(a) Find $\frac{dy}{dx}$, if $(\cos x)^y = (\cos y)^x$. OR (b) If $\sqrt{1-x^2} + \sqrt{1-y^2} = a(x-y)$, prove that $\frac{dy}{dx} = \sqrt{\frac{1-y^2}{1-x^2}}$.
CBSE CLASS 12 • MATHEMATICS • CHAPTER 05
Continuity and Differentiability
154 distinct questions across 205 board paper set appearances (2022–2026).
If $(\sin x)^y = y^{\cos x}$, then find $\frac{dy}{dx}$.
The general solution for the differential equation $\frac{dy}{dx} = e^{3x-y}$ is : (A) $3e^y = e^{3x} + C$ (B) $\log (3x-y) = C$ (C) $e^{3x-y} = C$ (D) $-e^y + 3e^{3x} = C$
(a) Differentiate $x^x$ with respect to $x \log x$. OR (b) If $y = P \cos ux + Q \sin ux$, show that $\frac{d^2y}{dx^2} + u^2y = 0$.
If $f(x) = | x | + | x - 1 |$, then which of the following is correct ?
(A) $f(x)$ is both continuous and differentiable, at $x = 0$ and $x = 1$.
(B) $f(x)$ is differentiable but not continuous, at $x = 0$ and $x = 1$.
(C) $f(x)$ is continuous but not differentiable, at $x = 0$ and $x = 1$.
(D) $f(x...
The solution for the differential equation $\log \left$ \frac{dy}{dx} \right$ = 3x + 4y$ is : (A) $3e^{4y} + 4e^{-3x} + C = 0$ (B) $e^{3x+4y} + C = 0$ (C) $3e^{-3y} + 4e^{4x} + 12C = 0$ (D) $3e^{-4y} + 4e^{3x} + 12C = 0$
A function $f(x) = |1 - x + |x||$ is :
(A) discontinuous at $x = 1$ only
(B) discontinuous at $x = 0$ only
(C) discontinuous at $x = 0, 1$
(D) continuous everywhere
Degree of the differential equation $\sin x + \cos \left(\frac{dy}{dx}\right) = y^2$ is
(A) 2
(B) 1
(C) not defined
(D) 0
The function $f(x) = x|x|$ is
(A) continuous and differentiable at $x = 0$.
(B) continuous but not differentiable at $x = 0$.
(C) differentiable but not continuous at $x = 0$.
(D) neither differentiable nor continuous at $x = 0$.
If $y = \frac{\cos x - \sin x}{\cos x + \sin x}$, then $\frac{dy}{dx}$ is :
(a) $-\sec^2\left$\frac{\pi}{4} - x\right$$
(b) $\sec^2\left$\frac{\pi}{4} - x\right$$
(c) $\log \left| \sec \left$ \frac{\pi}{4} - x \right$ \right|$
(d) $-\log \left| \sec \left$ \frac{\pi}{4} - x \right$ \right|$
(a) A die is rolled. Consider events : A = {1, 2, 5}, B = {3, 5}, C = {2, 3, 4, 5} and hence find : (i) $P(A|C)$ and $P(C|A)$ (ii) $P(A \cap B|C)$ and $P(A \cup B|C)$ OR (b) A box contains 6 cards numbered 1 to 6. A student is asked to pick up two cards, one by one after replacement and note down th...
(a) If $y\sqrt{x^2+1} = \log\sqrt{x^2+1}-x$, show that $(x^2+1)\frac{dy}{dx} + xy + 1 = 0$. OR (b) Find the differential of $x^{\cot x} + \frac{2x^2-3}{2x^2-x+2}$ with respect to x.
Assertion (A) : $f(x) = [x]$, $x \in \mathbb{R}$, the greatest integer function is not differentiable at $x = 2$.
Reason (R) : The greatest integer function is not continuous at any integral value.
Which of the following statements is true for the function $f(x) = \begin{cases} x^2 + 3, & x \neq 0 \\ 1 &, x = 0 \end{cases}$ ?
(A) $f(x)$ is continuous and differentiable $\forall \text{ x} \in \mathbb{R}$
(B) $f(x)$ is continuous $\forall \text{ x} \in \mathbb{R}$
(C) $f(x)$ is continuous and di...
If A denotes the set of continuous functions and B denotes set of differentiable functions, then which of the following depicts the correct relation between set A and B ?
(A) \begin{picture}(40,30)\put(0,0){\framebox(40,30){}}\put(20,15){\circle{20}}\put(20,15){\circle{12}}\put(18,13){A}\put(18,7){B...
If $f(x) = [x], x \in \mathbb{R}$ is the greatest integer function, then the correct statement is : (A) $f$ is continuous but not differentiable at $x = 2$. (B) $f$ is neither continuous nor differentiable at $x = 2$. (C) $f$ is continuous as well as differentiable at $x = 2$. (D) $f$ is not continu...
