public relations firm to promote its candidates in three ways − telephone, house calls and letters. The cost per contact (in paise) is given in matrix A as
The number of contacts of each type made in two cities X and Y is given in the matrix B as
Find the total amount spent by the party in the two cities.
What should one consider before casting his/her vote − party's promotional activity or their social activities ?
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- Question 18
Find the point on the curve 9
y2 =
x3, where the normal to the curve makes equal intercepts on the axes.
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- Question 20
Find the minimum value of (ax + by), where xy = c
2.
OR
Find the coordinates of a point of the parabola y = x
2 + 7x + 2 which is closest to the straight line y = 3x − 3.
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- Question 21
Maximise
z = 8
x + 9
y subject to the constraints given below :
2
x + 3
y ≤ 6
3
x − 2
y ≤6
y ≤ 1
x,
y ≥ 0
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- Question 22
Find the distance of the point (1, −2, 3) from the plane
x − y + z = 5 measured parallel to the line whose direction cosines are proportional to 2, 3, −6.
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- Question 23
Let
f : N → ℝ be a function defined as
f(
x) = 4
x2 + 12
x + 15. Show that
f : N → S, where S is the range of
f, is invertible. Also find the inverse of
f.
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- Question 24
Using integration, find the area of the region bounded by the line
x –
y + 2 = 0, the curve
x =
and
y-axis.
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- Question 25
Find the probability distribution of the number of doublets in four throws of a pair of dice. Also find the mean and variance of this distribution.
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- Question 26
Solve the following differential equation :
OR
Solve the following differential equation :
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