Let A = {1, 2, 3, 4, 5} and f:A→A is defined by f(x)=x, what is the range of the function?

• A) {1}
• B) {1,2,3}
• C) {1,2,3,4,5}
• D) {5}

Let A = {a, b, c}, B = {1, 2, 3, 4} and f: A → B such that f= {(a, 1), (b, 2), (c, 3)} be a function. What is f(b)?

• A) 2
• B) 3
• C) 4
• D) 1

In an equivalence relation, if (a, b) is in R, what can be said about (b, a)?

• A) It is not related to (a, b).
• B) It implies asymmetry.
• C) It is in R due to symmetry.
• D) It violates the transitivity property.

Which property ensures that for any element a in set S, [a]=[a]?

• A) Transitivity
• B) Reflexivity
• C) Equivalence
• D) Symmetry

For f(x) = 2x +1 and g(x) = 4x, the composite function (fog)(x) is __________.

• A) 8x-1
• B) 2x+1
• C) 8x+1
• D) 2x-1

For f(x) = 2x - 3 and g(x) = x^2 - 1, the composite function (fog)(x) is __________.

• A) 2x^2 -1
• B) 2x^2 +1
• C) 2x^2 - 5
• D) (i) 2x^2 +5

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• A) data:image/png;base64,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 .
• B) data:image/png;base64,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 .
• C) data:image/png;base64,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 .
• D) None of these

For f(x) = 4x^2file:///C:/Users/jamshed/AppData/Local/Temp/msohtmlclip1/01/clip_image002.png and g(x) = 3x - 1, the composite function (fog)(x) is __________.

• A) 36x^2 + 4 + 4x
• B) - 36x^2 + 4 + 24x
• C) 36x^2 - 4 + 4x
• D) 36x^2 + 4 - 24x

Let λ and μ be ordinal numbers and let A and B be two well-ordered sets such that λ = ord(A) and μ = ord(B). Then λ < μ if A is similar to an initial segment of B.

• A) true
• B) false
• C)
• D)

Vector valued function has ---------

• A) none of these
• B) domain consists of real numbers and range consists of vectors.
• C) domain and range consists of real numbers.
• D) domain consists of vectors and range consists of real numbers

$$\begin{gathered} {\text{If}}\,\,\vec r(t) = x(t)\hat i + y(t)\hat j\,\,{\text{is}}\,\,{\text{a}}\,\,{\text{vector - valued}}\,\,{\text{function}}\,\,{\text{in}}\,\,{\text{2 - space,}}\,\,{\text{and}}\,\,{\text{if}}\,\,x(t)\,\,{\text{and}}\,\,y(t)\,\,{\text{are}}\,\,{\text{differentiable,}}\,\, \hfill \\ {\text{then}}\,\,\,\frac{d}{{dt}}\,\left[ {\vec r(t)} \right] = \_\_\_\_\_\_\_\_\_. \hfill \\\ \end{gathered}$$

• A) $$x'(t)\hat i + y'(t)\hat j + z'(t)\hat k$$
• B) $$x'(t)\hat i + y'(t)\hat j$$
• C) $$x(t)\hat i + y'(t)\hat j$$
• D) $$x'(t)\hat i + y(t)\hat j$$

$\begin{gathered} {\text{Let G be the rectangular box defined by the inequalities }}a \leqslant x \leqslant b,\,\,\,c \leqslant y \leqslant d,\,\,\,\,\,e \leqslant z \leqslant f. \hfill \ {\text{If }}f\,\,{\text{is continuous on G, then}}\,\,\int\limits_a^b {\int\limits_c^d {\int\limits_e^f {f(x,\,\,y,\,\,z)} } } \,dz\,\,dy\,\,dx = \,\,\, - - - - - - - - \hfill \\\ \end{gathered}$

• A) ${\text{All}}\,{\text{three}}\,{\text{options}}\,{\text{are}}\,{\text{true}}{\text{.}}$
• B) $\int\limits_a^b {\int\limits_e^f {\int\limits_c^d {f(x,\,\,y,\,\,z)} } } \,dy\,\,dz\,\,dx$
• C) $\int\limits_e^f {\int\limits_a^b {\int\limits_c^d {f(x,\,\,y,\,\,z)} } } \,dy\,\,dx\,\,dz$
• D) $\int\limits_c^d {\int\limits_a^b {\int\limits_e^f {f(x,\,\,y,\,\,z)} } } \,dz\,\,dx\,\,dy\,$

$$\begin{gathered} {\text{In}}\,\,{\text{3 - space,}}\,\,\vec r(t) = x(t)\hat i + y(t)\hat j + z(t)\hat k,\,\,{\text{is}}\,\,{\text{smooth}}\,\,{\text{function}}\,\,{\text{of}}\,\,t\,\,{\text{if}}\,\,x'(t),\,\,\,y'(t)\,\,{\text{and}}\,\,z'(t)\,\,{\text{are}}\,\, \hfill \\ {\text{_________}}\,\,{\text{and}}\,\,{\text{there}}\,\,{\text{is}}\,\,{\text{no}}\,\,{\text{value}}\,\,{\text{of}}\,\,{\text{t}}\,\,{\text{at}}\,\,{\text{which}}\,\,{\text{all}}\,\,{\text{three}}\,\,{\text{derivatives}}\,\,{\text{are}}\,\,{\text{zero}}{\text{.}} \hfill \\\ \end{gathered}$$

