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Recent questions tagged isi2014-dcg

2 votes

2 answers

1

ISI2014-DCG-1
Let $(1+x)^n = C_0+C_1x+C_2x^2+ \dots + C_nx^n$, $n$ being a positive integer. The value of $\left( 1+\dfrac{C_0}{C_1} \right) \left( 1+\dfrac{C_1}{C_2} \right) \cdots \left( 1+\dfrac{C_{n-1}}{C_n} \right)$ is $\left( \frac{n+1}{n+2} \right) ^n$ $ \frac{n^n}{n!} $ $\left( \frac{n}{n+1} \right) ^n$ $ \frac{(n+1)^n}{n!} $

Arjun asked in Combinatory Sep 23, 2019

by Arjun

744 views

3 votes

1 answer

2

ISI2014-DCG-2
Let $a_n=\bigg( 1 – \frac{1}{\sqrt{2}} \bigg) \cdots \bigg( 1 – \frac{1}{\sqrt{n+1}} \bigg), \: n \geq 1$. Then $\underset{n \to \infty}{\lim} a_n$ equals $1$ does not exist equals $\frac{1}{\sqrt{\pi}}$ equals $0$

Arjun asked in Calculus Sep 23, 2019

by Arjun

811 views

4 votes

4 answers

3

ISI2014-DCG-3
$\underset{x \to \infty}{\lim} \left( \frac{3x-1}{3x+1} \right) ^{4x}$ equals $1$ $0$ $e^{-8/3}$ $e^{4/9}$

Arjun asked in Calculus Sep 23, 2019

by Arjun

1.6k views

3 votes

4 answers

4

ISI2014-DCG-4
$\underset{n \to \infty}{\lim} \dfrac{1}{n} \bigg( \dfrac{n}{n+1} + \dfrac{n}{n+2} + \cdots + \dfrac{n}{2n} \bigg)$ is equal to $\infty$ $0$ $\log_e 2$ $1$

Arjun asked in Calculus Sep 23, 2019

by Arjun

1.2k views

1 vote

1 answer

5

ISI2014-DCG-5
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

Arjun asked in Set Theory & Algebra Sep 23, 2019

by Arjun

1.2k views

2 votes

2 answers

6

ISI2014-DCG-6
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 $-15$ $22$ $11$ $0$

Arjun asked in Calculus Sep 23, 2019

by Arjun

514 views

2 votes

3 answers

7

ISI2014-DCG-7
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 these

Arjun asked in Calculus Sep 23, 2019

by Arjun

564 views

3 votes

3 answers

8

ISI2014-DCG-8
If $M$ is a $3 \times 3$ matrix such that $\begin{bmatrix} 0 & 1 & 2 \end{bmatrix}M=\begin{bmatrix}1 & 0 & 0 \end{bmatrix}$ and $\begin{bmatrix}3 & 4 & 5 \end{bmatrix} M = \begin{bmatrix}0 & 1 & 0 \end{bmatrix}$ ... $\begin{bmatrix} -1 & 2 & 0 \end{bmatrix}$ $\begin{bmatrix} 9 & 10 & 8 \end{bmatrix}$

Arjun asked in Linear Algebra Sep 23, 2019

by Arjun

754 views

3 votes

1 answer

9

ISI2014-DCG-9
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 $\eta=1, -2$ $\eta=-1, -2$ $\eta=3, -3$ $\eta=1, 2$

Arjun asked in Linear Algebra Sep 23, 2019

by Arjun

512 views

6 votes

3 answers

10

ISI2014-DCG-10
The number of divisors of $6000$, where $1$ and $6000$ are also considered as divisors of $6000$ is $40$ $50$ $60$ $30$

Arjun asked in Quantitative Aptitude Sep 23, 2019

by Arjun

997 views

1 vote

1 answer

11

ISI2014-DCG-11
Let $x_1$ and $x_2$ be the roots of the quadratic equation $x^2-3x+a=0$, and $x_3$ and $x_4$ be the roots of the quadratic equation $x^2-12x+b=0$. If $x_1, x_2, x_3$ and $x_4 \: (0 < x_1 < x_2 < x_3 < x_4)$ are in $G.P.,$ then $ab$ equals $64$ $5184$ $-64$ $-5184$

