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Test: Basic Concepts Of Differential And Integral Calculus- 1 - GRE MCQ


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30 Questions MCQ Test - Test: Basic Concepts Of Differential And Integral Calculus- 1

Test: Basic Concepts Of Differential And Integral Calculus- 1 for GRE 2024 is part of GRE preparation. The Test: Basic Concepts Of Differential And Integral Calculus- 1 questions and answers have been prepared according to the GRE exam syllabus.The Test: Basic Concepts Of Differential And Integral Calculus- 1 MCQs are made for GRE 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Basic Concepts Of Differential And Integral Calculus- 1 below.
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Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 1

Choose the most appropriate option (a) (b) (c) or (d)

The gradient of the curve y = 2x3 –3x2 – 12x +8 at x = 0 is

Detailed Solution for Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 1

ANSWER :- a

Solution :- 2x^3 - 3x^2 -12x + 8 = 0 

dy/dx = 6x^2 - 6x -12 = 0

(At x=0) = 6(0)^2 - 6(0) - 12 

        = -12

Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 2

The gradient of the curve y = 2x3 –5x2 – 3x at x = 0 is

Detailed Solution for Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 2

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Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 3

The derivative of y = 

Detailed Solution for Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 3

we know that y = √(x+1) = (x+1)1/2

dy/dx = 1/2 . (x+1)-1/2 = 1 / 2√(x+1)
hence option c is correct

Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 4

Detailed Solution for Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 4

df/dx = * d/dx(ax2 + bx + c)

Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 5

Detailed Solution for Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 5
  • Hence, option A is correct.

 

Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 6

If y = x (x –1) (x – 2) then  is

Detailed Solution for Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 6
  • firstly multiply all three equations 
  • we get , y = x (x2 - 3x + 2)
  • y = x3 - 3x2 + 2x 
  • dy/dx = 3x2 - 6x + 2
Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 7

The gradient of the curve y – xy + 2px + 3qy = 0 at the point (3, 2 ) is and q are

Detailed Solution for Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 7



Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 8

The curve y2 = ux3 + v passes through the point P(2, 3) and  = 4 at P. The values of u and v are 

Detailed Solution for Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 8

Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 9

The gradient of the curve y + px +qy = 0 at (1, 1) is 1/2. The values of p and q are

Detailed Solution for Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 9

y+px+qy=0
y+qy=-px
(1+q)y=-px
y=-[p/(1+q)]x
This is a linear function, so has constant gradient at all points on the curve. Hence
-p/(1+q)=1/2
2p=-(1+q)
But there is an issue: you have stated that the curve's gradient is 1/2 at the point (1,1) but the curve does not cross through this point! Regardless of our choices for p and q satisfying the expressions above this paragraph, the equation of the curve will always simplify to y=0.5x, which crosses through the origin (0,0), as well as (1,0.5) and (2,1) - but not (-1,1).
For your curve to pass through (-1,1), we would need to add a constant term, like so:
y+px+qy=1/2
 

Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 10

If xy = 1 then y2 + dy/dx is equal to

Detailed Solution for Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 10
  • xy = 1
  • y = 1/x
  • y2 = 1/x2
  • dy/dx = -1/ x2 
  • now , y2 + dy/dx
  • 1/x- -1/ x2 = 0
Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 11

The derivative of the function 

Detailed Solution for Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 11

y = 1 / 2√(x+√x). (1+1/2√x)
Hence, option c is correct
 

Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 12

Given e-xy –4xy = 0,  can be proved to be

Detailed Solution for Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 12

Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 13

Detailed Solution for Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 13

Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 14

If log (x / y) = x + y,   may be found to be

Detailed Solution for Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 14


Therefore, Option A 

Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 15

If f(x, y) = x3 + y3 – 3axy = 0,  can be found out as

Detailed Solution for Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 15

 if we take minus common , dy/dx = (ay - x2 )/ (y2 - ax)

Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 16

Given x = at2, y = 2at; is calculated as

Detailed Solution for Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 16

Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 17

Given x = 2t + 5, y = t2 – 2;  is calculated as 

Detailed Solution for Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 17

x = 2t + 5
dx/dt = 2
dy/dx = dy/dt / dx/dt
= 2t / 2 = t

Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 18

If y = 

Detailed Solution for Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 18
  • as we know, 
  • y = x-1/2 
  • dy/dx = -1/2. x-1/2-1
  • therefore, the correct and is -1/2(x-3/2) which is equal to option c as 
  • -1/2. 1/(x√x)
Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 19

If x = 3t2 –1, y = t3 –t, then   is equal to

Detailed Solution for Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 19
  • dx / dt = 6t
  • dy/dx = dy/dt / dx/dt = (3t2 - 1)/6t
  • option A is correct.
Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 20

The slope of the tangent to the curve y = at the point, where the ordinate and the abscissa are equal, is

Detailed Solution for Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 20

so the point is (√2,√2)

= -1

Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 21

The slope of the tangent to the curve y = x2 –x at the point, where the line y = 2 cuts the curve in the Ist quadrant, is

Detailed Solution for Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 21

Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 22

For the curve x2 + y2 + 2gx + 2hy = 0, the value of  at (0, 0) is

Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 23

 

Detailed Solution for Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 23

hence , the ans will be none of these

Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 24

If xy.yx = M, M is constant then is equal to

Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 25

Given x = t + t–1 and y = t – t–1 the value of  at t = 2 is

Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 26

If x3 –2x2 y2 + 5x +y –5 =0 then  at x = 1, y = 1 is equal to

Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 27

The derivative of x2 log x is

Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 28

The derivative of 

Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 29

Test: Basic Concepts Of Differential And Integral Calculus- 1 - Question 30

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