Among five objects, M, N, P, Q and R, P is twice as heavy as M and N is thrice as heavy as R. Q is one and half times as heavy as M and twice as heavy as R. Which of the five objects is the heaviest?

If \( p \) and \( q \) are numbers such that the pair of linear equations \( (p + 2)x + (q - 1)y = 10 \) and \( (q + 2)x + (p - 1)y = 10 \) have infinite solutions for \( x \) and \( y \), then \( p = q \).

NPAT - 2025
Linear Equations

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If \( x \), \( y \), and \( z \) are positive integers and \( p = \left( \left( (x - 1)^2 / |x| \right) + 2 \right) + \left( \left( (y - 1)^2 / |y| \right) + 2 \right) + \left( \left( (z - 1)^2 / |z| \right) + 2 \right), \) then \( p<6 \).

NPAT - 2025
Algebra

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Passage: Toru Dutt is considered the earliest Indian female writer in English. She travelled extensively in Europe from a young age with her family. She and her sister Aru became fascinated with Paris and French literature. In London, they came in contact with such august personages such as Sir Bartle Frere, the Gover- nor of Bombay from 1862 to 1867, and Sir Edward Ryan, the Chief Justice of the Supreme Court, Calcutta, from 1837 to 1843. Toru Dutt was greatly influenced in her writings by French Romantic poets like Victor Hugo and English writers like Elizabeth Browning, John Keats, Charlotte Bronte and Jane Austen. She was also intrigued by the legends and myths of India, and even learned Sanskrit. Her writings were marked by romantic melancholia and an obsession and preoccupation with death. This was partly due to her suffering and pain following the early tragic deaths of her siblings, especially her older sister Aru, with whom she was quite close. Her chosen subjects often portrayed separation, loneliness, captivity, dejec- tion, declining seasons and untimely death. She led an ”Ivory Tower existence” and her own death came quite early, at the age of 21, in the full bloom of her talent and on the eve of the awakening of her genius. Toru Dutt’s most famous work is A Sheaf Gleaned in French Fields, an anthology of poems translated from French to English. It also contained a few original poems that showcase her vast insight into French literature. She used to publish poems in the Bengal Magazine, under the pseudonym ”TD”. But most of her powerful work was published posthumously, in- cluding the French novel Le Journal de Mademoiselle D’Arvers and the unfinished English novel Bianca, or, the Young Spanish Maiden. Her work Ancient Ballads and Legends of Hindustan depicts a shrewd knowledge of Hindu mythology and an instinctive empathy with the conditions of life they represent. An assimilation of the Occident and the Orient nourished Toru’s poetic skills; in her, we find a tripartite influence of a French education, lectures at Cambridge and the study of Sanskrit literature.