Cosine Rule

Q. If the angles of a triangle ABC be in A.P., then

1. $c^2=a^2+b^2-ab$
2. $b^2=a^2+c^2-ac$
3. $a^2=b^2+c^2-ac$
4. $b^2=a^2+c^2$

Q.

If the sides of a triangle are in the ratio $2:\sqrt{6}:(\sqrt{3}+1)$, then the largest angle of the triangle will be

1. ${60}^{\circ }$

2. ${75}^{\circ }$

3. ${90}^{\circ }$

4. ${120}^{\circ }$

Q.

If the line segment joining the points $A(a,b)$ and $B(c,d)$ subtends an angle $\theta$ at the origin, then $\cos \theta$ is equal to

1. $\frac{ab+cd}{\sqrt{(a^2+b^2)(c^2+d^2)}}$

2. $\frac{ac+bd}{\sqrt{(a^2+b^2)(c^2+d^2)}}$

3. $\frac{ac-bd}{\sqrt{(a^2+b^2)(c^2+d^2)}}$

4. None of these

Q.

Simplify$\sin (90-x)$

Q. In a triangle $ABC$ with usual notation, which of the following is (are) CORRECT?

1. If the angles $A,B,C$ are in A.P., then $b^2=c^2+a^2-ca$
2. $\frac{\sin (B-C)}{\sin (B+C)}=\frac{b^2-c^2}{a^2}$
3. $\sum \frac{b^2-c^2}{\cos B+\cos C}=0$
4. $\frac{1+\cos (A-B)\cos C}{1+\cos (A-C)\cos B}=\frac{a^2+b^2}{a^2+c^2}$

Q. Point $P$ lies on the diagonal $AC$ of square $ABCD$ with $AP>CP.$ Let $O_1$ and $O_2$ be the circumcentres of $△ABP$ and $△CDP$ respectively. Given that $AB=12$ and $\angle O_{1}P{O}_2={120}^{\circ },$ then $AP=\sqrt{a}+\sqrt{b},$ where $a$ and $b$ are positive integers. Find $a+b.$
(correct answer + 5, wrong answer 0)

Q.

In $△$ ABC, if (a+b+c)(a-b+c)=3ac, then
[AMU 1996]

1. $\angle B={30}^o$

2. $\angle B={60}^o$

3. $\angle A+\angle C={90}^o$

4. $\angle C={60}^o$

Q. Let $D$ be the middle point of the side $BC$ of a triangle $ABC$. If the triangle $ADC$ is equilateral, then $a^2:b^2:c^2$ is equal to

1. $4:1:2$
2. $4:1:3$
3. $4:3:1$
4. $3:4:1$

Q.

If in a triangle ABC, $\angle C={60}^o,$ then $\frac{1}{a+c}+\frac{1}{b+c}=$
[IIT 1975]

1. $\frac{1}{a+b+c}$

2. $\frac{2}{a+b+c}$

3. $\frac{3}{a+b+c}$

4. None of these

Q. In a $△$ ABC, if $c^2+a^2-b^2=ac,$ then $\angle B=$
[MP PET 1983, 89, 90]

1. $\frac{\pi }{6}$
2. $\frac{\pi }{4}$
3. $\frac{\pi }{3}$
4. None of these

Q. For a triangle ABC it is given that $cosA+cosB+cosC=\frac{3}{2}$ Prove that the triangle is equilateral

1. True
2. False

Q.

If the lengths of the sides of a triangle be $7,4\sqrt{3}and\sqrt{13}cm$, then the smallest angle is

1. ${15}^{\circ }$

2. ${30}^{\circ }$

3. ${60}^{\circ }$

4. ${45}^{\circ }$

Q. In $△ABC,\frac{sin(A-B)}{sin(A+B)}=$
[MP PET 1986]

1. $\frac{a^2-b^2}{c^2}$
2. $\frac{a^2+b^2}{c^2}$
3. $\frac{c^2}{a^2-b^2}$
4. $\frac{c^2}{a^2+b^2}$

Q. In a $\Delta PQR$, P is the largest angle and $cosP=\frac{1}{3}$.
Further incircle of the triangle touches the sides PQ, QR and RP at N, L and M respectively, such that the lengths of PN, QL and RM are consecutive even integers. Then, possible length (s) of the side (s) of the triangle is (are)

1. 16
2. 18
3. 24
4. 22

Q. In a triangle ABC if a =13, b = 8 and c = 7, then find sin A.

1. $\frac{1}{2}$
   
2. $\frac{\sqrt{3}}{2}$
   
3. $\frac{\sqrt{3}}{4}$
   
4. $\frac{1}{4}$

Q.

In $\Delta ABC,ifa=3,b=4,c=5,thensin2B=$

1. $\frac{4}{5}$

2. $\frac{3}{20}$

3. $\frac{24}{25}$

4. $\frac{1}{50}$

Q.

In $△$ ABC, if cot A, cot B, cot C be in A. P. then $a^2,b^2,c^2$ are in

1. H.P.

2. G.P.

3. A.P.

4. None of these