JEE Main Mathematics - Properties Of Triangles, Height and Distances
Exam Duration: 60 Mins Total Questions : 30
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
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)
\(\frac { 1 }{ 2 } cot\left( \frac { \pi }{ 2n } \right) \)
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
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 \(\Delta ABC,\quad if\quad sin\left( \frac { C }{ 2 } \right) =\frac { a-b }{ 2\sqrt { ab } } tan\theta ,\quad then\quad (a-b)sec\theta \) is equal to
- (a)
2a+b
- (b)
2c
- (c)
c
- (d)
c+2
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
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 \(\Delta PQR,\angle R=\frac { \pi }{ 2 } ,if\quad tan\left( \frac { P }{ 2 } \right) \quad and\quad tan\left( \frac { Q }{ 2 } \right) \) 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
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
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
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