Problem 18: Solve the differential equation

(IC/M)

Problem 17: Determine the vaue of n.

(IC/M)

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CONDITION OF LINE BEING NORMAL TO ELLIPSE

What is the condition for line y = mx + c to normal a standard ellipse ? (asked by VANI RAVIJA)

PROBLEM 16: FIND THE VALUE a+b+c+d

(ALG/E) 

PROBLEM: 15 WHICH OF THE EXPRESSION EXISTS

( V&3D/E)

SOLUTION OF PROBLEM: 14

(IC/M)

SOLUTION OF PROBLEM: 13

(DC/CH) 

SOLUTION OF PROBLEM: 12

Find the values of a, b, c and d for the given limit.  (DC/M)
SOLUTION:

SOLUTION OF PROBLEM: 11

Plot the curve of y = |tan(x)|   (DC/M)
SOLUTION : We already know how to plot the curve of tan(x). |tanx| will always yield positive values, which means wherever tan(x) is negative we have to make it positive in order get |tan(x)|  curve.

Therefore we have to invert the negative strip of tan(x) curve.

IIT PROBLEMS ON VIDEOS

PROBLEM: 14 FIND THE VALUE OF DEFINITE INTEGRAL

                                                                                             (IC/M)

PROBLEM 13: FIND THE VALUE OF LIMIT

                                                                                           (DC/CH)

SOLUTION OF PROBLEM: 10

Let function f is from R to R be defined by f(x) =2x + sin(x) for x belongs to R. Then f is

a) one-to-one and onto

b) one-to-one but not onto.

c) onto but not one-to-one

d) neither one-to-one nor onto.      ( DC/CH)

SOLUTION: The given function has co-domain as R( set of all real numbers)

For the function f(x) = 2x + sin(x) , 2x ranges from negative infinity to positive infinity whereas sin(x) varies from [-1,1] so ultimately the f(x) will have its range as R. Therefore, the f(x) is Onto function.

Also, f'(x) = 2 + cos(x) , cos(x) will vary from [-1,1], So 2+cos(x) varies from [1,3].

Therefore, f'(x) is always positive. Hence f(x) will always be an increasing function.Therefore, f(x) will be one-to-one.

So, the correct option is (a)

SOLUTION OF PROBLEM: 9

Find the area bounded by the curve y = e(x) and the lines x = 0 and y = e. ( IC/E)
SOLUTION: 

NOTE: e(x) is e to the power x

SOLUTION OF PROBLEM: 8

Prove that the f(x) = 2(x)cube + 3(x)sq+6x +10 will always be an increasing function. (DC/M)

Solution: If we differentiate the given function we will get f'(x) = 6(x)sq + 6x +6 = 6( (x)sq +x +1)

Now, we can write f'(x) = 6[(x + 1/2)sq +3/4] which is always positive.So f(x) will always be an increasing function.

Alternatively, we can also say the for the quadratic expression (x)sq +x +1 coefficient of (x)sq is positive and discriminant is negative (-3), so the expression is always positive