Advanced Organic Chemistry Solutions PDF | Carey & Sundberg Problem Guide

EXAMINATION PAPERS – 2008
MATHEMATICS CBSE (All India)
CLASS – XII

Time allowed: 3 hours Maximum marks: 100

General Instructions: As given in CBSE Examination paper (Delhi) – 2008.

Set–I
SECTION–A
3x – 2
1. If f (x) is an invertible function, find the inverse of f (x) = .
5
1–x 1
2. Solve for x : tan – 1 = tan – 1 x; x > 0
1+x 2
「 x + 3y yù 「 4 – 1ù
3. If | =
7 – x 4| |0 4|
, find the values of x and y.
L ] L ]
4. Show that the points (1, 0), (6, 0), (0, 0) are collinear.
x + cos 6x
dx
5. Evaluate : ∫ 3x + sin 6x
2

ax 4x 3x 2
+ bx) dx = 4e + , find the values of a and b.
6. If ∫(e 2
→ → → → → →
7. If| a |= 3 ,| b |= 2 and angle between a and b is 60°, find a . b .
→
8. Find a vector in the direction of vector a = i‸ – 2j‸, whose magnitude is 7.
x– 3 y+2 z–5
9. If the equation of a line AB is = = , find the direction ratios of a line parallel to AB.
1 –2 4
x+2 3
x + 5 4 = 3, find the value of x.
10. If


SECTION–B
11. Let T be the set of all triangles in a plane with R as relation in T given by R = {(T1 , T2 ) : T1 m T2}.
Show that R isfanπ equivalence
1 a ⎞ relation.
fπ 1 a ⎞ 2b
12. Prove that tan + cos– 1 + tan – cos– 1 = .
| | | |
⎩4 2 b ⎠ ⎩ 4 2 b ⎠ a

,Examination Papers – 2008 33
OR
–1 –1 –1 8
Solve tan (x + 1) + tan (x – 1) = tan
31
13. Using properties of determinants, prove that following:
a + b + 2c a b
c b + c + 2a b = 2(a + b + c) 3
c a c + a + 2b
14. Discuss the continuity of the following function at x = 0 :
⎧ x 4 + 2x 3 + x 2
| , xs0
f (x) ={ tan – 1 x
| x=0
î| 0,
OR
Verify Lagrange’s mean value theorem for the following function:
f (x) = x 2 + 2x + 3, for [4, 6].
f ⎞
15. If f (x) = sec x – 1 , find f ' (x). Also find f ’ π .
| |
sec + 1 ⎩ 2⎠
OR
dy
If x 1 + y + y 1 + x = 0, find .
dx
π/ 2
tan x + cot x = 2π
Show that ∫
16. 0
17. Prove that the curves x = y 2 and xy = k intersect at right angles if 8k 2 = 1.
18. Solve the following differential equation:
dy
x + y = x log x; x s 0
dx
19. Form the differential equation representing the parabolas having vertex at the origin and
axis along positive direction of x-axis.
OR
Solve the following differential equation:
( 3xy + y 2 )dx + (x 2 + xy)dy = 0
20. If i‸ + j‸ + k‸, 2i‸ + 5j‸, 3i‸ + 2j‸ –3k‸ and i‸ – 6j‸ – k‸ are the position vectors of the points A, B, C and
—→ —→ —→ —→
D, find the angle between AB and CD . Deduce that AB and CD are collinear.
21. Find the equation of the line passing through the point P(4, 6, 2) and the point of intersection
x–1 y z+1
of the line = = and the plane x + y – z = 8.
3 2 7
22. A and B throw a pair of die turn by turn. The first to throw 9 is awarded a prize. If A starts the
9
game, show that the probability of A getting the prize is .
17

,34 Xam idea Mathematics – XII


SECTION–C
23. Using matrices, solve the following system of linear equations:
2x – y + z = 3
–x + 2y – z = – 4
x – y + 2z = 1
OR
Using elementary transformations, find the inverse of the following matrix:
「 2 – 1 4ù
|4 0 2|
| |
|L 3 – 2 7 |]
x2 y2
24. Find the maximum area of the isosceles triangle inscribed in the ellipse + = 1, with its
a2 b2
vertex at one end of major axis.
OR
Show that the semi–vertical angle of the right circular cone of given total surface area and
1
maximum volume is sin – 1 .
3
25. Find the area of that part of the circle x 2 + y 2 = 16 which is exterior to the parabola y 2 = 6x.
π x tan x
dx
Evaluate: ∫ sec x + tan x
26. 0
x + 2 2y + 3 3z + 4
27. Find the distance of the point (– 2, 3, – 4) from the line = = measured
3 4 5
parallel to the plane 4x + 12y – 3z + 1 = 0.
28. An aeroplane can carry a maximum of 200 passengers. A profit of Rs. 400 is made on each
first class ticket and a profit of Rs. 300 is made on each second class ticket. The airline
reserves at least 20 seats for first class. However, at least four times as many passengers
prefer to travel by second class then by first class. Determine how many tickets of each type
must be sold to maximise profit for the airline. Form an LPP and solve it graphically.
29. A man is known to speak truth 3 out of 4 times. He throws a die and report that it is a 6. Find
the probability that it is actually 6.

Set–II
Only those questions, not included in Set I, are given.
20. Using properties of determinants, prove the following:
a a +b a + 2b
a + 2b a a + b = 9b 2 (a + b)
a+b a + 2b a

, Examination Papers – 2008 35
π/ 2
log sin x dx
Evaluate: ∫
21. 0
22. Solve the following differential equation:
dy
(1 + x 2 ) + y = tan – 1 x
dx
27. Using matrices, solve the following system of linear equations:
3x – 2y + 3z = 8
2x + y – z = 1
4x – 3y + 2z = 4
OR
Using elementary transformations, find the inverse of the following matrix:
「 2 5 3ù
|3 4 1|
| |
|L1 6 2|]
28. An insurance company insured 2000 scooter drivers, 3000 car drivers and 4000 truck drivers.
The probabilities of their meeting with an accident respectively are 0.04, 0.06 and 0.15. One of
the insured persons meets with an accident. Find the probability that he is a car driver.
29. Using integration, find the area bounded by the lines x + 2y = 2, y – x = 1 and 2x + y = 7.

Set–III
Only those questions, not included in Set I and Set II are given.
20. If a, b and c are all positive and distinct, show that
a b c
Δ = b c a has a negative value.
c a b
cot– 1 (1 – x + x 2 ) dx
1
21. Evaluate:
∫0
22. Solve the following differential equation:
dy
x log x + y = 2 log x
dx
27. Using matrices, solve the following system of linear equations:
x+y+z=4
2x + y – 3z = – 9
2x – y + z = – 1
OR
Using elementary transformations, find the inverse of the following matrix: