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+2
votes
0
answers
1
ISI2014DCG31
For real $\alpha$, the value of $\int_{\alpha}^{\alpha+1} [x]dx$, where $[x]$ denotes the largest integer less than or equal to $x$, is $\alpha$ $[\alpha]$ $1$ $\dfrac{[\alpha] + [\alpha +1]}{2}$
asked
Sep 23, 2019
in
Calculus

27
views
isi2014dcg
calculus
integration
definiteintegrals
+2
votes
0
answers
2
ISI2014DCG32
Consider $30$ multiplechoice questions, each with four options of which exactly one is correct. Then the number of ways one can get only the alternate questions correctly answered is $3^{15}$ $2^{31}$ $2 \times \begin{pmatrix} 30 \\ 15 \end{pmatrix}$ $2 \times 3^{15}$
asked
Sep 23, 2019
in
Combinatory

78
views
isi2014dcg
permutationandcombination
0
votes
0
answers
3
ISI2014DCG33
Let $f(x)$ be a continuous function from $[0,1]$ to $[0,1]$ satisfying the following properties. $f(0)=0$, $f(1)=1$, and $f(x_1)<f(x_2)$ for $x_1 < x_2$ with $0 < x_1, \: x_2<1$. Then the number of such functions is $0$ $1$ $2$ $\infty$
asked
Sep 23, 2019
in
Calculus

27
views
isi2014dcg
calculus
functions
limits
+1
vote
1
answer
4
ISI2014DCG34
The following sum of $n+1$ terms $2 + 3 \times \begin{pmatrix} n \\ 1 \end{pmatrix} + 5 \times \begin{pmatrix} n \\ 2 \end{pmatrix} + 9 \times \begin{pmatrix} n \\ 3 \end{pmatrix} + 17 \times \begin{pmatrix} n \\ 4 \end{pmatrix} + \cdots$ up to $n+1$ terms is equal to $3^{n+1}+2^{n+1}$ $3^n \times 2^n$ $3^n + 2^n$ $2 \times 3^n$
asked
Sep 23, 2019
in
Combinatory

49
views
isi2014dcg
permutationandcombination
binomialtheorem
summation
+1
vote
1
answer
5
ISI2014DCG35
Let $A$ and $B$ be disjoint sets containing $m$ and $n$ elements respectively, and let $C=A \cup B$. Then the number of subsets $S$ (of $C$) which contains $p$ elements and also has the property that $S \cap A$ contains $q$ ... $\begin{pmatrix} m \\ pq \end{pmatrix} \times \begin{pmatrix} n \\ q \end{pmatrix}$
asked
Sep 23, 2019
in
Set Theory & Algebra

33
views
isi2014dcg
sets
disjointsets
+1
vote
1
answer
6
ISI2014DCG36
Consider any integer $I=m^2+n^2$, where $m$ and $n$ are odd integers. Then $I$ is never divisible by $2$ $I$ is never divisible by $4$ $I$ is never divisible by $6$ None of the above
asked
Sep 23, 2019
in
Numerical Ability

40
views
isi2014dcg
numericalability
numbersystem
remaindertheorem
+1
vote
1
answer
7
ISI2014DCG37
Let $f: \bigg( – \dfrac{\pi}{2}, \dfrac{\pi}{2} \bigg) \to \mathbb{R}$ be a continuous function, $f(x) \to +\infty$ as $x \to \dfrac{\pi^}{2}$ and $f(x) \to – \infty$ as $x \to \dfrac{\pi^+}{2}$. Which one of the following functions satisfies the above properties of $f(x)$? $\cos x$ $\tan x$ $\tan^{1} x$ $\sin x$
asked
Sep 23, 2019
in
Calculus

27
views
isi2014dcg
calculus
functions
limits
continuity
+1
vote
1
answer
8
ISI2014DCG38
Suppose that $A$ is a $3 \times 3$ real matrix such that for each $u=(u_1, u_2, u_3)’ \in \mathbb{R}^3, \: u’Au=0$ where $u’$ stands for the transpose of $u$. Then which one of the following is true? $A’=A$ $A’=A$ $AA’=I$ None of these
asked
Sep 23, 2019
in
Linear Algebra

59
views
isi2014dcg
linearalgebra
matrices
+1
vote
1
answer
9
ISI2014DCG39
The function $f(x) = x^{1/x}, \: x \neq 0$ has a minimum at $x=e$; a maximum at $x=e$; neither a maximum nor a minimum at $x=e$; None of the above
asked
Sep 23, 2019
in
Calculus

