Class 12th

$\frac{1}{2lo{g}_{a}10}+\frac{1}{2lo{g}_{b}10}+\frac{1}{2lo{g}_{c}10}+\frac{1}{2lo{g}_{d}10}$

$=\frac{lo{g}_{10}a}{2}+\frac{lo{g}_{10}b}{2}+\frac{lo{g}_{10}c}{2}+\frac{lo{g}_{10}d}{2}$

$=\frac{lo{g}_{10}a+lo{g}_{10}b+lo{g}_{10}c+lo{g}_{10}d}{2}$

$=\frac{lo{g}_{10}abcd}{2}$

$=\frac{lo{g}_{10}10^8}{2}$

=8 * ½ = 4

x (x – 6) > 2x – 12

x² – 6x > 2x – 12

x² – 8x > 12x > 0

(x – 2) (x – 6) > 0

x < 2 or x >6

The work function in electron emission is the minimum amount of energy needed for removing an electron from the surface of a material. This function is denoted as? , and it is measured in electron volts. An electron requires at least the work function as energy to escape the surface of a solid material into the vacuum. This represents the difference between energy of an electron at rest in vacuum and fermi level of the material. Work function varies from material to material.

Given equation is

x² – 25 <5

–5 < x² – 25 < 5

20 < x² < 30

$\sqrt[]{20}$ < x < $\sqrt[]{30}$

Hence x = 5

x = 5

Sum of square of even natural number n = 2² + 4² + 6² + ……………. +n²

= 2² [1² + 2²+ 3²+ ……………+ (n/2)²]

$4.\frac{(n/2)\left(\frac{n}{2}+1\right)\left(\frac{2n}{2}+1\right)}{6}=51*n$

(n+1) (n+2) = 51*6

n² + 3n +2 = 306

(n–16) (n+19) = 0

n=16

CP of wheat be Rs.100/kg So, when shopkeeper pays for 1kg wheat, he get 1.1kg of wheat.

CP of shopkeeper is Rs. $\frac{100}{1.1}perkg$

While selling, he sells 10% less

So, 5p = Rs. $\frac{100}{0.9kg}$

Profit percent = $\frac{\frac{100}{0.9}-\frac{100}{1.1}}{\frac{100}{1.1}}$

=22.22percent

Total volume = 2800 cubic feet

Area of ceiling = 2800/28 = 100sqfeet

Total time taken = $\frac{100}{0.125}$ = 800min

= 13hr 20min

Time taken to paint four walls= 40 – 13* $\frac{1}{3}$ = 26hr 40min

Let us find the pattern

10⁴ – 21 = 9979

10⁶ – 21 = 999979

10⁸ – 21 = 99999979

.         .         .

.         .         .

.         .         .

10ⁿ – 21 = 999 (n–2times) 79

When, n = 78

10⁷⁸ – 21 = 999 (76times) 79

Sun of all digit = 77 * 9 + 7

= 693 + 7 = 700

Let the CP of apple be p

Then, px + (p + 4)y – 8 = p * 3x + (p+4) (y–6)

px +py +4y = 3px + py – 6p + 4y – 24 + 8

2px – 6p = 16

p (x–3) = 8

The maximum value of p is 8 at x = 4

Then, CP of an apple and mango is 8 + 12 = 20 Rs.