Board Paper of Class 12-Science 2017 Maths Delhi(SET 3) - Solutions
General Instructions:
(i) All questions are compulsory.
(ii) This question paper contains 29 questions.
(iii) Questions 1- 4 in Section A are very short-answer type questions carrying 1 mark each.
(iv) Questions 5-12 in Section B are short-answer type questions carrying 2 marks each.
(v) Questions 13-23 in Section C are long-answer I type questions carrying 4 marks each.
(vi) Questions 24-29 in Section D are long-answer II type questions carrying 6 marks each.
- Question 1
If a line makes angles 90° and 60° respectively with the positive directions of
x and
y axes, find the angle which it makes with the positive direction of
z-axis.
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- Question 3
Determine the value of the constant 'k' so that function
is continuous at
x = 0.
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- Question 4
If A is a 3 × 3 invertible matrix, then what will be the value of k if det(A
–1) = (det A)
k.
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- Question 5
Prove that if E and F are independent events, then the events E and F' are also independent.
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- Question 6
A small firm manufactures necklaces and bracelets. The total number of necklaces and bracelets that it can handle per day is at most 24. It takes one hour to make a bracelet and half an hour to make a necklace. The maximum number of hours available per day is 16. If the profit on a necklace is Rs 100 and that on a bracelet is Rs 300. Formulate on L.P.P. for finding how many of each should be produced daily to maximize the profit?
It is being given that at least one of each must be produced.
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- Question 8
Find the vector equation of the line passing through the point A(1, 2, –1) and parallel to the line 5
x – 25 = 14 – 7
y = 35
z.
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- Question 9
Show that the function
is always increasing on
.
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- Question 10
The volume of a sphere is increasing at the rate of 3 cubic centimeter per second. Find the rate of increase of its surface area, when the radius is 2 cm.
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- Question 11
Show that all the diagonal elements of a skew symmetric matrix are zero.
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- Question 13
Let
and
then
(a) Let c
1 = 1 and c
2 = 2, find c
3 which makes
and
coplanar.
(b) If c
2 = –1 and c
3 = 1, show that no value of c
1 can make
and
coplanar.
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- Question 14
If
are mutually perpendicular vectors of equal magnitudes, show that the vector
is equally inclined to
. Also, find the angle which
makes with
.
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- Question 15
The random variable X can take only the values 0, 1, 2, 3. Give that P(X = 0) = P(X = 1) = p and P(X = 2) = P(X = 3) such that
, find the value of p.
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- Question 16
Often it is taken that a truthful person commands, more respect in the society. A man is known to speak the truth 4 out of 5 times. He throws a die and reports that it is a six. Find the probability that it is actually a six.
Do you also agree that the value of truthfulness leads to more respect in the society?
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- Question 17
Using properties of determinants, prove that
OR
Let
, find a matrix D such that CD − AB = O.
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- Question 18
Differentiate the function
with respect to
x.
OR
If
, prove that
.
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- Question 19
Evaluate :
OR
Evaluate :
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- Question 20
Solve the following L.P.P graphically:
Maximise |
Z = 20x + 10y |
Subject to the following constraints |
x + 2y ≤ 28,
|
|
3x + y ≤ 24,
|
|
x ≥ 2,
|
|
x, y ≥ 0
|
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- Question 21
Show that the family of curves for which
, is given by
x2 –
y2 =
cx.
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- Question 23
Solve the following equation for
x:
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- Question 24
Using integration, find the area of region bounded by the triangle whose vertices are (–2, 1), (0, 4) and (2, 3).
OR
Find the area bounded by the circle
x2 + y
2 = 16 and the line
in the first quadrant, using integration.
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- Question 25
Find the equation of the plane through the line of intersection of
and
and perpendicular to the plane
. Hence find whether the plane thus obtained contains the line
x − 1 = 2
y − 4 = 3
z − 12.
OR
Find the vector and Cartesian equations of a line passing through (1, 2, –4) and perpendicular to the two lines
and
.
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- Question 26
Consider f : R
+ → [−5, ∞), given by
f(
x) = 9
x2 + 6
x − 5. Show that f is invertible with
.
Hence Find
(i) f
−1(10)
(ii)
y if
where R
+ is the set of all non-negative real numbers.
OR
Discuss the commutativity and associativity of binary operation '*' defined on A = Q − {1} by the rule
a *
b =
a −
b + ab for all, a, b ∊ A. Also find the identity element of * in A and hence find the invertible elements of A.
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- Question 27
If the sum of lengths of the hypotenuse and a side of a right angled triangle is given, show that the area of the triangle is maximum, when the angle between them is
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- Question 28
If
, find A
–1 and hence solve the system of equations 2
x +
y – 3
z = 13, 3
x + 2
y +
z = 4,
x + 2
y –
z = 8.
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- Question 29
Find the particular solution of the differential equation
given that y = 0 when
.
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