Mathematics - Properties of Triangles, Height and Distances

Question - 1

If in a \(\Delta ABC\)  , A=30o , B=45o and a=1, then the values of b and c are respectively

  • A \(\sqrt { 2 } ,\frac { \sqrt { 3 } +1 }{ \sqrt { 2 } } \)
  • B \(\sqrt { 2 } ,\frac { \sqrt { 3 } -1 }{ \sqrt { 2 } } \)
  • C \(\sqrt { 3 } ,\frac { \sqrt { 3 } -1 }{ \sqrt { 2 } } \)
  • D \(\sqrt { 2 } ,\frac { \sqrt { 3 } +2 }{ \sqrt { 2 } } \)

Question - 2

If A=75o , B=45o , then \(b+c\sqrt { 2 } \) is equal to 

  • A 2a
  • B 2a+1
  • C 3a
  • D 2a-1

Question - 3

If a2,b2 and c2 are in AP, then cotA,cotB and cotC are in 

  • A AP
  • B GP
  • C HP
  • D AGP

Question - 4

In \(\Delta ABC,\left( \frac { b }{ c } +\frac { c }{ b } \right) cosA+\left( \frac { a }{ b } +\frac { b }{ a } \right) cosC+\left( \frac { a }{ c } +\frac { c }{ a } \right) cosB\) is equal to 

  • A 4
  • B 5
  • C 3
  • D 2

Question - 5

If in a \(\Delta ABC\) ,  the tangent of half the difference of two angles is one-third the tangent of half the sum of the angles. Then, the ratio of the sides opposite to the angles is 

  • A 2:1
  • B 1:2
  • C 3:1
  • D 1:1

Question - 6

In a \(\Delta ABC,\)  if 2s=a+b+c, then the value of \(\frac { s(s-a) }{ bc } -\frac { (s-b)(s-c) }{ bc } \) is equal to 

  • A sin A
  • B cos A
  • C tan A
  • D None of these

Question - 7

If in \(\Delta ABC,\) \(\Delta ={ a }^{ 2 }-{ \left( b-c \right) }^{ 2 },\) then the value of tanA is 

  • A \(\frac { 8 }{ 14 } \)
  • B \(\frac { 8 }{ 13 } \)
  • C \(\frac { 8 }{ 15 } \)
  • D \(\frac { 8 }{ 17 } \)

Question - 8

If in \(\Delta ABC,\) r1=r2+r3+r, then triangle is 

  • A a right-angled triangle
  • B equilateral triangle 
  • C isoscles triangle
  • D None of the above 

Question - 9

The sum of the radii of the circles, which are respectively inscribed and circumscribed about a polygon of n sides, whose side length is a, is 

  • A \(\frac { 1 }{ 2 } atan\left( \frac { \pi }{ 2n } \right) \)
  • B \(\frac { 1 }{ 2 } acot\left( \frac { \pi }{ 2n } \right) \)
  • C \(\frac { 1 }{ 2 } cot\left( \frac { \pi }{ 3n } \right) \)
  • D \(\frac { 1 }{ 2 } cot\left( \frac { \pi }{ 2n } \right) \)

Question - 10

Three vertical poles of height h1, h2 and h3 at the vertices A,B and C of a \(\Delta ABC\) subtend angles \(\alpha ,\beta \quad and\quad \gamma \) respectively, at the circumcentre of the triangle. If \(cot\alpha ,cot\beta ,cot\gamma \) are in AP, then h1, h2 and h3 are in 

  • A AP
  • B GP
  • C AGP
  • D HP
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