Board Paper of Class 12-Science 2017 Maths All India(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
Determine the value of '
k' for which the following function is continuous at
x = 3:
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- Question 2
If for any 2 × 2 square matrix A, A(adj A) =
, then write the value of |A|.
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- Question 3
Find the distance between the planes 2x – y + 2z = 5 and 5x – 2.5y + 5z = 20.
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- Question 6
Two tailors, A and B earn Rs 300 and Rs 400 per day respectively. A can stitch 6 shirts and 4 pairs of trousers while B can stitch 10 shirts and 4 pairs of trousers per day. To find how many days should each of them work and if it is desired to produce at least 60 shirts and 32 pairs of trousers at a minimum labour cost, formulate this as an LPP.
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- Question 7
A die, whose faces are marked 1, 2, 3, in red and 4, 5, 6 in green, is tossed. Let A be the event "number obtained is even" and B be the event "number obtained is red". Find if A and B are independent events.
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- Question 8
The
x-coordinate of a point on the line joining the points P(2, 2, 1) and Q(5, 1, –2) is 4. Find its
z-coordinate.
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- Question 9
Show that the function
f(
x) =
x3 – 3
x2 + 6
x – 100 is increasing on
ℝ. VIEW SOLUTION
- Question 10
Find the value of
c in Rolle's theorem for the function
f(
x) =
x3 – 3
x in
.
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- Question 11
If A is a skew-symmetric matrix of order 3, then prove that det A = 0.
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- Question 12
The volume of a sphere is increasing at the rate of 8 cm
3/s. Find the rate at which its surface area is increasing when the radius of the sphere is 12 cm.
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- Question 13
There are 4 cards numbered 1, 3, 5 and 7, one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on the two drawn cards. Find the mean and variance of X.
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- Question 14
Show that the points A, B, C with position vectors
,
and
respectively, are the vertices of a right-angled triangle. Hence find the area of the triangle.
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- Question 15
Of the students in a school, it is known that 30% have 100% attendance and 70% students are irregular. Previous year results report that 70% of all students who have 100% attendance attain A grade and 10% irregular students attain A grade in their annual examination. At the end of the year, one student is chosen at random from the school and he was found to have an A grade. What is the probability that the student has 100% attendance? Is regularity required only in school? Justify your answer.
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- Question 16
If
, then find the value of x.
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- Question 17
Using properties of determinants, prove that
OR
Find matrix A such that
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- Question 18
If x
y + y
x = a
b, then find
OR
If e
y(x + 1) = 1, then show that
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- Question 19
Evaluate :
OR
Evaluate :
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- Question 20
Solve the following linear programming problem graphically :
Maximise Z = 7x + 10y
subject to the constraints
4x + 6y ≤ 240
6x + 3y ≤ 240
x ≥ 10
x ≥ 0, y ≥ 0
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- Question 22
If
, then express
in the form of
, where
is parallel to
is perpendicular to
.
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- Question 23
Find the general solution of the differential equation
.
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- Question 24
Using the method of integration, find the area of the triangle ABC, coordinates of whose vertices are A(4, 1), B(6, 6) and C(8, 4).
OR
Find the area enclosed between the parabola 4
y = 3
x2 and the straight line 3
x – 2
y + 12 = 0.
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- Question 25
Find the particular solution of the differential equation
, given that
y = 0 when
x = 1.
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- Question 26
Find the coordinates of the point where the line through the points (3, –4, –5) and (2, –3, 1), crosses the plane determined by the points (1, 2, 3), (4, 2, –3) and (0, 4, 3).
OR
A variable plane which remains at a constant distance 3
p from the origin cuts the coordinate axes at A, B, C. Show that the locus of the centroid of triangle ABC is
.
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- Question 27
Consider
. Show that f is bijective. Find the inverse of f and hence find f
–1 (0) and x such that f
–1 (x) = 2.
OR
Let
and let * be a binary operation on A defined by (a, b) * (c, d) = (ac, b + ad) for (a, b), (c, d) ∊ A. Determine, whether * is commutative and associative. Then, with respect to * on A
(i) Find the identity element in A.
(ii) Find the invertible elements of A.
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- Question 28
If
, then find A
–1 and hence solve the system of linear equations
.
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- Question 29
A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.
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