Recent questions tagged non-gate

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1

ISI2014-DCG-57
If a focal chord of the parabola $y^2=4ax$ cuts it at two distinct points $(x_1,y_1)$ and $(x_2,y_2)$, then $x_1x_2=a^2$ $y_1y_2=a^2$ $x_1x_2^2=a^2$ $x_1^2x_2=a^2$

asked Sep 23, 2019 in Others by Arjun Veteran (435k points) | 18 views

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2

ISI2014-DCG-59
The equation $5x^2+9y^2+10x-36y-4=0$ represents an ellipse with the coordinates of foci being $(\pm3,0)$ a hyperbola with the coordinates of foci being $(\pm3,0)$ an ellipse with the coordinates of foci being $(\pm2,0)$ a hyperbola with the coordinates of foci being $(\pm2,0)$

asked Sep 23, 2019 in Others by Arjun Veteran (435k points) | 24 views

+1 vote

0 answers

3

ISI2014-DCG-65
The sum $\dfrac{n}{n^2}+\dfrac{n}{n^2+1^2}+\dfrac{n}{n^2+2^2}+ \cdots + \dfrac{n}{n^2+(n-1)^2} + \cdots \cdots$ is $\frac{\pi}{4}$ $\frac{\pi}{8}$ $\frac{\pi}{6}$ $2 \pi$

asked Sep 23, 2019 in Numerical Ability by Arjun Veteran (435k points) | 69 views

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1 answer

4

ISI2015-MMA-1
Let $\{f_n(x)\}$ be a sequence of polynomials defined inductively as $ f_1(x)=(x-2)^2$ $f_{n+1}(x) = (f_n(x)-2)^2, \: \: \: n \geq 1$ Let $a_n$ and $b_n$ respectively denote the constant term and the coefficient of $x$ in $f_n(x)$. Then $a_n=4, \: b_n=-4^n$ $a_n=4, \: b_n=-4n^2$ $a_n=4^{(n-1)!}, \: b_n=-4^n$ $a_n=4^{(n-1)!}, \: b_n=-4n^2$

asked Sep 23, 2019 in Combinatory by Arjun Veteran (435k points) | 75 views

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1 answer

5

ISI2015-MMA-2
If $a,b$ are positive real variables whose sum is a constant $\lambda$, then the minimum value of $\sqrt{(1+1/a)(1+1/b)}$ is $\lambda \: – 1/\lambda$ $\lambda + 2/\lambda$ $\lambda+1/\lambda$ None of the above

asked Sep 23, 2019 in Numerical Ability by Arjun Veteran (435k points) | 88 views

+1 vote

1 answer

6

ISI2015-MMA-3
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

asked Sep 23, 2019 in Numerical Ability by Arjun Veteran (435k points) | 64 views

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1 answer

7

ISI2015-MMA-12
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 $(3-2i)$ are two two roots of this polynomial then the value of $a$ is $-524/65$ $524/65$ $-1/65$ $1/65$

asked Sep 23, 2019 in Numerical Ability by Arjun Veteran (435k points) | 43 views

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1 answer

8

ISI2015-MMA-14
Consider the following system of equivalences of integers, $x \equiv 2 \text{ mod } 15$ $x \equiv 4 \text{ mod } 21$ The number of solutions in $x$, where $1 \leq x \leq 315$, to the above system of equivalences is $0$ $1$ $2$ $3$

asked Sep 23, 2019 in Numerical Ability by Arjun Veteran (435k points) | 36 views

0 votes

1 answer

9

ISI2015-MMA-15
The number of real solutions of the equations $(9/10)^x = -3+x-x^2$ is $2$ $0$ $1$ none of the above

asked Sep 23, 2019 in Numerical Ability by Arjun Veteran (435k points) | 46 views

+1 vote

2 answers

10

ISI2015-MMA-16
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

asked Sep 23, 2019 in Numerical Ability by Arjun Veteran (435k points) | 71 views

+1 vote

1 answer

11

ISI2015-MMA-18
The set of complex numbers $z$ satisfying the equation $(3+7i)z+(10-2i)\overline{z}+100=0$ represents, in the complex plane, a straight line a pair of intersecting straight lines a point a pair of distinct parallel straight lines

asked Sep 23, 2019 in Numerical Ability by Arjun Veteran (435k points) | 40 views