Arjun asked in Quantitative Aptitude Sep 23, 2019

by Arjun

440 views

2 votes

1 answer

12

ISI2014-DCG-12
The integral $\int _0^{\frac{\pi}{2}} \frac{\sin^{50} x}{\sin^{50}x +\cos^{50}x} dx$ equals $\frac{3 \pi}{4}$ $\frac{\pi}{3}$ $\frac{\pi}{4}$ none of these

Arjun asked in Calculus Sep 23, 2019

by Arjun

710 views

2 votes

1 answer

13

ISI2014-DCG-13
Let the function $f(x)$ be defined as $f(x)=\mid x-1 \mid + \mid x-2 \:\mid$. Then which of the following statements is true? $f(x)$ is differentiable at $x=1$ $f(x)$ is differentiable at $x=2$ $f(x)$ is differentiable at $x=1$ but not at $x=2$ none of the above

Arjun asked in Calculus Sep 23, 2019

by Arjun

546 views

1 vote

1 answer

14

ISI2014-DCG-14
$x^4-3x^2+2x^2y^2-3y^2+y^4+2=0$ represents A pair of circles having the same radius A circle and an ellipse A pair of circles having different radii none of the above

Arjun asked in Others Sep 23, 2019

by Arjun

350 views

2 votes

1 answer

15

ISI2014-DCG-15
Let $\mathbb{N}=\{1,2,3, \dots\}$ be the set of natural numbers. For each $n \in \mathbb{N}$, define $A_n=\{(n+1)k, \: k \in \mathbb{N} \}$. Then $A_1 \cap A_2$ equals $A_3$ $A_4$ $A_5$ $A_6$

Arjun asked in Set Theory & Algebra Sep 23, 2019

by Arjun

471 views

2 votes

2 answers

16

ISI2014-DCG-16
The sum of the series $\dfrac{1}{1.2} + \dfrac{1}{2.3}+ \cdots + \dfrac{1}{n(n+1)} + \cdots $ is $1$ $1/2$ $0$ non-existent

Arjun asked in Quantitative Aptitude Sep 23, 2019

by Arjun

667 views

2 votes

3 answers

17

ISI2014-DCG-17
$\underset{x \to 2}{\lim} \dfrac{1}{1+e^{\frac{1}{x-2}}}$ is $0$ $1/2$ $1$ non-existent

Arjun asked in Calculus Sep 23, 2019

by Arjun

614 views

2 votes

3 answers

18

ISI2014-DCG-18
$^nC_0+2^nC_1+3^nC_2+\cdots+(n+1)^nC_n$ equals $2^n+n2^{n-1}$ $2^n-n2^{n-1}$ $2^n$ none of these

Arjun asked in Combinatory Sep 23, 2019

by Arjun

740 views

3 votes

1 answer

19

ISI2014-DCG-19
It is given that $e^a+e^b=10$ where $a$ and $b$ are real. Then the maximum value of $(e^a+e^b+e^{a+b}+1)$ is $36$ $\infty$ $25$ $21$

Arjun asked in Calculus Sep 23, 2019

by Arjun

508 views

0 votes

1 answer

20

ISI2014-DCG-20
If $A(t)$ is the area of the region bounded by the curve $y=e^{-\mid x \mid}$ and the portion of the $x$-axis between $-t$ and $t$, then $\underset{t \to \infty}{\lim} A(t)$ equals $0$ $1$ $2$ $4$

Arjun asked in Geometry Sep 23, 2019

by Arjun

328 views

1 vote

1 answer

21

ISI2014-DCG-21
Suppose that the function $h(x)$ is defined as $h(x)=g(f(x))$ where $g(x)$ is monotone increasing, $f(x)$ is concave, and $g’’(x)$ and $f’’(x)$ exist for all $x$. Then $h(x)$ is always concave always convex not necessarily concave None of these

Arjun asked in Calculus Sep 23, 2019

by Arjun

461 views

2 votes

2 answers

22

ISI2014-DCG-22
The conditions on $a$, $b$ and $c$ under which the roots of the quadratic equation $ax^2+bx+c=0 \: ,a \neq 0, \: b \neq 0 $ and $c \neq 0$, are unequal magnitude but of the opposite signs, are the following: $a$ and $c$ have the same sign while $b$ has the ... $c$ has the opposite sign. $a$ and $c$ have the same sign. $a$, $b$ and $c$ have the same sign.

