Joint Entrance Examination

Graduate Aptitude Test in Engineering

Geotechnical Engineering

Transportation Engineering

Irrigation

Engineering Mathematics

Construction Material and Management

Fluid Mechanics and Hydraulic Machines

Hydrology

Environmental Engineering

Engineering Mechanics

Structural Analysis

Reinforced Cement Concrete

Steel Structures

Geomatics Engineering Or Surveying

General Aptitude

1

Let ABC be a triangle whose circumcentre is at P. If the position vectors of A, B, C and P are $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c $$ and $${{\overrightarrow a + \overrightarrow b + \overrightarrow c } \over 4}$$ respectively, then the position vector of the orthocentre of this triangle, is :

A

$${\overrightarrow a + \overrightarrow b + \overrightarrow c }$$

B

$$ - \left( {{{\overrightarrow a + \overrightarrow b + \overrightarrow c } \over 2}} \right)$$

C

$$\overrightarrow 0 $$

D

$$\left( {{{\overrightarrow a + \overrightarrow b + \overrightarrow c } \over 2}} \right)$$

Given,

Position vector of circumcentre, $$\overrightarrow C = {{\overrightarrow a + \overrightarrow b + \overrightarrow c } \over 4}$$

We know, position vector of centroid, $$\overrightarrow G = {{\overrightarrow a + \overrightarrow b + \overrightarrow c } \over 3}$$

Now, let $$\overrightarrow R $$ be the orthocentre of the triangle.

We know, $$\overrightarrow G $$ $$ = {{2\overrightarrow C + \overrightarrow R } \over 3}$$

$$ \Rightarrow $$ 3$$\overrightarrow G $$ $$ = 2\overrightarrow C + \overrightarrow R $$

$$ \Rightarrow $$ $$\overrightarrow R = 3\overrightarrow G - 2\overrightarrow C $$

= $$\left( {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right) - 2\left( {{{\overrightarrow a + \overrightarrow b + \overrightarrow c } \over 4}} \right)$$

= $${{\overrightarrow a + \overrightarrow b + \overrightarrow c } \over 2}$$

2

If the image of the point P(1, –2, 3) in the plane, 2x + 3y – 4z + 22 = 0 measured parallel to the line,

$${x \over 1} = {y \over 4} = {z \over 5}$$ is Q, then PQ is equal to:

$${x \over 1} = {y \over 4} = {z \over 5}$$ is Q, then PQ is equal to:

A

$$2\sqrt {42} $$

B

$$\sqrt {42} $$

C

$$6\sqrt 5 $$

D

$$3\sqrt 5 $$

Equation of line PQ is $${{x - 1} \over 1} = {{y + 2} \over 4} = {{z - 3} \over 5}$$

Let F be ($$\lambda $$ + 1, 4$$\lambda $$ $$-$$ 2, 5$$\lambda $$ + 3)

Since F lies on the plane

$$ \therefore $$ 2($$\lambda $$ + 1) + 3(4$$\lambda $$ $$-$$ 2) $$-$$ 4(5$$\lambda $$ + 3) + 22 $$=$$ 0

$$ \Rightarrow $$ $$-$$ 6$$\lambda $$ + 6 = 0 $$ \Rightarrow $$ $$\lambda $$ = 1

$$ \therefore $$ F is (2, 2, 8)

PQ = 2 PF = 2$$\sqrt {{1^2} + {4^2} + {5^2}} $$ = 2$$\sqrt {42} $$

Let F be ($$\lambda $$ + 1, 4$$\lambda $$ $$-$$ 2, 5$$\lambda $$ + 3)

Since F lies on the plane

$$ \therefore $$ 2($$\lambda $$ + 1) + 3(4$$\lambda $$ $$-$$ 2) $$-$$ 4(5$$\lambda $$ + 3) + 22 $$=$$ 0

$$ \Rightarrow $$ $$-$$ 6$$\lambda $$ + 6 = 0 $$ \Rightarrow $$ $$\lambda $$ = 1

$$ \therefore $$ F is (2, 2, 8)

