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 A=75o , B=45o , then \(b+c\sqrt { 2 } \) is equal to
- (a)
2a
- (b)
2a+1
- (c)
3a
- (d)
2a-1
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
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)
\(\frac { 8 }{ 17 } \)
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
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
In a \(\Delta ABC,\) a3 cos(B - C)+b3cos(C - A)+c3cos(A - B) is equal to
- (a)
3abc
- (b)
2abc
- (c)
abc
- (d)
5abc
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)
\({ tan }^{ 2 }\left( \frac { \pi }{ 8 } \right) \)
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 \(\Delta PQR,\angle R=\frac { \pi }{ 2 } ,if\quad tan\left( \frac { P }{ 2 } \right) \quad and\quad tan\left( \frac { Q }{ 2 } \right) \) are the roots of ax2 + bx + c = 0, \(a\neq 0,\) then
- (a)
b = a + c
- (b)
b = c
- (c)
c = a + b
- (d)
a = 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
The sides of a triangle are \(sin\alpha ,cos\alpha \quad and\quad \sqrt { 1+sin\alpha cos\alpha } \) for some \(0<\alpha <\frac { \pi }{ 2 } \) . Then, the greatest angle of the triangle is
- (a)
60o
- (b)
90o
- (c)
120o
- (d)
150o
In a \(\Delta ABC,2asin\left( \frac { A-B+C }{ 2 } \right) \) is equal to
- (a)
a2 + b2 - c2
- (b)
c2 + a2 - b2
- (c)
b2 - c2 - a2
- (d)
c2 - a2 - b2