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Higher Engineering Mathematics

Grewal B.S.

Chapter 4

Differential Calculus & Its Applications - all with Video Answers

Educators

Chapter Questions

03:38

Problem 1

The radius of curvature of the catenary $y=e$ cosh alc at the point whero it cressen the yuxis is

Tanishq Gupta
Tanishq Gupta
Numerade Educator
01:09

Problem 2

The envelope of the family of straipht lines $y=m x+a m^{2}$, (an being the parameter) is

Jeff Vermeire
Jeff Vermeire
Numerade Educator
01:52

Problem 3

The curvature of the circle $x^{2}+y^{2}=25$ at the point $(3,4)$ is

Jeff Vermeire
Jeff Vermeire
Numerade Educator
01:07

Problem 4

Tbe value of I.t $\frac{\log \sin x}{(\pi / 2-x)^{2}}$ id
(a) aere
(b) $1 / 2$
(c) $-1 / 2$
$(d)-2$,

Gaurav Kalra
Gaurav Kalra
Numerade Educator
01:54

Problem 5

\text { Faylor's expansion of the function } f x)=\frac{1}{1+x^{2}} \text { is }(a) $\sum_{n=0}^{\infty}(-1)^{n} x^{2 n}$ for $-1<x<1$
(b) $\sum_{n=0}^{\infty} x^{2 n}$ for $-1<x<1$
(c) $\sum_{n=0}^{\infty}(-1)^{n} x^{2 n}$ for any real $x$.
(d) $\sum_{n=0}^{-}(-1)^{n} x^{n}$ for $-1<x \leq 1$.

Jack Chen
Jack Chen
Numerade Educator
06:18

Problem 6

A triangle of maximum area inscribed in a circle of radius $r$
(a) is a right angled triangle with hypotenuse measuring $2 r$
(b) is an equilateral triangle
(c) is an isosceles triangle of height $r$
(d) does not exist.

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
01:20

Problem 7

The extreme value of $(x)^{1 / x}$ is
(a) $\boldsymbol{e}$
(b) (1/e)
(c) (e) $^{\text {te }}$
(d) 1 .

Jeff Vermeire
Jeff Vermeire
Numerade Educator
01:49

Problem 8

The percentage error in computing the aren of an ellipse when an error of 1 per cent is made in measuring the major und rainor axes is
(a) 0.2'ie
(b) $2 \%$
(c) 0.02*.

Robert Daugherty
Robert Daugherty
Numerade Educator
01:11

Problem 9

The length of subtangent of the reetangular hyperbola $x^{2}-y^{2}=a^{2}$ at the point $(\alpha, \sqrt{2} a)$ is
(a) $\sqrt{2} a$
(b) $2 \mathrm{a}$
(c) $\frac{1}{2 a}$
(d) $\frac{a^{3 / 2}}{\sqrt{2}}$.

Dilip Paruchuri
Dilip Paruchuri
Numerade Educator
07:25

Problem 10

The length of subnermal to the curve $y=x^{2}$ at $(2, b)$ is
(a) 28
(b) 32
(c) 96
(d) 64 .

Carlos Pinilla
Carlos Pinilla
Numerade Educator
02:04

Problem 11

If the nermal to the curve $y^{2}=5 x-1$ at the point $(1,-2)$ is of the form $a x-5 y+b=0$, then $a$ and $b$ are
(a) 4,14
(b) $4,-14$
(c) $-4,14$ $(d)-4,-14$

Jeff Vermeire
Jeff Vermeire
Numerade Educator
View

Problem 12

The radius of curvature of the curve $y=e^{x}$ at the point where it crosses the $y$-axis is
(a) 2
(b) $\sqrt{2}$
(c) $2 \sqrt{2}$
(d) $\frac{1}{2} \sqrt{2}$.

Ankur S
Ankur S
Numerade Educator
01:01

Problem 13

The equation of the asymptotes of $x^{3}+y^{3}=3 a x y$, is
(a) $x+y-a=0$
(b) $x-y+a=0$
(c) $x+y+a=0$
(d) $x-\dot{y}-a=0$.

