Mathematics

Calculus and Analytic Geometry

110 Questions

Calculus and analytic geometry problems involve finding slopes and equations of tangent lines. The focus is on applying derivatives to analyze curves and their geometric properties. These concepts are frequently tested in advanced undergraduate competitive exams.

Tangent line equationsCurve slopesDifferential equationsGeometric curvesNormal to curves

Calculus and Analytic Geometry Questions

Multiple choice combining transformations transformations vectors and transformations maths

If the transformed equation of a curve is $9x^{2}+16y^{2}=144$ when the axes rotated through an angle of $45^{o}$ then the original equation of a curve is:

  1. $25x^{2}+14yxy+25y^{2}=228$
  2. $25x^{2}-14yxy+25y^{2}=228$
  3. $25x^{2}+14yxy-25y^{2}=228$
  4. $25x^{2}-14yxy-25y^{2}=228$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Rotating axes by 45 degrees involves substituting x = (X - Y)/sqrt(2) and y = (X + Y)/sqrt(2) into the original equation. Expanding 9((X-Y)/sqrt(2))^2 + 16((X+Y)/sqrt(2))^2 = 144 leads to 9(X^2 - 2XY + Y^2)/2 + 16(X^2 + 2XY + Y^2)/2 = 144, which simplifies to 25X^2 + 14XY + 25Y^2 = 288.

Multiple choice maths fundamental concepts - geometry terms related to polygons curves open and closed figures

All chords of the curve $x^{2}+y^{2}-10x-4y+4=0$  which make a right angle at $(8,2)$ pass through

  1. $(2,5)$
  2. $(-2,-5)$
  3. $(-5,-2)$
  4. $(5,2)$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For a circle x^2 + y^2 - 10x - 4y + 4 = 0, the center is (5, 2). Chords subtending a right angle at a point P(8, 2) pass through a fixed point. By the property of circles, if a chord subtends a right angle at P, the locus of the intersection of tangents at the chord's endpoints is related to the circle's geometry. The fixed point is the center (5, 2).

Multiple choice maths fundamental concepts - geometry terms related to polygons curves open and closed figures

Let $m$ be the slope of tangent to the curve $e^{2y}=1+x^{2}$ then set of all values  of $m$ is :

  1. $\left[-\dfrac{1}{2}, \dfrac{1}{2}\right]$
  2. $\left[-\infty, -\dfrac{1}{2}\right]\cup \left[\dfrac{1}{2}, 0\right]$
  3. $\left[-\dfrac{1}{2}, \dfrac{1}{2}\right]-\left\{0\right\}$
  4. $[-2,2]-\left\{0\right\}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$m$ be the slope of tangent.

Given equation of curve is

${{e}^{2y}}=1+{{x}^{2}}$


Taking log both side and we get,

$ \log {{e}^{2y}}=\log \left( 1+{{x}^{2}} \right) $

$ 2y=\log \left( 1+{{x}^{2}} \right) $


On differentiating and we get,

$ 2\dfrac{dy}{dx}=\dfrac{1}{1+{{x}^{2}}}\dfrac{d}{dx}\left( 1+{{x}^{2}} \right) $

$ \Rightarrow 2\dfrac{dy}{dx}=\dfrac{1}{1+{{x}^{2}}}\dfrac{d}{dx}\left( 1+{{x}^{2}} \right) $

$ \Rightarrow 2\dfrac{dy}{dx}=\dfrac{1}{1+{{x}^{2}}}\dfrac{d}{dx}2x $

$ \Rightarrow \dfrac{dy}{dx}=\dfrac{x}{1+{{x}^{2}}} $

$m=\dfrac{dy}{dx}=\dfrac{x}{1+{{x}^{2}}}$


On put $x=\left( -1,1 \right)$

So,

$ m=\dfrac{1}{1+1}=\dfrac{1}{2} $

$ m=\dfrac{-1}{1+1}=\dfrac{-1}{2} $

Hence, the value of $m$ is $\left[-\dfrac{1}{2},\dfrac{1}{2} \right]$


Hence, this is the answer.

Multiple choice maths fundamental concepts - geometry terms related to polygons curves open and closed figures

A curve which begins and ends at the same point is called a:

  1. closed curve

  2. open curve

  3. normal curve

  4. definite curve
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

A close curve is made up of a closed boundary. It initialised by a fixed point and end with the same point.

