Cosine Rule

Q. if z=4+3\sqrt{20}i, i=\sqrt{-1}, then the positive value of {(\sqrt z+\sqrt{\overline z} )}^2 is equal to

Q. If P(at^2, 2at), Q(a/t^2, -2a/t)andS(a, 0)be any three points, show that1/SP+1/SQ is independent of t?

Q. In a triangle ABC, if the sides a, b and c are the roots of $x^3-11x^2+38x-40=0,\mathrm{then}\mathrm{the}\mathrm{value}\mathrm{of}\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}=$ .

1. $\frac{9}{16}$
2. -$\frac{9}{16}$
3. $\frac{3}{4}$
4. -$\frac{3}{4}$

Q.

In a $\Delta ABC,if\angle C={30}^{\circ },a=47cmandb=94cm$, then the triangle is

1. Right angled

2. Right angled isosceles

3. Isosceles

4. Obtuse angled

Q. the domain of the fuction f(x)=\sqrt{(2-2x-x^2}) is

Q.

2(bc cos A+ ca cos B + ab cos C)=

1. abc

2. a²+b²+c²

3. a²+b²-c²

4. 2abc

Q.

In any $\Delta ABC,$ prove that

$(b^2-c^2)cotA+(c^2-a^2)cotB+(A^2-B^2)cotC=0$

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 $\Delta ABC,a(bcosC-ccosB)$ is equal to -

1. b + c
2. b - c
3. $b^2-c^2$
4. $b^2+c^2$

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

Q. 32. If α and β are two distinct roots of equation a cosx + b sinx = 10, then tan(α +β )/2 is equal to

Q. $\underset{x\to a}{\lim }\frac{x^n-a^n}{x-a}$ is equal at
​
(a) naⁿ
(b) naⁿ⁻¹
(c) na
(d) 1

Q.

If in a $\Delta$ ABC, $2b^2=a^2+c^2$, then $\frac{sin3B}{sinB}$ is equal to

1. $\frac{c^2-a^2}{2ca}$

2. $\frac{c^2-a^2}{2ca}$

3. ${\left(\frac{c^2-a^2}{ca}\right)}^2$

4. ${\left(\frac{c^2-a^2}{2ca}\right)}^2$

Q. If α, β are the roots of the equation $x^2+px+1=0;\gamma ,\delta$ the roots of the equation $x^2+qx+1=0,then(\mathrm{\alpha }-\mathrm{\gamma })(\mathrm{\alpha }+\mathrm{\delta })(\mathrm{\beta }-\mathrm{\gamma })(\mathrm{\beta }+\mathrm{\delta })=$
(a) $q^2-p^2$
(b) $p^2-q^2$
(c) $p^2+q^2$
(d) none of these.

Q. ​let a, b and c be the sides opposite to the angles A, B and C respectively of a triangle ABC. find the value of k such that a + b = kc

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.

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

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

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

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

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

Q. The sides of $△ABC$ satisfy the equation $2a^2+4b^2+c^2=4ab+2ac$. Then

1. The triangle is isosceles
2. The triangle is obtuse
3. $B={\cos }^{-1}\left(\frac{7}{8}\right)$
4. $A={\cos }^{-1}\left(\frac{1}{4}\right)$

Q.

A ring, 10 cm in diameter, is suspended from a point 12 cm above its centre by 6 equal strings attached to its circumference at equal intervals. The cosine of the angle between consecutive strings is

1. $\frac{213}{236}$

2. $\frac{313}{338}$

3. $\frac{119}{345}$

4. None of these

Q. If x>0 then prove that log{x+√(x²+1)} > tan-¹x

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 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. If the roots of the equation $a{x}^2+2bx+c=0$ be alpha and beta and those of the equation $A{x}^2+2Bx+C=0$ be alpha+delta and beta+delta prove that,
$\frac{b^2-ac}{a^2}=\frac{B^2-AC}{A^2}$