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Hard Type


Q1. Prove that a2 + b2 + c2 + 2abc < 2 where a, b, c, are sides of ABC such that a + b + c = 2

Ans: a + b + c = 2
1 - a + 1 - b + 1 - c = 1
x + y + z = 1   where   x = 1 - a, y = 1 - b, z = 1 - c
Sine in
Sum of two sides > third side
a + b > c   0 < c < 1
Similarly   0 < a, b < 1
hence   0 < x, y, z < 1
Now,
a2 + b2 + c2 + 2abc = (1 - x)2 + (1 - y)2 + (1 - z)2 + 2(1 - x)(1 - y)(1 - z)

                         = 3 - 2(x + y + z) + x2 + y2 + z2 + 2{1 - (x + y + z) + (xy + yz + zx) - xyz}

                         = 1 + x2 + y2 + z2 - 2xyz + 2(xy + yz + zx)

                         = 1 + (x + y + z)2 - 2xyz

                         = 2 - 2xyz < 2   as   0 < x, y, z < 1
a2 + b2 + c2 + 2abc < 2


Dumb Question: How   0 < c < 1 ?

Ans: Since gn ,
Sum of two sides > third side
   a + b > c ............................................. (i)
or a + b + c > 2c
   2 > 2c   c < 1 ..................................... (ii)
Since c is side of
c > 0 ............................................... (iii)
From (ii) & (iii)
0 < c < 1


Q.2. A function R R is f(x) = . Find integral value of for which f is onto.

Ans: Since f : R R is onto mapping
  Range = codomain = R
  assumes all real values of x
Let y =
x2( + 8y) + 6x(1 - y) - (8 - y) = 0 v y R
D 0
    36(1 - y)2 + 4( + 8y)(8 + y) 0
[y2(8 + 9) + y(2 + 46) + (8 + 9) 0
    ax2 + bx + c 0 v x R   if   a > 0 & D 0
    (2 + 46)2 - 4(8 + 9)(8 + 9) 0   &   (8 + 9) > 0
(2 + 46)2) - [2(8 + 9)]2 0   &   > -
(2 + 46 - 16 - 18)(2 + 46 + 16 + 18) 0   &   > -
( - 14)( - 2)( + 8)2 0   and   > -
[2, 14] {-8}   and   > -
  [2, 14]


Q.3. Solve the eq.   [x] = x

Ans: x = [x] + .................................. (i)
[x] = [x] +   = ............................. (ii)
    [x] 1   in eq. (ii)
But if [x] = 1, then
= x   which is donje only when,
x [0, 1) ................................................. (iii)
& [x] = 1   [1, 2) ................................. (iv)
From (iii) & (iv) no value of x,
when [x] = 1
As we know
     [0, 1)
  0 < 1
     < 1   &   0
< 0   &   {[x] 0   or   [x] > 1}
[x] < 1   &   {[x] 0   or   [x] > 1}
[x] 0
  x = [x] + = [x] +
    x = , where x takes values less than 1.
  x = , where x < 1
    x = , where [x] is any non positive integer.


Q4. Sketch   y = (x - 1)(x - 2)

Ans: y = (x - 1)(x - 2)
(i) put   y = 0   x = 1, 2
(ii) y = x2 - 3x + 2
= 2x - 3   &  
as > 0
minimaq at x = 3/2.

(iii) Increases when x > 3/2 & decreases when x < 3/2.
y = x2 - 3x + 2
when x = 3/2
y =




Q5. Sketch curve   y = (x - 1)(x - 2)(x - 3)

Ans: y = (x - 1)(x - 2)(x - 3)
(i) Put y = 0   x = 1, 2, 3
(ii) y = x3 - 6x2 + 11x - 6
= 3x2 - 12x + 11   &   = 6x - 12
when = 0   x =
Maxima when   x =   as   = - 2
Minima when   x =   as   = 2

(iii) = 3x2 - 12x + 11
          = 3(x - )(x - )
> 0 or Increases when
x <   or   x >
Decreases when   < 0
or < x <

(iv) Concave upwards when x > 2 & concave down when x < 2.



x = 2 is point where concavity of curve changes.


Key Words:

* Function.
* Domain.
* Codomain.
* Range.
* Identity Function.
* Constant Function.
* Logarithmic Function.
* Modulus Function.
* Signum Function.
* Greatest Integer Function.
* Fractional Part Function.
* Odd & Even Function.
* Periodic Function.
* Composite Function.
* Mapping.
* Bijective.
* Surjective.
* Injective.
* Inverse of Function.






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