Mathematics - Application of Derivatives 1
Exam Duration: 45 Mins Total Questions : 30
The coordinates of the point on the curve \((x^2+1)(y-3)=x\) where a tangent to the curve has the greatest slope are
- (a)
\((\sqrt3, - \sqrt3 + 2)\)
- (b)
\((3,-2)\)
- (c)
\((0,3)\)
- (d)
\(-2\sqrt3,\sqrt3\)
If y = f (x) is the equation of a parabola which is touched by the line y = x at the point where x = 1, then
- (a)
2f'(0) = 3f'(1)
- (b)
f'(1) = 1
- (c)
f(0) + f'(1) + f''(1) = 2
- (d)
2f(0) = 1 + f'(0)
For the function \(y = x^{1\over x},x > 0,\) which one of the following is true?
- (a)
\(e^\pi=\pi^{e}\)
- (b)
\(e^\pi<\pi^{e}\)
- (c)
\(e^\pi>\pi^{e}\)
- (d)
\(e^\pi\le\pi^{e}\)
If \(3(a+2c)=4(b+3d),\) then the equation \(ax^3+bx^2+cx+d=0\) will have
- (a)
no real solution
- (b)
atleast one real root in (-1, 0)
- (c)
atleast one real root in (0,1)
- (d)
None of the above
A lizard, at an initial distance of 21 cm behind an insect, moves from rest with an acceleration of \(2cm/s^2\) and pursues the insect which is crawling uniformly along a straight line at a speed of 20 cm/s. Then, the lizard will catch the insect after
- (a)
24 s
- (b)
21 s
- (c)
1 s
- (d)
20 s
A spherical baloon is pumped at the rate of 10 inch/min the rate of increase of itys radius if iis radius is 15 inch is
- (a)
\(\frac { 1 }{ 30\pi } \) inch/min
- (b)
\(\frac { 1 }{ 60\pi } \) inch/min
- (c)
\(\frac { 1 }{ 90\pi } \)inch/min
- (d)
\(\frac { 1 }{ 120\pi } \)inch/min
The approximate value of square root of 25.2 is
- (a)
5.01
- (b)
5.02
- (c)
5.03
- (d)
5.04
x and yare the sides of two squares such that y = x - x2 The rate of change of area of the second square with respect to that of the first square is
- (a)
2X2 + 3x + 1
- (b)
3x2 + 2x-l
- (c)
2X2 - 3x + 1
- (d)
3x2 + 2x + 1
The slope of the tangent to the curve y = \(f^{ x }_{ 0 }\frac { dx }{ 1+x^{ 3 } } \) at the point where x = 1 is
- (a)
1/4
- (b)
1/2
- (c)
1
- (d)
none of these
The subtangent, ordinate and subnormal to the parabola y24ax at a point (different from the origin) are in
- (a)
AP
- (b)
GP
- (c)
HP
- (d)
none of these
If the subnormal at any point on y = a1-nis of constant length, then the value of n is
- (a)
-2
- (b)
1/2
- (c)
1
- (d)
2
The value of parameter a so that the line (3 - a)x + ay + (a2 -1) = 0 is normal to the curve.xy = 1, may Ii'e I,II thee imterval
- (a)
\((\infty ,0)\bigcup (3,\infty )\)
- (b)
(1,3)
- (c)
(-3,3)
- (d)
none of these
The parametric equation of given curve are x=a(2cost+cos2t) y=a(2sint-sin2t)
The length of sub tangent at any point 't' is
- (a)
IY tan tl
- (b)
|ycot t|
- (c)
| y cot(t/2)|
- (d)
| y tan (t/2)|
If length of perpendiculars from origin on tangent and normal at 't' are p and PI' respectively, then the value of 9p2 + p1 is equal to
- (a)
9a2
- (b)
9a2sin2(t/2)
- (c)
