Mathematics_Today_-_October_2016

44. (c) : S
    Q

ab

ab

a

b

SQ Q QS

S

=

−

(^222) ⇒ = +± +
2
()

45. (a) : Note that sides are ()sinab− α
    2

and

()cosab− α

2

46. A → q ; B → p, q, r ; C → r

(A) ∵ max | |,| |{}xy= 12 /

|| /, || /

|| /, || /

xy

yx

=<

=<

⎧

⎨

⎩

12 12

12 12

if

if

Required area = 1 × 1 



= –1/

= –1/

= 1/

= 1/





= 1 sq. unit
(B) The line y = x cuts
the lines |x + y| = 6
i.e, x + y = ± 6
At x = ± 3, ⇒ (x, y) ≡ (–3, –3) and (3, 3)
then –3 < a < 3
∴ 0 ≤ |a| < 3
⇒ [|a|] = 0, 1, 2
(C) Since (0, 0) and (1, 1) lie on the same side.
So, a^2 + ab + 1 > 0
∵ Coefficient of a^2 > 0 ∴ D < 0
b^2 – 4 < 0 or – 2 < b < 2
⇒ b = –1, 0, 1
∴ Number of non-zero integral values of b are 2.
(b = –1 and b = 1)

47. (2) : If x ∈ (0, 1)
    Then –1 ≤ x + y < 0
    And if x ∈ [1, 2)
    0 ≤ x + y < 1

2

1

(^123)

+=–1

+= 0

+= 1

Required area 

= 4 ⎛⎝⎜^1 ⋅⋅ ⎞⎠⎟

2

12

4

sinπ

= 2 sq. units

48. (3) : Let, D = (2, –1) be
    the reflection of A in x-axis,
    and let E = (1, 2) be the
    reflection in the line y = x.
    Then AB = BD and AC = CE,
    so the perimeter of ABC is
    DB + BC + CE ≥ DE =^10


49. (4) : 2(a + b) = x (a constant)
    



  



(^) 


Area of PQRS = (bsinθ + acosθ)(asinθ + bcosθ)
=+ab ab+ ≤ ab x x+ = ∴ = ⇒x=
22 2 2 2
2
2
288
sin θ () 32 16

50. (1) ””













  
 

 
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