Dilip Paruchuri
Dilip Paruchuri
Numerade Educator
04:05

Problem 14

If o be the angle between the tangent and radius vector at any point on the curve $r=f(8)$, then sin $\phi$ equals to
(a) $\frac{d r}{d s}$
(b) $r \frac{d \theta}{d s}$
(c) $r \frac{d \theta}{d r}$.

Ben Brown
Ben Brown
Numerade Educator
00:18

Problem 15

Fnvelope of the family of lines $x=m y+1 / m$ is $\ldots$

AG
Ankit Gupta
Numerade Educator
03:50

Problem 16

The chord of curvature parallel to $y$. axis for the curve $y=a$ log sec $x / a$ in w..

James Kiss
James Kiss
Numerade Educator
02:27

Problem 17

$\sinh x=\ldots x+\ldots x^{3}+\ldots x^{5}+\ldots$

Linda Hand
Linda Hand
Numerade Educator
02:18

Problem 18

The $n$th derivative of $(\cos x \operatorname{con} 2 x$ ons $3 x)=\ldots$.

John Nicolle
John Nicolle
Numerade Educator
02:25

Problem 19

If $x^{3}+y^{3}-\operatorname{sex} y=0$, then $d^{2} y / d x^{2}$ at $(3 a / 2,3 e / 2)=\ldots \ldots$

Adriano Chikande
Adriano Chikande
Numerade Educator
02:06

Problem 20

When the tangent at a point on a curve is parallel to x-axis, then the curvature at that point is same as the second derivative at that point. or F'alse.

David Nguyen
David Nguyen
Numerade Educator
02:30

Problem 21

If $x=a t^{2}, y=2 a t, t$ being the parameter, then $x y d^{2} y / d x^{2}=\ldots \ldots . .$

Jeff Vermeire
Jeff Vermeire
Numerade Educator
02:01

Problem 22

The radius of curvature for the parabola $x=a, y=2 a t$ at any point $t=\ldots \ldots \ldots .$

Malika Singh
Malika Singh
Numerade Educator
03:56

Problem 23

If $(a, b)$ are the coordinates of the centre of curvature whese curvature is $k$, then the equation of the circle of curvature in

WZ
Wen Zheng
Numerade Educator
03:05

Problem 24

\text { Kvolute is defined as the of the normals for a given curve. }

Dorcas Attuabea Addo
Dorcas Attuabea Addo
Numerade Educator
01:32

Problem 25

Envelope of the fumily of lines $\frac{x}{t}+y t=2 \mathrm{c}$ (where $t$ is the parameter) is $\ldots$

Jeff Vermeire
Jeff Vermeire
Numerade Educator
03:08

Problem 26

The angle between the radios vector and tangent for the curve $r=$ at $^{\text {"nte }}$ is wum

Susanna T.
Susanna T.
Numerade Educator
00:56

Problem 27

The subnormal of the piarabola $y^{2}=4 a x$ is

Jeff Vermeire
Jeff Vermeire
Numerade Educator
00:59

Problem 28

The fourth derivative of (e $\left.^{-8} x^{3}\right)$ is

Jeff Vermeire
Jeff Vermeire
Numerade Educator
03:14

Problem 29

If $y^{3}=P(x)$, a polynomial of degree 3 , then $\frac{2 d}{d x}\left(y^{3} \frac{d^{2} y}{d x^{2}}\right)$ equale
(a) $I^{\mu *}(x)+P(x)$
(b) $P^{\prime}(x)+P^{-2}(x)$
(c) $P(x) P^{\prime \prime}(x)$.

Aman Gupta
Aman Gupta
Numerade Educator
01:23

Problem 30

The envelope of the family of straight line $\frac{x}{d} \cos \theta+\frac{y}{b}$ sin $\theta=1$.

