12th Maths Paper Solutions Set 2 : CBSE Abroad Previous Year 2011

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.
Q1 :

What are the direction cosines of a line that makes equal angles with the co-ordinate axes?

Suppose a line l makes an angle Î± with the co-ordinate axes.

l = m = n = cosÎ±

âˆµ l2 + m2 + n2= 1

cos2Î±+cos2Î±+cos2Î±=13cos2Î± = 1cosÎ± =Â± 13âˆšl = m = n = Â± 13âˆš

Q2 :

If aâƒ— â‹…aâƒ— =0 and aâƒ— â‹…bâƒ— =0, then what can be concluded about the vector bâƒ— ?

Given, aâƒ— â‹…aâƒ— =0 and aâƒ— â‹…bâƒ— =0

âˆ£âˆ£aâƒ— âˆ£âˆ£2=0 and |a|-â†’âˆ£âˆ£bâƒ— âˆ£âˆ£cosÎ¸=0âˆ£âˆ£aâƒ— âˆ£âˆ£=0 and |a|-â†’âˆ£âˆ£bâƒ— âˆ£âˆ£cosÎ¸=0

Therefore, we can conclude that bâƒ—   can be any vector which makes an angle Î¸ with the zero vector aâƒ— .

Q3 :

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

The position vector of the mid-point R of the vector joining the points P(2, 3, 4) and Q(4, 1, -2),

OR-â†’-= (2iË† + 3jË† + 4kË†) + (4iË† + jË† - 2kË†)2       = (2 + 4)iË† + (3 + 1)jË† + (4 - 2)kË†2        = 6iË† + 4jË† + 2kË†2        = 3iË† + 2jË† + kË†

Q4 :

Evaluate :

13âˆšdx1+x2

Q5 :

If âˆ£âˆ£âˆ£x1xxâˆ£âˆ£âˆ£=âˆ£âˆ£âˆ£3142âˆ£âˆ£âˆ£, write the positive value of x.

Q6 :

Write the order of the product matrix:

âŽ¡âŽ£123âŽ¤âŽ¦ [234]

Q7 :

Write the values of x - y + z from the following equation:
âŽ¡âŽ£x+xyy+++zzzâŽ¤âŽ¦=âŽ¡âŽ£957âŽ¤âŽ¦

Q8 :

Write the principal value of tan−1(−1).

Q9 :

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

Q10 :

Evaluate:

e2x-e-2xe2x+e-2xdx

Q11 :

Find the mean number of heads in three tosses of a fair coin.

Q12 :

If vectors aâƒ— =2iË†+2jË†+3kË†, bâƒ— =-iË†+2jË†+kË† and câƒ— =3iË†+jË† are such that aâƒ— +Î»

Q13 :

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

Q14 :

Evaluate:
âˆ« e2x sin x dx

OR

Evaluate:

Q15 :

Prove that :

OR

Q16 :

Find the intervals in which the function f given by
f(x) = sin x + cos x, 0 â‰¤ x â‰¤ 2Ï€
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.

Q17 :

Prove the following :

OR

Solve the following equation for x :

Q18 :

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.

Q19 :

Prove using properties of determinants :

Q20 :

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

is continuous at x = Ï€.

Q21 :

Solve the following differential equation:

given that y = 0 when .

Q22 :

Find the shortest distance between the given lines:

Q23 :

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.

Q24 :

Evaluate as a limit of sums.

OR

Evaluate :

Q25 :

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

Q26 :

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.

Q27 :

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 :

Q28 :

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.