(a) If $x = e^{x/y}$, prove that $\frac{dy}{dx} = \frac{\log x - 1}{(\log x)^2}$ OR (b) Check the differentiability of $f(x) = \begin{cases} x^2 + 1, & 0 \le x < 1 \\ 3 - x, & 1 \le x \le 2 \end{cases}$ at $x = 1$.
(a) If $x \cos(p+y) + \cos p \sin(p+y) = 0$, prove that $\cos p \frac{dy}{dx} = - \cos^2(p+y)$, where $p$ is a constant. OR (b) Find the value of $a$ and $b$ so that function $f$ defined as: $f(x) = \begin{cases} \frac{x-2}{|x-2|} + a, & \text{if } x < 2 \\ a+b, & \text{if } x = 2 \\ \frac{x-2}{|x-2...
(a) If $y = \cos^{3}(\sec^{2} 2t)$, find $\frac{dy}{dt}$. OR (b) If $x^{y} = e^{x-y}$, prove that $\frac{dy}{dx} = \frac{\log x}{(1 + \log x)^{2}}$.
If $x e^y = 1$, then the value of $\frac{dy}{dx}$ at $x = 1$ is : (A) $-1$ (B) $1$ (C) $-e$ (D) $-\frac{1}{e}$
The order of the differential equation $\frac{d^4y}{dx^4} - \sin\left(\frac{d^2y}{dx^2}\right) = 5$ is : (A) $4$ (B) $3$ (C) $2$ (D) not defined
If $x = a \sin^3 \theta$, $y = b \cos^3 \theta$, then find $\frac{d^2y}{dx^2}$ at $\theta = \frac{\pi}{4}$.
If $\sin(xy) = 1$, then $\frac{dy}{dx}$ is equal to : (A) $\frac{x}{y}$ (B) $-\frac{x}{y}$ (C) $\frac{y}{x}$ (D) $-\frac{y}{x}$
The order of the following differential equation $\frac{d^3y}{dx^3} + x \left$ \frac{dy}{dx} \right$^5 = 4 \log \left$ \frac{d^4y}{dx^4} \right$$ is :
(A) not defined
(B) 3
(C) 4
(D) 5
If y = (\tan^{-1}x)^2, show that (x^2 + 1)^2 \frac{d^2y}{dx^2} + 2x(x^2 + 1) \frac{dy}{dx} = 2.
Let $y = f\left(\frac{1}{x}\right)$ and $f'(x) = x^3$. What is the value of $\frac{dy}{dx}$ at $x = \frac{1}{2}$ ? (A) $-\frac{1}{64}$ (B) $-\frac{1}{32}$ (C) $-32$ (D) $-64$
If $y = \log \sqrt{\sec \sqrt{x}}$, then the value of $\frac{dy}{dx}$ at $x = \frac{\pi^2}{16}$ is : (A) $\frac{1}{\pi}$ (B) $\pi$ (C) $\frac{1}{2}$ (D) $\frac{1}{4}$
(a) If $y = (\sin^{-1} x)^2$, then find $(1 - x^2) \frac{d^2 y}{dx^2} - x \frac{dy}{dx}$. OR (b) If $y^x = x^y$, then find $\frac{dy}{dx}$.
The sum of the order and the degree of the differential equation $\frac{d^2y}{dx^2} + \left$ \frac{dy}{dx} \right$^3 = \sin y$ is :
(a) 5
(b) 2
(c) 3
(d) 4
The equation of the path traced by a roller-coaster is given by the polynomial $f(x) = a(x + 9)(x + 1)(x - 3)$. If the roller-coaster crosses y-axis at a point $(0, -1)$, answer the following :
(i) Find the value of 'a'.
(ii) Find $f''(x)$ at $x = 1$.
If $\tan \left$ \frac{x+y}{x-y} \right$ = k$, then $\frac{dy}{dx}$ is equal to
(A) $\frac{-y}{x}$
(B) $\frac{y}{x}$
(C) $\sec^2 \left$ \frac{y}{x} \right$$
(D) $-\sec^2 \left$ \frac{y}{x} \right$$
(a) If $y = x^{\frac{1}{x}}$, then find $\frac{dy}{dx}$ at $x = 1$. OR (b) If $x = a \sin 2t$, $y = a(\cos 2t + \log \tan t)$, then find $\frac{dy}{dx}$.
A volleyball player serves the ball which takes a parabolic path given by the equation $h(t) = -\frac{7}{2}t^2 + \frac{13}{2}t + 1$, where $h(t)$ is the height of ball at any time $t$ (in seconds), $(t \geq 0)$. Based on the above information, answer the following questions: (i) Is $h(t)$ a continuo...
What is the product of the order and degree of the differential equation $\frac{d^2y}{dx^2} \sin y + \left$ \frac{dy}{dx} \right$^3 \cos y = \sqrt{y}$ ?
(a) 3
(b) 2
(c) 6
(d) not defined
The function $f(x) = |x|$ is
(a) continuous and differentiable everywhere.
