Napier’s Analogy

Q. If in a triangle ABC, (s-a)(s-b) = s(s-c), then angle C is equal to [MP PET 1986]

1. ${90}^o$
2. ${45}^o$
3. ${30}^o$
4. ${60}^o$

Q.

Evaluate: $\underset{x\to 2}{\lim }\left(\frac{\left(\sqrt{1-\cos 2\left(x-2\right)}\right)}{\left(x-2\right)}\right)$

1. $\sqrt{2}$

2. $-\sqrt{2}$

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

4. does not exist

Q.

If $5({\tan }^{2}x-{\cos }^{2}x)=2{\cos }^{2}x+9,$ then the value of $\cos 4x$ is

Q.

The set of extreme values of the function $f\left(x\right)=\tan x,x\in R-\left\{(2k+1)\left(\frac{\mathrm{\pi }}{2}\right),k\in I\right\}$

1. $ɸ$

2. $R–(-1,1)$

3. $R^{+}\cup \left\{0\right\}$

4. $R$

Q. In a triangle ABC,
$\angle A=60°andb:c=\sqrt{3}+1:2,thenthevalueof\angle B-\angle C=$ .

1. $45°$
2. $60°$
3. $15°$
4. $30°$

Q.

In $∆ABC$, $\tan A$and $\tan B$ are the roots of$pq(x^2+1)=r^{2}x.$ Then$∆ABC$ is

1. A right-angled triangle

2. Acute-angled triangle

3. An obtuse-angled triangle

4. An equilateral triangle

Q. Convert the complex number z =(i-1 )/[(1/2)+(√3/2)i] in polar form. My doubt here is that how we can we identify that cos theta equal to (√3-1)/2 √2 and sin theta=(√3+1)/2√2 is which angle , (that is the value of theta ) ???

Q.

Compute area of region bounded by curves y=tanx and y=tan^2 x (-π/2<x<π/2)

Q. Prove that: acosA + bcosB + ccosC=∆/R
where a, b, c are the sides of a triangle and A, B, C are it's angles and ∆ is the area of the triangle ABC and R is the circumradius.

Q. IN TRIANGLE ABC , tanA/2=5/6 , tanB/2=20/37 , tanC/2 = ???

Q. The sum of n terms of the series $co{t}^{-1}3+co{t}^{-1}7+co{t}^{-1}13+co{t}^{-1}21+......$ is

1. $ta{n}^{-1}\frac{n-1}{n+2}$
2. $ta{n}^{-1}\frac{n}{n+2}$
3. $ta{n}^{-1}\frac{n+1}{n+2}$
4. none of these

Q.

In a triangle $ABC$ we define
$x=tan\frac{B-C}{2}tan\frac{A}{2},y=tan\frac{C-A}{2}tan\frac{B}{2}$ and $z=tan\frac{AB}{2}tan\frac{C}{2}$
Then the value of $x+y+z$ (in terms of $x,y,z$) is

1. xyz

2. -xyz

3. 2xyz

4. none

Q. If $tan\frac{B-C}{2}=xcot\frac{A}{2},thenx=$

1. $\frac{c-a}{c+a}$
2. $\frac{a-b}{a+b}$
3. $\frac{b-c}{b+c}$
4. None of these

Q. If $\hat{a}\mathrm{and}\hat{b}$are unit vectors inclined at an angle θ, prove that
(i) $\cos \frac{\mathrm{\theta }}{2}=\frac{1}{2}\left|\hat{a}+\hat{b}\right|$

(ii) $\tan \frac{\mathrm{\theta }}{2}=\frac{\left|\hat{a}-\hat{b}\right|}{\left|\hat{a}+\hat{b}\right|}$

Q. If ${\tan }^{-1}\left(\frac{a}{x}\right)+{\tan }^{-1}\left(\frac{b}{x}\right)=\frac{\pi }{2},$ then $x$ is equal to

1. $ab$
2. $\frac{a}{b}$
3. $\frac{b}{a}$
4. $\sqrt{ab}$

Q. If $f\left(x\right)=\left(lo{g}_{cotx}tanx\right)(lo{g}_{tanx}cotx{)}^{-1}+ta{n}^{-1}(\frac{x}{\sqrt{4-x^2}})$ , then f’(0) is equal to

