Download App

CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY3 Marks
Question
If $\text{x}=\text{a}(\cos2\text{t}+2\text{t}\sin2\text{t})\ \text{and}\ \text{y}=\text{a}(\sin2\text{t}-2\text{t}\cos2\text{t}),$ then find $\frac{\text{d}^2\text{y}}{\text{dx}^2}.$
✓

Answer

$\text{x}=\text{a}(\cos2\text{t}+2\text{t}\sin2\text{t})$
$\frac{\text{dx}}{\text{dt}}=-2\text{a}\sin2\text{t}+2\text{a}\sin2\text{t}+4\text{at}\cos2\text{t}=4\text{at}\cos2\text{t}$
$\text{y}=\text{a}(\sin2\text{t}-2\text{t}\cos2\text{t})$
$\frac{\text{dy}}{\text{dt}}=2\text{a}\cos2\text{t}-2\text{a}\cos2\text{t}+4\text{at}\sin2\text{t}=4\text{at}\sin2\text{t}$
$\frac{\text{dy}}{\text{dx}}=\tan2\text{t}$
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{\text{d}}{\text{dx}}(\tan2\text{t})$
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=2\sec^22\text{t}\frac{\text{d}}{\text{dx}}(\text{t})$
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=2\sec^22\text{t}\times\frac{1}{4\text{at}\cos2\text{t}}$
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{1}{2\text{a}}\sec^32\text{t}$

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 "3 Marks Question" 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

Four cards are successively drawn without replacement from a deck of $52$ playing cards. What is the probability that all the four cards are kings?
→
Vectors $\vec{a}, \vec{b}$ and $\vec{c}$ are such that $\vec{a}+\vec{b}+\vec{c}=0$ and $|\vec{a}|=3,|\vec{b}|=5$ and $|\vec{c}|=7$. Then find the angle between $\vec{a}$ and $\vec{b}$.
→
Given a non-empty set X, consider the binary operation *: P(X) × P(X) → P(X) given by $\text{A}*\text{B}=\text{A}\cap\text{B}\ \ \forall\ \text{A},\text{ B}$ in P(X), where P(X) is the power set of X. Show that X is the identity element for this operation and X is the only invertible element in P(X) with respect to the operation *.
→
If $\vec{\text{a}}=5\hat{\text{i}}-\hat{\text{j}}-3\hat{\text{k}}$ and $\vec{\text{b}}=\hat{\text{i}}+3\hat{\text{j}}-5\hat{\text{k}},$ then show that the vectores $\vec{\text{a}}+\vec{\text{b}}$ and $\vec{\text{a}}-\vec{\text{b}}$ are orthonal.
→
Evaluate the following intregals:
$\int\sqrt{\frac{1-\text{x}}{1+\text{x}}}\text{dx}$
→
Write the angle between the lines $\frac{\text{x}-5}{7}=\frac{\text{y}+2}{-5}=\frac{\text{z}-2}{1}$ and $\frac{\text{x}-1}{1}=\frac{\text{y}}{2}=\frac{\text{z}-1}{3}.$
→
Classify the following functions as injection, surjection or bijection:
f : R → R, defined by f(x) = |x|
→
Find the intervals in which the following functions are increasing or decreasing.
$f(x) = x^3 - 6x^2 - 36x + 2$
→
Find the intervals in which the function f given by $f(x) = 2x^2- 3x^2 - 36x + 7$ is $(a)$ strictly increasing, $(b)$ strictly decreasing.
→
If A is a square matrix of order 3 such that |A| = 3, then write the value of adj (adj A).
→