(b) continuous and differentiable nowhere.
(c) continuous everywhere, but differentiable everywhere except at $x = 0$.
(d) continuous everywhere, but differentiable nowhere.
If $y = \sin^2(x^3)$, then $\frac{dy}{dx}$ is equal to :
(a) $2 \sin x^3 \cos x^3$
(b) $3x^3 \sin x^3 \cos x^3$
(c) $6x^2 \sin x^3 \cos x^3$
(d) $2x^2 \sin^2(x^3)$
The order and degree (if defined) of the differential equation, $\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^3 = x \sin\left(\frac{dy}{dx}\right)$ respectively are :
(a) $2, 2$
(b) $1, 3$
(c) $2, 3$
(d) $2, \text{ degree not defined}$
If $(x^2 + y^2)^2 = xy$, then find $\frac{dy}{dx}$.
If $x = a \cos \theta + b \sin \theta$, $y = a \sin \theta - b \cos \theta$, then which one of the following is true ?
(a) $y^2 \frac{d^2y}{dx^2} - x \frac{dy}{dx} + y = 0$
(b) $y^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} + y = 0$
(c) $y^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} - y = 0$
(d) $y^2 \frac{d^2y...
(b) Find the value of $(2a - 3b)$, if a and b represent respectively the order and the degree of the differential equation $x \left[ y \left$ \frac{d^2y}{dx^2} \right$^3 + x \left$ \frac{dy}{dx} \right$^2 - \frac{y dy}{x dx} \right] = 0$.
(a) Check whether function $\text{f}(x)$ defined as
$$\text{f}(x) = \begin{cases} \frac{|x - 3|}{2(x - 3)}, & x < 3 \\ \frac{x - 6}{6}, & x \ge 3 \end{cases}$$
is continuous at $x = 3$ or not ?
OR
(b) If $\sqrt{3}(x^2 + y^2) = 4xy$, then find $\frac{\text{d}y}{\text{d}x}$ at $\left(\frac{1}{2}, \fra...
(a) Check whether function $f(x)$ defined as
$$f(x) = \begin{cases} \frac{|x - 3|}{2(x - 3)}, & x < 3 \\ \frac{x - 6}{6}, & x \ge 3 \end{cases}$ is continuous at $x = 3$ or not ?
OR
(b) If $\sqrt{3}(x^2 + y^2) = 4xy$, then find $\frac{dy}{dx}$ at $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$.
If $x = a(\sin t - t \cos t)$ and $y = b(\cos t + t \sin t)$, then find $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$.
If $e^{-x} + e^{-y} = 2$, then $\frac{dy}{dx}$ is
(A) $e^{x-y}$
(B) $e^{y-x}$
(C) $-e^{x-y}$
(D) $-e^{y-x}$
If $x = a \sin^3 t$, $y = b \cos^3 t$, then find $\frac{dy}{dx}$ at $t = \frac{\pi}{4}$.
(iii) (a) Test for continuity of $f(x)$ at $x = 3$.
OR
(iii) (b) Test for differentiability of $f(x)$ at $x = 3$.
If $e^{x + y} = 3x$, then $\frac{dy}{dx}$ is
(A) $\frac{3}{e^{x + y}}$
(B) $\frac{1}{e^{x + y}}$
(C) $\frac{1 - e^{x + y}}{e^{x + y}}$
(D) $\frac{3 - e^{x + y}}{e^{x + y}}$
Sports car racing is a form of motorsport which uses sports car prototypes. The competition is held on special tracks designed in various shapes.
The equation of one such track is given as follows :
$f(x) = \begin{cases} x^4 - 4x^2 + 4, & 0 \leq x < 3 \\ x^2 + 40 &, x \geq 3 \end{cases}$
Based on ...
If $x + y = xy$, then $\frac{dy}{dx}$ is
(A) $\frac{y}{x-1}$
(B) $\frac{1}{x-1}$
(C) $\frac{y-1}{x-1}$
(D) $\frac{1-y}{x-1}$
The equation of one such track is given as follows :
$f(x) = \begin{cases} x^4 - 4x^2 + 4, & 0 \le x < 3 \\ x^2 + 40, & x \ge 3 \end{cases}$
Based on given information, answer the following questions :
(i) Find $f'(x)$ for $0 < x < 3$.
(ii) Find $f'(4)$
(iii) (a) Test for continuity of $f(x)$ at $x ...
The probability of getting an odd prime number on each die, when a pair of dice is rolled, is :
(A) $\frac{1}{18}$
(B) $\frac{1}{9}$
(C) $\frac{1}{36}$
(D) $\frac{1}{6}$
(a) Solve the differential equation : $x\frac{dy}{dx} = y - x \tan \frac{y}{x}$
OR
(b) Solve the differential equation : $\frac{dy}{dx} = e^{x + y} + e^{x - y}$
A Math teacher of a school is teaching the derivative of function in parametric form, to her students.
