Jamia Millia Islamia 2018 Previous Year Questions (PYQs) – Page 2 of 10
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Two bus tickets from city A to B and three tickets from A to C cost Rs. 77, but three tickets from A to B and two tickets from A to C cost Rs. 73. What are the fares for cities B and C from A?
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Solution
$2x + 3y = 77,\ 3x + 2y = 73$ → solving gives $x = 17,\ y = 13.$
In the group G = {2, 4, 6, 8} under multiplication modulo 10, the identity element is
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Solution
$(a × e) \bmod 10 = a.$
For $e=6$, $2×6=12\equiv2,\ 4×6=24\equiv4,\ 8×6=48\equiv8.$
A partition of {1, 2, 3, 4, 5} is the family
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Solution
A partition divides a set into disjoint non-empty subsets covering all elements.
{{1,2},{3,4,5}} satisfies it.
Let P(S) denote the power set of set S. Which of the following is always TRUE?
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Solution
A partition divides a set into disjoint non-empty subsets covering all elements.
{{1,2},{3,4,5}} satisfies it.
Find the remainder when $67^{99}$ is divided by 7.
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Solution
$67 \equiv 4 \pmod7$, so $4^3 \equiv 1$.
$99$ is multiple of 3 → remainder $=1$.
G = {a, b, c} is an abelian group with ‘e’ as identity element. The order of other elements is
A. 2, 2, 3 B. 3, 3, 3 C. 2, 2, 4 D. 2, 2, 2
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Solution
With 3 elements, it must be cyclic of order 3.
Period of $3\sec x/3$ is
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Solution
Period of $\sec kx$ = $\dfrac{2\pi}{k}$ → here $k = 1/3$.
Hence period = $6\pi$.
Principal value of $\cos^{-1}(\cos 5)$ is
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Solution
For the principal range $[0, \pi]$,
$\cos^{-1}(\cos 5) = 2\pi - 5$.
If $\sin t = \dfrac{1}{5}$ and $0 < t < \dfrac{\pi}{2}$, then $\cos(4t)$ = ?
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Solution
$\cos^2 t = 1 - \sin^2 t = \dfrac{24}{25}$
$\cos(4t) = 8\cos^4 t - 8\cos^2 t + 1 = 8\left(\dfrac{24}{25}\right)^2 - 8\left(\dfrac{24}{25}\right) + 1 = 0.6928$
Find the value of $\displaystyle \int \dfrac{x,dx}{\sqrt{x^2 + 4}}$
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Solution
Let $x = 2\tan\theta$, $dx = 2\sec^2\theta,d\theta$
$\Rightarrow I = \int \dfrac{2\tan\theta \cdot 2\sec^2\theta}{2\sec\theta}d\theta = 2\int \tan\theta\sec\theta,d\theta = 2\sec\theta + C = \dfrac{1}{2}\sqrt{x^2+4}+C$
Hence equivalent to $\dfrac{1}{4}\sec^{-1}\left(\dfrac{x}{2}\right)$
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