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

The number of distinct real roots of the equation 3x^{4} + 4x^{3} $$-$$ 12x^{2} + 4 = 0 is _____________.

Your Input ________

Correct Answer is **4**

3x^{4} + 4x^{3} $$-$$ 12x^{2} + 4 = 0

So, let f(x) = 3x^{4} + 4x^{3} $$-$$ 12x^{2} + 4

$$\therefore$$ f'(x) = 12x(x^{2} + x $$-$$ 2)

= 12x (x + 2) (x $$-$$ 1)

$$ \therefore $$ f'(x) = 12x^{3} + 12x^{2} – 24x = 12x(x + 2) (x – 1)

Points of extrema are at x = 0, –2, 1

f(0) = 4

f(–2) = –28

f(1) = –1

So, 4 Real Roots

So, let f(x) = 3x

$$\therefore$$ f'(x) = 12x(x

= 12x (x + 2) (x $$-$$ 1)

$$ \therefore $$ f'(x) = 12x

Points of extrema are at x = 0, –2, 1

f(0) = 4

f(–2) = –28

f(1) = –1

So, 4 Real Roots

2

A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is k (meter), then $$\left( {{4 \over \pi } + 1} \right)k$$ is equal to _____________.

Your Input ________

Correct Answer is **36**

Let x + y = 36

x is perimeter of square and y is perimeter of circle side of square = x/4

radius of circle = $${y \over {2\pi }}$$

Sum Areas = $${\left( {{x \over 4}} \right)^2} + \pi {\left( {{y \over {2\pi }}} \right)^2}$$

$$ = {{{x^2}} \over {16}} + {{{{(36 - x)}^2}} \over {4\pi }}$$

For min Area :

$$x = {{144} \over {\pi + 4}}$$

$$\Rightarrow$$ Radius = y = 36 $$-$$ $${{144} \over {\pi + 4}}$$

$$\Rightarrow$$ k = $${{36\pi } \over {\pi + 4}}$$

$$\left( {{4 \over \pi } + 1} \right)k$$ = 36

x is perimeter of square and y is perimeter of circle side of square = x/4

radius of circle = $${y \over {2\pi }}$$

Sum Areas = $${\left( {{x \over 4}} \right)^2} + \pi {\left( {{y \over {2\pi }}} \right)^2}$$

$$ = {{{x^2}} \over {16}} + {{{{(36 - x)}^2}} \over {4\pi }}$$

For min Area :

$$x = {{144} \over {\pi + 4}}$$

$$\Rightarrow$$ Radius = y = 36 $$-$$ $${{144} \over {\pi + 4}}$$

$$\Rightarrow$$ k = $${{36\pi } \over {\pi + 4}}$$

$$\left( {{4 \over \pi } + 1} \right)k$$ = 36

3

Let f : [$$-$$1, 1] $$ \to $$ R be defined as f(x) = ax^{2} + bx + c for all x$$\in$$[$$-$$1, 1], where a, b, c$$\in$$R such that f($$-$$1) = 2, f'($$-$$1) = 1 for x$$\in$$($$-$$1, 1) the maximum value of f ''(x) is $${{1 \over 2}}$$. If f(x) $$ \le $$ $$\alpha$$, x$$\in$$[$$-$$1, 1], then the least value of $$\alpha$$ is equal to _________.