She explains that sometimes the relation between two variables, (say $x$ and $y$), is neither explicit nor implicit, but the two variables are expressed in a third variable, (say $t$). So we have ...
If $x = e^{t+\frac{1}{t}}$ and $y = e^{t-\frac{1}{t}}$, then find $\frac{dy}{dx}$ at $t = -2$.
The sum of the order and the degree of the differential equation $y = x \frac{dy}{dx} + 3 \left[ 2 - \left$ \frac{dy}{dx} \right$^2 \right]$ is :
(A) not defined
(B) $3$
(C) $1$
(D) $2$
(a) A die is rolled. Consider events : $A = \{1, 2, 5\}$, $B = \{3, 5\}$, $C = \{2, 3, 4, 5\}$ and hence find : (i) $P(A|C)$ and $P(C|A)$ (ii) $P(A \cap B|C)$ and $P(A \cup B|C)$
The order and degree of differential equation $y = \left(\frac{d^2y}{dx^2}\right)^2 - \lambda \frac{dy}{dx}$ is :
(A) Order = 2, Degree = 2
(B) Order = 2, Degree = 3
(C) Order = 1, Degree = 2
(D) Order = 2, Degree = 4
The function $f(x) = \frac{3 - x}{3x^2 - x^3}$ is :
(A) discontinuous at only one point
(B) discontinuous at exactly two points
(C) discontinuous at exactly three points
(D) continuous at every point
If $y = e^{\sin \sqrt{x}}$, then $\frac{dy}{dx}$ is equal to :
(A) $e^{\sin \sqrt{x}} \cdot \frac{1}{2\sqrt{x}}$
(B) $e^{\cos \sqrt{x}}$
(C) $e^{\sin \sqrt{x}} \cdot \frac{\cos \sqrt{x}}{2\sqrt{x}}$
(D) $\frac{e^{\cos \sqrt{x}}}{2\sqrt{x}}$
If $x = \sin t$ and $y = \sin pt$, then prove that $(1 - x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} + p^2y = 0$.
If $f(x) = \begin{cases} \frac{|x|}{x}, & x \neq 0 \\ 0, & x = 0 \end{cases}$, then which of the following statements is \textit{not} correct ?
(A) $f$ is continuous for every $x < 0$.
(B) $f$ is continuous for every $x \in \mathbb{R}$.
(C) $f$ is continuous for every $x > 0$.
(D) $f$ is continuous ...
If $y = \log \sqrt{\tan x}$, then the value of $\frac{dy}{dx}$ at $x = \frac{\pi}{4}$ is :
(A) $\frac{1}{\sqrt{2}}$
(B) $0$
(C) $1$
(D) $\frac{1}{2}$
If $f(x) = \begin{cases} \frac{|x|}{x}, & x \neq 0 \\ 0, & x = 0 \end{cases}$, then which of the following statements is not correct ?
(A) $f$ is continuous for every $x < 0$.
(B) $f$ is continuous for every $x \in \mathbb{R}$.
(C) $f$ is continuous for every $x > 0$.
(D) $f$ is continuous for each ...
If $y = \sqrt{y + \cos x}$, then $\frac{dy}{dx}$ is equal to :
(A) $\frac{1}{2\sqrt{y + \cos x}}$
(B) $\frac{\cos x}{2y - 1}$
(C) $\frac{-\sin x}{2\sqrt{y + \cos x}}$
(D) $\frac{\sin x}{1 - 2y}$
If $f(x) = \begin{cases} \frac{|x|}{x}, & x \neq 0 \\ 0, & x = 0 \end{cases}$, then which of the following statements is \textbf{not} correct ?
(A) $f$ is continuous for every $x < 0$.
(B) $f$ is continuous for every $x \in \mathbb{R}$.
(C) $f$ is continuous for every $x > 0$.
(D) $f$ is continuous ...
If $y = \log \tan \left$\frac{\pi}{4} + \frac{x}{2}\right$$, then prove that $\frac{dy}{dx} - \sec x = 0$
If $x^y = e^{x-y}$ prove that $\frac{dy}{dx} = \frac{\log x}{(\log(xe))^2}$ and hence find its value at $x = e$.
If $x = a$\theta - \sin \theta$$, $y = a$1 - \cos \theta$$ find $\frac{d^2y}{dx^2}$.
Solve the differential equation $y + x \frac{dy}{dx} = x - y \frac{dy}{dx}$.
Questions number 19 and 20 are Assertion and Reason based questions. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below.
(A) Both Assertion (A) and Reason (R) are true and Reason (R) i...
Let $f(x)$ be a continuous function on $[a, b]$ and differentiable on $(a, b)$. Then, this function $f(x)$ is strictly increasing in $(a, b)$ if
(A) $f'(x) < 0, \forall \text{ x} \in (a, b)$
(B) $f'(x) > 0, \forall \text{ x} \in (a, b)$
(C) $f'(x) = 0, \forall \text{ x} \in (a, b)$
(D) $f(x) > 0, \f...