0 votes

1 answer

12

ISI2015-MMA-19
The limit $\:\:\:\underset{n \to \infty}{\lim} \Sigma_{k=1}^n \begin{vmatrix} e^{\frac{2 \pi i k }{n}} – e^{\frac{2 \pi i (k-1) }{n}} \end{vmatrix}\:\:\:$ is $2$ $2e$ $2 \pi$ $2i$

asked Sep 23, 2019 in Calculus by Arjun Veteran (435k points) | 45 views

+1 vote

1 answer

13

ISI2015-MMA-20
The limit $\underset{n \to \infty}{\lim} \left( 1- \frac{1}{n^2} \right) ^n$ equals $e^{-1}$ $e^{-1/2}$ $e^{-2}$ $1$

asked Sep 23, 2019 in Calculus by Arjun Veteran (435k points) | 34 views

+1 vote

1 answer

14

ISI2015-MMA-21
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 $5$ $5 \omega$ $5(1+\omega)$ $0$

asked Sep 23, 2019 in Others by Arjun Veteran (435k points) | 37 views

0 votes

1 answer

15

ISI2015-MMA-22
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 $0$

asked Sep 23, 2019 in Calculus by Arjun Veteran (435k points) | 29 views

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0 answers

16

ISI2015-MMA-23
Let $X$ be a nonempty set and let $\mathcal{P}(X)$ denote the collection of all subsets of $X$. Define $f: X \times \mathcal{P}(X) \to \mathbb{R}$ by $f(x,A)=\begin{cases} 1 & \text{ if } x \in A \\ 0 & \text{ if } x \notin A \end{cases}$ Then $f(x, A \cup B)$ ... $f(x,A)+f(x,B)\: - f(x,A) \cdot f(x,B)$ $f(x,A)\:+ \mid f(x,A)\: - f(x,B) \mid $

asked Sep 23, 2019 in Set Theory & Algebra by Arjun Veteran (435k points) | 25 views

0 votes

1 answer

17

ISI2015-MMA-24
The series $\sum_{k=2}^{\infty} \frac{1}{k(k-1)}$ converges to $-1$ $1$ $0$ does not converge

asked Sep 23, 2019 in Numerical Ability by Arjun Veteran (435k points) | 38 views

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1 answer

18

ISI2015-MMA-25
The limit $\displaystyle{}\underset{x \to \infty}{\lim} \left( \frac{3x-1}{3x+1} \right) ^{4x}$ equals $1$ $0$ $e^{-8/3}$ $e^{4/9}$

asked Sep 23, 2019 in Calculus by Arjun Veteran (435k points) | 55 views

0 votes

0 answers

19

ISI2015-MMA-26
$\displaystyle{}\underset{n \to \infty}{\lim} \frac{1}{n} \bigg( \frac{n}{n+1} + \frac{n}{n+2} + \cdots + \frac{n}{2n} \bigg)$ is equal to $\infty$ $0$ $\log_e 2$ $1$

asked Sep 23, 2019 in Calculus by Arjun Veteran (435k points) | 28 views

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0 answers

20

ISI2015-MMA-27
Let $\cos ^6 \theta = a_6 \cos 6 \theta + a_5 \cos 5 \theta + a_4 \cos 4 \theta + a_3 \cos 3 \theta + a_2 \cos 2 \theta + a_1 \cos \theta +a_0$. Then $a_0$ is $0$ $1/32$ $15/32$ $10/32$

asked Sep 23, 2019 in Numerical Ability by Arjun Veteran (435k points) | 33 views

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1 answer

21

ISI2015-MMA-28
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 $\sqrt{125}$ $69/5$ $\sqrt{112}$ $\sqrt{864}/5$

asked Sep 23, 2019 in Numerical Ability by Arjun Veteran (435k points) | 29 views

