Here are few mathematical shorcuts to solve probs asked in IIT-JEE....
1. (IIT 04) 
then the interval in which 'a' lies is
(a)a<-5 (b)-5<a<2 (c)a>5 (d)2<a<5
Comment: the alternatives do no have a common value. Put a convenient value for a. Let us try : a=0.
Then 
Therefore, (b) is the correct answer!!!
2. (AIEEE04) The domain of the function 
is
(a) [1,2) (b) [2,3) (c) [1,2] (d) [2,3]
Comment: Look at the alternatives: Neither (a) nor (c) is the correct ans. bçoz x=1,
does not exist. now he prob reduces to whether 3 belongs to domain or not. When x=3, denominator = 0. Hence, (b) is the ans.
3. (IIT 2K) The domain of function y(x) given by 
is
(a) [0,1] (b) (0,1] (c) 
(d) 
Comment: When x=1, 2+2^y = 2 is not possible. Hence, (a) and (b) are not the ans. Now, put the value for x betn. (0,1) and check. That is cumbersome. Therefore don't think this kind of technique works for every prob.
4. (IIT 05) In any triangle ABC (with usual notations of the sides a,b,c)
(a) b-c/2 sin A = a cos A (b)(b-c) cos A/2 = a sin B-C/2
(c)b+c/2 cos A/2 = a sin B-C/2 (d)(b+c) cos A/2 = 2a sin B+C/2
Comment: Try an isosceles triangle in which b=c, i.e. B = C.
Take (b): 0 =0 ; no other alternatives satisfy them.
5. A hyperbola having transverse axis of length 
, is confocal with the ellipse 
Then its eqn. is
(a) 
(b) 
(c) 
(d) 
Comment: The hyperbola in (a) alone has 2 sin 0 as the length of the transverse axis and hence (a) is the correct ans.
6. (IIT 2K) If 
are complex numbers such that 
=1, then Iz,1 + z,2 + z,3I is
(a) equal to 1 (b)<1 (c) >3 (d) equal to 3
Comment: Put 
. It satisfies the given condition and hence (a) is the ans.
7. (IIT 01) If a+b = pi/2 and b+c=a, then tan a equals
(a) 2(tan b + tan c) (b) tan b + tan c (c) tan b + 2 tan c (d) 2 tan b+ tan c
Comment: Put a=b=pi/4, then c=0. Tan a = 1.
Hence, (b) or (c) is the correct ans.
Suppose a = 22 1/2, then b = 67 1/2. Then c=-45.
Then tan b + tan c = 
. Hence, (c) is the ans.
MORE TO FOLLOW.......
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shortcut to solve math probs of iit.......
