Applicartions Of Derivatives Test

Applicartions Of Derivatives

This is Applicartions of Derivatives Test-03 for CBSE class 12 Maths.. There are 15 questions in this test with each question having around four answer choices.

Questions & Answers

1
The equation of the tangent to the curve \(y = {e^{2x}}\) at the point (0, 1) is
  • A
    y + 1 = 2 x
  • B
    y – 1 = 2 x
    Correct
  • C
    none of these
  • D
    1 – y = 2 x
2
The equation of the tangent to the curve \(y = {(4 - {x^2})^{2/3}}\) at x = 2 is
  • A
    x = – 2
  • B
    x = 2
    Correct
  • C
    y = – 1.
  • D
    y = 2
3
The slope of the tangent to the curve x = a sin t, y = a \(\left\{ {\cos t + \log (\tan \frac{t}{2})} \right\}\) at the point ‘t’ is
  • A
    tan\(\tan \frac{t}{2}\)
  • B
    none of these.
  • C
    tan t
  • D
    cot t
    Correct
4
Tangents to the curve \(y = {x^3}\) at the points (1, 1) and ( – 1, – 1) are
  • A
    perpendicular
  • B
    parallel
    Correct
  • C
    intersecting but not at right angles
  • D
    none of these.
5
The line\(\frac{x}{a} + \frac{y}{b} = 1\) touches the curve\(y = b{e^{ - x/a}}\) at the point
  • A
    ( – a ,ba)
  • B
    none of these
    Correct
  • C
    (a,-a)
  • D
    \(\left( {a,\frac{a}{b}} \right)\)
6
Tangents to the curve \({x^2} + {y^2} = 2\) at the points (1, 1) and ( – 1, 1)
  • A
    at right angles
    Correct
  • B
    intersecting but not at right angles
  • C
    parallel
  • D
    none of these
7
Equation of the tangent to the curve \({\left( {\frac{x}{a}} \right)^n} + {\left( {\frac{y}{b}} \right)^n} = 2\) at the point (a, b) is
  • A
    \(\frac{x}{a} + \frac{y}{b} = 1\)
  • B
    \(\frac{x}{a} + \frac{y}{b} = 2\)
    Correct
  • C
    \(\frac{x}{a} + \frac{y}{b} = 0\)
  • D
    none of these
8
The points on the curve 4 y = \(\left| {{x^2} - 4} \right|\) at which tangents are parallel to x – axis, are
  • A
    (4, 3) and ( – 4, – 3)
  • B
    none of these
  • C
    (0, 1) only
    Correct
  • D
    (2, 0) and ( – 2, 0)
9
The slope of the normal to the curvex = a ( cos θ + θ sin θ),y = a (sin θ – θ cos θ) at any point ‘θ’ is
  • A
    – tan θ
  • B
    – cot θ.
  • C
    tan θ
    Correct
  • D
    cot θ
10
The curve y = \({x^{1/5}}\) has at (0, 0)
  • A
    oblique tangent
  • B
    a horizontal tangent
  • C
    a vertical tangent
    Correct
  • D
    no tangent.
11
The point on the curve \({y^2} = x\) where tangent makes an angle of \({45^o}\) with the X – axis is
  • A
    \(\left( {\frac{1}{4},\frac{1}{2}} \right)\)
    Correct
  • B
    (4, 2)
  • C
    (2, – 2)
  • D
    \(\left( {\frac{1}{2},\frac{1}{4}} \right)\)
12
If the curve \(ay + {x^2} = 7\) and \({x^3} = y\) cut orthogonally at (1, 1), then a is equal to
  • A
    6
    Correct
  • B
    – 6
  • C
    0
  • D
    1
13
The point on the curve \(y = {(x - 3)^2}\) where the tangent is parallel to the chord joining (3, 0) and (4, 1) is
  • A
    (\(\frac{-7}{2}\), \(\frac{1}{4}\)
  • B
    (\(\frac{-5}{2}\) \(\frac{1}{4}\)
  • C
    ( \(\frac{7}{2}\) , \(\frac{1}{4}\) )
  • D
    (\(\frac{5}{2}\) \(\frac{1}{4}\)
    Correct
14
Tangents to the curve \(y = {x^3} + 3x\;at\;x = - 1\) and x = 1 are
  • A
    parallel
    Correct
  • B
    intersecting at an angle of \({45^o}\).
  • C
    intersecting at right angles
  • D
    intersecting obliquely but not at an angle of \({45^o}\)
15
The function f (x) = \({x^x}\) has a stationary point at
  • A
    x = e
  • B
    \(x = \frac{1}{e}\)
    Correct
  • C
    \(x = \sqrt e \)
  • D
    x = 1