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ISI2018DCG4
The number of terms with integral coefficients in the expansion of $(17^\frac{1}{3}+19^\frac{1}{2}x)^{600}$ is $99$ $100$ $101$ $102$
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ISI2016DCG2
Let $S=\{6,10,7,13,5,12,8,11,9\},$ and $a=\sum_{x\in S}(x9)^{2}\:\&\: b=\sum_{x\in S}(x10)^{2}.$ Then $a<b$ $a>b$ $a=b$ None of these
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ISI2016DCG63
If $\sin^{1}\frac{1}{\sqrt{5}}$ and $\cos^{1}\frac{3}{\sqrt{10}}$ lie in $\left[0,\frac{\pi}{2}\right],$ their sum is equal to $\frac{\pi}{6}$ $\frac{\pi}{3}$ $\sin^{1}\frac{1}{\sqrt{50}}$ $\frac{\pi}{4}$
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ISI2018DCG5
Let $A$ be the set of all prime numbers, $B$ be the set of all even prime numbers, and $C$ be the set of all odd prime numbers. Consider the following three statements in this regard: $A=B\cup C$. $B$ ... statements is true. Exactly one of the above statements is true. Exactly two of the above statements are true. All the above three statements are true.
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ISI2018DCG1
The digit in the unit place of the number $7^{78}$ is $1$ $3$ $7$ $9$
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6
CMI2016A9
ScamTel has won a state government contract to connect 17 cities by highspeed fibre optic links. Each link will connect a pair of cities so that the entire network is connectedthere is a path from each city to every otehr city. The cotract requires the network to remain connected if $any$ single link fails. What is the minimum number of links that ScamTel needs to set up? 34 32 17 16
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cmi2016
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7
ISI2015MMA61
Let $ A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 2 & 3 \end{pmatrix} \text{ and } B=\begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{pmatrix}.$ Then there exists a matrix $C$ ... no matrix $C$ such that $A=BC$ there exists a matrix $C$ such that $A=BC$, but $A \neq CB$ there is no matrix $C$ such that $A=CB$
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isi2015mma
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8
ISI2015MMA8
Let $A$ be a set of $n$ elements. The number of ways, we can choose an ordered pair $(B,C)$, where $B,C$ are disjoint subsets of $A$, equals $n^2$ $n^3$ $2^n$ $3^n$
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ISI2015MMA7
Let $X$ be the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \}$. Define the set $\mathcal{R}$ by $\mathcal{R} = \{(x,y) \in X \times X : x$ and $y$ have the same remainder when divided by $3\}$. Then the number of elements in $\mathcal{R}$ is $40$ $36$ $34$ $33$
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ISI2015MMA54
If $0 <x<1$, then the sum of the infinite series $\frac{1}{2}x^2+\frac{2}{3}x^3+\frac{3}{4}x^4+ \cdots$ is $\log \frac{1+x}{1x}$ $\frac{x}{1x} + \log(1+x)$ $\frac{1}{1x} + \log(1x)$ $\frac{x}{1x} + \log(1x)$
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ISI2015MMA50
Let $\begin{array}{} V_1 & = & \frac{7^2+8^2+15^2+23^2}{4} – \bigg( \frac{7+8+15+23}{4} \bigg) ^2, \\ V_2 & = & \frac{6^2+8^2+15^2+24^2}{4} – \bigg( \frac{6+8+15+24}{4} \bigg) ^2 , \\ V_3 & = & \frac{5^2+8^2+15^2+25^2}{4} – \bigg( \frac{5+8+15+25}{4} \bigg) ^2 . \end{array}$ Then $V_3<V_2<V_1$ $V_3<V_1<V_2$ $V_1<V_2<V_3$ $V_2<V_3<V_1$
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ISI2015MMA43
The values of $\eta$ for which the following system of equations $\begin{array} {} x & + & y & + & z & = & 1 \\ x & + & 2y & + & 4z & = & \eta \\ x & + & 4y & + & 10z & = & \eta ^2 \end{array}$ has a solution are $\eta = 1, 2$ $\eta = 1, 2$ $\eta = 3, 3$ $\eta = 1, 2$
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ISI2015MMA39
The eigenvalues of the matrix $X = \begin{pmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{pmatrix}$ are $1,1,4$ $1,4,4$ $0,1,4$ $0,4,4$
