A massless string connects two pulley of masses ' ' and '
' respectively as shown in the figure.
The heavier pulley is fixed and free to rotate about its central axis while the other is free to rotate as well as translate. Find the acceleration of the lower pulley if the system was released from the rest. [Given, ]
Not understanding sir
View Full Answer(2)Calculate the acceleration of block of the following diagram. Assume all surfaces are frictionless . Here m1 = 100kg and m2 = 50kg
0.33m/s2
0.66m/s2
1m/s2
1.32m/s2
1.32
View Full Answer(4)No. of transition state in given figure
1
2
3
4
2
View Full Answer(2)
When cell has stalled DNA replication fork, which checkpoint should be predominantly activated?
G1/S
G2/M
M
Both G2 M and M
G2/M should be activated as the cell has stalled DNA replication fork.
View Full Answer(1)Study 40% syllabus and score up to 100% marks in JEE
Given : and
Then the area (in sq. units) of the region bounded b the curves,
and
between the lines,
is :
Option: 1
Option: 2
Option: 3
Option: 4
Area Bounded by Curves When Intersects at More Than One Point -
Area bounded by the curves y = f(x), y = g(x) and intersect each other in the interval [a, b]
First find the point of intersection of these curves y = f(x) and y = g(x) , let the point of intersection be x = c
Area of the shaded region
When two curves intersects more than one point
rea bounded by the curves y=f(x), y=g(x) and intersect each other at three points at x = a, x = b amd x = c.
To find the point of intersection, solve f(x) = g(x).
For x ∈ (a, c), f(x) > g(x) and for x ∈ (c, b), g(x) > f(x).
Area bounded by curves,
-
Required area = Area of trapezium ABCD -
Correct Option (1)
View Full Answer(1)Let a function be continuous,
and F be defined as :
, where
Then for the function F, the point
is :
Option: 1 a point of infection.
Option: 2 a point of local maxima.
Option: 3 a point of local minima.
Option: 4 not a critical point.
Integration as Reverse Process of Differentiation -
Integration is the reverse process of differentiation. In integration, we find the function whose differential coefficient is given.
For example,
In the above example, the function cos x is the derived function of sin x. We say that sin x is an anti derivative (or an integral) of cos x. Similarly, x2 and ex are the anti derivatives (or integrals) of 2x and ex respectively.
Also note that the derivative of a constant (C) is zero. So we can write the above examples as:
Thus, anti derivatives (or integrals) of the above functions are not unique. Actually, there exist infinitely many anti derivatives of each of these functions which can be obtained by selecting C arbitrarily from the set of real numbers.
For this reason C is referred to as arbitrary constant. In fact, C is the parameter by varying which one gets different anti derivatives (or integrals) of the given function.
If the function F(x) is an antiderivative of f(x), then the expression F(x) + C is the indefinite integral of the function f(x) and is denoted by the symbol ∫ f(x) dx.
By definition,
-
Maxima and Minima of a Function -
Maxima and Minima of a Function
Let y = f(x) be a real function defined at x = a. Then the function f(x) is said to have a maximum value at x = a, if f(x) ≤ f(a) ∀ a ≥∈ R.
And also the function f(x) is said to have a minimum value at x = a, if f(x) ≥ f(a) ∀ a ∈ R
Concept of Local Maxima and Local Minima
The function f(x) is said to have a maximum (or we say that f(x) attains a maximum) at a point ‘a’ if the value of f(x) at ‘a’ is greater than its values for all x in a small neighborhood of ‘a’ .
In other words, f(x) has a maximum at x = ‘a’, if f(a + h) ≤ f(a) and f(a - h) ≤ f(a), where h ≥ 0 (very small quantity).
The function f(x) is said to have a minimum (or we say that f(x) attains a minimum) at a point ‘b’ if the value of f(x) at ‘b’ is less than its values for all x in a small neighborhood of ‘b’ .
In other words, f(x) has a maximum at x = ‘b’, if f(a + h) ≥ f(a) and f(a - h) ≥ f(a), where h ≥ 0 (very small quantity).
-
Correct Option (1)
View Full Answer(1)In the expansion of , if
is the least value of the term independent of
when
and
is the least value of the term independent of
when
, then the ratio
is equal to :
Option: 1
Option: 2
Option: 3
Option: 4
General Term of Binomial Expansion
Term independent of x: It means term containing x0,
Now,
Correct option 1
View Full Answer(1)The value of is equal to :
Option: 1
Option: 2
Option: 3
Option: 4
Properties of the Definite Integral (Part 2) - King's Property -
Property 4 (King's Property)
This is one of the most important properties of definite integration.
-
Application of Periodic Properties in Definite Integration -
Property 9
If f(x) is a periodic function with period T, then the area under f(x) for n periods would be n times the area under f(x) for one period, i.e.
-
At 300 K and 1 atm, 15 mL of a gaseous hydrocarbon requires 375 mL air containing 20% O2 by volume for complete combustion. After combustion, the gases occupy 330 mL. Assuming that the water formed is in liquid form and the volumes were measured at the same temperature and pressure, the formula of the hydrocarbon is :
Option: 1 C4H8
Option: 2 C4H10
Option: 3 C3H6
Option: 4 C3H8
Volume of N2 in air = 375 × 0.8 = 300 ml
Volume of O2 in air = 375 × 0.2 = 75 ml
15ml
0 0 15x -
After combustion total volume
330 = 300 + 15x
x = 2
Volume of O2 used
y = 12
So hydrocarbon is = C2H12
None of the options matches it therefore it is a BONUS.
----------------------------------------------------------------------
Alternatively Solution
15ml
0 0 15x -
Volume of O2 used
If further information (i.e., 330 ml) is neglected, option (C3H8 ) only satisfy the above equation.
View Full Answer(1)is equal to
Option: 1
Option: 2
Option: 3
Option: 4
As we learnt
Walli's Method -
Definite integral by first principle
where
- wherein
View Full Answer(1)
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