The value of \( 5^{2 \times \frac{1}{4}} - \left( \frac{5}{32} \times \frac{3}{5} \times \frac{7}{8} \times \frac{3}{16} \right) \) is:

To solve the expression, we first evaluate \( 5^{2 \times \frac{1}{4}} \) and then calculate the product of the other fractions. First, compute \( 5^{2 \times \frac{1}{4}} \): \[ 5^{2 \times \frac{1}{4}} = 5^{\frac{1}{2}} = \sqrt{5} \] Next, compute the product of the fractions: \[ \frac{5}{32} \times \frac{3}{5} \times \frac{7}{8} \times \frac{3}{16} = \frac{5 \times 3 \times 7 \times 3}{32 \times 5 \times 8 \times 16} = \frac{315}{20480} = \frac{67}{32} \] Thus, the value of the expression is \( \frac{67}{32} \).

The value of 5 - left( 5/32 x 3/5 x 7/8 x 3/16 right) is:

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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Manjeet, Rahim, Ram and Divya are four friends who live in a village. It is known that:
1. Each friend lives in a differently coloured house - orange, yellow, green and blue.
II. Each friend owns a different pet - Bee, Dee, Cee and Tee.
III. The pets each feed on different things - leaves, fish, vegetables and insects.
IV. Manjeet does not live in the blue house.
V. Tee eats leaves but does not live with Rahim or Divya.
VI. Dee does not eat insects.
VII. Ram lives in an orange house but does not own Cee or Tee.
VIII. The pet that feeds on insects does not live in the green or blue house.
IX. Rahim does not feed vegetables or insects to his pet, which is neither Dee nor Cee.

NPAT - 2025
Logical Puzzle

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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.