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JEE 2011 Question on Logarithm

Let (x0,y0) be the solution of the following equations: (2x)ln2 = (3y)ln3 3lnx = 2lny
Then x0 is
(A)
1 6
(B)
1 3
(C)
1 2
(D) 6
Answer and Comments: (C). We can treat the data as a system of equations in the unknowns lnx and lny instead of x and y. This is permissible because a (positive) real number is uniquely determined by its logarithm. So, taking logarithms, we get
(ln2)2 + (ln2)(lnx) = (ln3)2 + (ln3)(lny) (1) (ln3)(lnx) = (ln2)(lny) (2)
which can be treated as a system of linear equations in the unknowns lnx and lny. Eliminating lny from the two equations gives (ln2)3 + (ln2)2(lnx)−(ln3)2(ln2) = (ln3)2(lnx) (3)
5
which simpliï¬?es to [(ln3)2 −(ln2)2](lnx) =−(ln2)[(ln3)2 −(ln2)2] (4) As (ln3)6= ±(ln2), we cancel the bracketed factor and get lnx = −ln2 (5) which gives x = 1/2. Once the key idea strikes, viz. to take lnx and lny as the variables, the problem is straightforward. The properties of logarithms needed are elementary and standard. Nowhere we have to convert logarithms w.r.t. one base to those w.r.t. another.

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1. ABCD is a parallelogram in which ∠A = 110°. Find the measure of each of the angles ∠B, ∠C and ∠D. 

2. Two adjacent angles of a parallelogram are equal. What is the measure of each of these angles? 

3. Two adjacent angles of a parallelogram are in the ratio 4 : 5. Find the measure of each of its angles. 

4. Two adjacent angles of a parallelogram are (3x - 4)° and (3x + 16)°. Find the value of x and hence find the measure of each of its angles. 

5. The sum of two opposite angles of a parallelogram is 130°. Find the measure of each of its angles. 

6. Two sides of a parallelogram are in the ratio 5 : 3. If its perimeter is 64 cm, find the lengths of its sides.

7. The perimeter of a parallelogram is 140 cm. If one of the sides is longer than the other by 10 cm, find the length of each of its sides.

Question given below:

Solution 1:

Solution 2:

Solution 3:

Solution 4:

Draw a line from D parallel to BE cutting AC at P.

CDP and CBE are similar.

EP/EC = BD/BC = 3:5

Also, AFE and ADP are similar.

AE/EP = AD/FD = 2:1

Multiply both the equation from the top, so AE/EC = 6:5.

As we know in decimal system ten digits are used. I.e 0 - 9. Subtracting a number from base 10, 100, 1000. An easy way to calculate is just subtracting all digits of a digit from 9 and last digit by 10.

For an example, subtracting 56 from 100:

    100 - 56 = (9 - 5) | (10 - 6) = 44

Let's check it using our conventional method, in units place it is 6 there. So carry from tens place, there also 0, so so take carry from hundred places, I.e in units place has value 10 so subtracting 6 from 10 we get 4 and now in tens place there is digit 9. Now subtract 5 from 9 we get 9 - 5 = 4. So our answer is 44. So lengthy using carry.

If we use the first method we have to do just only subtract 9 from all digits and last digit from 10. This is one of the powerful sutra "All From 9 And Last From 10" Of Vedic Mathematics. We also know that complement of 56 is 44. As if we add 56 with 44 we get 100.

Next, try for subtracting 65 from 10000:

10000 - 65 = 10000 - 0065 = 9935.