Vector Algebra Questions

MathematicsVector Algebra152 Questions Found

MathematicsScalar Triple ProductJEE Advanced 2009Moderate

Match the statements / expressions given in Column-I with the values given in Column-II.

List-I

(P)

Root(s) of the equation

2 sin^{2} θ + sin^{2} 2 θ = 2

(Q)

Points of discontinuity of the function

f (x) = [\frac{6 x}{π}] cos [\frac{3 x}{π}]

, where

[y]

denotes the largest integer less than or equal to

y

(R)

Volume of the parallelopiped with its edges represented by the vectors

\hat{i} + \hat{j}, \hat{i} + 2 \hat{j}

and

\hat{i} + \hat{j} + π \hat{k}

(S)

Angle between vector

a

and

b

where

a, b

and

c

are unit vectors satisfying

a + b + 3 c = 0

List-II

(1)

π /6

(2)

π /4

(3)

π /3

(4)

π /2

(5)

π

Answer:Q → 4, Q → NaN, R → 5, S → 3

MathematicsScalar Triple ProductJEE Advanced 2013Moderate

View in:English Hindi

Match List I with List II:

List-I

(P)

Volume of parallelepiped determined by vectors

a, b

and

c

is 2. Then the volume of the parallelepiped determined by vectors

2 (a \times b), 3 (b \times c)

and

2 (c \times a)

(Q)

Volume of parallelepiped determined by vectors

a, b

and

c

is 5. Then the volume of the parallelepiped determined by vectors

3 (a + b), 3 (b + c)

and

2 (c + a)

(R)

Area of a triangle with adjacent sides determined by vectors

a

and

b

is 20. Then the area of the triangle with adjacent sides determined by vectors

(2 a + 3 b)

and

(a - b)

(S)

Area of a parallelogram with adjacent sides determined by vectors

a

and

b

is 30. Then the area of the parallelogram with adjacent sides determined by vectors

(a + b)

and

a

List-II

(1)

100

(2)

(3)

(4)

Answer:P → 3, Q → 4, R → 1, S → 2

MathematicsVector (Cross) ProductJEE Advanced 2014Difficult

View in:English Hindi

Match List I with List II:

List-I

(P)

Let

y (x) = cos (3 cos^{- 1} x), x \in [- 1, 1], x \neq = \pm \frac{3}{2}

. Then

\frac{1}{y ( x )} {(x^{2} - 1) \frac{d ^{2} y ( x )}{d x ^{2}} + x \frac{d y ( x )}{d x}}

equals

(Q)

Let

A_{1}, A_{2}, \dots, A_{n} (n > 2)

be the vertices of a regular polygon of

n

sides with its centre at the origin. Let

a_{k}

be the position vector of the point

A_{k}, k = 1, 2, \dots, n

. If

∣ \sum_{k = 1}^{n - 1} (a_{k} \times a_{k + 1}) ∣ = ∣ \sum_{k = 1}^{n - 1} (a_{k} \cdot a_{k + 1}) ∣

, then the minimum value of

n

(R)

If the normal from the point

P (h, 1)

on the ellipse

\frac{x ^{2}}{6} + \frac{y ^{2}}{3} = 1

is perpendicular to the line

x + y = 8

, then the value of

h

(S)

Number of positive solutions satisfying the equation

tan^{- 1} (\frac{1}{2 x + 1}) + tan^{- 1} (\frac{1}{4 x + 1}) = tan^{- 1} (\frac{2}{x ^{2}})

List-II

(1)

(2)

(3)

(4)

Answer:P → 4, Q → 3, R → 2, S → 1

MathematicsComponents of a VectorJEE Advanced 2015Difficult

View in:English Hindi

Match the following:

List-I

(P)

In a triangle

Δ X Y Z

, let

a, b

, and

c

be the lengths of the sides opposite to the angles

X, Y

and

Z

, respectively. If

2 (a^{2} - b^{2}) = c^{2}

and

λ = \frac{s i n ( X - Y )}{s i n Z}

, then possible values of

n

for which

cos (nπ λ) = 0

is (are)

