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+1
vote
2
answers
1
ISI2014DCG1
Let $(1+x)^n = C_0+C_1x+C_2x^2+ \dots + C_nx^n$, $n$ being a positive integer. The value of $\left( 1+\dfrac{C_0}{C_1} \right) \left( 1+\dfrac{C_1}{C_2} \right) \cdots \left( 1+\dfrac{C_{n1}}{C_n} \right)$ 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!} $
asked
Sep 23
in
Combinatory

129
views
isi2014dcg
permutationandcombination
binomialtheorem
+2
votes
1
answer
2
ISI2014DCG2
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 $0$
asked
Sep 23
in
Calculus

92
views
isi2014dcg
calculus
limits
+3
votes
4
answers
3
ISI2014DCG3
$\underset{x \to \infty}{\lim} \bigg( \frac{3x1}{3x+1} \bigg) ^{4x}$ equals $1$ $0$ $e^{8/3}$ $e^{4/9}$
asked
Sep 23
in
Calculus

127
views
isi2014dcg
calculus
limits
+3
votes
2
answers
4
ISI2014DCG4
$\underset{n \to \infty}{\lim} \dfrac{1}{n} \bigg( \dfrac{n}{n+1} + \dfrac{n}{n+2} + \cdots + \dfrac{n}{2n} \bigg)$ is equal to $\infty$ $0$ $\log_e 2$ $1$
asked
Sep 23
in
Calculus

94
views
isi2014dcg
calculus
limits
summation
series
+1
vote
1
answer
5
ISI2014DCG5
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 nonempty none of the above
asked
Sep 23
in
Calculus

79
views
isi2014dcg
calculus
functions
sets
+2
votes
1
answer
6
ISI2014DCG6
If $f(x)$ is a real valued function such that $2f(x)+3f(x)=154x$, for every $x \in \mathbb{R}$, then $f(2)$ is $15$ $22$ $11$ $0$
asked
Sep 23
in
Calculus

52
views
isi2014dcg
calculus
functions
+2
votes
3
answers
7
ISI2014DCG7
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
asked
Sep 23
in
Calculus

40
views
isi2014dcg
calculus
functions
range
+1
vote
3
answers
8
ISI2014DCG8
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 $\begin{bmatrix}3 & 4 & 5 \end{bmatrix} M = \begin{bmatrix}0 & 1 & 0 \end{bmatrix}$ ... $\begin{bmatrix} 1 & 2 & 0 \end{bmatrix}$ $\begin{bmatrix} 9 & 10 & 8 \end{bmatrix}$
asked
Sep 23
in
Linear Algebra

74
views
isi2014dcg
linearalgebra
matrix
+1
vote
1
answer
9
ISI2014DCG9
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 $\eta=1, 2$ $\eta=1, 2$ $\eta=3, 3$ $\eta=1, 2$
asked
Sep 23
in
Linear Algebra

39
views
isi2014dcg
linearalgebra
systemofequations
+3
votes
3
answers
10
ISI2014DCG10
The number of divisors of $6000$, where $1$ and $6000$ are also considered as divisors of $6000$ is $40$ $50$ $60$ $30$
asked
Sep 23
in
Numerical Ability

65
views
isi2014dcg
numericalability
numbersystem
divisors
+1
vote
1
answer
11
ISI2014DCG11
Let $x_1$ and $x_2$ be the roots of the quadratic equation $x^23x+a=0$, and $x_3$ and $x_4$ be the roots of the quadratic equation $x^212x+b=0$. 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 $64$ $5184$ $64$ $5184$
asked
Sep 23
in
Numerical Ability

26
views
isi2014dcg
quadraticequations
roots
geometricprogression
0
votes
1
answer
12
ISI2014DCG12
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
asked
Sep 23
in
Calculus

63
views
isi2014dcg
calculus
definiteintegrals
integration
+1
vote
1
answer
13
ISI2014DCG13
Let the function $f(x)$ be defined as $f(x)=\mid x1 \mid + \mid x2 \:\mid$. 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
asked
Sep 23
in
Calculus

