GATE Questions

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Let the two-digit number be 10a + b
a + b = 12 --- (1)
If a>b, a - b = 6
If b>a, b - a = 6
If a - b = 6, adding it to equation (1), we get
2a = 18 => a =9
so b = 12 - a = 3
Number would be 93.
if b - a = 6, adding it to the equation (1), we get
2b = 18 => b = 9
a = 12 - b = 3.
Number would be 39.
There fore, Number would be 39 or 93.

All of us, believe that the Solar System evolved from a giant cloud of dust and gas. They believe that this dust and gas began to collapse under the weight of its own gravity. As it did so, the matter contained within this could begin moving in a giant circle, much like the water in a drain moves around the center of the drain in a circle.

At the center of this spinning cloud, a small star began to form. This star grew larger and larger as it collected more and more of the dust and gas that collapsed into it.

Further away from the center of this mass where the star was forming, there were smaller clumps of dust and gas that were also collapsing. The star in the center eventually ignited forming our Sun, while the smaller clumps became the planets, minor planets, moons, comets, and asteroids.

Let the actual distance be t kms

Now, According to the data,

$$\begin{gathered} 5t + 860 = 6t \\ => 560t + 860 = 6t \\ => 60t + 8 = 72t \\ => 8 = 12t \\ => t = 8/12 = 2/3 hrs \end{gathered}$$

Now, the total distance = 6t = 6 x 2/3 = 2 x 2 = 4 kms.

Time taken by both Meghana and Ganesh to work together is given by =

t1 x t2

Where

$$\begin{gathered} t1 = 32 hrs \\ t2 = 12 12 hrs = 252 hrs \end{gathered}$$

Therefore, time took by both to work together = 

32 x 252 = 16 x 25 = 4 x 5 = 20 hrs

The pattern in the given series is +4, +9, +16, +25, ... i.e. 22,32,42,52...

The next number is 284+25 = 309.

Let the five consecutive odd numbers be x-4, x-2, x, x+2, x+4

According to the question,

Difference between square of the average of first two odd number and the of the average last

two odd numbers is 396

i.e, x+3 and x-3

$$\begin{gathered} ⇒x + 32 - x - 32 = 396 \\ ⇒ x2 + 9 + 6x - x2 + 6x - 9 = 396 \\ ⇒ 12x = 396 \\ ⇒ x = 33 \end{gathered}$$

Hence, the smallest odd number is 33 - 4 = 29.

Given length of the rectangle = 3 cm

Breadth of the rectangle = 4 cm

Then, the diagonal of the rectangle D = 32 + 42 = 25 = 5

Then, it implies side of square = 5 cm

We know that Area of square = S x S = 5 x 5 = 25 sq.cm.

Number of words with 5 letters from given 9 alphabets formed = 95

Number of words with 5 letters from given 9 alphabets formed such that no letter is repeated is = 9P5

Number of words can be formed which have at least one letter repeated =  95 - 9P5

= 59049 - 15120

= 43929