Implicit Differentiation

Q. if y=2/sin theta+ root 3 cos theta, then minimum value of y is

Q. Find the derivative of cos x from first principle.

Q. 1.If tan x=n tan y and sin x=m sin y then prove that (m^2 - 1 )/ (n^2 - 1)=cos^2 x ? 2.If x sin^3 A + y cos^3 A=sinAcosA and x sinA=y cosA.Prove that x^2+y^2=1 ?

Q. If $\mathrm{cosec}x-\sin x=a^3,\sec x-\cos x=b^3$, then prove that $a^2{b}^2\left(a^2+b^2\right)=1$.

Q. 10. If cos (pi/4-x)cos2x+sinxsin2xsecx= cosxsin2xsecx+cos (pi/4+x)cos2x then posSible value of secx

Q.

Let f(x) = $x^2$ -bx+c, b is odd positive integer, f(x) = 0 have two prime

numbers as roots and b+c=35.Then the global minimum value of f(x) is

1. 

2. 

3. 

4. 

Q. 36. If sin alpha + sin beta is equal to a and cos alpha + cos beta is equal to b then find value of cos alpha + beta and sin alpha + beta

Q. 37. sin(B+A) + cos(B-A)/ sin(B-A) + cos(B+A) is equal to 1. cosB + sinB/cosB - sinB 2. cosA + sinA/cosA - sinA 3. cosA - sinA/cosA + sinA 4. cosA - sinA/cosA

Q. If A + B + C = ${180}^{\circ }$ then sin 2A + sin 2B + sin 2C =

1. sin A sin B sin C
2. 4 sin A sin B sin C
3. 3 sin A sin B sin C
4. 2 sin A sin B sin C

Q. (b)If x=a cos θ + bsin θ and Yb cos θ-a sin θ, then eliminate θ from these two relations.

Q. Sketch the graphs of the following pairs of functions on the same axes:
(i) $f\left(x\right)=\sin x,g\left(x\right)=\sin \left(x+\frac{\pi }{4}\right)$
(ii) f(x) = sin x, g(x) = sin 2x
(iii) f(x) = sin 2x, g(x) = 2 sin x
(iv) $f\left(x\right)=\sin \frac{x}{2},g\left(x\right)=\sin x$

Q. 11. If cos(a-b)-1=0, show that cos a+cos b=0 and sin a +sin b=0

Q. 4.Effect on inequality between L.H.S and R.H.S when divided/multiplied by -ve or +ve term on both sides

Q. If $y=\sqrt{x+\sqrt{y+\sqrt{x+\sqrt{y+....\infty ,}}}}$ then $\frac{dy}{dx}$ is equal to

1. $\frac{1}{2y-1}$
2. $\frac{y^2-x}{2y^3-2xy-1}$
3. $(2y-1)$
4. None of these

Q. 64.FIND X IF COS 3X + SIN(2x - 7pi/6)= -2

Q.

Sin(n+1)A sin(n+2)A + cos(n+1)A cos(n+2)A=

1. sin2A

2. cosA

3. sinA

4. cos2A

Q. Find the value of θ when \sqrt3 sin theta +cos theta = 2

Q. Prove the following identities (1-16)
$\mathrm{cosec}x\left(\sec x-1\right)-\cot x\left(1-\cos x\right)=\tan x-\sin x$

Q.

In $△$ ABC, a sin (B - C) + b sin (C - A) + c sin (A-B) =
[ISM Dhanbad 1973]

1. 0

2. a+b+c

3. $a^2+b^2+c^2$

4. $2(a^2+b^2+c^2)$

Q.

The angles A, B and C of a triangle ABC are in AP, if b : c = : $\sqrt{3}:\sqrt{2}$, then the angle A is

1. 30°

2. 15°

3. 75°

4. 45°

Q. Let $y=y\left(x\right)$ be a function of $x$ satisfying $y\sqrt{1-x^2}=k-x\sqrt{1-y^2}$ where $k$ is a constant and $y\left(\frac{1}{2}\right)=-\frac{1}{4}$. Then $\frac{dy}{dx}$ at $x=\frac{1}{2}$ is equal to :

1. $-\frac{\sqrt{5}}{2}$
2. $\frac{\sqrt{5}}{2}$
3. $-\frac{\sqrt{5}}{4}$
4. $\frac{2}{\sqrt{5}}$

Q. If α and β are two solutions of the equation a tan x + b sec x = c, then find the values of sin (α + β) and cos (α + β).

Q. Prove that:
(i) $\frac{\cos (A+B+C)+\cos (-A+B+C)+\cos (A-B+C)+\cos (A+B-C)}{\sin (A+B+C)+\sin (-A+B+C)+\sin (A-B+C)-\sin (A+B-C)}=\cot C$

(ii) sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 0

Q.

$\text{If cos}\left(\frac{x}{2}\right)cos\left(\frac{x}{2^2}\right)cos\left(\frac{x}{2^3}\right)...............to\infty =\frac{sinx}{x}then\frac{1}{2^2}se{c}^2\left(\frac{x}{2}\right)\frac{1}{2^4}se{c}^2\left(\frac{x}{2^2}\right)+.......=$

1. 

2. 

3. 

4. 

Q. derivative of cos^2 x

Q. 47. Prove that cosθ .cos(θ /2) - cos3θ .cos(9θ /2) = sin7θ .sin8θ

Q. 37. If sec(theta)cos(alpha) = \sqrt{}2 and tan(theta)cot(alpha) = \sqrt{}3 , then find the values of tan(theta) and tan(alpha) that satisfy the equations.

Q.

$If{2}^x-2^y=2^{x+y}then\frac{dy}{dx}=$

1. 2^y-x

2. 2^y/x

3. -2^y-x

4. 2^x/y

Q. 10. What is number of soln y=cos4x and y=sinx/2 where x[0 2]

Q. The value of $y^{\prime \prime }\left(1\right)$ if $x^3-2x^2{y}^2+5x+y-5=0$ when $y\left(1\right)=1$, is equal to

1. $\frac{22}{7}$
2. $-7\frac{21}{28}$
3. $8$
4. $-8\frac{22}{27}$