Polynomial Functions

Q.

Express $\left[\right(x,y):x^2+y^2=25,\text{where}x,yϵW]$ as a set of ordered pairs.

Q.

f(x) = $\frac{(x-b)(x-c)}{(x-a)}$, where a, b, c are distinct real numbers, will assume all real values provided :

1. c lies between a and b

2. a lies between b and c

3. b lies between a and c

4. None of these

Q. A copper rod $2\text{m}$ long is stretched by $1\text{mm}$. Then, strain will be -

1. ${10}^{-4}$, volumetric
2. $5\times {10}^{-4}$, volumetric
3. $5\times {10}^{-4}$, longitudinal
4. $5\times {10}^{-3}$, volumetric

Q. A steel wire $2\text{m}$ long is suspended from the ceiling. When a mass is hung at its lower end, the increase in length recorded is $1\text{cm}$. Determine the strain in the wire.

1. $5\times {10}^{-2}$
2. $5\times {10}^{-3}$
3. $2.5\times {10}^{-3}$
4. $2.5\times {10}^{-2}$

Q. A metal wire is stretched beyond its elastic limit and released, then wire

1. loses its elastic property completely and it will not contract.
2. will contract to its original length.
3. will contract till it achieves its elastic limit.
4. will contract but its final length will be greater than original length.

Q. The dimensions of four wires of the same material are given below. In which wire the increase in length will be maximum when the same tension is applied

1. Length 100 cm, diameter 1 mm
2. Length 200 cm, diameter 2 mm
3. Length 300 cm, diameter 3 mm
4. Length 50 cm, diameter 0.5 mm

Q. Which of the following is (are) polynomial function(s)?

1. $f\left(x\right)=x^3-2$
2. $f\left(x\right)=5$
3. $f\left(x\right)=a{x}^2+4x-4$, where $a\in R$
4. $f\left(x\right)=\frac{1}{x}-2$

Q.

If $a$ and $b$ are two integers such that $10a+b=20$ and $g\left(x\right)=x^2+ax+b$. Then, integer n such that $g\left(10\right).g\left(11\right)=g\left(n\right)$ is

1. $15$

2. $65$

3. $130$

4. $165$

Q. Let $a,b,cϵR$. If $f\left(x\right)=a{x}^2+bx+c$ be such that $a+b+c=3$ and $f(x+y)=f\left(x\right)+f\left(y\right)+xy,\forall x,yϵR,\text{then}{\sum }_{n=1}^{10}f\left(n\right)$ is equal to

1. 330
2. 165
3. 255
4. 190

Q. Evaluate:
(i) $\sqrt{-25}\times \sqrt{-49}$
(ii) $\sqrt{-36}\times \sqrt{16}$

Q. A wooden wheel of radius $R$ is made of two semicircular parts (see figure). The two parts are held together by a ring made of a metal strip of cross - sectional area $S$ and length $L$. $L$ is slightly less than $2\pi R$. To fit the ring on the wheel, it is heated so that its temperature rises by $\Delta T$ and it just steps over the wheel. As it cools down to surrounding temperature, it presses the semicircular parts together. If the coefficient of linear expansion of the metal is $\alpha$ and its Young's modulus is $Y$, the force that one part of the wheel applies on the other part is-

1. $2\pi SY\alpha \Delta T$
2. $SY\alpha \Delta T$
3. $\pi SY\alpha \Delta T$
4. $2SY\alpha \Delta T$

Q. Polynomial function is a superset of which of the following functions?

1. Constant function
2. Linear function
3. Quadratic function
4. Modulus function

Q. A block is fastened at one end of a wire and is rotated in a vertical circle of radius $R$. Determine the ratio of change in length of the wire at the lowest point to that at the highest point of the circle. Assume that speed of the block at highest and lowest points is the same ($v$).

1. $\frac{v+gR}{v-gR}$
2. $\frac{v-gR}{v+gR}$
3. $\frac{v^2+gR}{v^2-gR}$
4. $\frac{v^2-gR}{v^2+gR}$

Q. The Young’s modulus of steel is twice as that of brass. Two wires of same length and of same area of cross section, one of steel and another of brass are suspended from the same roof. If we want the lower ends of the wires to be at the same level, then the weights added to the steel and brass wires must be in the ratio of

1. $2:1$
2. $4:1$
3. $1:1$
4. $1:2$

Q. If f ( x ) = x 2 , find .

Q. A wire of cross-section area $2{\text{mm}}^2$ is stretched by $0.1\text{mm}$ by a certain force. How far (length) will the wire of same material and length but area $8{\text{mm}}^2$ stretch under the action of same force?

1. $0.25\text{mm}$
2. $0.005\text{mm}$
3. $0.025\text{mm}$
4. $0.05\text{mm}$

Q. Let $P\left(x\right)$ be a polynomial of degree $2010$. Suppose $P\left(n\right)=\frac{n}{1+n}$ for all $n=0,1,2,\dots ,2010.$ Find $P\left(2012\right).$
(correct answer + 5, wrong answer 0)

Q. A steel wire and a copper wire of equal length and equal cross-sectional area are joined end to end and the combination is subjected to a tension. Find the ratio of the stresses developed in steel and copper wire
$Y$ of steel is $2\times {10}^{11}{\text{N/m}}^2$ and $Y$ of copper is $1.3\times {10}^{11}{\text{N/m}}^2$.

Q. A wire elongates by $1.0\text{mm}$ when a load $W$ is hanged from it. If this wire goes over a pulley, and two weights W each are hung at the two ends, the elongation of the wire will be

1. $0.5\text{mm}$
2. $1.0\text{mm}$
3. $2.0\text{mm}$
4. $4.0\text{mm}$

Q. A block of weight $10N$ is fastened to one end of a wire of cross-sectional area $3m{m}^2$ and is rotated in a vertical circle of radius $20cm$. The speed of the block at the bottom of the circle is $2m/s$. Find the elongation of the wire when the block is at the bottom of the circle. Young’s modulus of the material of the wire $=2.0\times {10}^{11}N/m^2$.

1. $0.002cm$
2. $0.003cm$
3. $0.001cm$
4. $0.004cm$

Q. A force of $5\times {10}^3\text{N}$ is applied on the rod of length $1\text{m}$. Find the linear strain if elongation in the rod is $10\text{mm}$.

1. $0.1$
2. $1$
3. $0.01$
4. $0.001$

Q. A steel wire of length $1\text{m}$ and cross sectional area $A$ is stretched by $1\text{cm}$ under a given load. When the same load is applied to another steel wire of double length and half the cross-sectional area of the first one, the amount of stretching (extension) is

1. $0.5\text{cm}$
2. $2\text{cm}$
3. $4\text{cm}$
4. $1.5\text{cm}$

Q. A wire of area of cross-section $3.0m{m}^2$ and natural length $50cm$ is fixed at one end and a mass of $2.1kg$ is hung from the other end. Find the elastic potential energy stored in the wire in steady state. Young’s modulus of the material of the wire $=1.9\times {10}^{11}N{m}^{-2}$. Take $g=10m{s}^{-2}$.

1. $1.9\times {10}^{-4}J$
2. $2.9\times {10}^{-4}J$
3. $3.9\times {10}^{-4}J$
4. $4.9\times {10}^{-4}J$