AMU MCA Vector Previous Year Question Paper and Solution with PDF

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AMU MCA Previous Year Questions (PYQs)

AMU MCA Vector PYQ

Qus : 1 AMU MCA PYQ

The shortest distance from the point $(1,0,-2)$ to the plane $x+2y+z=4$ is

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AMU MCA Previous Year PYQ AMU MCA AMU MCA 2020 PYQ

Solution

Distance formula:

$d=\frac{|ax_1+by_1+cz_1-d|}{\sqrt{a^2+b^2+c^2}}$

Plane: $x+2y+z-4=0$

$d=\frac{|1(1)+2(0)+1(-2)-4|}{\sqrt{1^2+2^2+1^2}}$

$=\frac{|1-2-4|}{\sqrt6}=\frac{5}{\sqrt6} =\frac{5\sqrt6}{6}$

Qus : 2 AMU MCA PYQ

Two bodies of mass 100 g and 20 g are moving with velocities $(2\hat{i}-7\hat{j}+3\hat{k})$ cm/s and $(-10\hat{i}+35\hat{j}-3\hat{k})$ cm/s respectively. What is the velocity of centre of mass of this two body system?

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Solution

Velocity of centre of mass:

$\vec{V}=\frac{m_1\vec{v}_1+m_2\vec{v}_2}{m_1+m_2}$

$=\frac{100(2\hat{i}-7\hat{j}+3\hat{k})+20(-10\hat{i}+35\hat{j}-3\hat{k})}{120}$

$=\frac{(200-200)\hat{i}+(-700+700)\hat{j}+(300-60)\hat{k}}{120}$

$=\frac{240\hat{k}}{120}=2\hat{k}$

Qus : 3 AMU MCA PYQ

If $\alpha, \beta, \gamma$ be the angles which a line subtends with the positive direction of coordinate axes, then $\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma$ equals

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AMU MCA Previous Year PYQ AMU MCA AMU MCA 2022 PYQ

Solution

Qus : 4 AMU MCA PYQ

If Laplace transform $L{F(x)}=\frac1{s}e^{-3s}$, then $L{e^{-x}F(3x)}$ is

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Solution

Qus : 5 AMU MCA PYQ

If the centre of mass of three particles of masses 10, 20 and 30 gram be at the point $(1,-2,3)$, where should a fourth particle of 40 gram be placed so that the combined centre of mass may be $(1,1,1)$ ?

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Solution

Qus : 6 AMU MCA PYQ

Which of the following is a 2-dimensional subspace of $\mathbb{R}^3$?

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AMU MCA Previous Year PYQ AMU MCA AMU MCA 2025 PYQ

Solution

Set ${(0,x,z)}$ contains two independent parameters and satisfies subspace properties.

Qus : 7 AMU MCA PYQ

If $U={(x,y)\in\mathbb R^2\mid y=mx}$ and $W={(x,y)\in\mathbb R^2\mid y=tx,\ t\ne m}$ Then $\dim(U+W)$ is

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Solution

Both are distinct lines through origin. Their sum spans $\mathbb R^2$. So dimension = 2.

Qus : 8 AMU MCA PYQ

Let $\mathbb{R}^3={(x,y,z)}$ and $W$ be the subspace generated by ${(1,2,-3)}$. Geometrically, $W$ represents

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AMU MCA Previous Year PYQ AMU MCA AMU MCA 2021 PYQ

Solution

Single vector span ⇒ line through origin Direction ratios: $1,2,-3$ Direction cosines: $\frac{1}{\sqrt{14}},\frac{2}{\sqrt{14}},\frac{-3}{\sqrt{14}}$

Qus : 9 AMU MCA PYQ

A particle of mass 2 kg is moving such that its position is given by $P(t)=5\hat{i}-2t^2\hat{j}$ Find angular momentum at $t=2$ sec about origin.

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Solution

Position at $t=2$:

$\vec{r}=5\hat{i}-8\hat{j}$

Velocity:

$\vec{v}=\frac{d\vec{r}}{dt}=-4t\hat{j}$

At $t=2$:

$\vec{v}=-8\hat{j}$

Momentum:

$\vec{p}=m\vec{v}=2(-8\hat{j})=-16\hat{j}$

Angular momentum:

$\vec{L}=\vec{r}\times\vec{p}$

$=(5\hat{i}-8\hat{j})\times(-16\hat{j})$

$= -80\hat{k}$

Qus : 10 AMU MCA PYQ

If $A \cap B = {0}$, $\dim A = 2$, $\dim B = 1$ in 3D space, then

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AMU MCA Previous Year PYQ AMU MCA AMU MCA 2020 PYQ

Solution

$\dim(A + B) = 2 + 1 - 0 = 3$ So $V = A + B$

Qus : 11 AMU MCA PYQ

Let $V$ be a 3-dimensional vector space with $A$ and $B$ its subspaces of dimensions $2$ and $1$ respectively. If $A \cap B = {0}$, then

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AMU MCA Previous Year PYQ AMU MCA AMU MCA 2020 PYQ

Solution

$\dim(A + B) = 2 + 1 - 0 = 3$ So $V = A + B$

Qus : 12 AMU MCA PYQ

The direction cosines of the line which is equally inclined to the axes is

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Solution

If equally inclined then $l = m = n$

$l^2 + m^2 + n^2 = 1$

$3l^2 = 1$

$l = \pm\frac{1}{\sqrt{3}}$

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