(a) Check whether the function $f(x) = x^2 |x|$ is differentiable at $x = 0$ or not.
OR
(b) If $y = \sqrt{\tan \sqrt{x}}$, prove that $\sqrt{x} \frac{dy}{dx} = \frac{1 + y^4}{4y}$.
(a) If $\sqrt{1 - x^2} + \sqrt{1 - y^2} = a(x - y)$, prove that $\frac{dy}{dx} = \frac{\sqrt{1 - y^2}}{\sqrt{1 - x^2}}$.
OR
(b) If $y = (\tan x)^x$, then find $\frac{dy}{dx}$.
The order and degree of differential function $\left[ 1 + \left$ \frac{dy}{dx} \right$^2 \right]^5 = \frac{d^2y}{dx^2}$ are
(A) order $1$, degree $1$
(B) order $1$, degree $2$
(C) order $2$, degree $1$
(D) order $2$, degree $2$
(a) If $\sqrt{1 - x^2} + \sqrt{1 - y^2} = a(x - y)$, then prove that $\frac{dy}{dx} = \sqrt{\frac{1 - y^2}{1 - x^2}}$.
OR
(b) If $x = a \left$\cos \theta + \log \tan \frac{\theta}{2}\right$$ and $y = \sin \theta$, then find $\frac{d^2y}{dx^2}$ at $\theta = \frac{\pi}{4}$.
(a) If $x = e^{\frac{x}{y}}$, then prove that $\frac{dy}{dx} = \frac{x - y}{x \log x}$.
(a) Solve the differential equation $2(y + 3) - xy \frac{dy}{dx} = 0$; given $y(1) = -2$.
OR
(b) Solve the following differential equation:
$(1 + x^2) \frac{dy}{dx} + 2xy = 4x^2$.
(a) If $y = \log \left$ \sqrt{x} + \frac{1}{\sqrt{x}} \right$^2$, then show that $x(x + 1)^2 y_2 + (x + 1)^2 y_1 = 2$.
OR
(b) If $x \sqrt{1 + y} + y \sqrt{1 + x} = 0, -1 < x < 1, x \neq y$, then prove that $\frac{dy}{dx} = \frac{-1}{(1 + x)^2}$.
(a) If $x = e^{\frac{x}{y}}$, then prove that $\frac{dy}{dx} = \frac{x - y}{x \log x}$.
OR
(b) If $f(x) = \begin{cases} 2x - 3, & -3 \le x \le -2 \\ x + 1, & -2 < x \le 0 \end{cases}$
Check the differentiability of $f(x)$ at $x = -2$.
If $y = x + \tan x$, then prove that
$\cos^2 x \frac{d^2y}{dx^2} - 2y + 2x = 0$
If $y = \sin^{-1} x$, then $(1 - x^2) \frac{d^2y}{dx^2}$ is equal to: (A) $x \frac{dy}{dx}$ (B) $-x \frac{dy}{dx}$ (C) $x^2 \frac{dy}{dx}$ (D) $-x^2 \frac{dy}{dx}$
The order and degree of the differential equation $\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^2 = x \sin \left(\frac{dy}{dx}\right)$ are :
(A) order 2, degree 2
(B) order 2, degree 1
(C) order 2, degree not defined
(D) order 1, degree not defined
If $y = \log_{2x} (\sqrt{2x})$, then $\frac{dy}{dx}$ is equal to :
(A) 0
(B) 1
(C) $\frac{1}{x}$
(D) $\frac{1}{\sqrt{2x}}$
The solution of the differential equation $\frac{dy}{dx} = \frac{-x}{y}$ represents family of :
(A) parabolas
(B) circles
(C) ellipses
(D) hyperbolas
OR Find $\frac{dy}{dx}$ if $y^x + x^y + x^x = a^b$, where a and b are constants.
(a) For a positive constant 'a', differentiate $a^{t+\frac{1}{t}}$ with respect to $\left$t+\frac{1}{t}\right$^a$, where t is a non-zero real number. OR (b) Find $\frac{dy}{dx}$ if $y^x + x^y + x^x = a^b$, where a and b are constants.
If f(x) = [x], x ∈ R} is the greatest integer function, then the correct statement is : (A) f is continuous but not differentiable at x = 2. (B) f is neither continuous nor differentiable at x = 2. (C) f is continuous as well as differentiable at x = 2. (D) f is not continuous but differentiable at ...
Solve the differential equation $x \cos \left(\frac{y}{x}\right) \frac{dy}{dx} = y \cos \left(\frac{y}{x}\right) + x$.
Find $\frac{dy}{dx}$ if $y^x + x^y + x^x = a^b$, where $a$ and $b$ are constants.
Let f(x) = |x|, $x \in R$. Then, which of the following statements is incorrect ? (A) f has a minimum value at x = 0. (B) f has no maximum value in R. (C) f is continuous at x = 0. (D) f is differentiable at x = 0.
