GATE Questions

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We know that,

$$\begin{gathered} i2 = -1 \\ i3 = i2 x i = -1 x i = -i \\ i4 = i2 x i2 = -1 x -1 = 1 \\ Hence, i233 =i4 x 58 + 1 = i232 x i1 = 1 x i = i \end{gathered}$$

In geometry, we can define plane as a flat surface with no thickness.

The surface extends with no ends.

A plane does not have any edges even if we draw it on paper with edges, it does't mean it has edges.

Hence, A plane is an undefined term because it is described generally, not using a formal definition.

As given Kapil's 8th birthday was celebrated 2 years ago,

Kapil’s present age = 8 + 2 = 10 years.

Father's age is twice of Kapil’s age after 10 years.
=> F + 10 = 2 (K + 10) = 2 (10+ 10)
F = 30years,

Avanthi's age =1/6(F) = 30/6 = 5 years.

P = Rs. 15225, n = 9 months = 3 quarters, R = 16% p.a. per quarter.

Amount = 15225x1+41003

= (15225 x 26/25 x 26/25 x 26/25) = Rs. 17126.05

=> C.I. = 17126 - 15625 = Rs. 1901.05.

You have to actually count all the balls in the pyramid.
Simple way is to count the balls in each level.

So lower most level lets say 1st level has 4 x 4 balls; i.e. 16 balls
Similarly 2nd level has 3 x 3 balls; i.e. 9 balls.
3rd level has 2 x 2 balls; i.e. 4 balls.
The top most level has just 1 ball.

So the total number of balls is 16+9+4+1 = 30.

So the image has 30 Balls in all.

Length of the thread is 36 cms.
length wise twice breath wise once then totally thrice.
so 36/3 = 12.
so length is 12 x 2 = 24 cms
breath is 12 cms
Area of the book = length x breath = 24 x 12 = 288 sq.cm

Let the length of the train be 'L' mts

let the speed of the train be 'S' m/s

Given it crosses a pole in 10 sec=> L/S = 10 ......(1)
Given it takes 20 sec (double of pole) to cross a platform of length 200 mts

=> (L + 200)/S = 20

=> L/S + 200/S = 20

But from (1) L/S = 10

=> 200/S = 20 - 10

=> S = 20 m/s

Then, from (1)

=> L = 10 x 20 = 200 mts.

Hence, the length of the train = 200 mts.

According to the given data,

P + Q = 41 ......(1)

R - 1 = P + 2

R = P + 3

and

P + 4 = Q - 1

=> Q = P + 5 ....(2)

From (1)&(2)

P = 18

Q = 18 + 5 = 23

R = 18 + 3 = 21

=> P/S = 3/4

=> S = 4/3 x 18 = 24

Required Difference = S - R = 24 - 21 = 3.