### 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