(Q)

In a triangle

Δ X Y Z

, let

a, b

and

c

be the lengths of the sides opposite to the angles

X, Y

, and

Z

respectively. If

1 + cos 2 X - 2 cos 2 Y = 2 sin X sin Y

, then possible value(s) of

\frac{a}{b}

is (are)

(R)

R^{2}

, let

3 \hat{i} + \hat{j}, \hat{i} + 3 \hat{j}

and

β \hat{i} + (1 - β) \hat{j}

be the position vectors of

X, Y

and

Z

with respect to the origin

O

, respectively. If the distance of

Z

from the bisector of the acute angle of

O X

with

O Y

\frac{3}{2}

, then possible value(s) of

∣ β ∣

is (are)

(S)

Suppose that

F (α)

denotes the area of the region bounded by

x = 0, x = 2, y^{2} = 4 x

and

y = ∣ α x - 1∣ + ∣ α x - 2∣ + α x

, where

α \in {0, 1}

. Then the value(s) of

F (α) + \frac{8}{3} 2

, when

α = 0

and

α = 1

, is (are)

List-II

(1)

(2)

(3)

(4)

(5)

Answer:P → NaN, Q → 1, P → 2, S → 5

MathematicsScalar (Dot) ProductJEE Advanced 2015Difficult

View in:English Hindi

Match the following:

List-I

(P)

R^{2}

, if the magnitude of the projection vector of the vector

α \hat{i} + β \hat{j}

3 \hat{i} + \hat{j}

3

and if

α = 2 + 3 β

, then possible value of

∣ α ∣

is/are

(Q)

Let

a

and

b

be real numbers such that the function

f (x) = {- 3 a x^{2} - 2, b x + a^{2}, x < 1 x \geq 1

if differentiable for all

x \in R

. Then possible value of

a

is (are)

(R)

Let

ω \neq = 1

be a complex cube root of unity. If

(3 - 3 ω + 2 ω^{2})^{4 n + 3} + (2 + 3 ω - 3 ω^{2})^{4 n + 3} + (- 3 + 2 ω + 3 ω^{2})^{4 n + 3} = 0

, then possible value (s) of

n

is (are)

(S)

Let the harmonic mean of two positive real numbers

a

and

b

be 4. If

q

is a positive real number such that

a, 5, q, b

is an arithmetic progression, then the value(s) of

∣ q - a ∣

is (are)

List-II

(1)

(2)

(3)

(4)

(5)

Answer:P → 2, P → 2, R → NaN, Q → 5

MathematicsScalar (Dot) ProductJEE Advanced 1978Difficult

View in:English Hindi

From a point $O$ inside a triangle $A B C$ , perpendiculars $O D, O E, O F$ are drawn to the sides $B C, C A, A B$ respectively. Prove that the perpendiculars from $A, B, C$ to the sides $E F, F D, D E$ are concurrent.

MathematicsVector (Cross) ProductJEE Advanced 1982Moderate

View in:English Hindi

$A_{1}, A_{2}, \dots, A_{n}$ are the vertices of a regular plane polygon with $n$ sides and $O$ is its centre. Show that $\sum_{i = 1}^{n - 1} (O A_{i} \times O A_{i + 1}) = (1 - n) (O A_{2} \times O A_{1})$ .

MathematicsVector (Cross) ProductJEE Advanced 1987Moderate

View in:English Hindi

If $A, B, C, D$ are any four points in space, prove that $∣ A B \times C D + B C \times A D + C A \times B D ∣ = 4 (area of triangle A B C)$ .

MathematicsScalar Triple ProductJEE Advanced 1989Moderate

View in:English Hindi

If vectors $a, b, c$ are coplanar, show that $a \cdot a b \cdot a c \cdot a a \cdot b b \cdot b c \cdot b a \cdot c b \cdot c c \cdot c = 0$ .

