Applicartions Of Derivatives Test

Applicartions Of Derivatives

This is Applicartions of Derivatives Test-02 for CBSE class 12 Maths.. There are 15 questions in this test with each question having around four answer choices.

Questions & Answers

1
Let f (x) = \({x^3} + \frac{3}{2}{x^2} + 3x + 3,\) then f (x) is
  • A
    an odd function
  • B
    an increasing function
    Correct
  • C
    a decreasing function.
  • D
    an even function
2
Let f (x) \( = \frac{x}{{1 + x}}\) – log(1 + x), where x > 0, then f is
  • A
    a decreasing function
    Correct
  • B
    neither increasing nor decreasing
  • C
    none of these
  • D
    an increasing function
3
Let f (x) = \({x^4}\) – 4x, then
  • A
    f is increasing in \(\left[ {1,\infty } \right)\)
    Correct
  • B
    f is decreasing in \(\left[ {1,\infty } \right)\)
  • C
    none of these.
  • D
    f is increasing in \(\left[ { - \infty ,1} \right)\)
4
Let f (x) = \({x^3} - 6{x^2} + 9x + 18,\) then f (x) is strict decreasing in
  • A
    \(\left[ {3,\infty } \right)\)
  • B
    \(\left[ { - \infty ,1} \right)\)
  • C
    [1, 3]
    Correct
  • D
    \(\left( { - \infty ,1} \right]\)
5
Let g (x) be continuous in a neighbourhood of ‘a’ and g (a) ≠ 0. Let f be a function such that f ‘ (x) = g(x) \({(x - a)^2}\) , then
  • A
    none of these
  • B
    f is increasing at a if g (a) > 0
    Correct
  • C
    f is decreasing at a if g (a) >
  • D
    f is increasing at a if g (a) < 0
6
The function f (x) = a x + b is strict increasing for all \(x \in {\mathbf{R}}\;\) iff
  • A
    a< 0
  • B
    a> 0
    Correct
  • C
    none of these.
  • D
    a = 0
7
The function f (x ) = a x + b is strict decreasing for all \(x \in {\mathbf{R}}\;\) iff
  • A
    a< 0
  • B
    a> 0
    Correct
  • C
    none of these.
  • D
    a = 0
8
The function f (x) = \({x^2} - 2x\) is strict decreasing in the interval
  • A
    \( ({\text{ }}--\infty ,{\text{ }}1]\)
    Correct
  • B
    \( [1,\infty ) \)
  • C
    R
  • D
    none of these
9
For the curve \(x = {t^2} - 1,y = {t^2} - t\) tangent is parallel to X – axis where
  • A
    \(t = \frac{1}{2}\)
    Correct
  • B
    \(t = \frac{1}{{\sqrt 3 }}\)
  • C
    t = 0
  • D
    \(t = - \frac{1}{{\sqrt 3 }}\)
10
The equation of the normal to the curve y = sinx at (0, 0) is
  • A
    x – y = 0
  • B
    y = 0
  • C
    x + y = 0
    Correct
  • D
    x = 0
11
The tangent to the parabola \({x^2} = 2y\) at the point \(\left( {1,\frac{1}{2}} \right)\) makes with the X – axis an angle of
  • A
    \({60^\circ }\)
  • B
    \({0^\circ }\)
  • C
    \({45^\circ }\)
    Correct
  • D
    \({30^\circ }\)
12
The curvey = \(a\;{x^3} + b{x^2} + c\;x\) is inclined at \({45^\circ }\) to the X – axis at (0, 0) but it touches X – axis at (1, 0) , then the values of a, b, c, are given by
  • A
    a = – 2, b = 1, c = 1
  • B
    a = 1, b = – 2, c = 1
    Correct
  • C
    a = 1, b = 1, c = – 2
  • D
    a = – 1, b = 2, c = 1.
13
The normal to the curvex = a \(\left( {\cos \theta + \theta \sin \theta } \right),\)y = a \(\left( {\sin \theta - \theta \cos \theta } \right)\)at any point \(\theta \) is such that
  • A
    none of these
  • B
    it passes through the origin
  • C
    it is at a constant distance from the origin
    Correct
  • D
    it makes a constant angle with X – axis
14
The normal to the curve 2 y = 3 – \({x^2}\;\)at (1, 1) is
  • A
    x – y + 1 + 0
  • B
    – y = 0
  • C
    x – y = 0
    Correct
  • D
    x + y + 1 = 0
15
The equation of the tangent to the curve \({y^2} = 4\;a\;x\) at the point \(\left( {a\;{t^2},2\;a\;t} \right)\) is
  • A
    ty = x + \(a\;{t^2}\)
    Correct
  • B
    tx + y = \(a\;{t^3}\)
  • C
    ty = x – \(a\;{t^2}\)
  • D
    none of these