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 position of point wrt ellipse ellipse maths

The value of $\alpha$ for which the point $(\alpha,\alpha+2)$ is an interior point of smaller segment of the curve $x^{2}+y^{2}-4=0$ made by the chord of the curve whose equation is $3x+4y+12=0$ is

  1. $\left(-\infty,\dfrac {-20}{7}\right)$
  2. $(-2,0)$
  3. $\left(-\infty,\dfrac {20}{7}\right)$
  4. $\alpha\ \epsilon\ \phi$
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice position of point wrt ellipse ellipse maths

Let a curve satisfying the differential equation $y^2dx+\left(x-\dfrac{1}{y}\right)dy=0$ which passes through $(1, 1)$. If the curve also passes through $(k, 2)$, then value of k is?

  1. $\dfrac{1}{2}-\dfrac{1}{\sqrt{e}}$
  2. $\dfrac{3}{2}+\dfrac{1}{\sqrt{e}}$
  3. $\dfrac{3}{2}-\dfrac{1}{\sqrt{e}}$
  4. $\dfrac{1}{2}+\dfrac{1}{\sqrt{e}}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$y^2dx+\left(x-\dfrac{1}{y}\right)dy=0$
$\Rightarrow \dfrac{dx}{dy}+\dfrac{x}{y^2}=\dfrac{1}{y^3}$
Integrating factor (I.F.)$=e^{-\dfrac{1}{y}}$
Now $x.e^{-\dfrac{1}{y}}=\displaystyle\int e^{-\dfrac{1}{y}}\dfrac{1}{y^3}dy$
Put $-\dfrac{1}{y}=y$
$x.e^t=\displaystyle\int e^t(-t)dt$
$\Rightarrow x.e^t=-(t.e^t-e^t)+c$
$\Rightarrow e^{-\dfrac{1}{y}}=e^{-\dfrac{1}{y}}\left(1+\dfrac{1}{y}\right)+c$
$\Rightarrow x=1+\dfrac{1}{y}+c.e^{\dfrac{1}{y}}$
it passes through point $(1, 1)$
$\therefore c=-\dfrac{1}{e}$
Equation of curve is
$x=1+\dfrac{1}{y}-e^{\dfrac{1}{y}-1}$
It passes through $(k, 2)$
$\therefore k=1+\dfrac{1}{2}-e^{-\dfrac{1}{2}}=\dfrac{3}{2}-\dfrac{1}{\sqrt{e}}$.

Multiple choice position of point wrt ellipse ellipse maths

The minimum distance of origin from the curve $\frac{a^2}{x^2}+\frac{b^2}{y^2}=1$ is $(a>0,b>0)$

  1. a-b

  2. a+b

  3. 2a+2b

  4. 2(a-b)
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Let $(a sec\theta,b cosec \theta)$ be a point on the curve, then its diastance  from the origin
$=\sqrt{a^2 sec^2 \theta + b^2 cosec ^2 \theta}$
$\therefore f(\theta)=a^2 sec^2 \theta +b^2 cosec ^2 \theta$
$=a^2+a^2 tan ^2 \theta + b^2 + b^2 cot ^ 2 \theta$
$=a^2+b^2 + a^2 tan ^2 \theta + b^2 cot ^2 \theta$
$\geq a^2+b^2+2 \sqrt{a^2 b^2}=(a+b)^2$
$\therefore$ minimum value of $f(\theta)=(a+b)^2$
$\therefore$ minimum value of $\sqrt{a^2 sec^2 \theta + b^2 cosec^2 \theta }$ is $a+b$

Multiple choice maths geometric sequences introduction to geometric progression understanding geometric progressions introduction to geometric progressions

Tangent at a point ${P _1}$ (other than (0, 0) on the curve $y = {x^3}$ meets the curve again at ${P _2}$. The tangent at ${P _2}$ meets the curve again at ${P _3}$ and so on. Show that the abscissae of ${P _1},{P _2},..........,{P _n}$ form a G.P. Also find the ratio $\left[ {area\,\left( {\Delta {P _1}.{P _2}.{P _3}} \right)/area\,\left( {\Delta {P _2}{P _3}{P _4}} \right)} \right].$

  1. $\dfrac{1}{2}$
  2. $\dfrac{1}{4}$
  3. $\dfrac{1}{8}$
  4. $\dfrac{1}{16}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For the curve y = x^3, the tangent at x1 meets the curve at x2 = -2x1. This forms a geometric progression with common ratio -2. The area of the triangle formed by points on the curve scales with the coordinates, leading to a ratio of 1/16 for consecutive triangles.

Multiple choice

What is a tangent space at a point on a manifold?

  1. The set of all tangent vectors at that point

  2. The set of all normal vectors at that point

  3. The set of all curves passing through that point

  4. The set of all surfaces passing through that point
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The tangent space at a point on a manifold is the vector space of all tangent vectors to the manifold at that point.