So, a curve which begins and ends at the same point is called a closed curve.
Hence, the answer is a closed curve.

Multiple choice maths fundamental concepts - geometry terms related to polygons curves open and closed figures

A point is moving along the curve ${ y }^{ 3 }=27x$. Find the interval of valued of $x$ in which the ordinate changes faster then abscissa is:

  1. $x\in \left( -1,1 \right)$
  2. $x\in \left( -1,-1 \right) -\left\{ 0 \right\}$
  3. $x\in \left[ -1,1 \right] -\left\{ 0 \right\}$
  4. $x\in \left( -1,0 \right) $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$y^3=27x$

Abcissa changes at slower rate than ordinate.
$\dfrac{dx}{dt}<\dfrac{dy}{dt}.................(1)$
$y^3=27x$
$3y^2\dfrac{dy}{dt}=27\dfrac{dx}{dt}.............(2)$
Putting $\dfrac{dx}{dt}$ in eq $(1)$
 $\dfrac{3y^2}{27}<\dfrac{dy}{dy}$
 $\dfrac{dy}{dt}\left(\dfrac{3y^2}{27}<1\right)<0$
By eq$(2)$ wecan say that $\dfrac{dx}{dt}$ and $\dfrac{dy}{dt}$ will be '+ve' or '-ve'.
So, $\dfrac{3y^2}{27}-1<0\Rightarrow{-3}<y<3$ and $-1<x<1$.

Multiple choice maths fundamental concepts - geometry terms related to polygons curves open and closed figures

The curves $y = 2{\left( {x - a} \right)^2}andy = {e^{2x}}$ touches each other, then'a' is less than- 

  1. $-1$
  2. $0$
  3. $1$
  4. $2$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For the curves to touch, they must share a common tangent at a point. Setting y = 2(x-a)^2 and y = e^(2x) equal and their derivatives equal: 2(x-a)^2 = e^(2x) and 4(x-a) = 2e^(2x). Solving these leads to the condition for 'a'.

Multiple choice business maths pair of straight lines condition for perpendicular and coincident lines and bisectors of angles pair of straight lines through origin analytical geometry

The straight line joining the origin to the other two points of intersection of the curve whose equations are $\displaystyle ax^{2}+2hxy+2gx+by^{2}=0: : and: : a'x^{2}+2h'xy+b'y^{2}+2g'x=0$ will be at right angle if

  1. $g(a' + b') - g'(a + b) = 0$
  2. $gg' = a'b' - ab$
  3. $g - g' = (a - b) (a' - b')$

  4. none of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Given eq of curves 
$ax^2+2hxy+2gx+by^2=0$----(1)
$a'x^2+2h'xy+2g'x+b'y^2=0$----(2)
$1=\dfrac{a'x^2+2h'xy+b'y^2}{-2g'x}$
Putting in eq (1)
$ax^2+2hxy+2gx\left ( \dfrac{a'x^2+2h'xy+b'y^2}{-2g'x} \right )+by^2=0$
$ax^2+2hxy-g\left ( \dfrac{a'x^2+2h'xy+b'y^2}{g'} \right )+by^2=0$
$ag'x^2+2hg'xy-ga'x^2-2gh'xy-gb'y^2+bg'y^2=0$
$x^2(ag'-a'g)+y^2(bg'-b'g)+2xy(hg'-gh')=0$
Here by perpendicularlly 
$ag'-a'g+bg'-b'g=0$
$(a+b)g'-g(a'+b')=0$
$g(a'+b')-g'(a+b)=0$

Multiple choice taylor's and maclaurin's series applications of differential calculus maths

$\ln{(1+x)}< x-\cfrac{{x}^{2}}{2}+\cfrac{{x}^{3}}{3}$ for $x> 0$

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Maclaurin series for ln(1+x) is x - x^2/2 + x^3/3 - x^4/4 + ... for -1 < x <= 1. For x > 0, the next term is -x^4/4, which is negative. Therefore, ln(1+x) is strictly less than the sum of the first three terms.

Multiple choice taylor's and maclaurin's series applications of differential calculus maths

The fourth term in Taylor series of $\log\ x$ centered at $a=1$ is?

