Joint Entrance Examination

Graduate Aptitude Test in Engineering

Geomatics Engineering Or Surveying

Engineering Mechanics

Hydrology

Transportation Engineering

Strength of Materials Or Solid Mechanics

Reinforced Cement Concrete

Steel Structures

Irrigation

Environmental Engineering

Engineering Mathematics

Structural Analysis

Geotechnical Engineering

Fluid Mechanics and Hydraulic Machines

General Aptitude

1

If the mean of the data : 7, 8, 9, 7, 8, 7, $$\lambda $$, 8 is 8, then the variance of this data is :

A

$${7 \over 8}$$

B

1

C

$${9 \over 8}$$

D

2

$$\overline x $$ = $${{7 + 8 + 9 + 7 + 8 + 7 + \lambda + 8} \over 8}$$ = 8

$$ \Rightarrow $$$$\,\,\,$$ $${{54 + \lambda } \over 8}$$ = 8 $$ \Rightarrow $$ $$\lambda $$ = 10

Now variance = $$\sigma $$^{2}

= $${{{{\left( {7 - 8} \right)}^2} + {{\left( {8 - 8} \right)}^2} + {{\left( {9 - 8} \right)}^2} + {{\left( {7 - 8} \right)}^2} + {{\left( {8 - 8} \right)}^2} + {{\left( {7 - 8} \right)}^2} + {{\left( {10 - 8} \right)}^2} + {{\left( {8 - 8} \right)}^2}} \over 8}$$

$$ \Rightarrow $$ $$\sigma $$^{2} = $${{1 + 0 + 1 + 1 + 0 + 1 + 4 + 0} \over 8}$$ = $${8 \over 8}$$ = 1

Hence, the variance is 1.

$$ \Rightarrow $$$$\,\,\,$$ $${{54 + \lambda } \over 8}$$ = 8 $$ \Rightarrow $$ $$\lambda $$ = 10

Now variance = $$\sigma $$

= $${{{{\left( {7 - 8} \right)}^2} + {{\left( {8 - 8} \right)}^2} + {{\left( {9 - 8} \right)}^2} + {{\left( {7 - 8} \right)}^2} + {{\left( {8 - 8} \right)}^2} + {{\left( {7 - 8} \right)}^2} + {{\left( {10 - 8} \right)}^2} + {{\left( {8 - 8} \right)}^2}} \over 8}$$

$$ \Rightarrow $$ $$\sigma $$

Hence, the variance is 1.

2

The mean and the standarddeviation(s.d.) of five observations are9 and 0, respectively. If one of the observations is changed such that the mean of the new set of five observations becomes 10, then their s.d. is :

A

0

B

1

C

2

D

4

Here mean = $$\overline x $$ = 9

$$ \Rightarrow $$ $$\overline x $$ = $${{\sum {{x_i}} } \over n}$$ = 9

$$ \Rightarrow $$ $${\sum {{x_i}} }$$ = 9 $$ \times $$ 5 = 45

Now, standard deviation = 0

$$\therefore\,\,\,$$ all the five terms are same i.e.; 9

Now for changed observation

$${\overline x _{new}}$$ = $${{36 + {x_5}} \over 5} = 10$$

$$ \Rightarrow $$ x_{5} = 14

$$\therefore\,\,\,$$ $$\sigma $$_{new} = $$\sqrt {{{\sum {{{\left( {{x_i} - {{\overline x }_{new}}} \right)}^2}} } \over n}} $$

= $$\sqrt {{{4{{\left( {9 - 10} \right)}^2} + {{\left( {14 - 10} \right)}^2}} \over 5}} $$ = 2

$$ \Rightarrow $$ $$\overline x $$ = $${{\sum {{x_i}} } \over n}$$ = 9

$$ \Rightarrow $$ $${\sum {{x_i}} }$$ = 9 $$ \times $$ 5 = 45

Now, standard deviation = 0

$$\therefore\,\,\,$$ all the five terms are same i.e.; 9

Now for changed observation

$${\overline x _{new}}$$ = $${{36 + {x_5}} \over 5} = 10$$

$$ \Rightarrow $$ x

$$\therefore\,\,\,$$ $$\sigma $$

= $$\sqrt {{{4{{\left( {9 - 10} \right)}^2} + {{\left( {14 - 10} \right)}^2}} \over 5}} $$ = 2

3

5 students of a class have an average height 150 cm and variance 18 cm^{2}. A new student, whose height is 156 cm, joined them. The variance (in cm^{2}) of the height of these six students is :

A

16

B

22

C

20

D

18

Average height of 5 students,

$$\overline x = {{{x_1} + {x_2} + {x_3} + {x_4} + {x_5}} \over 5} = 150$$

$$ \Rightarrow \,\,\,\sum\limits_{i = 1}^5 {{x_i}} = 750$$

We know,

Variance $$\left( \sigma \right) = {{\sum {x_i^2} } \over 5} - {\left( {\overline x } \right)^2}$$

given that,

$${{\sum {x_i^2} } \over 5} - {\left( {150} \right)^2} = 18$$

$$ \Rightarrow \,\,\,\sum {x_i^2} = 112590$$

Height of new student, x_{6} $$=$$ 156 cm

New average height $$\left( {{{\overline x }_{new}}} \right) = {{750 + 156} \over 6} = 151$$

