Cosine Rule

Q.

What is the integration of $\tan 2x$?

Q. Given $\frac{b+c}{11}=\frac{c+a}{12}=\frac{a+b}{13}$ for a $\Delta ABC$ with usual notation. If $\frac{\cos A}{\alpha }=\frac{\cos B}{\beta }=\frac{\cos C}{\gamma }$, then the ordered triad $(\alpha ,\beta ,\gamma )$ has a value :

1. $(3,4,5)$
2. $(19,7,25)$
3. $(7,19,25)$
4. $(5,12,13)$

Q. $\text{Prove that}\cos (\frac{\pi }{4}+x)+\cos (\frac{\pi }{4}-x)=\sqrt{2}\cos x$

Q.

The range of ${\sec }^{-1}x$ is

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 triangle ABC , A is obtuse , sinA=3/5 , sinB=5/13 then sinC=?

Q.

$Ina\Delta ABC,ifa=\sqrt{2},b=\sqrt{3}andc=\sqrt{5},\text{show that its area is}\frac{1}{2}\sqrt{6}sq.units.$

Q.

Two ships leave a port at the same time. One goes 24 $km/hr$ in the direction N ${38}^{\circ }$ E and other travels 32 $km/hr$ in the direction S ${52}^{\circ }$ E. Find the distance between the ships at fine end of 3 hrs.

Q.

Answer each of the following questions in one word or one sentence or as per exact requirement of for question:

$Ina\Delta ABC,ifb=20,c=21andsinA=\frac{3}{5},finda.$

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

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

Q. In a \triangle ABC, if $b^2+c^2=3a^2,$ then cot B + cot C = cot A =

1. 1
2. $\frac{ab}{4△}$
3. \N
4. $\frac{ac}{4△}$

Q.

If a is any real number the number of roots of cot x-tanx=ain the first quadrant is

Q. (Pythagorass Theorem) Prove by vector method that in a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Q.

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

1. Right angled

2. Obtuse angled

3. Right angled isosceles

4. Isosceles

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. None of these

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

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

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

Q. Let $ABC$ be a triangle with $\angle BAC={120}^{\circ }$ and $\left(AB\right)\cdot \left(AC\right)=1.$ Also, let $AD$ be the length of the angle bisector of angle $A$ of the triangle. Then

1. Minimum value of $AD$ is $\frac{1}{2}$
2. Maximum value of $AD$ is $\frac{1}{2}$
3. $AD$ is minimum when $△ABC$ is isosceles.
4. $AD$ is maximum when $△ABC$ is isosceles.

Q. If the area of the triangle $ABC$ is $\Delta ,$ such that $b^2\sin 2C+c^2\sin 2B=k\Delta ,$ then the value of $k$ is

Q.

What is the circumference of a circle when the diameter is 18?

Q. y=2/root(a^2-b^2)arc tan(a-b/a+b tan(x/2)), a>b>0, then (a) y1=-1/(a+bcosx) (b) y2=bsinx/(a+bcosx)^2 (c) y1= 1/(a-bcosx) (d) y2=-bsinx/(a-bcosx)^2

Q.

Answer each of the following questions in one word or one sentence or as per exact requirement of for question:

$Ifina\Delta ABC,\frac{cosA}{a}=\frac{cosB}{b}=\frac{cosC}{c},\text{then find the measures of angles A, B, C.}$

Q.

$(c^2-a^2+b^2)tanA=(a^2-b^2+c^2)tanB=(b^2-c^2+a^2)tanC$

Q.

A jet of enemy is flying along the path given by y=x^2+7 , Find the nearest distance from where a soldier placed at (3, 7) can shoot down the helicopter

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.

In reference to a circle the value of $\mathrm{\pi }$ is equal to?

1. $\frac{\mathrm{area}}{\mathrm{circumference}}$

2. $\frac{\mathrm{area}}{\mathrm{diameter}}$

3. $\frac{\mathrm{circumference}}{\mathrm{diameter}}$

4. $\frac{\mathrm{circumference}}{\mathrm{radius}}$

Q. Are the following equivalent?
1)x belongs to R-(-1, 3)
2)x belongs to (- infinity to -1) union of [3, infinity)
3)x <= to -1 or x >=3.
can someone provide me detail explanation

Q. If ${\cot }^{2}x=\cot (x-y)\cot (x-z)$, then the value of $\cot 2x$ is

1. $\frac{1}{2}(\tan y+\tan z)$
2. $\frac{1}{2}(\cot y+\cot z)$
3. $\frac{1}{2}(\sin y+\sin z)$
4. $\frac{1}{2}(\cot y-\cot z)$

Q. if the three positive numbers a, b, c are in AP and 1/a^2, 1/b^2, 1/c^2 also in AP then

Q. If the sides $a,b,c$ of a triangle $ABC$ form successive terms of G.P. with common ratio $r(>1)$ then which of the following is correct?

1. $A>\frac{\pi }{3}$
2. $B\ge \frac{\pi }{3}$
3. $C<\frac{\pi }{3}$
4. $A<B<\frac{\pi }{3}$

Q. In any triangle ABC, prove the following: $b\sin B-c\sin C=a\sin (B-C)$.

Q.

$\text{In a triangle ABC, if}\angle A={60}^0,a=5,b=4,\text{then c is a root of the equation}$

1. c² - 5c - 9 = 0

2. c² - 4c - 9 = 0

3. c² - 10c + 25 = 0

4. c² - 5c - 41 = 0