Nick Johnson
Nick Johnson
Numerade Educator
03:56

Problem 31

Curvature of a straight line in
$\begin{array}{llll}\text { (A) }- & \text { (B) zere } & \text { (C) Both }(A) \text { and }(B) & \text { (D) None ef these }\end{array}$

WZ
Wen Zheng
Numerade Educator
06:15

Problem 32

'The value of 'c'ef the Cauchy's Mean value theorein for $/(x)=e^{\prime}$ and $B(x)=e^{*}$ in 12, al is

Ahmed Ibrahim
Ahmed Ibrahim
Numerade Educator
00:31

Problem 33

If the equation of a eurve remains unchanged when $x$ and $y$ are interchanged, then the carve is eymmetrical abeut

Hossam Mohamed
Hossam Mohamed
Numerade Educator
00:24

Problem 34

For the curve $y^{2}(1+x)=x^{2}(1-x)$, the origin is a $\ldots$ inode/cusp/cenjugate point).

Sahil Patel
Sahil Patel
Numerade Educator
01:56

Problem 35

The number of loeps of $r=\alpha$ yin 20 are $\ldots \ldots . . .$ and these of $r=a \cos 3 \theta$ are ..........

Robert Leedy
Robert Leedy
Numerade Educator
00:24

Problem 36

Tengents at the origin for the curve $y^{2}\left(x^{2}+y^{2}\right)+a^{2}\left(x^{2}-y^{2}\right)=0$ are.

Sahil Patel
Sahil Patel
Numerade Educator
04:58

Problem 37

The aryanptote to the curve $y^{2}(4-x)=x^{3}$ is, it.

WZ
Wen Zheng
Numerade Educator
01:05

Problem 38

The curve $r=a A 1+\cos \theta$ ) intersects orthogonally with the curve $\begin{array}{lll}\text { (A) } r=b / 1-\cos \theta) & \text { (B) } r=b /(1+\operatorname{lin} \theta) & \text { (C) } r=b\left(1+\sin ^{2} \theta\right)\end{array}$
$\begin{array}{ll} \left.\text { (D) } r=b / 11+\cos ^{2} \theta\right) & \left(\mathrm{V}, T, U_{4}, 2010\right)\end{array}$

AG
Ankit Gupta
Numerade Educator
01:05

Problem 39

The curve $r=a / 1+\cos \theta$ ) intersects orthogonally with the curve
$\begin{array}{llll}\text { LA) } r=b / 1-\cos \theta) & \text { (B) } r-b /(1+\sin \theta) & \text { (C) } r=b\left(1+\sin ^{2} \theta\right) & \left.\text { (D) } r=b / 1+\cos ^{2} \theta\right)\end{array}$
(V.T.U., 2010)

AG
Ankit Gupta
Numerade Educator
03:13

Problem 40

\text { The regpion where the curve } r=a \sin \theta \text { doen not lie is }

Kelly Brooks
Kelly Brooks
Numerade Educator
01:05

Problem 41

If $f(x)$ in continuous in the clesed interval $l a, b 1$, differentiable in $(n, b)$ and $f(\alpha)=f(b)$, then there esikts at least one value c of $x$ in $(a, b)$ such that $f^{\prime}(c)$ is equal to
(A) 1
(B) $-1$
(C) 2 $\begin{array}{ll}\text { (D) } 0 . & \left(V, \tau, U_{.}, 2009\right)\end{array}$

Nick Johnson
Nick Johnson
Numerade Educator
05:23

Problem 42

If twe curves intersect orthogonally in cartesian form, then the angle betireen the same two eurves in polar form i:
(A) $\pi / 4$
(B) Zero
(C) $\mathbf{1}$ radian
(D) None of these.

Charles Machakwa
Charles Machakwa
Numerade Educator
04:10

Problem 43

If the angle between the radius vector and the tangent is censtant, then the curve is, $\begin{array}{lll}\text { (A) } r=a \cos \theta & \text { (B) } r^{2}=\alpha^{2} \cos ^{2} \theta & \text { (C) } r \text { al } 6 e^{\prime \prime}\end{array}$

Malika Singh
Malika Singh
Numerade Educator