Mathematics - Properties of Triangles, Height and Distances
Exam Duration: 45 Mins Total Questions : 30
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)
\(\sqrt { 2 } ,\frac { \sqrt { 3 } +2 }{ \sqrt { 2 } } \)
If a2,b2 and c2 are in AP, then cotA,cotB and cotC are in
- (a)
AP
- (b)
GP
- (c)
HP
- (d)
AGP
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
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
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
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
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
If the angle of elevation of the top of a hill from each of the vertices A,B and C of a horizontal triangle is \(\alpha \) . Then, the height of the hill is
- (a)
\(\frac { 1 }{ 2 } btan\alpha .secB\)
- (b)
\(\frac { 1 }{ 2 } btan\alpha .cosecA\)
- (c)
\(\frac { 1 }{ 2 } ctan\alpha .sinC\)
- (d)
\(\frac { 1 }{ 2 } atan\alpha .cosecA\)
Match the Columns
Column I | Column II | ||
---|---|---|---|
A. | \(cot\frac { A }{ 2 } =\frac { b+c }{ a } \) | p. | always right angles |
B. | \(atanA+btanB=(a+b)tan\left( \frac { A+B }{ 2 } \right) \) | q. | always isosceles |
C. | \(acosA=bcosB\) | r. | may be right angled |
D. | \(cosA=\frac { sinB }{ 2sinC } \) | s. | may be right-angled isosceles |
- (a)
A B C D pr, qrs, rs, qrs - (b)
A B C D qp, prs, qs, qsp - (c)
A B C D qr, qps, ps, qpr - (d)
None of the above
The ratio of the areas of two regular octagons, which are respectively inscribed and circumscribed to a circle of radius r is equal to
- (a)
\(sin\left( \frac { \pi }{ 8 } \right) \)
- (b)
\({ sin }^{ 2 }\left( \frac { 3\pi }{ 8 } \right) \)
- (c)
\({ cos }^{ 2 }\left( \frac { \pi }{ 8 } \right) \)
- (d)
\({ tan }^{ 2 }\left( \frac { \pi }{ 8 } \right) \)
If in a \(\Delta ABC,\) the angles are in AP and the lengths of two larger sides are 10 and 9 respectively, then the length of the third side can be
- (a)
\(5+\sqrt { 6 } \)
- (b)
\(2\sqrt { 6 } \)
- (c)
\(3-\sqrt { 6 } \)
- (d)
\(3+\sqrt { 6 } \)
In \(\Delta ABC,\) A=15o , \(b=10\sqrt { 2 } \) cm, the value of 'a' for which this will be a unique triangle meeting these requirement is
- (a)
\(10\sqrt { 2 } cm\)
- (b)
\(5cm\quad \)
- (c)
\(5\left( \sqrt { 2 } +1 \right) cm\)
- (d)
\(5\left( \sqrt { 2 } -1 \right) cm\)
In a \(\Delta PQR,\) if 3sinP+4cosQ=6 and 4sinQ+3cosP = 1, then the angle R is equal to
- (a)
150o
- (b)
30o
- (c)
45o
- (d)
135o
In a \(\Delta ABC,\angle C=\frac { \pi }{ 2, } \) if r is the inradius and R is the circumradius of the \(\Delta ABC,\) then 2(r+R) is equal to
- (a)
c + a
- (b)
a + b + c
- (c)
a + b
- (d)
b + c
If in a \(\Delta ABC,\) the altitudes from the vertices A,B and C on opposite sides are in HP, then sin A, sin B and sin C are in
- (a)
HP
- (b)
AGP
- (c)
AP
- (d)
GP
If in a \(\Delta ABC,\) \(a{ cos }^{ 2 }\left( \frac { C }{ 2 } \right) +c{ cos }^{ 2 }\left( \frac { A }{ 2 } \right) =\frac { 3b }{ 2 } ,\) Then the sides a, b and c
- (a)
are in AP
- (b)
are in GP
- (c)
are in HP
- (d)
satisfy a + b = c
If in a \(\Delta ABC,a=4,b=3,\angle A={ 60 }^{ o },\) then c is the root of the equation
- (a)
c2-3c-7=0
- (b)
c2+3c+7=0
- (c)
c2-3c+7=0
- (d)
c2+3c-7=0
In a \(\Delta ABC,tan\frac { A }{ 2 } =\frac { 5 }{ 6 } ,tan\frac { C }{ 2 } =\frac { 2 }{ 5 } ,\) then
- (a)
a,c and b are in AP
- (b)
a,b and c are in AP
- (c)
b,a and c are in AP
- (d)
a,b and c are in GP