1. $2$
2. $-2$
3. $\frac{1}{2}$
4. $0$

Q. If $tan\frac{B-C}{2}=xcot\frac{A}{2},thenx=$

1. $\frac{c-a}{c+a}$
2. $\frac{a-b}{a+b}$
3. $\frac{b-c}{b+c}$
4. None of these

Q. If in a triangle ABC, (s-a)(s-b) = s(s-c), then angle C is equal to [MP PET 1986]

1. ${90}^o$
2. ${45}^o$
3. ${30}^o$
4. ${60}^o$

Q. If in a triangle a = $\sqrt{3}+1,b=\sqrt{3}-1,C={60}^{\circ }$ then the value of A is ?

1. ${75}^{\circ }$
2. ${90}^{\circ }$
3. ${105}^{\circ }$
4. ${120}^{\circ }$

Q. The angle between the curves y² = x and x² = y at (1, 1) is

(a) ${\tan }^{-1}\frac{4}{3}$
(b) ${\tan }^{-1}\frac{3}{4}$
(c) 90°
(d) 45°

Q. If y = x + tan x, show that ${\cos }^{2}x\frac{d^{2}y}{d{x}^2}-2y+2x=0$.

Q. Write the value of θ ϵ for which area of the triangle formed by points O (0, 0), A (a cos θ, b sin θ) and B (a cos θ, − b sin θ) is maximum.

Q. If $3f\left(\tan x\right)+2f\left(\cot x\right)=x+3$, $x\in (0,\frac{\pi }{2})$, then $15f\left(\sqrt{3}\right)=$
(correct answer + 1, wrong answer - 0.25)

1. $4\pi +1$
2. $4\pi +7$
3. $2\pi +3$
4. $2\pi +9$

Q. If in a triangle a = $\sqrt{3}+1,b=\sqrt{3}-1,C={60}^{\circ }$ then the value of A is ?

1. ${75}^{\circ }$
2. ${90}^{\circ }$
3. ${105}^{\circ }$
4. ${120}^{\circ }$

Q. The vectors $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$satisfy the equations $2\overrightarrow{a}+\overrightarrow{b}=\overrightarrow{p}\mathrm{and}\overrightarrow{a}+2\overrightarrow{b}=\overrightarrow{q},\mathrm{where}\overrightarrow{p}=\hat{i}+\hat{j}\mathrm{and}\overrightarrow{q}=\hat{i}-\hat{j}.$ If θ is the angle between $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$, then
(a) $\cos \mathrm{\theta }=\frac{4}{5}$

(b) $\sin \mathrm{\theta }=\frac{1}{\sqrt{2}}$

(c) $\cos \mathrm{\theta }=-\frac{4}{5}$

(d) $\cos \mathrm{\theta }=-\frac{3}{5}$

Q. If in a triangle $ABC,a=5,b=4$ and $\cos (A-B)=\frac{31}{32}$. then the third side $c$ is equal to

1. 3
2. 6
3. 7
4. 9

Q. If $a,b,c$ are the sides opposite to angles $A,B,C$ of a triangle $ABC,$ respectively and $\angle A=\frac{\pi }{3},b:c=\sqrt{3}+1:2,$ then the value of $\angle B-\angle C$ is

1. $\frac{\pi }{12}$
2. $\frac{\pi }{6}$
3. $\frac{\pi }{4}$
4. $\frac{\pi }{2}$

Q. If in a triangle $ABC,a=5,b=4$ and $\cos (A-B)=\frac{31}{32}$. then the third side $c$ is equal to

1. 6
2. 3
3. 9
4. 7

Q. The least value of the product $xyz(x,y,z\ne 1)$ for which the determinant $\left|\begin{array}{ccc}x& 1& 1\\ 1& y& 1\\ 1& 1& z\end{array}\right|$ is non-negative is:

1. $-8$
2. $-1$
3. $-2\sqrt{2}$
4. $-16\sqrt{2}$
5. None of these

Q.

In a triangle $ABC$ we define
$x=tan\frac{B-C}{2}tan\frac{A}{2},y=tan\frac{C-A}{2}tan\frac{B}{2}$ and $z=tan\frac{AB}{2}tan\frac{C}{2}$
Then the value of $x+y+z$ (in terms of $x,y,z$) is

1. xyz

2. -xyz

3. 2xyz

4. none