Mathematics Test - 3

Solution

m[ -3 4 ] + n[ 4 -3 ] = [ -3m 4m ] + [ 4n -3n ]

= [ (-3m)+4n 4m+(-3n) ] = [ 10 -11 ]

• we can compare both sides,

-3m + 4n = 10 .... (1)

And 4m - 3n = -11 .... (2)

• Solving both equations

multiply equation (1) by 4 and equation (2) by 3

-12m + 14n = 40

and 12m - 9n = -33

• Add equation (1) and (2)

-12m + 14n + 12m - 9n = 40 - 33

5n = 7

n = 7/5

• Substitute value of n in equation (1)

-3m + 4×(7/5) = 10

-3m + 28/5 = 10

-15m + 28 = 50

-15m = 50 - 28

m = -22/15

• Substitute value of m and n in 3m + 7n

3(-22/15) + 7(7/5) = -22/5 + 49/5

= 27/5

• Hence, value of 3m + 7n is 27/5

Solution

Between (n + 1) white balls there are (n + 2) gaps in which (n + 1) black ball are to arranged.

No. of reqd arrangements = (n + 1)! (n + 1)! = [(n + 1)!]²

Now between (n + 1) black balls (n + 1) white balls are to be filled no. of ways

= (n + 1)! (n + 1)!

∴ Reqd ways = 2[(n + 1)!]²

Solution

Let's denote the height of the pole as h and the horizontal distance from the bottom of the pole to the point where the rope touches the ground as b. We can use trigonometry in the right-angled triangle formed by the pole, the ground, and the rope.

The given angle is 30°, so we can use the sine and cosine functions to find the height (h) and base (b) of the triangle.

sin(30°) = h/5

h = 5 * sin(30°)

h = 5 * 1/2

h = 5/2 m

So, the height of the pole is 5/2 m. The correct answer is A. 5/2m.