38. (a, b, c) : Here tanA, tanB are the roots of the
    equation abx^2 – c^2 x + ab = 0

Therefore, tanABtan c tan tan
ab

+= AB=

2

and 1

Now tanA tanB = 1 ⇒ cos(A + B) = 0
(as A, B are both acute)

∴ AB+=ππieC=

22

..

Also cot C = cot π
2

= 0

So ΔABC is right angled at C and

sinA = cosB ⇒ sin^2 A + sin^2 B = 1

Also tan( ) tan tan

tan tan

AB AB.

AB

a

b

b

a ab

ab

− = −

+

=

−

+

= −

1112

22

39. (a, c)


40. (b, c) : Let x – 3y + k = 0 touch the circle at (x 1 , y 1 )
    Then xx 1 + yy 1 – 2(x + x 1 ) + y + y 1 – 5 = 0

and x – 3y + k = 0 are identical

Hence, xy xy

k

112 11

1

1

3

− = + 25

−

=− + −

= −− ++− + −

× − ×− +

22112 5

211 3

()()(11 11)

()

xy xy

k

∴ − = +

−

=−

+

⇒ =−

+

+=

+

xy

k

x

k

k

k

(^1121)
1
1
3
10
5
10
5
2 2
5
and y
k
k
(^1) k
30
5
1 25
5

• − = −


• 
  

  (x 1 , y 1 ) is on the circle.
  So,
  2
  5
  25
  5
  4 2
  5
  2
  25
  5
  22
  k
  k
  k
  k
  k
  k
  k

  
  


• k
  ⎛
  ⎝⎜
  ⎞
  ⎠⎟


• 
  

  −

  
  


• 
  

  ⎛
  ⎝⎜
  ⎞
  ⎠⎟
  −⋅

  
  


• 
  

  ⎛
  ⎝⎜
  ⎞
  ⎠⎟

  
  


• ⋅
  −


• 
  

  ⎛
  ⎝⎜
  ⎞
  ⎠⎟
  − 550 =
  On simplification, k^2 + 10k – 75 = 0 ⇒ k = 5, –15
  ∴ == = =
  −
  −

  
  


• 
  

  −
  xy x y 1110 1 =−
  10
  1 20
  10
  30
  10
  3
  25 15
  515
  ,,.=2or 4
  So, (x 1 , y 1 ) = (1, 2), (3, –4).
  ””
  The IITs are struggling to get the best
  students, so much so that cutoff marks
  have been lowered in the past two years
  to fill seats in the country’s premier
  engineering institutes.
  Marks as low as 75 out of 372 in the joint
  entrance examination (JEE-Advance)
  this year were good enough to land
  a general category student a seat in
  one of the 23 IITs. Reserved category
  students —Dalits, tribals, and people
  with disability — had to get only 38 to
  secure admission.
  This comes at a time the IIT Council
  decided to increase the number
  of students from around 70,000 to
  100,000 over the next three years,
  enrolling even day scholars for the
  first time.
  The lowered benchmark has raised
  questions about academic quality.
  IIT-Roorkee expelled 11 BTech students
  early this year for poor academic
  performance. Last year, the premier
  institute expelled 72 students for dismal
  academic record, but re-admitted them
  later.
  Experts said tough
  question papers and
  negative marking in the
  entrance examination
  had resulted in a
  shortfall of candidates
  meeting the benchmark,
  thus forcing the IIT
  management to lower
  the bar.
  Doubts over a fall in
  standard have been
  dismissed.
  “The teaching-learning
  process at IITs is intense,
  and many students who
  come with low ranks in
  JEE turn out to be gems
  when they pass the BTech programme,”
  said the director of an IIT who didn’t
  wish to be named.
  A former director suggested improving
  the education standard at schools so that
  students could pass the entrance exam
  for IITs with higher grades. Of 147,678
  students who appeared for this year’s JEE
  (Advanced), 36,566 qualified.
  “Maintaining quality is a major issue
  which is why IITs coach such students
  so that they are on par with the rest.
  Students who were expelled by IIT-
  Roorkee in 2015 have done much better
  now with extra help, coaching from other
  students and faculty.”
  Courtesy : Hindustan Times

  
  

Declining IIT cutoffs spark fears of falling standards

LOWERING THE BAR

Cutoffs over the years show a steady drop in merit

GENERAL OBC

35%

24.5%

31.39%

17.5%

12.25%

22.05%

18%

10%

2014 2015 2016 2014 2015 2016 2014 2015 2016

20%

SC/ST