RRB NTPC Graduate Level (25th March 2026 Shift 3) — RRB NTPC 2025

• Given: Center of circle is O, External point is A. $OA = 6$ cm. Radius $OB = OC = 3$ cm. Tangents are AB and AC. Find $\angle OBC$.

• Basic Method (Solution):
Step 1: The radius is perpendicular to the tangent at the point of contact. So, $\angle OBA = 90^\circ$ and $\angle OCA = 90^\circ$.
Step 2: In right-angled triangle OBA, we can use trigonometry to find $\angle OAB$.
$\sin(\angle OAB) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{OB}{OA} = \frac{3}{6} = \frac{1}{2}$.
Since $\sin(30^\circ) = 1/2$, we have $\angle OAB = 30^\circ$.
Step 3: In the right triangle OBA, the sum of angles is $180^\circ$. So, $\angle AOB = 180^\circ - 90^\circ - 30^\circ = 60^\circ$.
Step 4: By symmetry (triangles OBA and OCA are congruent), $\angle AOC = 60^\circ$.
Step 5: The total angle $\angle BOC = \angle AOB + \angle AOC = 60^\circ + 60^\circ = 120^\circ$.
Step 6: Now consider the triangle OBC. It is an isosceles triangle because $OB = OC$ (both are radii of the same circle).
Therefore, the angles opposite to these sides are equal: $\angle OBC = \angle OCB$.
Step 7: The sum of angles in $\Delta OBC$ is $180^\circ$.
$\angle OBC + \angle OCB + \angle BOC = 180^\circ$
$2\angle OBC + 120^\circ = 180^\circ$
$2\angle OBC = 60^\circ \implies \angle OBC = 30^\circ$.

• Shortcut Method: Notice $OA=6$ and radius $OB=3$. This is a $30^\circ-60^\circ-90^\circ$ right triangle because the hypotenuse is exactly double the leg. This makes the central half-angle $\angle AOB = 60^\circ$. Total central angle $\angle BOC = 120^\circ$. Since $\Delta OBC$ is isosceles, the base angles are $(180-120)/2 = 30^\circ$.

• Final Answer: $30^\circ$

• Key Fact: Jalaluddin Khalji, the founder of the Khalji dynasty, is historically noted for his mild, tolerant, and conciliatory policies. He actually put forward the progressive view that the state should be based on the willing support of the governed.

• Supporting Detail: Because the large majority of the population in India was Hindu, he explicitly argued that a truly Islamic state could not be established in India and instead advocated for a policy of tolerance to win the goodwill of both the nobility and the masses.

• Why others wrong: Statement B says he believed the state should never be based on willing support, which directly contradicts his actual historical philosophy of governance through appeasement and tolerance.

• Exam Trick: Jalaluddin was the 'soft' Khalji who avoided bloodshed (even forgiving the rebel Malik Chhajju). His nephew, Alauddin Khalji, was the exact opposite (ruthless and harsh).

• Given conditions: Equation is $D \times N \% P + K \# E$. Find D's relation to E.
Operators: $\times$ = Wife, $\%$ = Father, $+$ = Mother, $\#$ = Daughter.

• Deduction process:
Let's decode the chain from left to right:
1) $D \times N$: D is the wife of N. (D is Female, N is Male. Married couple in Gen 1).
2) $N \% P$: N is the father of P. (Since D is N's wife, D is the mother of P).
3) $P + K$: P is the mother of K. (P is Female, Gen 2. K is Gen 3).
4) $K \# E$: K is the daughter of E. (K is Female. Since P is already K's mother, E must be K's father. Thus, P is the wife of E).
Now, we need the relation of D to E.
E's wife is P.
P's mother is D (deduced from step 2).
Therefore, D is the mother of E's wife (Mother-in-law).

• Exam Trick: Trace backwards from E. E is the father of K. P is the mother of K. So E and P are married. P's father is N. N's wife is D. So D is P's mother. Since P is E's wife, D is E's Wife's mother.

• Final Answer: Wife's mother

• Given: Men 1 ($M_1$) = 9, Work 1 ($W_1$) = 117, Time 1 ($H_1$) = 3. Work 2 ($W_2$) = 208, Time 2 ($H_2$) = 1. Find Men 2 ($M_2$).

• Basic Method (Solution):
Step 1: Use the standard Chain Rule formula: $\frac{M_1 \times H_1}{W_1} = \frac{M_2 \times H_2}{W_2}$.
Step 2: Substitute the known values into the formula.
$\frac{9 \times 3}{117} = \frac{M_2 \times 1}{208}$
Step 3: Simplify the left side of the equation.
$9 \times 3 = 27$.
$\frac{27}{117}$. Both are divisible by 9. $27/9 = 3$. $117/9 = 13$.
So, $\frac{3}{13} = \frac{M_2}{208}$.
Step 4: Cross-multiply to solve for $M_2$.
$13 \times M_2 = 3 \times 208$.
$M_2 = \frac{624}{13}$.
Step 5: Perform the division.
$624 \div 13 = 48$.
Therefore, 48 students are needed.

• Shortcut Method: 1 student makes $117 / (9 \times 3) = 117 / 27 = 13/3$ charts per hour. To make 208 charts in 1 hour, we need $208 / (13/3) = (208 \times 3) / 13$. Since $208 / 13 = 16$, the answer is $16 \times 3 = 48$ students.

• Final Answer: 48 students

• Key Fact: The file extension '.wma' stands for 'Windows Media Audio', a proprietary series of audio codecs and their corresponding audio coding formats developed by Microsoft.

