Vector Algebra - Algebra - Mathematics Previous Year Questions

1

2024 · Mathematics · Algebra · Vector Algebra

TG EAPCET 2024 ONLINE 10TH MAY EVENING SHIFT

$\mathbf{a}$ is a vector perpendicular to the plane containing non zero vectors $\mathbf{b}$ and $\mathbf{c}$. If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are such that

$|\mathbf{a}+\mathbf{b}+\mathbf{c}|=\sqrt{|\mathbf{a}|^{2}+|\mathbf{b}|^{2}+|\mathbf{c}|^{2}}$, then

$|(\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c}|+|(\mathbf{a} \times \mathbf{b}) \times \mathbf{c}|=$

A

$|\mathbf{a}|+|\mathbf{b}|+|\mathbf{c}|$

B

$|\mathbf{a}\|\mathbf{b}\| \mathbf{c}|$

C

$|a|^{2}+|b|^{2}+\mid d^{2}$

D

$|\mathbf{a}|^{2}|\mathbf{b}|^{2}|\mathbf{c}|^{2}$

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2

2024 · Mathematics · Algebra · Vector Algebra

TG EAPCET 2024 ONLINE 10TH MAY EVENING SHIFT

$\mathbf{a}, \mathbf{b}, \mathbf{c}$ are three vectors each having $\sqrt{2}$ magnitude such that $(\mathbf{a}, \mathbf{b})=(\mathbf{b}, \mathbf{c})=(\mathbf{c}, \mathbf{a})=\frac{\pi}{3}$. If $\mathbf{x}=\mathbf{a} \times(\mathbf{b} \times \mathbf{c})$ and $\mathbf{y}=\mathbf{b} \times(\mathbf{c} \times \mathbf{a})$, then

A

$|\mathbf{x}|=|y|$

B

$|x|=\sqrt{2}|y|$

C

$|x|=2|y|$

D

$|x|+|y|=2$

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3

2024 · Mathematics · Algebra · Vector Algebra

TG EAPCET 2024 ONLINE 10TH MAY EVENING SHIFT

If $\mathbf{a}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+3 \hat{\mathbf{k}}, \mathbf{c}=-\hat{\mathbf{k}}$ are position vectors of two points and $\mathbf{b}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\lambda \hat{\mathbf{k}}, \mathbf{d}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ are two vectors, then the lines $\mathbf{r}=\mathbf{a}+t \mathbf{b}, \mathbf{r}=\mathbf{c}+s \mathbf{d}$ are

A

skew lines, when $\lambda=\frac{19}{3}$

B

coplanar, $\forall \lambda \in R$

C

skew lines when $\lambda \neq \frac{19}{3}$

D

coplanar, when $\lambda \neq \frac{19}{3}$

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4

2024 · Mathematics · Algebra · Vector Algebra

TG EAPCET 2024 ONLINE 10TH MAY EVENING SHIFT

If $\mathbf{a}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=3(\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}})$ and $\mathbf{c}$ is a vector such that $\mathbf{a} \times \mathbf{c}=\mathbf{b}$ and $\mathbf{a} . \mathbf{c}=3$, then $\mathbf{a} \cdot(\mathbf{c} \times \mathbf{b}-\mathbf{b}-\mathbf{c})=$

A

32

B

24

C

20

D

36

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5

2024 · Mathematics · Algebra · Vector Algebra

TG EAPCET 2024 ONLINE 10TH MAY MORNING SHIFT

The position vector of the point of intersection of the line joining the points $\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and the line joining the points $2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-6 \hat{\mathbf{k}}, 3 \hat{\mathbf{i}}-\hat{\mathbf{j}}-7 \hat{\mathbf{k}}$ is

A

$\hat{\mathbf{i}}-3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$

B

$4 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}-8 \hat{\mathbf{k}}$

C

$\hat{\mathbf{i}}+3 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}$

D

$\hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$

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6

2024 · Mathematics · Algebra · Vector Algebra

TG EAPCET 2024 ONLINE 10TH MAY MORNING SHIFT

$P$ and $Q$ are the points of trisection of the segment $A B$. If $2 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and $4 \hat{\mathbf{i}}+\hat{\mathbf{j}}-6 \hat{\mathbf{k}}$ are the position vectors of $A$ and $B$ respectively, then the position vector of the point which divides $P Q$ in the ratio $2: 3$ is

