IIT-JEE 2010 - Career Point

29. Describe how potassium dichromate is made from
chromite ore and give the equations for the chemical
reactions involved.
Write balanced ionic equations for reacting ions to
represent the action of acidified potassium
dichromate solution on :
(i) Potassium iodide solution
(ii) Acidified ferrous sulphate solution
Write two uses of potassium dichromate.
30. Give appropriate reasons for each of the following
observations :
(i) Sulphur vapour exhibits some paramagnetic
behaviour.
(ii) Silicon has no allotropic form analogous to
graphite.
(iii) Of the noble gases only xenon is known to form
real chemical compounds.
(iv) Nitrogen shows only a little tendency for
catenation, whereas phosphorus shows a clear
tendency for catenation.
MATHEMATICS
Section A
1. Show that the relation R in the set {1, 2, 3} is given
by R = {(1, 2), (2, 1)} is symmetric.
−1
2. Evaluate : x.tan x dx.
∫
3. Find the differential equation of the family of curves
given by- x 2 + y 2 = 2ax.
→
→
8. If a = î + ĵ ; b = ĵ+
kˆ ; c = kˆ + î find a unit vector
in the direction of
→
→
→
→
a + b+
c .
9. What is the angle between vector → aand
magnitude 3 and 2 respectively.
→
bwith
10. Find the direction cosines of a line which make equal
angles with the co ordinate axes.
Section B
11. Consider f : N → N, g : N → N and h : N → R
defined as f (x) = 2x, g (y) = 3y + 4 and h (z) = sin
z ∀ x, y and z in N. Show the ho(gof) = (hog)of.
12. Differentiate cot –1 ⎛1−
x ⎞
⎜ ⎟ w.r.t. x.
⎝1+
x ⎠
13. Solve the differential equations :
(1 + e 2x ) dy + e x (1 + y 2 ) dx = 0. Give that y = 1,
when x = 0.
or
dy
Solve the differential equation : x − y − 2x
3 = 0
dx
π
14. Evaluate :
∫
/ 4
0
or
sin
π / 4
+
Evaluate :
∫
log( 1
0
2x sin 3x
dx.
tan x)dx
4. Find the principle value of tan –1 (–1).
5. Find a matrix C such that 2A – B + C = 0
⎡3
1⎤
⎡− 2 1⎤
Where A = ⎢ ⎥ and B =
⎣0
2
⎢ ⎥ ⎦ ⎣ 0 3 ⎦
6. If A is a square matrix of order 3 such that
| adj A | = 64, find | A |.
7. Find the value of x if the matrix A =
singular.
⎡ 4
⎢
⎢
3
⎢⎣
10
3
− 2
−1
5⎤
7
⎥
⎥
x⎥⎦
is
3x + 1
15. Evaluate :
∫
dx .
2
2x − 2x + 3
16. If x = a (θ – sin θ) and y = a (1 – cos θ),
find
d
2
dx
y
2
at θ
= 2
π .
17. Find the value of K so that the function,
⎧Kx
+ 1, if x ≤ π
f (x) = ⎨
⎩cos x , if x > π
or
⎪⎧
3
x + 3, if x ≠ 0
Show that the function f (x) = ⎨
⎪⎩ 1 , if x = 0
XtraEdge for IIT-JEE 73 DECEMBER 2009