54
views
isi2014dcg
maximaminima
calculus
0
votes
1
answer
10
ISI2014DCG40
Let the following two equations represent two curves $A$ and $B$. $A: 16x^2+9y^2=144\:\: \text{and}\:\: B:x^2+y^210x=21$ Further, let $L$ and $M$ be the tangents to these curves $A$ and $B$, respectively, at the point $(3,0)$. Then the angle between these two tangents, $L$ and $M$, is $0^{\circ}$ $30^{\circ}$ $45^{\circ}$ $90^{\circ}$
asked
Sep 23, 2019
in
Others

15
views
isi2014dcg
curves
+1
vote
1
answer
11
ISI2014DCG41
The number of permutations of the letters $a, b, c$ and $d$ such that $b$ does not follow $a,c$ does not follow $b$, and $c$ does not follow $d$, is $11$ $12$ $13$ $14$
asked
Sep 23, 2019
in
Combinatory

46
views
isi2014dcg
permutationandcombination
0
votes
0
answers
12
ISI2014DCG42
Let $f(x)=\sin x^2, \: x \in \mathbb{R}$. Then $f$ has no local minima $f$ has no local maxima $f$ has local minima at $x=0$ and $x=\pm\sqrt{(k+\frac{1}{2} ) \pi}$ for odd integers $k$ and local maxima at $x=\pm\sqrt{(k+\frac{1}{2} ) \pi}$ for even integers $k$ None of the above
asked
Sep 23, 2019
in
Calculus

22
views
isi2014dcg
calculus
maximaminima
0
votes
0
answers
13
ISI2014DCG43
Let $f(x) = \begin{cases}\mid \:x \mid +1, & \text{ if } x<0 \\ 0, & \text{ if } x=0 \\ \mid \:x \mid 1, & \text{ if } x>0. \end{cases}$ Then $\underset{x \to a}{\lim} f(x)$ exists if $a=0$ for all $a \in R$ for all $a \neq 0$ only if $a=1$
asked
Sep 23, 2019
in
Calculus

14
views
isi2014dcg
calculus
functions
limits
0
votes
1
answer
14
ISI2014DCG44
The function $f(x)=\sin x(1+ \cos x)$ which is defined for all real values of $x$ has a maximum at $x= \pi /3$ has a maximum at $x= \pi$ has a minimum at $x= \pi /3$ has neither a maximum nor a minimum at $x=\pi/3$
asked
Sep 23, 2019
in
Calculus

16
views
isi2014dcg
calculus
maximaminima
0
votes
0
answers
15
ISI2014DCG45
Which of the following is true? $\log(1+x) < x \frac{x^2}{2} + \frac{x^3}{3} \text{ for all } x>0$ $\log(1+x) > x \frac{x^2}{2} + \frac{x^3}{3} \text{ for all } x>0$ $\log(1+x) > x \frac{x^2}{2} + \frac{x^3}{3} \text{ for some } x>0$ $\log(1+x) < x \frac{x^2}{2} + \frac{x^3}{3} \text{ for some } x>0$
asked
Sep 23, 2019
in
Calculus

23
views
isi2014dcg
calculus
functions
logarithms
0
votes
1
answer
16
ISI2014DCG46
The maximum value of the real valued function $f(x)=\cos x + \sin x$ is $2$ $1$ $0$ $\sqrt{2}$
asked
Sep 23, 2019
in
Calculus

26
views
isi2014dcg
calculus
maximaminima
0
votes
0
answers
17
ISI2014DCG47
The value of the definite integral $\int_0^{\pi} \mid \frac{1}{2} + \cos x \mid dx$ is $\frac{\pi}{6} + \sqrt{3}$ $\frac{\pi}{6}  \sqrt{3}$ $0$ $\frac{1}{2}$
asked
Sep 23, 2019
in
Calculus

19
views
isi2014dcg
calculus
integration
definiteintegrals
0
votes
0
answers
18
ISI2014DCG48
If $x$ is real, the set of real values of $a$ for which the function $y=x^2ax+12a^2$ is always greater than zero is $ \frac{2}{3} < a \leq \frac{2}{3}$ $ \frac{2}{3} \leq a < \frac{2}{3}$ $ \frac{2}{3} < a < \frac{2}{3}$ None of these
asked
Sep 23, 2019
in
Calculus

12
views
isi2014dcg
calculus
functions
quadraticequations
0
votes
1
answer
19
ISI2014DCG49
Let $f(x) = \dfrac{x}{(x1)(2x+3)}$, where $x>1$. Then the $4^{th}$ derivative of $f, \: f^{(4)} (x)$ is equal to $ \frac{24}{5} \bigg[ \frac{1}{(x1)^5}  \frac{48}{(2x+3)^5} \bigg]$ ... $\frac{64}{5} \bigg[ \frac{1}{(x1)^5} + \frac{48}{(2x+3)^5} \bigg]$
asked
Sep 23, 2019
in
Others

