Download App

CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY5 Marks
Question
If $\text{x}=2\cos\text{t}-\cos2\text{t},\text{y}=2\sin\text{t}-\sin2\text{t},$ find $\frac{\text{d}^2\text{y}}{\text{dx}^2}\ \text{at}\ \text{t}=\frac{\pi}{2}.$
✓

Answer

Given,
$\text{x}=2\cos\text{t}-\cos\text{2}\text{t}$
$\text{y}=2\sin\text{t}-\sin\text{2t}$
Differentiating w.r.t. t,
$\frac{\text{dy}}{\text{dx}}=2(-\sin\text{t})-2(-\sin\text{2t})$
$\Rightarrow\frac{\text{dy}}{\text{dt}}=2\cot-2\cos\text{2t}$
Dividing both:
$\frac{\text{dy}}{\text{dx}}=\frac{2(\cos\text{t}-\cos\text{2t})}{2(\sin\text{2t}-\sin\text{t})}$
Differentiating w.r.t. t,
$\Rightarrow\frac{\text{d}\frac{\text{dy}}{\text{dx}}}{\text{dt}}=\frac{(\sin\text{2t}-\sin\text{t})(-\sin\text{t}+2\sin\text{2t})-(\cos\text{t}-\cos\text{2t})(2\cos\text{2t}-\cos\text{t})}{(\sin\text{2t}-\sin\text{t})^2}$
Dividing:
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{(\sin\text{2t}-\sin\text{t})(2\sin\text{t}-\sin\text{t})-(\cos\text{t}-\cos\text{2t})(2\cos\text{2t}-\cos\text{t})}{2(\sin\text{2t}-\sin)^3}$
Putting: $\text{t}=\frac{\pi}{2}$
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{1+2}{-2}=-\frac{3}{2}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Solve the following equation:
$\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}=\text{y}^2$
→
Find the approximate value of $\log_{10} 1005,$ given that $\log_{10} e = 0.4343.$
→
Using differentials, find the approximate values of the following:
$\sqrt{25.02}$
→
Find the equations of the tangent and the normal to the following curves at the indicated points.
$y = x^2$ at $(0, 0)$
→
Evaluate the following integrals:
$\int^\limits{\pi}_0\sin^3\text{x}(1+2\cos\text{x})(1+\cos\text{x})^2\text{ dx}$
→
Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}} = \tan(\text{x}+\text{y})$
→
If A and B are two events such that $2\text{P(A)}=\text{P(B)}=\frac{5}{13}$ and $\text{P}\Big(\frac{\text{A}}{\text{B}}\Big)=\frac{2}{5}$ find $\text{P}(\text{A}\cap\text{B}).$
→
In a hospital, there are 20 kidney dialysis machines and that the chance of any one of them to be out of service during a day is 0.02. Determine the probability that exactly 3 machines will be out of service on the same day.
→
The probability distribution of a random variable X is given below:
$\text{X}$ $0$ $1$ $2$ $3$
$\text{P}(\text{X})$ $\text{k}$ $\frac{\text{k}}{2}$ $\frac{\text{k}}{4}$ $\frac{\text{k}}{8}$
  1. Determine the value of k.
  2. Determine $\text{P}(\text{X}\leq2)$ and $\text{P}(\text{X}\geq2)$
  3. Find $\text{P}(\text{X}\leq2)+\text{P}(\text{X}\geq2)$
→
$\int\sin^{-1}\sqrt{\frac{\text{x}}{\text{a}+\text{x}}}\text{dx}$
Hint: Put $\text{x} = \text{a}\tan^2θ$
→