New variance $$ = {{\,\sum\limits_{i = 1}^6 {x_i^2} } \over 6} - {\left( {{{\overline x }_{new}}} \right)^2}$$

$$ = {{112590 + {{\left( {156} \right)}^2}} \over 6} - {\left( {151} \right)^2}$$

$$ = 22821 - 22801$$

$$ = 20$$

$$\overline x = {{{x_1} + {x_2} + {x_3} + {x_4} + {x_5}} \over 5} = 150$$

$$ \Rightarrow \,\,\,\sum\limits_{i = 1}^5 {{x_i}} = 750$$

We know,

Variance $$\left( \sigma \right) = {{\sum {x_i^2} } \over 5} - {\left( {\overline x } \right)^2}$$

given that,

$${{\sum {x_i^2} } \over 5} - {\left( {150} \right)^2} = 18$$

$$ \Rightarrow \,\,\,\sum {x_i^2} = 112590$$

Height of new student, x

New average height $$\left( {{{\overline x }_{new}}} \right) = {{750 + 156} \over 6} = 151$$

New variance $$ = {{\,\sum\limits_{i = 1}^6 {x_i^2} } \over 6} - {\left( {{{\overline x }_{new}}} \right)^2}$$

$$ = {{112590 + {{\left( {156} \right)}^2}} \over 6} - {\left( {151} \right)^2}$$

$$ = 22821 - 22801$$

$$ = 20$$

4

A data consists of n observations : x_{1}, x_{2}, . . . . . . ., x_{n}.

If $$\sum\limits_{i = 1}^n {{{\left( {{x_i} + 1} \right)}^2}} = 9n$$ and

$$\sum\limits_{i = 1}^n {{{\left( {{x_i} - 1} \right)}^2}} = 5n,$$

then the standard deviation of this data is :

If $$\sum\limits_{i = 1}^n {{{\left( {{x_i} + 1} \right)}^2}} = 9n$$ and

$$\sum\limits_{i = 1}^n {{{\left( {{x_i} - 1} \right)}^2}} = 5n,$$

then the standard deviation of this data is :

A

2

B

$$\sqrt 5 $$

C

5

D

$$\sqrt 7 $$

$$\sum\limits_{i = 1}^n {{{\left( {{x_i} + 1} \right)}^2}} = 9n $$

$$\Rightarrow \sum\limits_{i = 1}^n {x_i^2} + 2\sum\limits_{i = 1}^n {{x_i}} + n = 9n\,\,\,\,\,...\,(1)$$

$$\sum\limits_{i = 1}^n {{{\left( {{x_i} - 1} \right)}^2}} = 5n $$

$$\Rightarrow \sum\limits_{i = 1}^n {x_i^2} - 2\sum\limits_{i = 1}^n {{x_i}} + n = 5n\,\,\,\,\,...\,(2)$$

Performing (1) + (2), we get

$$2\sum\limits_{i = 1}^n {x_i^2} + 2n = 14n$$

$$\sum\limits_{i = 1}^n {x_i^2} = 6n$$

Performing (1) $$-$$ (2), we get

$$ \Rightarrow 4\sum\limits_{i = 1}^n {{x_i}} = 4n$$

$$ \Rightarrow $$$$ \Rightarrow \sum\limits_{i = 1}^n {{x_i}} = n$$

S.D($$\sigma $$)$$ = \sqrt {{{\sum {x_i^2} } \over n} - {{\left( {\overline x } \right)}^2}} $$

$$\sigma $$ $$ = \sqrt {{{6n} \over n} - \left( 1 \right)} $$

$$\sigma $$ $$ = \sqrt 5 $$

$$\Rightarrow \sum\limits_{i = 1}^n {x_i^2} + 2\sum\limits_{i = 1}^n {{x_i}} + n = 9n\,\,\,\,\,...\,(1)$$

$$\sum\limits_{i = 1}^n {{{\left( {{x_i} - 1} \right)}^2}} = 5n $$

$$\Rightarrow \sum\limits_{i = 1}^n {x_i^2} - 2\sum\limits_{i = 1}^n {{x_i}} + n = 5n\,\,\,\,\,...\,(2)$$

Performing (1) + (2), we get

$$2\sum\limits_{i = 1}^n {x_i^2} + 2n = 14n$$

$$\sum\limits_{i = 1}^n {x_i^2} = 6n$$

Performing (1) $$-$$ (2), we get

$$ \Rightarrow 4\sum\limits_{i = 1}^n {{x_i}} = 4n$$

$$ \Rightarrow $$$$ \Rightarrow \sum\limits_{i = 1}^n {{x_i}} = n$$

S.D($$\sigma $$)$$ = \sqrt {{{\sum {x_i^2} } \over n} - {{\left( {\overline x } \right)}^2}} $$

$$\sigma $$ $$ = \sqrt {{{6n} \over n} - \left( 1 \right)} $$

$$\sigma $$ $$ = \sqrt 5 $$

Number in Brackets after Paper Name Indicates No of Questions

AIEEE 2002 (1) *keyboard_arrow_right*

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JEE Main 2017 (Online) 8th April Morning Slot (1) *keyboard_arrow_right*

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JEE Main 2018 (Offline) (1) *keyboard_arrow_right*

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JEE Main 2019 (Online) 12th April Morning Slot (2) *keyboard_arrow_right*

JEE Main 2020 (Online) 8th January Morning Slot (1) *keyboard_arrow_right*

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JEE Main 2021 (Online) 16th March Morning Shift (1) *keyboard_arrow_right*

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Trigonometric Functions & Equations *keyboard_arrow_right*

Properties of Triangle *keyboard_arrow_right*

Inverse Trigonometric Functions *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*

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*

Straight Lines and Pair of Straight Lines *keyboard_arrow_right*

Circle *keyboard_arrow_right*

Conic Sections *keyboard_arrow_right*