• Supporting Detail: It was originally designed to compete with the popular MP3 format, offering better audio compression and quality at lower bitrates, and is natively supported by the Windows operating system and Windows Media Player.

• Why others wrong: WiFi, Wired, and Wireless have absolutely nothing to do with standard file extension naming conventions for audio formats. These are fictitious distractors.

• Exam Trick: WMA = Windows Media Audio. Similarly, WMV = Windows Media Video.

• Given: Two numbers are $x$ and $36$. Their mean proportional is $18$. Find $x$.

• Basic Method (Solution):
Step 1: The mean proportional (or geometric mean) between two numbers $a$ and $b$ is given by the formula $\sqrt{a \times b}$.
Step 2: Let the two numbers be $a = x$ and $b = 36$.
According to the problem, $\sqrt{x \times 36} = 18$.
Step 3: Square both sides to remove the square root.
$x \times 36 = 18^2$
Step 4: Calculate $18^2$.
$18 \times 18 = 324$.
$36x = 324$.
Step 5: Solve for $x$.
$x = \frac{324}{36}$.
We know $36 \times 10 = 360$, so $36 \times 9 = 360 - 36 = 324$. Therefore, $x = 9$.

• Shortcut Method: The formula can also be written as $x/18 = 18/36$. Simplifying the right side gives $1/2$. So $x/18 = 1/2$. Therefore, $x = 18/2 = 9$. This avoids squaring 18 entirely and solves it in 3 seconds.

• Final Answer: 9

• Given conditions: Find the odd letter cluster out of JNT, LPU, WAF, DHM based on alphabetical shift patterns.

• Deduction process:
Let's map the numerical positional values of each letter (A=1...Z=26) and find the differences:
A) JNT: J(10) $\rightarrow$ N(14) $\rightarrow$ T(20). Pattern is: 10 + 4 = 14; 14 + 6 = 20. So, (+4, +6).
B) LPU: L(12) $\rightarrow$ P(16) $\rightarrow$ U(21). Pattern is: 12 + 4 = 16; 16 + 5 = 21. So, (+4, +5).
C) WAF: W(23) $\rightarrow$ A(1) $\rightarrow$ F(6). Pattern is: 23 + 4 = 27 (A); 1 + 5 = 6 (F). So, (+4, +5).
D) DHM: D(4) $\rightarrow$ H(8) $\rightarrow$ M(13). Pattern is: 4 + 4 = 8; 8 + 5 = 13. So, (+4, +5).
Options B, C, and D all follow a uniform logical shift of +4 then +5. Option A shifts by +4 then +6, breaking the pattern.

• Exam Trick: Find the first gap: J-N (4), L-P (4), W-A (4), D-H (4). All same. Find second gap: N-T (6), P-U (5), A-F (5), H-M (5). Option A has a noticeably larger jump at the end. JNT is the odd one out.

• Final Answer: JNT

• Given conditions: Find the pattern: [Letter] $\rightarrow$ [Symbol] $\rightarrow$ [Number].

• Deduction process:
Let's scan the series, locating every symbol (&, %, @, &, #, *) and checking their immediate left and right neighbors:
Series: G & R E 5 K % 7 @ 2 S & 6 C T # 3 4 Y * D
1) & : Preceded by G (Letter), followed by R (Letter). (Fails)
2) % : Preceded by K (Letter), followed by 7 (Number). MATCH! (1)
3) @ : Preceded by 7 (Number). (Fails)
4) & : Preceded by S (Letter), followed by 6 (Number). MATCH! (2)
5) # : Preceded by T (Letter), followed by 3 (Number). MATCH! (3)
6) * : Preceded by Y (Letter), followed by D (Letter). (Fails)
There are exactly three symbols (%, &, #) that satisfy the condition.

• Exam Trick: Find the symbols first. Skip any symbol preceded by a number (like @). Check the rest. K%7, S&6, T#3 all work perfectly. Total = 3.

• Final Answer: Three

• Key Fact: Eastern India (comprising states like Bihar, Jharkhand, Odisha, and West Bengal) shows the least proportional contribution to India's formal tertiary (services) sector output when compared to its massive demographic size and the presence of historic mega-cities like Kolkata.

• Supporting Detail: While Southern and Western India heavily dominate the IT, finance, and organized services sectors, Eastern India remains largely agrarian and mineral-extraction dependent. Its urban centers suffer from significant informal sector employment and have not successfully transitioned to high-value formal service hubs at the same scale as cities like Bengaluru, Mumbai, or Gurgaon.

• Why others wrong: Southern and Western India are the undisputed powerhouses of India's formal tertiary sector (IT, banking, healthcare). Northern India (specifically Delhi NCR) also has a massive formal services footprint.

• Exam Trick: East India = High Population, High Minerals, but low IT/Formal Services.

• Key Fact: Under the Indian parliamentary system, the Prime Minister holds office during the "pleasure of the President", but this is not an arbitrary power. As long as the PM enjoys the majority support in the Lok Sabha, the President cannot dismiss him.

• Supporting Detail: Statement 1 is correct (Article 75(4) explicitly requires the President to administer the oaths of office and secrecy). Statement 3 is also correct; if the PM loses the floor test (confidence of the Lok Sabha), he is legally bound to resign, failing which the President has the constitutional authority to dismiss the government.

• Why others wrong: Statement 2 is entirely incorrect. The President does not have dictatorial or arbitrary powers to dismiss a Prime Minister who holds a democratic majority in the Lok Sabha.

• Exam Trick: President's "pleasure" = Subject to Lok Sabha majority. It is never arbitrary.