Your Answer : (Not Answered)

Correct Answer : C

The smallest such number is 123456789.

The digits 0 to 9 add up to 45, so distinct digits summing to 45 must be all ten digits or all of 1 to 9 (leaving out 0, which adds nothing). A nine-digit number is smaller than any ten-digit one, and arranging 1 to 9 in increasing order gives the smallest: 123456789.

123555789 repeats the digit 5, and 99999 repeats 9, so neither has distinct digits. 123457869 uses the right digits but is larger than 123456789, because its sixth digit is 7 instead of 6.

Your Answer : (Not Answered)

Correct Answer : C

The degree of a polynomial is the highest total degree among its terms.

• \( x^{3} \) → degree 3
• \( -x^{2}y^{2} \) → degree \( 2 + 2 = 4 \)
• \( -8y^{2} \) → degree 2
• \( 2 \) → degree 0

The largest is 4. In a term with several variables the exponents are added, which is what puts \( x^{2}y^{2} \) above \( x^{3} \).

Your Answer : (Not Answered)

Correct Answer : E

Correct answer: Fourth to the left

Facing the centre, each person’s left is the next seat clockwise and their right the next seat anticlockwise. Fix B; D sits two to B’s left, F three beyond D, A two to F’s left and C two to A’s left - a single chain filling five seats. G three to E’s right then fits the remaining three in only one way. Clockwise the circle reads B → C → D → G → H → F → E → A.

Counting four seats to B’s left - clockwise - gives C, D, G and then H. In a circle of eight, four seats the other way reach the same person, so H is equally fourth to B’s right - but of the descriptions offered only fourth to the left appears.

Your Answer : (Not Answered)

Correct Answer : B

Correct answer: DBM

Facing the centre, each person’s left is the next seat clockwise and their right the next seat anticlockwise. Fix T; A sits two to T’s left, M three beyond A, H two to T’s right and D two to H’s right. Of the two ways of placing B with R two seats to B’s right, only one keeps B off T’s shoulder, and P takes the seat left over. Clockwise the circle reads T → R → A → B → D → M → H → P.

In DBM the third person named, M, sits two seats clockwise of the second, B - that is, second to B’s left. Checking the other groups the same way, none of them satisfies the condition.

Your Answer : (Not Answered)

Correct Answer : B

Correct answer: Only Conclusion II is true.

The statements are \( P < A \le C \ge K \), \( C \le O < L \) and \( R \le N \le C \).

II. \( R < L \): assemble the run through C. \( R \le N \le C \le O < L \) points one way throughout, and the final link is strict, so \( R < L \). True.

I. \( A = K \): A and K both hang below C - \( A \le C \) and \( K \le C \) - on separate branches. That permits them to be equal but does not require it: with \( C = 10 \), \( A = 10, K = 3 \) satisfies every statement, and so does \( A = 3, K = 10 \). Not determined.

Only II follows. Conclusion I is worth pausing on, because an equality conclusion can only be established by an unbroken run of \(=\) or \(\le\)-and-\(\ge\) links that pin the two values together; two elements merely capped by the same third are never forced into equality.

Your Answer : (Not Answered)

Correct Answer : A

The statement that must be false is Aman knows Bimal.

Chris knows both of the others: he plays cards with Aman and is a good friend of Bimal. The lawyer and the businessman do not know each other, so Chris cannot be either of them, since he would then know the other one. Chris must therefore be the doctor, and Aman and Bimal are the lawyer and the businessman, who do not know each other.

So 'Aman knows Bimal' is necessarily false. The other statements are in fact true: Chris is the doctor, and since Bimal did not study beyond class X he cannot be the lawyer, which makes Bimal the businessman and Aman the lawyer.

Your Answer : (Not Answered)

Correct Answer : D

The missing number is 65.

Each term is one more than a cube:

\[ \begin{aligned} 2 &= 1^{3} + 1,\quad 9 = 2^{3} + 1,\quad 28 = 3^{3} + 1 \\ 65 &= 4^{3} + 1,\quad 126 = 5^{3} + 1 \end{aligned} \]

so the gap is filled by \(4^{3} + 1 = 65\). 63 is \(4^{3} - 1\), and 57 and 53 do not fit the pattern; the differences 7, 19, 37, 61 also confirm 65, since they rise by 12, 18, 24.