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0 answers

22

ISI2015-MMA-29
The set $\{x \: : \begin{vmatrix} x+\frac{1}{x} \end{vmatrix} \gt6 \}$ equals the set $(0,3-2\sqrt{2}) \cup (3+2\sqrt{2}, \infty)$ $(- \infty, -3-2\sqrt{2}) \cup (-3+2 \sqrt{2}, \infty)$ $(- \infty, 3-2\sqrt{2}) \cup (3+2\sqrt{2}, \infty)$ $(- \infty, -3-2\sqrt{2}) \cup (-3+2 \sqrt{2},3-2\sqrt{2}) \cup (3+2 \sqrt{2}, \infty )$

asked Sep 23, 2019 in Numerical Ability by Arjun Veteran (435k points) | 25 views

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23

ISI2015-MMA-30
Suppose that a function $f$ defined on $\mathbb{R} ^2$ satisfies the following conditions: $\begin{array} &f(x+t,y) & = & f(x,y)+ty, \\ f(x,t+y) & = & f(x,y)+ tx \text{ and } \\ f(0,0) & = & K, \text{ a constant.} \end{array}$ Then for all $x,y \in \mathbb{R}, \:f(x,y)$ is equal to $K(x+y)$ $K-xy$ $K+xy$ none of the above

asked Sep 23, 2019 in Calculus by Arjun Veteran (435k points) | 24 views

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24

ISI2015-MMA-31
Consider the sets defined by the real solutions of the inequalities $A = \{(x,y):x^2+y^4 \leq 1 \} \:\:\:\:\:\:\:\: B = \{ (x,y):x^4+y^6 \leq 1\}$ Then $B \subseteq A$ $A \subseteq B$ Each of the sets $A – B, \: B – A$ and $A \cap B$ is non-empty none of the above

asked Sep 23, 2019 in Set Theory & Algebra by Arjun Veteran (435k points) | 34 views

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1 answer

25

ISI2015-MMA-32
If a square of side $a$ and an equilateral triangle of side $b$ are inscribed in a circle then $a/b$ equals $\sqrt{2/3}$ $\sqrt{3/2}$ $3/ \sqrt{2}$ $\sqrt{2}/3$

asked Sep 23, 2019 in Geometry by Arjun Veteran (435k points) | 23 views

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1 answer

26

ISI2015-MMA-33
If $f(x)$ is a real valued function such that $2f(x)+3f(-x)=15-4x,$ for every $x \in \mathbb{R}$, then $f(2)$ is $-15$ $22$ $11$ $0$

asked Sep 23, 2019 in Calculus by Arjun Veteran (435k points) | 28 views

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27

ISI2015-MMA-34
If $f(x) = \dfrac{\sqrt{3}\sin x}{2+\cos x}$, then the range of $f(x)$ is the interval $[-1, \sqrt{3}/2]$ the interval $[- \sqrt{3}/2, 1]$ the interval $[-1, 1]$ none of the above

asked Sep 23, 2019 in Calculus by Arjun Veteran (435k points) | 32 views

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28

ISI2015-MMA-35
If $f(x)=x^2$ and $g(x)= x \sin x + \cos x$ then $f$ and $g$ agree at no points $f$ and $g$ agree at exactly one point $f$ and $g$ agree at exactly two points $f$ and $g$ agree at more than two points

asked Sep 23, 2019 in Geometry by Arjun Veteran (435k points) | 40 views

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1 answer

29

ISI2015-MMA-36
For non-negative integers $m$, $n$ define a function as follows $f(m,n) = \begin{cases} n+1 & \text{ if } m=0 \\ f(m-1, 1) & \text{ if } m \neq 0, n=0 \\ f(m-1, f(m,n-1)) & \text{ if } m \neq 0, n \neq 0 \end{cases}$ Then the value of $f(1,1)$ is $4$ $3$ $2$ $1$

asked Sep 23, 2019 in Calculus by Arjun Veteran (435k points) | 29 views

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30

ISI2015-MMA-46
If the tangent at the point $P$ with coordinates $(h,k)$ on the curve $y^2=2x^3$ is perpendicular to the straight line $4x=3y$, then $(h,k) = (0,0)$ $(h,k) = (1/8, -1/16)$ $(h,k) = (0,0) \text{ or } (h,k) = (1/8, -1/16)$ no such point $(h,k)$ exists

asked Sep 23, 2019 in Geometry by Arjun Veteran (435k points) | 38 views

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