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# Board Paper of Class 12-Science 2011 Maths Abroad(SET 2) - Solutions

General Instructions:
i. All questions are compulsory.
ii. The question paper consists of 29 questions divided into three sections A, B and C. Section A comprises of 10 questions of one mark each, Section B comprises of 12 questions of four marks each, and Section C comprises of 7 questions of six marks each.
iii. All questions in section A are to be answered in one word, one sentence or as per the exact requirements of the question.
iv. There is no overall choice. However, internal choice has been provided in 4 questions of four marks each and 2 questions of six marks each. You have to attempt only one of the alternatives in all such questions.
v. Use of calculators is not permitted.

• Question 1
What are the direction cosines of a line that makes equal angles with the co-ordinate axes? VIEW SOLUTION

• Question 2
If , then what can be concluded about the vector $\stackrel{\to }{\mathrm{b}}$? VIEW SOLUTION

• Question 3
Write the position vector of the mid-point of the vector joining the points P(2, 3, 4) and Q(4, 1, −2). VIEW SOLUTION

• Question 4
Evaluate :

$\underset{1}{\overset{\sqrt{3}}{\int }}\frac{\mathrm{dx}}{1+{\mathrm{x}}^{2}}$ VIEW SOLUTION

• Question 5
If $\left|\begin{array}{cc}\mathrm{x}& \mathrm{x}\\ 1& \mathrm{x}\end{array}\right|=\left|\begin{array}{cc}3& 4\\ 1& 2\end{array}\right|$, write the positive value of x. VIEW SOLUTION

• Question 6
Write the order of the product matrix:

VIEW SOLUTION

• Question 7
Write the values of xy + z from the following equation:
$\left[\begin{array}{ccccc}x& +& y& +& z\\ & x& +& z& \\ & y& +& z& \end{array}\right]=\left[\begin{array}{c}9\\ 5\\ 7\end{array}\right]$ VIEW SOLUTION

• Question 8
Write the principal value of tan−1(−1). VIEW SOLUTION

• Question 9
Write fog if f : R → R and g : R → R are given by
f(x) = |x| and g(x) = |5x − 2|. VIEW SOLUTION

• Question 10
Evaluate:

$\int \frac{{e}^{2x}-{e}^{-2x}}{{e}^{2x}+{e}^{-2x}}dx$ VIEW SOLUTION

• Question 11
Find the mean number of heads in three tosses of a fair coin. VIEW SOLUTION

• Question 12
If vectors  are such that $\stackrel{\to }{a}+\mathrm{\lambda }\stackrel{\to }{b}$ is perpendicular to $\stackrel{\to }{c}$, find the value of λ. VIEW SOLUTION

• Question 13
Find the particular solution of the differential equation:
(1 + e2x) dy + (1 + y2) ex dx = 0, given that y = 1 when x = 0. VIEW SOLUTION

• Question 14
Evaluate:
e2x sin x dx

OR

Evaluate:
$\int \frac{3x+5}{\sqrt{{x}^{2}-8x+7}}dx$ VIEW SOLUTION

• Question 16
Find the intervals in which the function f given by
f(x) = sin x + cos x, 0 ≤ x ≤ 2π
is strictly increasing or strictly decreasing.
OR

Find the points on the curve y = x3 at which the slope of the tangent is equal to the y-coordinate of the point. VIEW SOLUTION

• Question 17
Prove the following :

OR

Solve the following equation for x :

VIEW SOLUTION

• Question 18
Consider  given by f(x) = x2 + 4. Show that f is invertible with the inverse (f−1) of f

given by  where R+ is the set of all non-negative real numbers. VIEW SOLUTION

• Question 19
Prove using properties of determinants :

VIEW SOLUTION

• Question 20
Find the value of k, so that the function f defined by

is continuous at x = π. VIEW SOLUTION

• Question 21
Solve the following differential equation:

given that y = 0 when $x=\frac{\mathrm{\pi }}{3}$. VIEW SOLUTION

• Question 22
Find the shortest distance between the given lines:

VIEW SOLUTION

• Question 23
A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of grinding/cutting machine and a sprayer. It takes 2 hours on the grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp. It takes one hour on the grinding/cutting machine and 2 hours on the sprayer to manufacture a shade. On any day, the sprayer is available for at most 20 hours and the grinding/cutting machine for at most 12 hours. The profit from the sale of a lamp is Rs 5 and that from a shade is Rs 3. Assuming that the manufacturer can sell all the lamps and shades that he produces, how should he schedule his daily production in order to maximise his profit? Make an L.P.P. and solve it graphically. VIEW SOLUTION

• Question 24
Evaluate as a limit of sums.
OR

Evaluate :

VIEW SOLUTION

• Question 25
Using the method of integration, find the area of the region bounded by the following lines :
2x + y = 4
3x − 2y = 6
x − 3y + 5 = 0 VIEW SOLUTION

• Question 26
A window is in the form of a rectangle surmounted by a semi-circular opening. The total perimeter of the window is 10 metres. Find the dimensions of the rectangle so as to admit maximum light through the whole opening. VIEW SOLUTION

• Question 27
Use product to solve the system of equation:
x − y + 2z = 1
2y − 3z = 1
3x − 2y + 4z = 2.

OR

Using elementary transformations, find the inverse of the matrix :
VIEW SOLUTION

• Question 28
Find the vector equation of the plane passing through the points A(2, 2, −1), B (3, 4, 2) and C (7, 0, 6) Also, find the Cartesian equation of the plane. VIEW SOLUTION

• Question 29
Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black balls. One ball is transferred from bag I to bag II and then a ball is drawn from bag II at random. The ball so drawn is found to be red in colour. Find the probability that the transferred ball is black. VIEW SOLUTION
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