• A) $${\text{continuous}}$$
• B) $${\text{discontinuous}}$$
• C)
• D)

$${\text{A vector valued function in 2 - D can be expressed as}}$$

• A) $$\vec r(t) = x(t)i + y(t)j$$
• B) $$\vec r(t) = x(t)j + y(t)i$$
• C) $$\vec r(t) = x(t) - y(t)$$
• D) $$\vec r(t) = x(t) + y(t)$$

$$\eqalign{ & {\text{If }}x'(t){\text{ and }}y'(t){\text{ are continuous for }}a \leqslant t \leqslant b{\text{, then the given}} \cr & {\text{parametric equations are}} \cr}$$

• A) $$x = x(t) + y(t){\text{ ;}} \left( {a \leqslant t \leqslant b} \right)$$
• B) $$x = x(t),y = y(t){\text{ ;}} \left( {a \leqslant t \leqslant b} \right)$$
• C) $$x = x'(t),y = y'(t){\text{ ;}} \left( {a \leqslant t \leqslant b} \right)$$
• D) $$x = x'(t) + y'(t){\text{ ;}} \left( {a \leqslant t \leqslant b} \right)$$

$$\begin{gathered} {\text{The}}\,\,{\text{lemniscates}}\,\,{\text{are}}\,\,{\text{centered}}\,\,{\text{at}}\,\,{\text{the}}\,\,{\text{origin,}}\,\,{\text{but}}\,\,{\text{the}}\,\,{\text{position}}\,\,{\text{relative}}\,\,{\text{to}}\,{\text{the}}\,\,{\text{polar}}\,\,{\text{axis}}\,\, \hfill \\ {\text{depends}}\,\,{\text{on}}\,\,{\text{the}}\,\,{\text{sign}}\,\,{\text{preceding}}\,\,{\text{the}}\,\,{a^2}\,\,{\text{and}}\,\,{\text{whether}}\,\,{\text{_________}}\,\,{\text{appears}}\,\,{\text{in}}\,\,{\text{the}}\,\,{\text{equation}}{\text{.}} \hfill \\\ \end{gathered}$$

• A) $${\text{(d)}}\,\,\,{\text{Both}}\,\,{\text{(a)}}\,\,{\text{or}}\,\,{\text{(b)}}{\text{.}}$$
• B) $$(b)\,\,\,\cos 2\theta$$
• C) $$(c)\,\,\,\tan \theta$$
• D) $$(a)\,\,\,\sin 2\theta$$

$${\text{A vector valued function in 3 - D can be expressed as}}$$

• A) $$\vec r(t) = x(t)i + y(t)j + z(t)k$$
• B) $$\vec r(t) = x(t) - y(t) - z(t)$$
• C) $$\vec r(t) = x(t)j + y(t)i + z(t)k$$
• D) $$\vec r(t) = x(t) + y(t) + z(t)$$

$${\text{If}}\,\,P = 1 + 4xy\,\,{\text{and}}\,\,Q = 5{x^2}\,\,{\text{then}}\,\,\_\_\_\_\_\_\_\_\_.$$

• A) $$\frac{{\partial P}}{{\partial y}} \ne \frac{{\partial Q}}{{\partial x}}$$
• B) $$\frac{{\partial Q}}{{\partial x}} = 5{x^2}$$
• C) $$\frac{{\partial P}}{{\partial y}} = \frac{{\partial Q}}{{\partial x}}$$
• D) $$\frac{{\partial P}}{{\partial y}} = 1 + 4x$$

$${\text{If}}\,\,\vec r(t) = t\,\hat i + 2t\,\hat j{\text{,}}\,\,{\text{then}}\,\,\,\frac{d}{{dt}}\,\left[ {\vec r(t)} \right] = \_\_\_\_\_\_\_\_\_.$$

• A) $$\,t\,\hat i + 2\hat j$$
• B) $$\,\hat i + 2t\,\hat j$$
• C) $$\,\hat i + 2\hat j$$
• D) $$\,\hat i + \hat j$$

$${\text{The}}\,\,{\text{differential}}\,\,{\text{equation,}}\,\,dz{\kern 1pt} = {\kern 1pt} 4xy{\kern 1pt} dx{\kern 1pt} {\kern 1pt} + {\kern 1pt} {\kern 1pt} \left( {2{x^2} + 3{y^2}} \right){\kern 1pt} dy,\,\,{\text{is}}\,\,{\kern 1pt} {\text{an exact differential equation}}{\text{.}}$$

• A) $${\text{True}}$$
• B) $${\text{False}}$$
• C)
• D)