Consider 5a²- 45a + 100 > 0

∴ a² -9a + 20 > 0

∴ (a-5) (a-4) > 0

This is possible when both (a - 5) and (a - 4) are either positive or negative.

This is true when a > 5 or a < 4. ∴ a > 5 or a < 4

So, except for this range all the positive integers satisfy this inequality.

∴ Col. A is more than Col. B

P is the centroid of the triangle.

∴ P divides the median AD in the ratio 2: 1.

∴ AP = ²⁄₃ × median AD

∴ AP = ²⁄₃ × 6 = 4 = radius of the circle.

Let distance traveled by the man upstream and downstream is d km each.

Speed of man upstream = 6 - 2 = 4 kmph

Speed of man downstream = 6 + 2 = 8 kmph

∴ ^d⁄₄ + ^d⁄₈ = 3 ∴ d = 8 kms.

∴ Total distance = 2d = 16 kms.

Consider the equilateral triangle ABC. Then the line joining the midpoints D and E is parallel to BC and is half the length of BC (Standard property).

∴ Triangle is an equilateral triangle with side exactly half of that of triangle ABC.

Let BC = 2 ∴ DE = 1

Area of triangle ABC = \(\frac{\sqrt{3}}{4}\times 2^{2}=\sqrt{3}\)

Area of triangle ADE = \(\frac{\sqrt{3}}{4}\times 1^{2}=\frac{\sqrt{3}}{4}\)

∴ Area of triangle ABC is 4 times area of triangle ADE.

∴ Col. A = Col. B

The three numbers could be (1, 4, 5) (10,2, 1). In both the cases, the average is more than 3.

Since Paul is faster than Mike, number of days required will be less if Paul starts first Similarly, number of days required will be more if Mike starts working first.

Col. A : \(\left [ -\left \{ 4+\sqrt{1} \right \} \right ]^{\frac{1}{2}}=[-5]^{\frac{1}{2}}=\sqrt{-5}\)

Square root of-5 is a complex number.

Col. B: -2.23 = real number.

Complex and real numbers cannot be compared.

x = angle of reflection for mirror RQ. ∴ x = 47°

Consider right angled triangle STQ.

∴ ∠STQ = 43°

∴ y = 43° (Angle of reflection = angle of incidence)

∴ Col. A = 47°, Co!. B = 43°

Volume of one pebble X number of pebbles = Volume of water level raised.

∴ ⁴⁄₃ × π × (14)³ × N = π x (70)² × 56

∴ N = \(\frac{70\times 70\times 56\times 3}{4\times 14\times 14\times 14}\) ∴ N = 75

Col. A:

The units digit of 12²³ will determine the remainder.

12²³ will have same digit in units place as does 2²³. For number 2, power cycle is 2¹,2²,2³,2⁴.... will end in 2, 4, S, 16 and same sequence again for further powers.

∴ 2²³ will give 3rd term in the units place i.e. 8.

∴ If we divided by 5, the remainder will be 3.

Col B:

For number 3, power cycle is 3,9,7, 1 and repeat.

∴ 3⁷⁵ will give 3rd term which is 7.

∴ If we divide by 5, the remainder will be 2.