Let $f(x) = x^2$, $x \in \mathbb{R}$. Then, which of the following statements is incorrect ? (A) Minimum value of f does not exist. (B) There is no point of maximum value of f in $\mathbb{R}$. (C) f is continuous at $x = 0$. (D) f is differentiable at $x = 0$.
The order and degree of the differential equation $\frac{d^2y}{dx^2} + 4 \left$ \frac{dy}{dx} \right$ = x \log \left$ \frac{d^2y}{dx^2} \right$$ are respectively :
(A) $0, 3$
(B) $2, 1$
(C) $2, \text{not defined}$
(D) $1, \text{not defined}$
Differentiate $y = \sqrt{\log \left\{\sin \left(\frac{x^3}{3} - 1\right)\right\}}$ with respect to x.
Solve the differential equation $\frac{dy}{dx} = \cos x - 2y$.
25. (a) Differentiate $\left(\frac{5^x}{x^5}\right)$ with respect to x.\nOR\n(b) If $l - 2x^2 - 5xy + y^3 = 76$, then find $\frac{\text{d}y}{\text{d}x}$.
If $x = at$, $y = \frac{a}{t}$, then $\frac{dy}{dx}$ is :
(A) $t^2$
(B) $-t^2$
(C) $\frac{1}{t^2}$
(D) $-\frac{1}{t^2}$
The solution of the differential equation $\frac{dy}{dx} = \frac{1}{\log y}$ is :
(A) $\log y = x + c$
(B) $y \log y - y = x + c$
(C) $\log y - y = x + c$
(D) $y \log y + y = x + c$
Let $f(x)$ be a continuous function on $[a, b]$ and differentiable on $(a, b)$. Then, this function $f(x)$ is strictly increasing in $(a, b)$ if
(A) $f'(x) < 0, \forall x \in (a, b)$
(B) $f'(x) > 0, \forall x \in (a, b)$
(C) $f'(x) = 0, \forall x \in (a, b)$
(D) $f(x) > 0, \forall x \in (a, b)$
The value of '$n$', such that the differential equation $x^n \frac{dy}{dx} = y(\log y - \log x + 1)$; (where $x, y \in R^+$) is homogeneous, is (A) $0$ (B) $1$ (C) $2$ (D) $3$
The function $f: R \rightarrow Z$ defined by $f(x)=[x]$; where $[.]$ denotes the greatest integer function, is
(A) Continuous at $x=2.5$ but not differentiable at $x=2.5$
(B) Not Continuous at $x=2.5$ but differentiable at $x=2.5$
(C) Not Continuous at $x=2.5$ and not differentiable at $x=2.5$
(D) C...
Assertion (A): Consider the function defined as $f(x) = |x| + |x - 1|$, $x \in \mathbb{R}$. Then $f(x)$ is not differentiable at $x = 0$ and $x = 1$.
Reason (R): Suppose $f$ be defined and continuous on $(a, b)$ and $c \in (a, b)$, then $f(x)$ is not differentiable at $x = c$ if $\lim_{h \to 0^-} \f...
If $(x - a)^2 + (y - b)^2 = c^2$, for some $c > 0$, prove that $\frac{[1+(\frac{dy}{dx})^2]^{3/2}}{\frac{d^2y}{dx^2}}$ is a constant independent of $a$ and $b$.
(a) If $x = e^{\cos 3t}$ and $y = e^{\sin 3t}$, prove that $\frac{dy}{dx} = -\frac{y \log x}{x \log y}$. OR (b) Show that : $\frac{d}{dx}(|x|) = \frac{x}{|x|}, x \neq 0$
(b) If $y = A \sin 2x + B \cos 2x$ and $\frac{d^2y}{dx^2} - ky = 0$, find the value of $k$.
If $x^{30} y^{20} = (x + y)^{50}$, prove that $\frac{dy}{dx} = \frac{y}{x}$.
Find $\frac{dy}{dx}$, if $5^x + 5^y = 5^{x+y}$.
The solution of the differential equation $\frac{dy}{dx} = 1 - x + y - xy$ is :
(A) $\log |1 + y| = x - \frac{x^2}{2} + c$
(B) $\log |1 + y| = -x + \frac{x^2}{2} + c$
(C) $e^y = x - \frac{x^2}{2} + c$
(D) $e^{(1+y)} = -x + \frac{x^2}{2} + c$
The degree of the differential equation $x \left$ \frac{d^2y}{dx^2} \right$^3 + y \left$ \frac{dy}{dx} \right$^4 + y^5 = 0$ is :
(A) 2
(B) 3
(C) 4
(D) 5
If $x = a \cos \theta$ and $y = b \sin \theta$, then prove that $\frac{d^2y}{dx^2} = -\frac{b^4}{a^2y^3}$.
(a) If $x = e^{x/y}$, prove that $\frac{dy}{dx} = \frac{\log x - 1}{(\log x)^2}$ \n\nOR \n\n(b) Check the differentiability of $f(x) = \begin{cases} x^2+1, & 0 \leq x < 1 \\ 3-x, & 1 \leq x \leq 2 \end{cases}$ at $x = 1$.
