Instantaneous Rate of Change as Slope of Tangent

Q. A ladder $5\text{m}$ long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of $2\text{cm/s}$. How fast is its height on the wall decreasing when the foot of the ladder is $4\text{m}$ away from the wall?

Q. A ladder 20 ft long has one end on the ground and the other end in contact with a vertical wall. The lower end slips along the ground. If the lower end of the ladder is 16 ft away from the wall, upper end is moving $\lambda$ times as fast as the lower end, then $\lambda$ is

1. $\frac{1}{3}$
2. $\frac{4}{3}$
3. $\frac{2}{3}$
4. $\frac{5}{3}$

Q.

The altitude of a cone is 20 cm and its semi-vertical angle is ${30}^{\circ }$. If the semi-vertical angle is increasing at the rate of $2^{\circ }$ per second, then the radius of the base is increasing at the rate of

1. 30 cm/sec

2. 10 cm/sec

3. 160 cm/sec

4. 160/3 cm/sec

Q.

$\mathrm{y}=\mathrm{x}{(\mathrm{x}-3)}^2$ decreases for the values of $\mathrm{x}$ given by:

1. $1<\mathrm{x}<3$

2. $\mathrm{x}<0$

3. $\mathrm{x}>0$

4. $0<\mathrm{x}<3/2$

Q.

A ladder20 ft long has one end on the ground and the other end in contact with a vertical wall. The lower end slips along the ground. If the lower end of the ladder is 16 ft away from the wall, upper end is moving $\lambda$ times as fast as the lower end, then $\lambda$ is

1. 2/3

2. 4/3

3. 5/3

4. 1/3

Q.

If $V=\frac{4}{3}\pi r^3$, at what rate in cubic units is V increasing when r = 10 and $\frac{dr}{dt}=0.01$ ?

1. $4\pi$

2. $\pi$

3. $4\frac{\pi }{3}$

4. $40\pi$

Q.

The angle between the lines whose direction cosines satisfy the equations $l+m+n=0,l^2+m^2-n^2=0$ is given by
[MP PET 1993; RPET 2001]

1. 
   

2. 
   

3. 
   

4. 
   

Q.

A ladder20 ft long has one end on the ground and the other end in contact with a vertical wall. The lower end slips along the ground. If the lower end of the ladder is 16 ft away from the wall, upper end is moving $\lambda$ times as fast as the lower end, then $\lambda$ is

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

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

3. $\frac{4}{3}$

4. $\frac{5}{3}$

Q. Let $A=\{X={(x,y,z)}^T:PX=O\text{and}{x}^2+y^2+z^2=1\},$ where $P=\left[\begin{array}{ccc}1& 2& 1\\ -2& 3& -4\\ 1& 9& -1\end{array}\right]$, then the set $A$:

1. contains more than two elements
2. is a singleton.
3. contains exactly two elements
4. is an empty set.

Q. Match List I with List II and select the correct answer.

List - I	List - II
A) Spring constant	I) [$M^1{L}^2{T}^{-2}$]
B) Pascal	II) [$M^0{L}^0{T}^{-1}$]
C) Hertz	III)[ $M^1{L}^0{T}^{-2}$]
D) Joule	IV) [$M^1{L}^{-1}{T}^{-2}$]

1. A-III, B-IV, C-II, D-I
2. A-IV, B-III, C-II, D-I
3. A-III, B-IV, C-I, D-II
4. A-IV, B-III, C-I, D-II

Q. The radius of the base of the right circular cone shown below is $5$ inches, and the height of the cone is $7$ inches. Solving which of the following equations gives the measure, $\theta$ of the angle formed by a slant height of the cone and a radius?

1. $\tan \theta =\frac{5}{7}$
2. $\tan \theta =\frac{7}{5}$
3. $\sin \theta =\frac{5}{7}$
4. $\sin \theta =\frac{7}{5}$

Q. If the curves $y=\frac{\ln x}{x}$ and $y=\lambda x^2$ (where $\lambda$ constant) touch each other, then $3e\lambda$ is

1. $e$
2. $e^2$
3. 1
4. 9

Q. If the curves $y=\frac{\ln x}{x}$ and $y=\lambda x^2$ (where $\lambda$ constant) touch each other, then $3e\lambda$ is

1. $e$
2. $e^2$
3. 1
4. 9

Q.

If $V=\frac{4}{3}\pi r^3$, at what rate in cubic units is V increasing when r = 10 and $\frac{dr}{dt}=0.01$ ?

1. 4 π

2. 40 π

3. π

4. 4 π/3

Q.

A ladder20 ft long has one end on the ground and the other end in contact with a vertical wall. The lower end slips along the ground. If the lower end of the ladder is 16 ft away from the wall, upper end is moving $\lambda$ times as fast as the lower end, then $\lambda$ is

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

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

3. $\frac{4}{3}$

4. $\frac{5}{3}$

Q.

The altitude of a cone is 20 cm and its semi-vertical angle is ${30}^{\circ }$. If the semi-vertical angle is increasing at the rate of $2^{\circ }$ per second, then the radius of the base is increasing at the rate of

1. 30 cm/sec

2. $\frac{160}{3}cm/sec$

3. 10 cm/sec

4. 160 cm/sec

Q. Solve the given quadratic equation by factorization method
$\frac{1}{a+b+x}=\frac{1}{x}+\frac{1}{a}+\frac{1}{b},a+b\ne 0$