Your Input ________

Correct Answer is **5**

$$f(x) = a{x^2} + bx + c$$

$$f'(x) = 2ax + b,$$

$$f''(x) = 2a$$

Given, $$f''( - 1) = {1 \over 2}$$

$$ \Rightarrow a = {1 \over 4}$$

$$f'( - 1) = 1 \Rightarrow b - 2a = 1$$

$$ \Rightarrow b = {3 \over 2}$$

$$f( - 1) = a - b + c = 2$$

$$ \Rightarrow c = {{13} \over 4}$$

Now, $$f(x) = {1 \over 4}({x^2} + 6x + 13),x \in [ - 1,1]$$

$$f'(x) = {1 \over 4}(2x + 6) = 0$$

$$ \Rightarrow x = - 3 \notin [ - 1,1]$$

$$f(1) = 5,f( - 1) = 2$$

$$f(x) \le 5$$

So, $$\alpha$$_{minimum} = 5

$$f'(x) = 2ax + b,$$

$$f''(x) = 2a$$

Given, $$f''( - 1) = {1 \over 2}$$

$$ \Rightarrow a = {1 \over 4}$$

$$f'( - 1) = 1 \Rightarrow b - 2a = 1$$

$$ \Rightarrow b = {3 \over 2}$$

$$f( - 1) = a - b + c = 2$$

$$ \Rightarrow c = {{13} \over 4}$$

Now, $$f(x) = {1 \over 4}({x^2} + 6x + 13),x \in [ - 1,1]$$

$$f'(x) = {1 \over 4}(2x + 6) = 0$$

$$ \Rightarrow x = - 3 \notin [ - 1,1]$$

$$f(1) = 5,f( - 1) = 2$$

$$f(x) \le 5$$

So, $$\alpha$$

4

Let a be an integer such that all the real roots of the polynomial

2x^{5} + 5x^{4} + 10x^{3} + 10x^{2} + 10x + 10 lie in the interval (a, a + 1). Then, |a| is equal to ___________.

2x

Your Input ________

Correct Answer is **2**

Let, $$f(x) = 2{x^5} + 5{x^4} + 10{x^3} + 10{x^2} + 10x + 10$$

$$ \Rightarrow f'(x) = 10({x^4} + 2{x^3} + 3{x^2} + 2x + 1)$$

$$ = 10\left( {{x^2} + {1 \over {{x^2}}} + 2\left( {x + {1 \over x}} \right) + 3} \right)$$

$$ = 10\left( {{{\left( {x + {1 \over x}} \right)}^2} + 2\left( {x + {1 \over x}} \right) + 1} \right)$$

$$ = 10{\left( {\left( {x + {1 \over x}} \right) + 1} \right)^2} > 0;\forall x \in R$$

$$ \therefore $$ f(x) is strictly increasing function. Since, it is an odd degree polynomial it will have exactly one real root.

Now, by observation.

$$f( - 1) = 3 > 0$$

$$f( - 2) = - 64 + 80 - 80 + 40 - 20 + 10$$

$$ = - 34 < 0$$

$$ \Rightarrow f(x)$$ has at least one root in $$( - 2, - 1) \equiv (a,a + 1)$$

$$ \Rightarrow a = - 2$$

$$ \Rightarrow $$ |a| = - 2

$$ \Rightarrow f'(x) = 10({x^4} + 2{x^3} + 3{x^2} + 2x + 1)$$

$$ = 10\left( {{x^2} + {1 \over {{x^2}}} + 2\left( {x + {1 \over x}} \right) + 3} \right)$$

$$ = 10\left( {{{\left( {x + {1 \over x}} \right)}^2} + 2\left( {x + {1 \over x}} \right) + 1} \right)$$

$$ = 10{\left( {\left( {x + {1 \over x}} \right) + 1} \right)^2} > 0;\forall x \in R$$

$$ \therefore $$ f(x) is strictly increasing function. Since, it is an odd degree polynomial it will have exactly one real root.

Now, by observation.

$$f( - 1) = 3 > 0$$

$$f( - 2) = - 64 + 80 - 80 + 40 - 20 + 10$$

$$ = - 34 < 0$$

$$ \Rightarrow f(x)$$ has at least one root in $$( - 2, - 1) \equiv (a,a + 1)$$

$$ \Rightarrow a = - 2$$

$$ \Rightarrow $$ |a| = - 2

Number in Brackets after Paper Name Indicates No of Questions

JEE Main 2021 (Online) 27th August Morning Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 26th August Morning Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 17th March Evening Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 26th February Evening Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 25th February Evening Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 24th February Morning Shift (1) *keyboard_arrow_right*

JEE Main 2020 (Online) 6th September Morning Slot (1) *keyboard_arrow_right*

JEE Main 2020 (Online) 5th September Evening Slot (1) *keyboard_arrow_right*

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

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*