A

$\frac{1}{15}(44 \hat{\mathbf{i}}-33 \hat{\mathbf{j}}-18 \hat{\mathbf{k}})$

B

$\frac{1}{5}(36 \hat{\mathbf{i}}-26 \hat{\mathbf{j}}-18 \hat{\mathbf{k}})$

C

$\frac{1}{5}(3 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}-9 \hat{\mathbf{k}})$

D

$\frac{1}{15}(-3 \hat{\mathbf{i}}-7 \hat{\mathbf{j}}+9 \hat{\mathbf{k}})$

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7

2024 · Mathematics · Algebra · Vector Algebra

TG EAPCET 2024 ONLINE 10TH MAY MORNING SHIFT

If $\mathbf{a}=4 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ and $\mathbf{b}=6 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ are two vectors, then the magnitude of the component of $\mathbf{b}$ parallel to $\mathbf{a}$ is

A

$2 \sqrt{2}$

B

$10 \sqrt{2}$

C

$4 \sqrt{2}$

D

$6 \sqrt{2}$

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8

2024 · Mathematics · Algebra · Vector Algebra

TG EAPCET 2024 ONLINE 10TH MAY MORNING SHIFT

$\mathbf{a}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}, \mathbf{b}=2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\mathbf{c}=2 \hat{\mathbf{k}}-\hat{\mathbf{i}}$ are three vectors and $\mathbf{d}$ is a unit vector perpendicular to $\mathbf{c}$. If $\mathbf{a}, \mathbf{b}$ and $\mathbf{d}$ are coplanar vectors, then $|\mathbf{d} \cdot \mathbf{b}|=$

A

0

B

$\frac{1}{\sqrt{14}}$

C

$\sqrt{\frac{2}{7}}$

D

$\sqrt{\frac{7}{2}}$

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9

2024 · Mathematics · Algebra · Vector Algebra

TG EAPCET 2024 ONLINE 11TH MAY MORNING SHIFT

$2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ are the position vectores of two points $A$ and $B$ respectively and $C$ divides $A B$ in the ratio $3: 2$ : If $3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ is the position of vector of a point $D$, then the unit vector in the direction of $C D$ is

A

$\frac{1}{7 \sqrt{2}}(8 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}-3 \hat{\mathbf{k}})$

B

$\frac{1}{\sqrt{266}}(4 \hat{\mathbf{i}}-13 \hat{\mathbf{j}}+9 \hat{\mathbf{k}})$

C

$\frac{1}{3 \sqrt{42}}(8 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}+17 \hat{\mathbf{k}})$

D

$\frac{1}{7 \sqrt{2}}(8 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})$

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10

2024 · Mathematics · Algebra · Vector Algebra

TG EAPCET 2024 ONLINE 11TH MAY MORNING SHIFT

A unit vector $\hat{\mathbf{e}}=a \hat{\mathbf{i}}+b \hat{\mathbf{j}}+c \hat{\mathbf{k}}$ is coplanar with the vectors $\hat{\mathbf{i}}-3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$, and $3 \hat{\mathbf{i}}+\hat{\mathbf{j}}-5 \hat{\mathbf{k}}$. If $\hat{\mathbf{e}}$ is perpendicular to the vector $\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$, then $2 a^{2}+3 b^{2}+4 c^{2}=$

A

1

B

3

C

-1

D

$\sqrt{2}$

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11

2024 · Mathematics · Algebra · Vector Algebra

TG EAPCET 2024 ONLINE 11TH MAY MORNING SHIFT

$\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}, \hat{\mathbf{b}}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\hat{\mathbf{c}}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}$ are three vectors. If $\hat{\mathbf{d}}$ is a normal to the plane of $\hat{\mathbf{a}}$ and $\hat{\mathbf{b}}$ and d. $\hat{\mathbf{c}}=2$, then $|\hat{\mathbf{d}}|=$

A

$\sqrt{6}$

B

$2 \sqrt{3}$

C

$\sqrt{3}$

D

2

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12

2024 · Mathematics · Algebra · Vector Algebra

TG EAPCET 2024 ONLINE 9TH MAY EVENING SHIFT

$\mathbf{a}, \mathbf{b}, \mathbf{c}$ are non-coplanar vectors. If the three points $\lambda a-2 b+c, 2 a+\lambda b-2 \mathbf{c}$ and $4 \mathbf{a}+7 \mathbf{b}-8 \mathbf{c}$ are collinear, then $\lambda=$

