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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
590
views
isi2014-dcg
combinatory
binomial-theorem
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
520
views
isi2014-dcg
calculus
limits
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
957
views
isi2014-dcg
calculus
limits
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
650
views
isi2014-dcg
calculus
limits
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
666
views
isi2014-dcg
set-theory
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
390
views
isi2014-dcg
calculus
functions
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
376
views
isi2014-dcg
calculus
functions
range
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
529
views
isi2014-dcg
linear-algebra
matrix
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
387
views
isi2014-dcg
linear-algebra
system-of-equations
5
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
717
views
isi2014-dcg
quantitative-aptitude
number-system
factors
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
298
views
isi2014-dcg
quadratic-equations
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
476
views
isi2014-dcg
calculus
definite-integral
integration
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
360
views
isi2014-dcg
calculus
differentiation
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
187
views
isi2014-dcg
circle
ellipse
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
338
views
isi2014-dcg
set-theory
algebra
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
392
views
isi2014-dcg
quantitative-aptitude
summation
2
votes
2
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
344
views
isi2014-dcg
calculus
limits
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
460
views
isi2014-dcg
combinatory
binomial-theorem
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
312
views
isi2014-dcg
calculus
maxima-minima
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
217
views
isi2014-dcg
calculus
definite-integral
area
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
294
views
isi2014-dcg
calculus
functions
maxima-minima
convex-concave
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
317
views
isi2014-dcg
quantitative-aptitude
quadratic-equations
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
362
views
isi2014-dcg
quantitative-aptitude
arithmetic-series
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
224
views
isi2014-dcg
calculus
functions
1
vote
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
319
views
isi2014-dcg
linear-algebra
determinant
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
277
views
isi2014-dcg
quantitative-aptitude
logarithms
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
203
views
isi2014-dcg
curves
1
vote
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
273
views
isi2014-dcg
calculus
area-under-the-curve
0
votes
1
answer
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
438
views
isi2014-dcg
calculus
continuity
differentiation
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
268
views
isi2014-dcg
quantitative-aptitude
quadratic-equations
roots
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Recent questions tagged isi2014-dcg
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Please upload 4th Mock Test, due date was 4th Dec.
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