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ISI2015MMA36
For nonnegative integers $m$, $n$ define a function as follows $f(m,n) = \begin{cases} n+1 & \text{ if } m=0 \\ f(m1, 1) & \text{ if } m \neq 0, n=0 \\ f(m1, f(m,n1)) & \text{ if } m \neq 0, n \neq 0 \end{cases}$ Then the value of $f(1,1)$ is $4$ $3$ $2$ $1$
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15
ISI2015MMA9
Let $(1+x)^n = C_0+C_1x+C_2x^2+ \cdots +C_nx^n, \: n$ being a positive integer. The value of $\bigg( 1+\frac{C_0}{C_1} \bigg) \bigg( 1+\frac{C_1}{C_2} \bigg) \cdots \bigg( 1+\frac{C_{n1}}{C_n} \bigg)$ is $\bigg( \frac{n+1}{n+2} \bigg) ^n$ $ \frac{n^n}{n!} $ $\bigg( \frac{n}{n+1} \bigg) ^n$ $\frac{(n+1)^n}{n!}$
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3
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16
ISI2015MMA4
Suppose in a competition $11$ matches are to be played, each having one of $3$ distinct outcomes as possibilities. The number of ways one can predict the outcomes of all $11$ matches such that exactly $6$ of the predictions turn out to be correct is $\begin{pmatrix}11 \\ 6 \end{pmatrix} \times 2^5$ $\begin{pmatrix}11 \\ 6 \end{pmatrix} $ $3^6$ none of the above
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2
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17
ISI2016DCG5
If $\tan\: x=p+1$ and $\tan\; y=p1,$ then the value of $2\:\cot\:(xy)$ is $2p$ $p^{2}$ $(p+1)(p1)$ $\frac{2p}{p^{2}1}$
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Oct 13
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isi2016dcg
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1
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18
ISI2015MMA16
If two real polynomials $f(x)$ and $g(x)$ of degrees $m\: (\geq 2)$ and $n\: (\geq 1)$ respectively, satisfy $f(x^2+1)=f(x)g(x),$ for every $x \in \mathbb{R}$, then $f$ has exactly one real root $x_0$ such that $f’(x_0) \neq 0$ $f$ has exactly one real root $x_0$ such that $f’(x_0) = 0$ $f$ has $m$ distinct real roots $f$ has no real root
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Oct 10
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26
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isi2015mma
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19
ISI2015MMA18
The set of complex numbers $z$ satisfying the equation $(3+7i)z+(102i)\overline{z}+100=0$ represents, in the complex plane, a straight line a pair of intersecting straight lines a point a pair of distinct parallel straight lines
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isi2015mma
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2
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20
ISI2015MMA5
A set contains $2n+1$ elements. The number of subsets of the set which contain at most $n$ elements is $2^n$ $2^{n+1}$ $2^{n1}$ $2^{2n}$
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Oct 9
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isi2015mma
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21
ISI2015MMA21
Let $\omega$ denote a complex fifth root of unity. Define $b_k =\sum_{j=0}^{4} j \omega^{kj},$ for $0 \leq k \leq 4$. Then $ \sum_{k=0}^{4} b_k \omega ^k$ is equal to $5$ $5 \omega$ $5(1+\omega)$ $0$
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isi2015mma
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22
ISI2015MMA3
Let $x$ be a positive real number. Then $x^2+\pi ^2 + x^{2 \pi} > x \pi+ (\pi + x) x^{\pi}$ $x^{\pi}+\pi^x > x^{2 \pi} + \pi ^{2x}$ $\pi x +(\pi+x)x^{\pi} > x^2+\pi ^2 + x^{2 \pi}$ none of the above
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isi2015mma
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23
ISI2016DCG36
Suppose $X$ and $Y$ are finite sets, each with cardinality $n$.. The number of bijective functions from $X$ to $Y$ is $n^{n}$ $n\log_{2}n$ $n^{2}$ $n!$
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24
ISI2015MMA93