A

-1

B

2

C

-2

D

1

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13

2024 · Mathematics · Algebra · Vector Algebra

TG EAPCET 2024 ONLINE 9TH MAY EVENING SHIFT

If $\mathrm{a}, \mathrm{b}$ are two vectors such that $|\mathrm{a}|=3,|\mathrm{~b}|=4$, $|\mathbf{a}+\mathbf{b}|=\sqrt{37},|\mathbf{a}-\mathbf{b}|=k$ and $(\mathbf{a}, \mathbf{b})=\theta$, then $\frac{4}{13}(k \sin \theta)^2=$

A

1

B

2

C

3

D

4

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14

2024 · Mathematics · Algebra · Vector Algebra

TG EAPCET 2024 ONLINE 9TH MAY EVENING SHIFT

$A(3,2,-1), B(4,1,0), C(2,1,4)$ are the vertices of a $\triangle A B C$. If the bisector of $B A C$ ! intersects the side $B C$ at $D(p, q, r)$, then $\sqrt{2 p+q+r}=$

A

3

B

4

C

1

D

2

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15

2024 · Mathematics · Algebra · Vector Algebra

TG EAPCET 2024 ONLINE 9TH MAY EVENING SHIFT

$(3,0,2)$ and $(0,2, k)$ are the direction ratios of two lines and $\theta$ is the angle between them. If $|\cos \theta|=\frac{6}{13}$, then $k=$

A

$\pm 2$

B

$\pm 3$

C

$\pm 5$

D

$\pm 7$

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16

2024 · Mathematics · Algebra · Vector Algebra

TG EAPCET 2024 ONLINE 9TH MAY EVENING SHIFT

$r$ is a vector perpendicular to the planet, determined by the vectors $2 \hat{\mathbf{i}}-\hat{\mathbf{j}}$ and $\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$, If the magnitude of the projection of $\mathbf{r}$ on the vector $2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ is l , then $|\mathbf{r}|=$

A

$\sqrt{6}$

B

$3 \sqrt{6}$

C

$\frac{2 \sqrt{6}}{3}$

D

$\frac{3 \sqrt{6}}{2}$

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17

2024 · Mathematics · Algebra · Vector Algebra

TG EAPCET 2024 ONLINE 9TH MAY EVENING SHIFT

$\mathbf{b}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \mathbf{k}, \quad \mathbf{c}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ are two vectors and $\mathbf{a}$ is a vector such that $\cos (\mathbf{a}, \mathbf{b} \times \mathbf{c})=\sqrt{\frac{2}{3}}$. If $\mathbf{a}$ is a unit vector, then $|\mathbf{a} \times(\mathbf{b} \times \mathbf{c})|=$

A

3

B

2

C

1

D

4

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18

2025 · Mathematics · Algebra · Vector Algebra

TG EAPCET 2025 ONLINE 2ND MAY EVENING SHIFT

If $a=|\mathbf{a}| ; b=|\mathbf{b}|$, then $\left(\frac{\mathbf{a}}{a^2}-\frac{\mathbf{b}}{b^2}\right)^2$

A

$\left(\frac{a-b}{a^2 b^2}\right)^2$

B

$\left(\frac{\mathbf{a}-\mathbf{b}}{\mathbf{a b}}\right)^{\mathbf{2}}$

C

$\left(\frac{b \mathbf{a}-a \mathbf{b}}{a b}\right)^2$

D

$\left(\frac{a \mathbf{a}-b \mathbf{b}}{a^2 b^2}\right)^2$

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19

2025 · Mathematics · Algebra · Vector Algebra

TG EAPCET 2025 ONLINE 2ND MAY EVENING SHIFT

The four points whose position vectors are given by $2 a+3 b-c, a-2 b+3 c, 3 a+4 b-2 c$ and $a-6 b+6 c$ are

A

collinear

B

coplanar

C

Vertices of a square

D

Vertices of a rectangle

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20

2025 · Mathematics · Algebra · Vector Algebra

TG EAPCET 2025 ONLINE 2ND MAY EVENING SHIFT

$A, B, C$ and $D$, are any four points. If $E$ and $F$ are mid-points of $A C$ and $B D$ respectively, then $\mathbf{A B}+\mathbf{C B}+\mathbf{C D}+\mathbf{A D}=$

A

EF

B

$2 E F$

C

3 EF

D

$4 E F$

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