If $x \cos(p + y) + \cos p \sin(p + y) = 0$, prove that $\cos p \frac{dy}{dx} = -\cos^2(p + y)$, where $p$ is a constant.
Given that $y = (\sin x)^{x} \cdot x^{\sin x} + a^{x}$, find $\frac{dy}{dx}$.
If e^{x^{2}y} = c, then \frac{dy}{dx} is : (A) \frac{xe^{x^{2}y}}{2y} (B) \frac{-2y}{x} (C) \frac{2y}{x} (D) \frac{x}{2y}
Given that $x^y + y^x = a^b$, where $a$ and $b$ are positive constants, find $\frac{dy}{dx}$.
The function $f(x) = |x| + |x - 2|$ is
(A) continuous, but not differentiable at $x = 0$ and $x = 2$.
(B) differentiable but not continuous at $x = 0$ and $x = 2$.
(C) continuous but not differentiable at $x = 0$ only.
(D) neither continuous nor differentiable at $x = 0$ and $x = 2$.
Find $\frac{dy}{dx}$, if $(\cos x)^y = (\cos y)^x$.
If $\sqrt{1-x^2} + \sqrt{1-y^2} = a(x-y)$, prove that $\frac{dy}{dx} = \sqrt{\frac{1-y^2}{1-x^2}}$.
Assertion (A) : The degree of the differential equation $\left(\frac{d^2y}{dx^2}\right)^3 + \left(\frac{dy}{dx}\right)^2 + \sin\left(\frac{dy}{dx}\right) + 1 = 0$ is 3. Reason (R) : The highest power of the highest order derivative involved in a differential equation, when it is written as a polynom...
If $y = \left(x + \sqrt{x^2 + 1}\right)^p$, then prove that $(x^2 + 1)y_2 + xy_1 - p^2y = 0$; where $y_1 = \frac{dy}{dx}$ and $y_2 = \frac{d^2y}{dx^2}$.
If $x = 3 \cos \theta$ and $y = 5 \sin \theta$, then $\frac{dy}{dx}$ is equal to : (A) $-\frac{3}{5} \tan \theta$ (B) $-\frac{5}{3} \cot \theta$ (C) $-\frac{5}{3} \tan \theta$ (D) $-\frac{3}{5} \cot \theta$
(a) If $x \sin(a + y) - \sin y = 0$, prove that $\frac{dy}{dx} = \frac{\sin^2(a + y)}{\sin a}$ OR (b) Find $\frac{dy}{dx}$, if $y = (\cos x)^x + \cos^{-1} \sqrt{x}$.
OR
Find $\frac{dy}{dx}$, if $y = x^{\cos x} - 2^{\sin x}$.
The sum of the order and the degree of the differential equation $\frac{d^{2}y}{dx^{2}} + (\frac{dy}{dx})^{3} = \sin y$ is : (a) 5 (b) 2 (c) 3 (d) 4
If $y = (x + \sqrt{x^2 - 1})^2$, then show that $(x^2 - 1) \left$ \frac{dy}{dx} \right$^2 = 4y^2$.
The function f(x) = |x| - x is :
(a) continuous but not differentiable at x = 0.
(b) continuous and differentiable at x = 0.
(c) neither continuous nor differentiable at x = 0.
(d) differentiable but not continuous at x = 0.
The degree of the differential equation $\left[ 1 + \left$ \frac{dy}{dx} \right$^2 \right]^3 = \left$ \frac{d^2y}{dx^2} \right$^2$ is (a) 4 (b) $\frac{3}{2}$ (c) 2 (d) Not defined
Let $f(x)$ be a polynomial function of degree 6 such that $\frac{d}{dx}(f(x)) = (x-1)^3(x-3)^2$, then
ASSERTION (A): $f(x)$ has a minimum at $x = 1$.
REASON (R): When $\frac{d}{dx}(f(x)) < 0, \forall x \in (a-h, a)$ and $\frac{d}{dx}(f(x)) > 0, \forall x \in (a, a+h)$; where 'h' is an infinitesimall...
If $\tan \left(\frac{x + y}{x - y}\right) = k$, then $\frac{dy}{dx}$ is equal to
(A) $\frac{-y}{x}$
(B) $\frac{y}{x}$
(C) $\sec^2(\frac{y}{x})$
(D) $-\sec^2(\frac{y}{x})$
If $y = \sqrt{ax + b}$, prove that $y \left$ \frac{d^2y}{dx^2} \right$ + \left$ \frac{dy}{dx} \right$^2 = 0$.
(b) If $\text{f}(x) = \begin{cases} ax + b &; 0 < x \le 1 \\ 2x^2 - x &; 1 < x < 2 \end{cases}$ is a differentiable function in $(0, 2)$, then find the values of $a$ and $b$.
