Mathematics Test

Question 1
1 / -0

What is \(\left[\begin{array}{lll}\mathrm{x} & \mathrm{y} & \mathrm{z}\end{array}\right]\left[\begin{array}{lll}\mathrm{a} & \mathrm{h} & \mathrm{g} \\ \mathrm{h} & \mathrm{b} & \mathrm{f} \\ \mathrm{g} & \mathrm{f} & \mathrm{c}\end{array}\right]\) equal to?

A
\(\left[\begin{array}{ccc}\mathrm{ax} & \mathrm{hy} & \mathrm{gz} \\ \mathrm{h} & \mathrm{b} & \mathrm{f} \\ \mathrm{g} & \mathrm{f} & \mathrm{c}\end{array}\right]\)

B
\(\left[\begin{array}{ccc}\mathrm{a} & \mathrm{h} & \mathrm{g} \\ \mathrm{hx} & \mathrm{by} & \mathrm{fz} \\ \mathrm{g} & \mathrm{f} & \mathrm{c}\end{array}\right]\)

C
\(\left[\begin{array}{ccc}\mathrm{ax} & \mathrm{hy} & \mathrm{gz} \\ \mathrm{hx} & \mathrm{by} & \mathrm{fz} \\ \mathrm{gx} & \mathrm{fy} & \mathrm{cz}\end{array}\right]\)

D
\(\mathrm{[a x+h y+g z \quad h x+b y+f z \quad g x+f y+c z]}\)

Question 2

1 / -0

In the following cases, find the distance of each of the given points from the corresponding given plane.

Point

Plane

(a)\((0,0,0)\)

(b)\((3,-2,1)\)

(c)\((2,3,-5)\)

(d)\((-6,0,0)\)

\(3 x-4 y+12 z=3\)

\(2 x-y+2 z+3=0\)

\(x+2 y-2 z=9\)

\(2 x-3 y+6 z-2=0\)

\((a)=\frac{3}{13},(b)=\frac{13}{3},(c)=3,(d)=2\)

\((a)=\frac{13}{3},(b)=\frac{3}{13},(c)=2,(d)=3\)

\((a)=-\frac{3}{13},(b)=-\frac{13}{3},(c)=-3,(d)=-2\)

\((a)=\frac{13}{3},(b)=\frac{3}{13},(c)=-2,(d)=-3\)

Question 3
1 / -0

If one end of a diameter of the circle \(x^{2}+y^{2}-4 x-6 y+11=0\) is \((3,4),\) then find the coordinates of the other end of the diameter.

A
\(1,2\)

B
\(3,4\)

C
\(3,2\)

D
None of these

Question 4
1 / -0

Find the value of \(\lim _{x \rightarrow-\infty}\left[\frac{x^{4} \sin \left(\frac{1}{x}\right)+x^{2}}{\left(1+|x|^{3}\right)}\right]\).

A
0

B
1

C
-1

D
2

Question 5
1 / -0

Let X be a square matrix. Consider the following two statements on X.

I. X is invertible.

II. Determinant of X is non-zero.

Which one of the following is TRUE?

A
I implies II; II does not imply I.

B
II implies I; I does not imply II.

C
I does not imply II; II does not imply I.

D
I and II are equivalent statements.

Question 6
1 / -0

Find the value of \(\frac{d y}{d x}=e^{x-y}+x^{2} e^{-y}\).

A
\(3 x^{3}\left(e^{y}-e^{x}\right)=c\)

B
\(3 x^{3} e^{-y}\left(e^{x}-1\right)=c\)

C
\(\left(e^{y}-e^{x}\right)=3 x^{3}+c^{\prime}\)

D
None of these

Question 7
1 / -0

Let \(f: \mathbf{R} \rightarrow \mathbf{R}\) be defined as \(f(x)=x^{4}\). Choose the correct answer.

A
\(f\) is one-one onto

B
\(f\) is many-one onto

C
\(f\) is one-one but not onto

D
\(f\) is neither one-one nor onto

Question 8
1 / -0

For a set of five true or false questions, no student has written the all correct answers and no students have given the same sequence of answers. What is the maximum number of students in the class for this to be possible?

A
32

B
31

C
30

D
None of the above

Question 9
1 / -0

If \(f: \mathrm{R} \rightarrow \mathrm{R}\) be given by \(f(x)=\left(3-x^{3}\right)^{\frac{1}{3}},\) then \(fof(x)\) is:

A
\(\frac{1}{x^{3}}\)

B
\(x^{3}\)

C
\(x\)

D
\(\left(3-x^{3}\right)\)

Question 10
1 / -0

What is the area of the region bounded by the parabolas y² = 6 (x - 1) and y² = 3x?

A
\(\frac{\sqrt{6}}{3}\)

B
\(\frac{2 \sqrt{6}}{3}\)

C
\(\frac{4 \sqrt{6}}{3}\)

D
\(\frac{5 \sqrt{6}}{3}\)

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Mathematics Tes...

Question 1

\(\left[\begin{array}{ccc}\mathrm{ax} & \mathrm{hy} & \mathrm{gz} \\ \mathrm{h} & \mathrm{b} & \mathrm{f} \\ \mathrm{g} & \mathrm{f} & \mathrm{c}\end{array}\right]\)

\(\left[\begin{array}{ccc}\mathrm{a} & \mathrm{h} & \mathrm{g} \\ \mathrm{hx} & \mathrm{by} & \mathrm{fz} \\ \mathrm{g} & \mathrm{f} & \mathrm{c}\end{array}\right]\)

\(\left[\begin{array}{ccc}\mathrm{ax} & \mathrm{hy} & \mathrm{gz} \\ \mathrm{hx} & \mathrm{by} & \mathrm{fz} \\ \mathrm{gx} & \mathrm{fy} & \mathrm{cz}\end{array}\right]\)

\(\mathrm{[a x+h y+g z \quad h x+b y+f z \quad g x+f y+c z]}\)

Question 2

\((a)=\frac{3}{13},(b)=\frac{13}{3},(c)=3,(d)=2\)

\((a)=\frac{13}{3},(b)=\frac{3}{13},(c)=2,(d)=3\)

\((a)=-\frac{3}{13},(b)=-\frac{13}{3},(c)=-3,(d)=-2\)

\((a)=\frac{13}{3},(b)=\frac{3}{13},(c)=-2,(d)=-3\)

Question 3

\(1,2\)

\(3,4\)

\(3,2\)

None of these

Question 4

0

1

-1

2

Question 5

I implies II; II does not imply I.

II implies I; I does not imply II.

I does not imply II; II does not imply I.

I and II are equivalent statements.

Question 6

\(3 x^{3}\left(e^{y}-e^{x}\right)=c\)

\(3 x^{3} e^{-y}\left(e^{x}-1\right)=c\)

\(\left(e^{y}-e^{x}\right)=3 x^{3}+c^{\prime}\)

None of these

Question 7

\(f\) is one-one onto

\(f\) is many-one onto

\(f\) is one-one but not onto

\(f\) is neither one-one nor onto

Question 8

32

31

30

None of the above

Question 9

\(\frac{1}{x^{3}}\)

\(x^{3}\)

\(x\)

\(\left(3-x^{3}\right)\)

Question 10

\(\frac{\sqrt{6}}{3}\)

\(\frac{2 \sqrt{6}}{3}\)

\(\frac{4 \sqrt{6}}{3}\)

\(\frac{5 \sqrt{6}}{3}\)

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