Arjun asked in Quantitative Aptitude Sep 23, 2019

by Arjun

503 views

3 votes

2 answers

23

ISI2014-DCG-23
The sum of the series $\:3+11+\dots +(8n-5)\:$ is $4n^2-n$ $8n^2+3n$ $4n^2+4n-5$ $4n^2+2$

Arjun asked in Quantitative Aptitude Sep 23, 2019

by Arjun

556 views

0 votes

1 answer

24

ISI2014-DCG-24
Let $f(x) = \dfrac{2x}{x-1}, \: x \neq 1$. State which of the following statements is true. For all real $y$, there exists $x$ such that $f(x)=y$ For all real $y \neq 1$, there exists $x$ such that $f(x)=y$ For all real $y \neq 2$, there exists $x$ such that $f(x)=y$ None of the above is true

Arjun asked in Calculus Sep 23, 2019

by Arjun

415 views

2 votes

1 answer

25

ISI2014-DCG-25
The determinant $\begin{vmatrix} b+c & c+a & a+b \\ q+r & r+p & p+q \\ y+z & z+x & x+y \end{vmatrix}$ equals $\begin{vmatrix} a & b & c \\ p & q & r \\ x & y & z \end{vmatrix}$ ... $3\begin{vmatrix} a & b & c \\ p & q & r \\ x & y & z \end{vmatrix}$ None of these

Arjun asked in Linear Algebra Sep 23, 2019

by Arjun

463 views

2 votes

1 answer

26

ISI2014-DCG-26
Let $x_1 > x_2>0$. Then which of the following is true? $\log \big(\frac{x_1+x_2}{2}\big) > \frac{\log x_1+ \log x_2}{2}$ $\log \big(\frac{x_1+x_2}{2}\big) < \frac{\log x_1+ \log x_2}{2}$ There exist $x_1$ and $x_2$ such that $x_1 > x_2 >0$ and $\log \big(\frac{x_1+x_2}{2}\big) = \frac{\log x_1+ \log x_2}{2}$ None of these

Arjun asked in Quantitative Aptitude Sep 23, 2019

by Arjun

423 views

0 votes

0 answers

27

ISI2014-DCG-27
Let $y^2-4ax+4a=0$ and $x^2+y^2-2(1+a)x+1+2a-3a^2=0$ be two curves. State which one of the following statements is true. These two curves intersect at two points These two curves are tangent to each other These two curves intersect orthogonally at one point These two curves do not intersect

Arjun asked in Geometry Sep 23, 2019

by Arjun

324 views

2 votes

2 answers

28

ISI2014-DCG-28
The area enclosed by the curve $\mid\: x \mid + \mid y \mid =1$ is $1$ $2$ $\sqrt{2}$ $4$

Arjun asked in Calculus Sep 23, 2019

by Arjun

502 views

0 votes

2 answers

29

ISI2014-DCG-29
If $f(x) = \sin \bigg( \dfrac{1}{x^2+1} \bigg),$ then $f(x)$ is continuous at $x=0$, but not differentiable at $x=0$ $f(x)$ is differentiable at $x=0$, and $f’(0) \neq 0$ $f(x)$ is differentiable at $x=0$, and $f’(0) = 0$ None of the above

Arjun asked in Calculus Sep 23, 2019

by Arjun

715 views

2 votes

1 answer

30

ISI2014-DCG-30
Consider the equation $P(x) =x^3+px^2+qx+r=0$ where $p,q$ and $r$ are all real and positive. State which of the following statements is always correct. All roots of $P(x) = 0$ are real The equation $P(x)=0$ has at least one real root The equation $P(x)=0$ has no negative real root The equation $P(x)=0$ must have one positive and one negative real root

Arjun asked in Quantitative Aptitude Sep 23, 2019

by Arjun

411 views

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