PQ = 2 PF = 2$$\sqrt {{1^2} + {4^2} + {5^2}} $$ = 2$$\sqrt {42} $$

3

The distance of the point (1, 3, – 7) from the plane passing through the point (1, –1, – 1), having normal
perpendicular to both the lines

$${{x - 1} \over 1} = {{y + 2} \over { - 2}} = {{z - 4} \over 3}$$

and

$${{x - 2} \over 2} = {{y + 1} \over { - 1}} = {{z + 7} \over { - 1}}$$ is

$${{x - 1} \over 1} = {{y + 2} \over { - 2}} = {{z - 4} \over 3}$$

and

$${{x - 2} \over 2} = {{y + 1} \over { - 1}} = {{z + 7} \over { - 1}}$$ is

A

$${{10} \over {\sqrt {83} }}$$

B

$${{5} \over {\sqrt {83} }}$$

C

$${{10} \over {\sqrt {74} }}$$

D

$${{20} \over {\sqrt {74} }}$$

Let the plane be

a(x $$-$$ 1) + b(y + 1) + c (z + 1) = 0

Normal vector

$$\left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr 1 & { - 2} & 3 \cr 2 & { - 1} & { - 1} \cr } } \right| = 5\widehat i + 7\widehat j + 3\widehat k$$

So plane is 5(x $$-$$ 1) + 7(y + 1) + 3(z + 1) = 0

$$ \Rightarrow $$ 5x + 7y + 3z + 5 = 0

Distance of point (1, 3, $$-$$ 7) from the plane is

$${{5 + 21 - 21 + 5} \over {\sqrt {25 + 49 + 9} }} = {{10} \over {\sqrt {83} }}$$

a(x $$-$$ 1) + b(y + 1) + c (z + 1) = 0

Normal vector

$$\left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr 1 & { - 2} & 3 \cr 2 & { - 1} & { - 1} \cr } } \right| = 5\widehat i + 7\widehat j + 3\widehat k$$

So plane is 5(x $$-$$ 1) + 7(y + 1) + 3(z + 1) = 0

$$ \Rightarrow $$ 5x + 7y + 3z + 5 = 0

Distance of point (1, 3, $$-$$ 7) from the plane is

$${{5 + 21 - 21 + 5} \over {\sqrt {25 + 49 + 9} }} = {{10} \over {\sqrt {83} }}$$

4

Let $$\overrightarrow a = 2\widehat i + \widehat j -2 \widehat k$$ and $$\overrightarrow b = \widehat i + \widehat j$$.

Let $$\overrightarrow c $$ be a vector such that $$\left| {\overrightarrow c - \overrightarrow a } \right| = 3$$,

$$\left| {\left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow c } \right| = 3$$ and the angle between be $$30^\circ $$.

Then $$\overrightarrow a .\overrightarrow c $$ is equal to

Let $$\overrightarrow c $$ be a vector such that $$\left| {\overrightarrow c - \overrightarrow a } \right| = 3$$,

$$\left| {\left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow c } \right| = 3$$ and the angle between be $$30^\circ $$.

Then $$\overrightarrow a .\overrightarrow c $$ is equal to

A

2

B

5

C

$${1 \over 8}$$

D

$${{25} \over 8}$$

Given:

$$\overrightarrow a = 2\widehat i + \widehat j - 2\widehat k,\,\,\overrightarrow b = \widehat i + \widehat j$$

$$ \Rightarrow $$ $$\left| {\overrightarrow a } \right| = 3$$

$$ \therefore $$ $$\overrightarrow a \times \overrightarrow b = 2\widehat i - 2\widehat j + \widehat k$$

$$\left| {\overrightarrow a \times \overrightarrow b } \right| = \sqrt {{2^2} + {2^2} + {1^2}} = 3$$

