Surds Trick:
When you see \(\sqrt{A+B} + \sqrt{A-B}\), try squaring the expression to simplify the radical terms.
Probability Rule:
Total outcomes for 4 coins = 16. Valid outcomes for “no consecutive heads” = 8.
Number Puzzles:
Always test vertical and horizontal relationships using squares (\(x^2\)) and cubes (\(x^3\)).
QT & DI | ID: 100020
The value of the following sum of terms:
\(\frac{5}{2^2 \cdot 3^2} + \frac{7}{3^2 \cdot 4^2} + \frac{9}{4^2 \cdot 5^2} + \frac{11}{5^2 \cdot 6^2} + \frac{13}{6^2 \cdot 7^2} + \frac{15}{7^2 \cdot 8^2} =\)
1/64
15/64
15/16
7/64
Correct Option: 2 (15/64)
Rationale: Telescoping series where \(\text{Term} = \frac{1}{n^2} – \frac{1}{(n+1)^2}\).
Sum = \((\frac{1}{4} – \frac{1}{9}) + \dots + (\frac{1}{49} – \frac{1}{64}) = \frac{1}{4} – \frac{1}{64} = \frac{15}{64}\).
QT & DI | ID: 100017
The sum of three consecutive odd numbers is always divisible by:
A. 2 | B. 3 | C. 5 | D. 6
A and B only
B and D only
B only
A and C only
Correct Option: 3 (B only)
Rationale: Sum = \((2n-1) + (2n+1) + (2n+3) = 6n + 3 = 3(2n + 1)\). Always divisible by 3.
QT & DI | ID: 100007
Four coins are tossed simultaneously. What is the probability that two consecutive heads never occur together?
7/8
1/4
1/3
1/2
Correct Option: 4 (1/2)
Rationale: Total outcomes = 16. Favorable (No HH) = {TTTT, HTTT, THTT, TTHT, TTTH, HTHT, THTH, HTTH} = 8. Probability = \(8/16 = 1/2\).
QT & DI | ID: 100009
The value of expression:
$$\frac{\sqrt{\sqrt{5}+2} + \sqrt{\sqrt{5}-2}}{\sqrt{\sqrt{5}+1}}$$
1
\(\sqrt{2}\)
4
\(\sqrt{5}\)
Correct Option: 2 (\(\sqrt{2}\))
Rationale: Let \(N\) be the numerator.
$$N^2 = (\sqrt{5}+2) + (\sqrt{5}-2) + 2\sqrt{(\sqrt{5}+2)(\sqrt{5}-2)}$$
$$N^2 = 2\sqrt{5} + 2\sqrt{5-4} = 2(\sqrt{5}+1)$$
Since the denominator squared is \(D^2 = \sqrt{5}+1\), the expression squared is \(\frac{2(\sqrt{5}+1)}{\sqrt{5}+1} = 2\). Result = \(\sqrt{2}\).
Logical Reasoning | Missing Number Puzzle
Study the given pattern carefully and select the number that can replace the question mark (?) in it.