Multiple choice

Which of the following is an eigenvalue of the linear transformation f(x) = 3x - 2?

  1. 1

  2. 2

  3. 3

  4. 4
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

An eigenvalue of a linear transformation is a scalar λ such that there exists a nonzero vector x such that f(x) = λx. In this case, f(3) = 3(3) - 2 = 7 and 3 is a nonzero scalar, so 3 is an eigenvalue of f(x) = 3x - 2.

Multiple choice

What is the equation of the tangent line to the curve $y = x^3 - 2x^2 + x - 1$ at the point $(1, -1)$?

  1. $y = 3x - 4$
  2. $y = 2x - 3$
  3. $y = x - 2$
  4. $y = -x + 2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation of the tangent line to a curve at a point is given by the formula $y - y_1 = m(x - x_1)$, where $m$ is the slope of the tangent line and $(x_1, y_1)$ is the point of tangency. In this case, the slope of the tangent line is $m = f'(1) = 3(1)^2 - 2(1) + 1 = 2$. So the equation of the tangent line is $y - (-1) = 2(x - 1)$, which simplifies to $y = 3x - 4$.

Multiple choice

What is the Picard group of the elliptic curve $y^2 = x^3 + x + 1$?

  1. $\mathbb{Z}$
  2. $\mathbb{Z}/2\mathbb{Z}$
  3. $\mathbb{Z}/3\mathbb{Z}$
  4. $\mathbb{Z}/4\mathbb{Z}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Picard group of an elliptic curve is a group that is related to the number of linearly independent holomorphic differentials on the curve. In this case, the elliptic curve $y^2 = x^3 + x + 1$ has one linearly independent holomorphic differential, so its Picard group is $\mathbb{Z}$.

Multiple choice

What is the genus of the curve defined by the equation $y^2 = x^5 + x^3 + x$?

  1. 0

  2. 1

  3. 2

  4. 3
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The genus of a curve is a topological invariant that is related to the number of holes in the curve. In this case, the curve defined by the equation $y^2 = x^5 + x^3 + x$ has one hole, so its genus is 1.

Multiple choice

What is the Picard group of the elliptic curve $y^2 = x^3 + 2x + 1$?

  1. $\mathbb{Z}$
  2. $\mathbb{Z}/2\mathbb{Z}$
  3. $\mathbb{Z}/3\mathbb{Z}$
  4. $\mathbb{Z}/4\mathbb{Z}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The Picard group of an elliptic curve is a group that is related to the number of linearly independent holomorphic differentials on the curve. In this case, the elliptic curve $y^2 = x^3 + 2x + 1$ has two linearly independent holomorphic differentials, so its Picard group is $\mathbb{Z}/2\mathbb{Z}$.

Multiple choice

What is the genus of the curve defined by the equation $y^2 = x^6 + x^4 + x^2$?

  1. 0

  2. 1

  3. 2

  4. 3
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The genus of a curve is a topological invariant that is related to the number of holes in the curve. In this case, the curve defined by the equation $y^2 = x^6 + x^4 + x^2$ has two holes, so its genus is 2.

Multiple choice

What is the Picard group of the elliptic curve $y^2 = x^3 + 3x + 2$?

  1. $\mathbb{Z}$
  2. $\mathbb{Z}/2\mathbb{Z}$
  3. $\mathbb{Z}/3\mathbb{Z}$
  4. $\mathbb{Z}/4\mathbb{Z}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The Picard group of an elliptic curve is a group that is related to the number of linearly independent holomorphic differentials on the curve. In this case, the elliptic curve $y^2 = x^3 + 3x + 2$ has three linearly independent holomorphic differentials, so its Picard group is $\mathbb{Z}/3\mathbb{Z}$.

Multiple choice

What is the name of the theorem that states that the derivative of a function is equal to the slope of the tangent line to the graph of the function at a given point?

  1. Mean value theorem

  2. Chain rule

  3. Product rule

  4. Tangent line theorem
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The tangent line theorem states that the derivative of a function is equal to the slope of the tangent line to the graph of the function at a given point.

Multiple choice

What is the name of the mathematical curve that is defined by the equation (r = a(1 + \cos \theta))?

  1. Cardioid

  2. Ellipse

  3. Hyperbola

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

The mathematical curve defined by the equation (r = a(1 + \cos \theta)) is called a cardioid, which has a heart-shaped shape.

Multiple choice

What is the fundamental theorem of calculus?

  1. The derivative of a function is equal to the slope of its tangent line.

  2. The integral of a function is equal to the area under its curve.

  3. The limit of a sequence is equal to the value it approaches as the number of terms approaches infinity.

  4. The sum of the interior angles of a triangle is 180 degrees.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The fundamental theorem of calculus states that the integral of a function is equal to the area under its curve. This theorem is used to find the area of regions, the volume of solids, and the length of curves.