\(9a^{ 2 }cos^{ 2 }\left( \frac { 3t }{ 2 } \right) \)
- (d)
a2
The value of b for which the function f(x) =sinx-bx+c is decreasing for x∈R is given by
- (a)
b < 1
- (b)
b > 1
- (c)
b > 1
- (d)
b < 1
The function f(x)=x3+6x2+(9+2k)x+1 is strictly increasing for all x, if
- (a)
\(k>{3\over2}\)
- (b)
\(k\ge{3\over2}\)
- (c)
\(k<{3\over2}\)
- (d)
\(k\le{3\over2}\)
the equation of the tangent to the y=e2x at the point (0, 1) is
- (a)
y+1=2x
- (b)
1-y=2x
- (c)
y-1=2x
- (d)
none of these
The equation of the tangent to the curve y=(4-x2)2/3 at x=2 is
- (a)
x=-2
- (b)
x=2
- (c)
y=2
- (d)
y=-1
If the tangent to the curve y=2x2-x+1 is parallel to the line y=3x+9 at the point
- (a)
(2,3)
- (b)
(2,-1)
- (c)
(2,1)
- (d)
(1,2)
The tangent to the curve y=x2+3x will pass through the point (0,-9) if it drawn at the point
- (a)
(0,1)
- (b)
(-3,0)
- (c)
(-4,4)
- (d)
(1,4)
The two curves x3-xy2 +5=0 and 3x2y-y3-7=0
- (a)
cut at right angles
- (b)
touch each other
- (c)
cut at an angle π/4
- (d)
cut at an angle π/3
Use differentials to approximate the cube root of 127
- (a)
5.05
- (b)
5.026
- (c)
6.02
- (d)
4.09
The combined resistance R of two resistors R1 and R2(R1,R2>0)is given by \({1\over R}={1\over R_1}+{1\over R_2}\)
If R2+R2=C (a constant) then maximum resistance R is obtained if
- (a)
R1>R2
- (b)
R1<R2
- (c)
R1=R2
- (d)
None of these
Fill in the blanks.
(i) The curves y = 4x2 + 2x - 8 and y = x3 - X + 10 touch each other at the point P
(ii) The values of a for which the function f(x) = sinx - ax + b increases on R are Q
(iii)The least value of the function \(f(x) = ax +{b\over x}\) (a>0,b>0, x>) is R
(iv) The equation of normal to the curve y = tanx at (0,0) is S
- (a)
P Q R S \(\left(-{1\over 3},-{74\over 9}\right)\) (-∞,1) \(\sqrt{ab}\) y-2x=0 - (b)
P Q R S (3,34) (-∞,1) 2\(\sqrt{ab}\) y+2x=0 - (c)
P Q R S (3,34) (-∞,-1) \(\sqrt{ab}\) y+x - (d)
P Q R S \(\left(-{1\over 3},-{74\over 9}\right)\) (-∞,1) \(2\sqrt{ab}\) x-2y=0
The sides of an equilateral triangle are increasing at the rate of 2 cm/see. The rate at which the area increases, when side is 10 em is
- (a)
10 cm2/s
- (b)
√3 cm2/s
- (c)
10√3 cm2/s
- (d)
10/3 cm2/s
The tangent to the curve y = e2x at the point (0, 1) meets x-axis at
- (a)
(0, 1)
- (b)
\(\left(-{1\over2},0\right)\)
- (c)
(2,0)
- (d)
(0,2)
The slope of tangent to the curve x = t2 + 3t - 8, y = 2t2 - 2t - 5 at the point (2, -1) is
- (a)
22/7
- (b)
6/7
- (c)
-6/7
- (d)
-6
The interval on which the funcl:ion f(x) = 2x3 + 9x2 + 12x - 1 is decreasing is
- (a)
[-1,∞)
- (b)
[-2, -1]
- (c)
(-∞, -2)
- (d)
[-1,1]
If x is real, the minimum value of X2 - 8x + 17 is
- (a)
-1
- (b)
0
- (c)
1
- (d)
2
The maximum value of sin x . cos x is
- (a)
\(1\over4\)
- (b)
\(1\over2\)
- (c)
√2
- (d)
2√2
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