(a) If $y = \sqrt{ax + b}$, then prove that $y \left$ \frac{d^2y}{dx^2} \right$ + \left$ \frac{dy}{dx} \right$^2 = 0$.
The sum of the order and the degree of the differential equation $\left(\frac{d^3y}{dx^3}\right)^2 + 3x\left(\frac{d^2y}{dx^2}\right)^4 = \log x$, is:
(a) 5
(b) 6
(c) 7
(d) 4
If $\cos y = x \cos(a + y)$, and $\cos a \neq \pm 1$, prove that $\frac{dy}{dx} = \frac{\cos^2(a + y)}{\sin a}$.
Case Study - 1
Let $f(x)$ be a real valued function. Then its
• Left Hand Derivative (L.H.D.) : $Lf'(a) = \lim_{h \to 0} \frac{f(a - h) - f(a)}{-h}$
• Right Hand Derivative (R.H.D.) : $Rf'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$
Also, a function $f(x)$ is said to be differentiable at $x = a...
If $y = \log (\sin e^x)$, then $\frac{dy}{dx}$ is :
(a) $\cot e^x$
(b) $\text{cosec } e^x$
(c) $e^x \cot e^x$
(d) $e^x \text{cosec } e^x$
The number of solutions of the differential equation $\frac{dy}{dx} = \frac{y + 1}{x - 1}$, when $y(1) = 2$, is :
(a) zero
(b) one
(c) two
(d) infinite
Let $f(x)$ be a real valued function. Then its
- Left Hand Derivative (L.H.D.) : $Lf'(a) = \lim_{h \to 0} \frac{f(a - h) - f(a)}{- h}$
- Right Hand Derivative (R.H.D.) : $Rf'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$
Also, a function $f(x)$ is said to be differentiable at $x = a$ if its L.H.D...
(iii) (a) Check if the function $f(x)$ is differentiable at $x = 1$.
OR
(iii) (b) Find the $f'(2)$ and $f'(-1)$.
The sum of the order and the degree of the differential equation $\frac{d}{dx} \left$ \frac{dy}{dx} \right$^3$ is
(a) 2
(b) 3
(c) 5
(d) 0
The sum of the order and the degree of the differential equation $\frac{d}{dx}\left(\left(\frac{dy}{dx}\right)^3\right)$ is
(a) 2
(b) 3
(c) 5
(d) 0
Assertion (A) : The slope of the normal to the curve $y = 2x^2 - 5x$ at $x = -1$ is $-1$.
Reason (R) : The slope of the normal to the curve $y = f(x)$ at point $(\alpha, \beta)$ is given by $(x - \alpha) + \left(\frac{dy}{dx}\right)_{(\alpha, \beta)} \cdot (y - \beta) = 0$.
The difference of the order and the degree of the differential equation $(\frac{d^2y}{dx^2})^2 + (\frac{dy}{dx})^3 + x^4 = 0$ is :
(a) $1$
(b) $2$
(c) $-1$
(d) $0$
A fair die is rolled. Events E and F are E = {1, 3, 5} and F = {2, 3} respectively. Value of P(E|F) is :
(a) $\frac{2}{3}$
(b) $\frac{1}{3}$
(c) $\frac{1}{6}$
(d) $\frac{1}{2}$
If y = $\log \left[ \tan \left$ \frac{\pi}{4} + \frac{x}{2} \right$ \right]$, then $\frac{dy}{dx}$ is :
(a) sec x
(b) cosec x
(c) tan x
(d) sec x tan x
If $y = 3 \log \sqrt{\sin x}$, then $\frac{dy}{dx}$ at $x = \frac{\pi}{4}$ is : (a) 3 (b) $\frac{2}{3}$ (c) $\frac{3}{2}$ (d) $\frac{-3}{2}$
Find the sum of the order and the degree of the differential equation : $\left$x + \frac{dy}{dx}\right$^{2} = \left(\frac{dy}{dx}\right)^{2} + 1$
Solve the differential equation : $\log \left$ \frac{dy}{dx} \right$ = x - y$.
Find the sum of the order and the degree of the differential equation: $$x + \frac{dy}{dx}$^2 = (\frac{dy}{dx})^2 + 1$
Solve the differential equation: $\frac{dy}{dx} = \frac{x^2 + y^2}{xy}$.
If $y \sqrt{1 - x^2} + x \sqrt{1 - y^2} = 1$, then prove that $\frac{dy}{dx} = - \sqrt{\frac{1 - y^2}{1 - x^2}}$
Find the product of the order and the degree of the differential equation $\left[\frac{d}{dx}(xy^2)\right].\frac{dy}{dx} + y = 0$.
Find the general solution of the following differential equation : $\frac{dy}{dx} = e^{x-y} + x^2 e^{-y}$
(b) Find the value of (2a - 3b), if a and b represent respectively the order and the degree of the differential equation x [ y (d^2y / dx^2)^3 + x (dy / dx)^2 - y/x dy/dx ] = 0.