X maro G Strategy | Inequalities | Easy
For real values of x, the range of the function f(x) = (2x – 3)/(2x² + 4x – 6)
1.(- oo , 1/8] U [1 , oo)
2.(-oo , 1/4] U [1/2 , oo)
3.(-oo , 1/8] U [1/2 , oo)
4.(-oo , 1/4] U [1 , oo)
Solution & Explanation
Correct Answer: 3
We can factor the denominator:
2x² + 4x – 6 = 2(x + 3)(x – 1)
So, the function becomes:
f(x) = (2x – 3) / (2(x + 3)(x – 1)) , thus x ≠ -3 , 1
Range of f(x)
f(0) = 1/2
f(1) = not possible
f(2) = 1/10
f(3) = 1/8
Only option 3 includes f(0) and f(3)
Thus option 3 is the answer
X Maro G Strategy | Surds and Indices | Moderate
If 12¹²x × 4²⁴x + ¹² × 5²y = 8⁴z × 20¹²x × 2433x-6 , where x, y and z are natural numbers, then x+y+z equals_____
Solution & Explanation
12¹²x × 4²⁴x + ¹² × 5²y = 8⁴z × 20¹²x × 2433x-6
12 = 2² × 3 , 4 = 2² , 8 = 2³ , 20 = 2² × 5 , 243 = 3⁵
Thus expression can be written as :
224x × 312x × 248x+24 × 52y = 212z × 2²⁴x × 512x × 315x – 30
Cancels out the same powers and comparing the powers
12x = 15x – 30 , x = 10
2y = 12x , 2y = 12(10) , y = 60
12 z = 48x + 24 , 12z = 504 , z = 42
x + y + z = 10 + 60 + 42 = 112
Answer 112