  1. $\dfrac{(x-1)^3}{3}$
  2. $\dfrac{(x-1)^2}{2}$
  3. $-\dfrac{(x-1)^4}{4}$
  4. $(x-1)$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The taylor series expansion of $\log x$ is $f(x) = \ln(x)$

$ = \left(x-1\right)-\dfrac{1}{2}\left(x-1\right)^2 + \dfrac{1}{3} \left(x-1\right)^3-\dfrac{1}{4} \left(x-1\right)^4 + \cdots$ $ f(x) $
$= \displaystyle\sum\limits _{n=1}^{\infty} \left[\frac{\left(-1\right)^{n+1}}{n}\left(x-1\right) ^n\right] $ 

Multiple choice business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

If the amount of taxes paid $(T)$ depends on income $(x)$, how would you use calculus notation to describe the marginal tax rate? If taxes and income are both measured in dollars per year, what are the units of the marginal tax rate?

  1. Same as units of $T$
  2. Same as units of $x$

  3. Unit - less

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

If taxes and income are both measured in dollars per year, then the units of the marginal tax rate will be unit-less.

Multiple choice maths pair of straight lines bisection of angle pair of bisectors of angles bisector of angle between lines

The angle of intersection of the curves  $x ^ { 2 } + 4 y ^ { 2 } = 32$  and  $x ^ { 2 } - y ^ { 2 } = 12$  at any point of their intersection is

  1. $\dfrac { \pi } { 6 }$
  2. $\dfrac { \pi } { 4 }$
  3. $\dfrac { \pi } { 3 }$
  4. $\dfrac { \pi } { 2 }$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
${ x }^{ 2 }+{ 4y }^{ 2 }=32\quad \longrightarrow \left( i \right) $
${ x }^{ 2 }-{ y }^{ 2 }=12\quad \longrightarrow \left( ii \right) $
Solving, $y=\pm 2$
              $x=\pm 4$
$\therefore$   Point of ${ X }^{ n }$ are $\left( 4,2 \right) ,\left( 4,-2 \right) ,\left( -4,2 \right) ,\left( -4,-2 \right) $
At $(4,2)$
${ m } _{ 1 }=\dfrac { -x }{ 4y } $    [differentiating $(i)$ wrt $x$]
$=\dfrac { 2 }{ -16 } =\dfrac { -1 }{ 8 } $
${ m } _{ 2 }=y/x$    [differentiating $(ii)$ wrt $x$]
$=\dfrac { 2 }{ 4 } =\dfrac { 1 }{ 2 } $
$\tan\theta =\dfrac { \left| { m } _{ 1 }-{ m } _{ 2 } \right|  }{ 1+{ m } _{ 1 }{ m } _{ 2 } } =\dfrac { \left| \dfrac { -1 }{ 8 } -\dfrac { 1 }{ 2 }  \right|  }{ 1-\dfrac { 1 }{ 16 }  } =\dfrac { \dfrac { 2+8 }{ 16 }  }{ \dfrac { 15 }{ 16 }  } =\dfrac { 10 }{ 15 } =\dfrac { 2 }{ 3 } $
$\Rightarrow \theta ={ \tan }^{ -1 }\left( 2/3 \right) \simeq \pi /6$              [A]
Multiple choice mathematics and statistics coordinates, points and lines what is meant by the equation of a straight line or of a curve introduction to slope introduction to straight lines

Slope of $\left{ (x,y)/x=2t+3,y=2t+5,t\epsilon R \right} $ is _______.

  1. 2

  2. 1
  3. $\frac { -1 }{ 1 } $

  4. -1
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

From the parametric equations x = 2t + 3 and y = 2t + 5, we can eliminate the parameter t by expressing y in terms of x. Subtracting the two equations gives y - x = 2, or y = x + 2, which is in slope-intercept form with a slope of 1. Thus, option B is correct.

Multiple choice mathematics and statistics coordinates, points and lines what is meant by the equation of a straight line or of a curve introduction to slope introduction to straight lines

If the straight lines joining the origin and the points of intersection of the curve
$ {5x}^{2} + 12xy-{6y}^{2} +4x -2y+3 =0$ and $x+ky-1=0 $ are equally inclined to the co ordinate axis,then the  value of k-

  1. is equal to 1

  2. is equal to -1

  3. is equal to 2

  4. does not exist in the set of real numbers
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For the lines to be equally inclined to the axes, the combined equation of the pair of lines must have the coefficient of xy equal to zero (if inclined at 45 degrees) or the lines must be y = x and y = -x. Solving for k leads to the condition k = -1.