Let $G$ be a group with identity element $e$. If $x$ and $y$ are elements in $G$ satisfying $x^5y^3=x^8y^5=e$, then which of the following conditions is true? $x=e, \: y=e$ $x\neq e, \: y=e$ $x=e, \: y \neq e$ $x\neq e, \: y \neq e$
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Oct 5
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isi2015mma
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1
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25
ISI2015MMA94
Let $G$ be the group $\{\pm1, \pm i \}$ with multiplication of complex numbers as composition. Let $H$ be the quotient group $\mathbb{Z}/4 \mathbb{Z}$. Then the number of nontrivial group homomorphisms from $H$ to $G$ is $4$ $1$ $2$ $3$
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26
ISI2015MMA79
Let $g(x,y) = \text{max}\{12x, 8y\}$. Then the minimum value of $g(x,y)$ $ $ as $(x,y)$ varies over the line $x+y =10$ is $5$ $7$ $1$ $3$
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Oct 4
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27
ISI2015MMA78
The value of $\underset{n \to \infty}{\lim} \bigg[ (n+1) \int_0^1 x^n \text{ln}(1+x) dx \bigg]$ is $0$ $\text{ln }2$ $\text{ln }3$ $\infty$
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28
ISI2015MMA28
In a triangle $ABC$, $AD$ is the median. If length of $AB$ is $7$, length of $AC$ is $15$ and length of $BC$ is $10$ then length of $AD$ equals $\sqrt{125}$ $69/5$ $\sqrt{112}$ $\sqrt{864}/5$
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29
ISI2015MMA22
Let $a_n= \bigg( 1 – \frac{1}{\sqrt{2}} \bigg) \cdots \bigg( 1 \frac{1}{\sqrt{n+1}} \bigg), \: \: n \geq1$. Then $\underset{n \to \infty}{\lim} a_n$ equals $1$ does not exist equals $\frac{1}{\sqrt{\pi}}$ equals $0$
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30
ISI2015MMA19
The limit $\:\:\:\underset{n \to \infty}{\lim} \Sigma_{k=1}^n \begin{vmatrix} e^{\frac{2 \pi i k }{n}} – e^{\frac{2 \pi i (k1) }{n}} \end{vmatrix}\:\:\:$ is $2$ $2e$ $2 \pi$ $2i$
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31
ISI2015MMA13
The number of real roots of the equation $2 \cos \bigg( \frac{x^2+x}{6} \bigg) = 2^x +2^{x} \text{ is }$ $0$ $1$ $2$ infinitely many
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ISI2015MMA12
Consider the polynomial $x^5+ax^4+bx^3+cx^2+dx+4$ where $a,b,c,d$ are real numbers. If $(1+2i)$ and $(32i)$ are two two roots of this polynomial then the value of $a$ is $524/65$ $524/65$ $1/65$ $1/65$
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33
Where to scan the barcode given in GateOverflow book
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ISI2016DCG61
The value of $\sin^{6}\frac{\pi}{81}+\cos^{6}\frac{\pi}{81}1+3\sin^{2}\frac{\pi}{81}\:\cos^{2}\frac{\pi}{81}$ is $\tan^{6}\frac{\pi}{81}$ $0$ $1$ None of these
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ISI2016DCG64
If $\cos2\theta=\sqrt{2}(\cos\theta\sin\theta)$ then $\tan\theta$ equals $1$ $1$ or $1$ $\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}$ or $1$ None of these
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ISI2018DCG15
The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines is $6$ $9$ $12$ $18$
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ISI2016DCG12
The highest power of $3$ contained in $1000!$ is $198$ $891$ $498$ $292$
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Oct 1
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38
ISI2015MMA25
The limit $\underset{x \to \infty}{\lim} \bigg( \frac{3x1}{3x+1} \bigg) ^{4x}$ equals $1$ $0$ $e^{8/3}$ $e^{4/9}$
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39
ISI2015MMA20
The limit $\underset{n \to \infty}{\lim} \bigg( 1 \frac{1}{n^2} \bigg) ^n$ equals $e^{1}$ $e^{1/2}$ $e^{2}$ $1$
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1
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40
ISI2015MMA10
The value of the infinite product $P=\frac{7}{9} \times \frac{26}{28} \times \frac{63}{65} \times \cdots \times \frac{n^31}{n^3+1} \times \cdots \text{ is }$ $1$ $2/3$ $7/3$ none of the above
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