JEE Maths - Properties of Triangle

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Question - 1

If the sines of the angles A and, B of a triangle ABC satisff the equation c2x2-c(a+b)x+ab=0,then the triangle is (a,b,c, are sides of D)

  • A acute angled
  • B obtuse angled
  • C right angled
  • D no such triangle is possible

Question - 2

ln \(\triangle A B C, a \geq b \geq c\) ,If \(\frac{a^{3}+b^{3}+c^{3}}{\sin ^{3} A+\sin ^{3} B+\sin ^{3} C}=7\) then the maximum possible value of a is

  • A 7
  • B 49
  • C \(\sqrt[3]{7}\)
  • D \(\frac{1}{\sqrt{7}}\)

Question - 3

If in a triangle ABC, sin l, sin B, sin C are in A.P then 

  • A the altitudes are in A.P.
  • B the reciprocals of altitudes are in A.P.
  • C the altitudes are in G.P.
  • D the medians are in A.P.

Question - 4

Let a, b and A are given and c, and c, are two values of c, then the value of \(c_{1}^{2}+c_{2}^{2}-2 c_{1} c_{2}\) cos2A  =

  • A the altitudes are in A.P.
  • B the reciprocals of altitudes are in A.P.
  • C the altitudes are in G.P.
  • D the medians are in A.P.

Question - 5

If a, b, c be the sides of a triangle and \(P=\frac{(a+b+c)^{2}}{a b+b c+c a}\) , then

  • A \(P \in[1,2]\)
  • B \(P \in[3,4)\)
  • C \(P \in(2,4]\)
  • D None of these

Question - 6

lf a, b, c are the sides of a triangle such that b.\(c=\lambda^{2}\) ,for some positive \(\lambda\) then

  • A \(a \geq 2 \lambda \sin \frac{A}{2}\)
  • B \(b \geq 2 \lambda \sin \frac{B}{2}\)
  • C \(c \geq 2 \lambda \sin \frac{C}{2}\)
  • D all are correct

Question - 7

In \(\triangle \mathrm{ABC}, \) a2(s-a)+b2(s-b)+c2(s-c)=

  • A 4R(cos A + cos B + cos C)
  • B \(4 R \Delta(\sin A+\sin B+\sin C)\)
  • C \(4 \mathrm{R} \Delta\left(1+4 \sin \frac{\mathrm{A}}{2} \sin \frac{\mathrm{B}}{2} \sin \frac{\mathrm{C}}{2}\right)\)
  • D None of these

Question - 8

In an obtuse angled triangle, the obtuse angle is \(\frac{3 \pi}{4}\) and the other two angles are equal to two values of \(\theta\) satisfying \(a \tan \theta+b \sec \theta=c\) ,where \(|b| \leq \sqrt{a^{2}+c^{2}}\) then a2-c2 is equal to

  • A ac
  • B 2ac
  • C \(\frac{a}{c}\)
  • D None of these

Question - 9

In a triangleABC, \(2 a c \sin \frac{1}{0}(A-B+C)=\) 

  • A a2+b2-c2
  • B c2+a2-b2
  • C b2 -c2 -a2
  • D c2-a2-b2

Question - 10

The sides of a triangle are \(\sin \alpha, \cos \alpha\) and \(\sqrt{1+\sin \alpha \cos \alpha}\) for some for some \(0<\alpha<\frac{\pi}{2}\). Then the greatest angle of the triangle is

  • A 150o
  • B 90o
  • C 120p
  • D 60o
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