We have $$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow c = \left| {\overrightarrow a \times \overrightarrow b } \right|\left| {\overrightarrow c } \right|\sin 30n$$

$$ \Rightarrow $$ $$\left| {\left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow c } \right| = 3\left| {\overrightarrow c } \right|.{1 \over 2}$$

$$ \Rightarrow $$ $$3 = 3\left| {\overrightarrow c } \right|.{1 \over 2}$$

$$ \therefore $$ $$\left| {\overrightarrow c } \right| = 2$$

Now $$\left| {\overrightarrow c - \overrightarrow a } \right| = 3$$

On squaring, we get

$$ \Rightarrow $$ $${c^2} + {a^2} - 2 - \overrightarrow c .\overrightarrow a = 9$$

$$ \Rightarrow $$ $$4 + 9 - 2 - \overrightarrow a .\overrightarrow c = 9$$

$$ \Rightarrow $$ $$\overrightarrow a .\overrightarrow c = 2$$ [$$ \because $$ $$\overrightarrow c .\overrightarrow a \,\, = \,\,\overrightarrow a .\overrightarrow c $$]

$$\overrightarrow a = 2\widehat i + \widehat j - 2\widehat k,\,\,\overrightarrow b = \widehat i + \widehat j$$

$$ \Rightarrow $$ $$\left| {\overrightarrow a } \right| = 3$$

$$ \therefore $$ $$\overrightarrow a \times \overrightarrow b = 2\widehat i - 2\widehat j + \widehat k$$

$$\left| {\overrightarrow a \times \overrightarrow b } \right| = \sqrt {{2^2} + {2^2} + {1^2}} = 3$$

We have $$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow c = \left| {\overrightarrow a \times \overrightarrow b } \right|\left| {\overrightarrow c } \right|\sin 30n$$

$$ \Rightarrow $$ $$\left| {\left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow c } \right| = 3\left| {\overrightarrow c } \right|.{1 \over 2}$$

$$ \Rightarrow $$ $$3 = 3\left| {\overrightarrow c } \right|.{1 \over 2}$$

$$ \therefore $$ $$\left| {\overrightarrow c } \right| = 2$$

Now $$\left| {\overrightarrow c - \overrightarrow a } \right| = 3$$

On squaring, we get

$$ \Rightarrow $$ $${c^2} + {a^2} - 2 - \overrightarrow c .\overrightarrow a = 9$$

$$ \Rightarrow $$ $$4 + 9 - 2 - \overrightarrow a .\overrightarrow c = 9$$

$$ \Rightarrow $$ $$\overrightarrow a .\overrightarrow c = 2$$ [$$ \because $$ $$\overrightarrow c .\overrightarrow a \,\, = \,\,\overrightarrow a .\overrightarrow c $$]

Number in Brackets after Paper Name Indicates No of Questions

AIEEE 2002 (9) *keyboard_arrow_right*

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Straight Lines and Pair of Straight Lines *keyboard_arrow_right*

Circle *keyboard_arrow_right*

Conic Sections *keyboard_arrow_right*

Complex Numbers *keyboard_arrow_right*

Quadratic Equation and Inequalities *keyboard_arrow_right*

Permutations and Combinations *keyboard_arrow_right*

Mathematical Induction and Binomial Theorem *keyboard_arrow_right*

Sequences and Series *keyboard_arrow_right*

Matrices and Determinants *keyboard_arrow_right*

Vector Algebra and 3D Geometry *keyboard_arrow_right*

Probability *keyboard_arrow_right*

Statistics *keyboard_arrow_right*

Mathematical Reasoning *keyboard_arrow_right*

Trigonometric Functions & Equations *keyboard_arrow_right*

Properties of Triangle *keyboard_arrow_right*

Inverse Trigonometric Functions *keyboard_arrow_right*

Functions *keyboard_arrow_right*

Limits, Continuity and Differentiability *keyboard_arrow_right*

Differentiation *keyboard_arrow_right*

Application of Derivatives *keyboard_arrow_right*

Indefinite Integrals *keyboard_arrow_right*

Definite Integrals and Applications of Integrals *keyboard_arrow_right*

Differential Equations *keyboard_arrow_right*