41
views
isi2014dcg
calculus
differentiation
functions
0
votes
0
answers
20
ISI2014DCG50
$\underset{x \to 0}{\lim} \dfrac{x \tan x}{1 \cos tx}$ is equal to $0$ $1$ $\infty$ $2$
asked
Sep 23, 2019
in
Calculus

24
views
isi2014dcg
calculus
limits
0
votes
1
answer
21
ISI2014DCG51
The function $f(x)$ defined as $f(x)=x^36x^2+24x$, where $x$ is real, is strictly increasing strictly decreasing increasing in $( \infty, 0)$ and decreasing in $(0, \infty)$ decreasing in $( \infty, 0)$ and increasing in $(0, \infty)$
asked
Sep 23, 2019
in
Calculus

28
views
isi2014dcg
calculus
maximaminima
0
votes
0
answers
22
ISI2014DCG52
The area under the curve $x^2+3x4$ in the positive quadrant and bounded by the line $x=5$ is equal to $59 \frac{1}{6}$ $61 \frac{1}{3}$ $40 \frac{2}{3}$ $72$
asked
Sep 23, 2019
in
Geometry

12
views
isi2014dcg
curves
area
0
votes
0
answers
23
ISI2014DCG53
The value of the integral $\displaystyle{}\int_{1}^1 \dfrac{x^2}{1+x^2} \sin x \sin 3x \sin 5x dx$ is $0$ $\frac{1}{2}$ $ – \frac{1}{2}$ $1$
asked
Sep 23, 2019
in
Calculus

30
views
isi2014dcg
calculus
integration
definiteintegrals
0
votes
0
answers
24
ISI2014DCG54
The number of real roots of the equation $1+\cos ^2x+\cos ^3 x – \cos^4x=5$ is equal to $0$ $1$ $3$ $4$
asked
Sep 23, 2019
in
Numerical Ability

39
views
isi2014dcg
numericalability
trigonometry
roots
0
votes
1
answer
25
ISI2014DCG55
If $a,b,c$ are sides of a triangle $ABC$ such that $x^22(a+b+c)x+3 \lambda (ab+bc+ca)=0$ has real roots then $\lambda < \frac{4}{3}$ $\lambda > \frac{5}{3}$ $\lambda \in \big( \frac{4}{3}, \frac{5}{3}\big)$ $\lambda \in \big( \frac{1}{3}, \frac{5}{3}\big)$
asked
Sep 23, 2019
in
Numerical Ability

33
views
isi2014dcg
numericalability
geometry
quadraticequations
0
votes
1
answer
26
ISI2014DCG56
Two opposite vertices of a rectangle are $(1,3)$ and $(5,1)$ while the other two vertices lie on the straight line $y=2x+c$. Then the value of $c$ is $4$ $3$ $4$ $3$
asked
Sep 23, 2019
in
Numerical Ability

22
views
isi2014dcg
numericalability
geometry
rectangles
lines
+1
vote
1
answer
27
ISI2014DCG57
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

13
views
isi2014dcg
parabola
nongate
+1
vote
1
answer
28
ISI2014DCG58
Consider a circle with centre at origin and radius $2\sqrt{2}$. A square is inscribed in the circle whose sides are parallel to the $X$ an $Y$ axes. The coordinates of one of the vertices of this square are $(2, 2)$ $(2\sqrt{2},2)$ $(2, 2\sqrt{2})$ $(2\sqrt{2}, 2\sqrt{2})$
asked
Sep 23, 2019
in
Numerical Ability

30
views
isi2014dcg
numericalability
geometry
circle
squares
0
votes
0
answers
29
ISI2014DCG59
The equation $5x^2+9y^2+10x36y4=0$ 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

14
views
isi2014dcg
hyperbola
ellipses
nongate
0
votes
1
answer
30
ISI2014DCG60
The equation of any circle passing through the origin and with its centre on the $X$axis is given by $x^2+y^22ax=0$ where $a$ must be positive $x^2+y^22ax=0$ for any given $a \in \mathbb{R}$ $x^2+y^22by=0$ where $b$ must be positive $x^2+y^22by=0$ for any given $b \in \mathbb{R}$
asked
Sep 23, 2019
in
Numerical Ability

24
views
isi2014dcg
numericalability
geometry
circle
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