MathematicsVector Triple ProductJEE Advanced 1994Moderate

View in:English Hindi

If the vectors $b, c, d$ are not coplanar, then prove that the vector $(a \times b) \times (c \times d) + (a \times c) \times (d \times b) + (a \times d) \times (b \times c)$ is parallel to $a$ .

...16

Match the statements / expressions given in Column-I with the values given in Column-II.

List-I

List-II

Match List I with List II:

List-I

List-II

Match List I with List II:

List-I

List-II

Match the following:

List-I

List-II

Match the following:

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A1​,A2​,…,An​ are the vertices of a regular plane polygon with n sides and O is its centre. Show that ∑i=1n−1​(OAi​×OAi+1​)=(1−n)(OA2​×OA1​).

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A,B,C,D are any four points in space, prove that ∣AB×CD+BC×AD+CA×BD∣=4(area of triangle ABC).

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If vectors a,b,c are coplanar, show that ​a⋅ab⋅ac⋅a​a⋅bb⋅bc⋅b​a⋅cb⋅cc⋅c​​=0.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the vectors b,c,d are not coplanar, then prove that the vector (a×b)×(c×d)+(a×c)×(d×b)+(a×d)×(b×c) is parallel to a.

Match the statements / expressions given in Column-I with the values given in Column-II.

List-I

List-II

Match List I with List II:

List-I

List-II

Match List I with List II:

List-I

List-II

Match the following:

List-I

List-II

Match the following:

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A1​,A2​,…,An​ are the vertices of a regular plane polygon with n sides and O is its centre. Show that ∑i=1n−1​(OAi​×OAi+1​)=(1−n)(OA2​×OA1​).

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A,B,C,D are any four points in space, prove that ∣AB×CD+BC×AD+CA×BD∣=4(area of triangle ABC).

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If vectors a,b,c are coplanar, show that ​a⋅ab⋅ac⋅a​a⋅bb⋅bc⋅b​a⋅cb⋅cc⋅c​​=0.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the vectors b,c,d are not coplanar, then prove that the vector (a×b)×(c×d)+(a×c)×(d×b)+(a×d)×(b×c) is parallel to a.

$A_{1}, A_{2}, \dots, A_{n}$ are the vertices of a regular plane polygon with $n$ sides and $O$ is its centre. Show that $\sum_{i = 1}^{n - 1} (O A_{i} \times O A_{i + 1}) = (1 - n) (O A_{2} \times O A_{1})$ .

If $A, B, C, D$ are any four points in space, prove that $∣ A B \times C D + B C \times A D + C A \times B D ∣ = 4 (area of triangle A B C)$ .

If vectors $a, b, c$ are coplanar, show that $a \cdot a b \cdot a c \cdot a a \cdot b b \cdot b c \cdot b a \cdot c b \cdot c c \cdot c = 0$ .

If the vectors $b, c, d$ are not coplanar, then prove that the vector $(a \times b) \times (c \times d) + (a \times c) \times (d \times b) + (a \times d) \times (b \times c)$ is parallel to $a$ .

$A_{1}, A_{2}, \dots, A_{n}$ are the vertices of a regular plane polygon with $n$ sides and $O$ is its centre. Show that $\sum_{i = 1}^{n - 1} (O A_{i} \times O A_{i + 1}) = (1 - n) (O A_{2} \times O A_{1})$ .

If $A, B, C, D$ are any four points in space, prove that $∣ A B \times C D + B C \times A D + C A \times B D ∣ = 4 (area of triangle A B C)$ .

If vectors $a, b, c$ are coplanar, show that $a \cdot a b \cdot a c \cdot a a \cdot b b \cdot b c \cdot b a \cdot c b \cdot c c \cdot c = 0$ .

If the vectors $b, c, d$ are not coplanar, then prove that the vector $(a \times b) \times (c \times d) + (a \times c) \times (d \times b) + (a \times d) \times (b \times c)$ is parallel to $a$ .