38
views
isi2014dcg
calculus
function
limitcontinuity
differentiable
+1
vote
1
answer
14
ISI2014DCG14
$x^43x^2+2x^2y^23y^2+y^4+2=0$ 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
asked
Sep 23
in
Others

14
views
isi2014dcg
circle
ellips
+1
vote
1
answer
15
ISI2014DCG15
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 $A_3$ $A_4$ $A_5$ $A_6$
asked
Sep 23
in
Set Theory & Algebra

31
views
isi2014dcg
settheory
algebra
+1
vote
2
answers
16
ISI2014DCG16
The sum of the series $\dfrac{1}{1.2} + \dfrac{1}{2.3}+ \cdots + \dfrac{1}{n(n+1)} + \cdots $ is $1$ $1/2$ $0$ nonexistent
asked
Sep 23
in
Numerical Ability

49
views
isi2014dcg
numericalability
summation
series
+2
votes
1
answer
17
ISI2014DCG17
$\underset{x \to 2}{\lim} \dfrac{1}{1+e^{\frac{1}{x2}}}$ is $0$ $1/2$ $1$ nonexistent
asked
Sep 23
in
Calculus

18
views
isi2014dcg
calculus
limits
+2
votes
3
answers
18
ISI2014DCG18
$^nC_0+2^nC_1+3^nC_2+\cdots+(n+1)^nC_n$ equals $2^n+n2^{n1}$ $2^nn2^{n1}$ $2^n$ none of these
asked
Sep 23
in
Combinatory

46
views
isi2014dcg
permutationandcombination
binomialtheorem
+2
votes
1
answer
19
ISI2014DCG19
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 $36$ $\infty$ $25$ $21$
asked
Sep 23
in
Calculus

27
views
isi2014dcg
calculus
maximaminima
maximumvalues
0
votes
1
answer
20
ISI2014DCG20
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 $0$ $1$ $2$ $4$
asked
Sep 23
in
Others

22
views
isi2014dcg
calculus
integration
definiteintegration
area
+1
vote
0
answers
21
ISI2014DCG21
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
asked
Sep 23
in
Calculus

16
views
isi2014dcg
calculus
functions
maximaminima
convexconcave
+1
vote
2
answers
22
ISI2014DCG22
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.
asked
Sep 23
in
Numerical Ability

27
views
isi2014dcg
numericalability
quadraticequations
+1
vote
1
answer
23
ISI2014DCG23
The sum of the series $\:3+11+\dots +(8n5)\:$ is $4n^2n$ $8n^2+3n$ $4n^2+4n5$ $4n^2+2$
asked
Sep 23
in
Numerical Ability

33
views
isi2014dcg
numericalability
summation
series
0
votes
1
answer
24
ISI2014DCG24
Let $f(x) = \dfrac{2x}{x1}, \: x \neq 1$. 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
asked
Sep 23
in
Calculus

12
views
isi2014dcg
calculus
functions
+1
vote
1
answer
25
ISI2014DCG25
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
asked
Sep 23
in
Linear Algebra

38
views
isi2014dcg
linearalgebra
determinant
+1
vote
1
answer
26
ISI2014DCG26
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
asked
Sep 23
in
Numerical Ability

34
views
isi2014dcg
numericalability
logarithms
0
votes
0
answers
27
ISI2014DCG27
Let $y^24ax+4a=0$ and $x^2+y^22(1+a)x+1+2a3a^2=0$ 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
asked
Sep 23
in
Others

20
views
isi2014dcg
curves
+1
vote
1
answer
28
ISI2014DCG28
The area enclosed by the curve $\mid\: x \mid + \mid y \mid =1$ is $1$ $2$ $\sqrt{2}$ $4$
asked
Sep 23
in
Calculus

12
views
isi2014dcg
calculus
areaunderthecurve
0
votes
1
answer
29
ISI2014DCG29
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
asked
Sep 23
in
Calculus

28
views
isi2014dcg
calculus
limits
continuitydifferentiability
+1
vote
1
answer
30
ISI2014DCG30
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
asked
Sep 23
in
Numerical Ability

21
views
isi2014dcg
numericalability
quadraticequations
roots
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