Your Answer : (Not Answered)

Correct Answer : A

Look at the gaps between the letters of each cluster in the alphabet. In EAI the letters go back 4 places and then forward 8 places, in KGO they go back 4 and then forward 8, and in UQY they go back 4 and then forward 8, so these three follow the same pattern. In HEL the letters go back 3 and then forward 7, which is a different pattern. So the odd one out is HEL, option (A), while EAI, KGO and UQY are alike.

Your Answer : (Not Answered)

Correct Answer : A

Correct answer: DFH

Each triple holds three letters two places apart, and each triple starts one place after the last: A, B, C. The next starts at D, so its letters are D, F and H, giving DFH. Two rules operate together - the internal spacing and the step between triples - and the answer must satisfy both. DEF keeps the correct start but collapses the internal spacing to one. Both the internal spacing and the step between triples must be honoured, and an answer that satisfies only one of them will always be among the options offered.

Your Answer : (Not Answered)

Correct Answer : B

Correct answer: 72

Each answer is the average of the two numbers: \( \tfrac{13+45}{2} = 29 \), \( \tfrac{24+58}{2} = 41 \), \( \tfrac{74+32}{2} = 53 \). So \[ \frac{97+47}{2} = \mathbf{72} \]

Your Answer : (Not Answered)

Correct Answer : A

Elections must be held within 6 months.

Under Article 243E(3)(b) of the Constitution, an election to constitute a Panchayat must be completed before the expiry of a period of six months from the date of its dissolution. Article 243U lays down the same rule for municipalities.

A newly constituted Panchayat after a dissolution serves only the remainder of the original five-year term. The periods of 3 months, 5 months and 1 year do not appear in the Constitution's rule.

Your Answer : (Not Answered)

Correct Answer : B

The Life Insurance Corporation Act was enacted in 1956.

It nationalised life insurance, transferring the business of 245 insurers to the newly created Life Insurance Corporation of India.

Your Answer : (Not Answered)

Correct Answer : B

The curie is a unit of radioactivity.

One curie is \(3.7\times10^{10}\) disintegrations per second, originally the activity of one gram of radium-226. The SI unit is the becquerel, one disintegration per second.

Your Answer : (Not Answered)

Correct Answer : D

Elmer Sperry invented the gyrocompass.

Patented in 1908, it uses a spinning gyroscope to find true north rather than magnetic north, and is unaffected by the iron of a ship's hull. It made accurate navigation in steel warships possible.

Your Answer : (Not Answered)

Correct Answer : B

Ammonium sulphate, (NH₄)₂SO₄, contains about 21 per cent nitrogen by mass:

\[ \frac{2 \times 14}{132} \times 100 = 21.2\% \]

It is a straight nitrogenous fertiliser, valued for also supplying sulphur, though urea at 46 per cent nitrogen is far more concentrated.

Your Answer : (Not Answered)

Correct Answer : D

Washing soda is hydrated sodium carbonate, Na₂CO₃·10H₂O.

It softens water by precipitating the calcium and magnesium as carbonates, so soap can lather. Baking soda is a different compound, sodium bicarbonate.

Your Answer : (Not Answered)

Correct Answer : A

The Gandhi Peace Prize carries the largest sum.

Instituted in 1995 for the 125th birth anniversary of Gandhi, it carries Rs 1 crore together with a citation and a plaque. The Bharat Ratna carries no money at all, and the Jnanpith Rs 11 lakh.

Your Answer : (Not Answered)

Correct Answer : C

Engineering is not recognised.

Nobel's will named physics, chemistry, medicine, literature and peace, and economics was added in 1968. Engineering, mathematics and the earth sciences have no Nobel; the Fields Medal and the Abel Prize serve mathematics.

Your Answer : (Not Answered)

Correct Answer : A

Stemflow is the part of intercepted rainfall that runs down branches and stems to reach the ground at the base of the trunk.

Water that drips from the leaves instead is throughfall or canopy drip, and interception is the catching of the rain in the first place. Stemflow concentrates water and dissolved nutrients right at the root collar.

Your Answer : (Not Answered)

Correct Answer : D

The National Council for Senior Citizens is headed by the Minister of Social Justice and Empowerment.

The council advises the government on policies and programmes for older persons under the National Policy on Older Persons. It was earlier the National Council for Older Persons and was reconstituted under its present name; its chairperson is the Union Minister for Social Justice and Empowerment, the ministry responsible for the welfare of senior citizens.

The Prime Minister heads bodies such as NITI Aayog's Governing Council and the National Development Council, and the Vice-Chairman of NITI Aayog runs that planning body, but neither leads this council.