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0
votes
1
answer
1
ISI2015MMA19
The limit $\:\:\:\underset{n \to \infty}{\lim} \Sigma_{k=1}^n \begin{vmatrix} e^{\frac{2 \pi i k }{n}} – e^{\frac{2 \pi i (k1) }{n}} \end{vmatrix}\:\:\:$ is $2$ $2e$ $2 \pi$ $2i$
asked
Sep 23, 2019
in
Calculus

32
views
isi2015mma
calculus
limits
nongate
+1
vote
1
answer
2
ISI2015MMA20
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

20
views
isi2015mma
calculus
limits
nongate
+1
vote
1
answer
3
ISI2015MMA21
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

25
views
isi2015mma
complexnumber
nongate
0
votes
1
answer
4
ISI2015MMA22
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

18
views
isi2015mma
calculus
limits
nongate
0
votes
0
answers
5
ISI2015MMA23
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

13
views
isi2015mma
sets
functions
nongate
0
votes
1
answer
6
ISI2015MMA24
The series $\sum_{k=2}^{\infty} \frac{1}{k(k1)}$ converges to $1$ $1$ $0$ does not converge
asked
Sep 23, 2019
in
Numerical Ability

21
views
isi2015mma
numbersystem
convergencedivergence
summation
nongate
+1
vote
1
answer
7
ISI2015MMA25
The limit $\displaystyle{}\underset{x \to \infty}{\lim} \left( \frac{3x1}{3x+1} \right) ^{4x}$ equals $1$ $0$ $e^{8/3}$ $e^{4/9}$
asked
Sep 23, 2019
in
Calculus

35
views
isi2015mma
calculus
limits
nongate
0
votes
0
answers
8
ISI2015MMA26
$\displaystyle{}\underset{n \to \infty}{\lim} \frac{1}{n} \bigg( \frac{n}{n+1} + \frac{n}{n+2} + \cdots + \frac{n}{2n} \bigg)$ is equal to $\infty$ $0$ $\log_e 2$ $1$
asked
Sep 23, 2019
in
Calculus

19
views
isi2015mma
calculus
limits
nongate
0
votes
0
answers
9
ISI2015MMA27
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

20
views
isi2015mma
numericalability
trigonometry
nongate
0
votes
1
answer
10
ISI2015MMA28
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

19
views
isi2015mma
numericalability
geometry
median
nongate
0
votes
0
answers
11
ISI2015MMA29
The set $\{x \: : \begin{vmatrix} x+\frac{1}{x} \end{vmatrix} \gt6 \}$ equals the set $(0,32\sqrt{2}) \cup (3+2\sqrt{2}, \infty)$ $( \infty, 32\sqrt{2}) \cup (3+2 \sqrt{2}, \infty)$ $( \infty, 32\sqrt{2}) \cup (3+2\sqrt{2}, \infty)$ $( \infty, 32\sqrt{2}) \cup (3+2 \sqrt{2},32\sqrt{2}) \cup (3+2 \sqrt{2}, \infty )$
asked
Sep 23, 2019
in
Numerical Ability

16
views
isi2015mma
numbersystem
nongate
0
votes
0
answers
12
ISI2015MMA30
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 $K(x+y)$ $Kxy$ $K+xy$ none of the above
asked
Sep 23, 2019
in
Calculus

16
views
isi2015mma
calculus
functions
nongate
0
votes
0
answers
13
ISI2015MMA31
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, 2019
in
Set Theory & Algebra

19
views
isi2015mma
sets
nongate
0
votes
1
answer
14
ISI2015MMA32
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

12
views
isi2015mma
triangles
nongate
+1
vote
1
answer
15
ISI2015MMA33
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, 2019
in
Calculus

17
views
isi2015mma
calculus
functions
nongate
0
votes
1
answer
16
ISI2015MMA34
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

26
views
isi2015mma
calculus
functions
range
trigonometry
nongate
0
votes
0
answers
17
ISI2015MMA35
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

23
views
isi2015mma
trigonometry
nongate
+1
vote
1
answer
18
ISI2015MMA36
For nonnegative integers $m$, $n$ define a function as follows $f(m,n) = \begin{cases} n+1 & \text{ if } m=0 \\ f(m1, 1) & \text{ if } m \neq 0, n=0 \\ f(m1, f(m,n1)) & \text{ if } m \neq 0, n \neq 0 \end{cases}$ Then the value of $f(1,1)$ is $4$ $3$ $2$ $1$
asked
Sep 23, 2019
in
Calculus

18
views
isi2015mma
calculus
functions
nongate
+1
vote
1
answer
19
ISI2015MMA37
Let $a$ be a nonzero real number. Define $f(x) = \begin{vmatrix} x & a & a & a \\ a & x & a & a \\ a & a & x & a \\ a & a & a & x \end{vmatrix}$ for $x \in \mathbb{R}$. Then, the number of distinct real roots of $f(x) =0$ is $1$ $2$ $3$ $4$
asked
Sep 23, 2019
in
Linear Algebra

53
views
isi2015mma
linearalgebra
determinant
functions
0
votes
0
answers
20
ISI2015MMA38
A real $2 \times 2$ matrix $M$ such that $M^2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \varepsilon \end{pmatrix}$ exists for all $\varepsilon > 0$ does not exist for any $\varepsilon > 0$ exists for some $\varepsilon > 0$ none of the above is true
asked
Sep 23, 2019
in
Linear Algebra

25
views
isi2015mma
linearalgebra
matrices
+2
votes
2
answers
21
ISI2015MMA39
The eigenvalues of the matrix $X = \begin{pmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{pmatrix}$ are $1,1,4$ $1,4,4$ $0,1,4$ $0,4,4$
asked
Sep 23, 2019
in
Linear Algebra

80
views
isi2015mma
linearalgebra
matrices
eigenvalue
0
votes
0
answers
22
ISI2015MMA40
Let $x_1, x_2, x_3, x_4, y_1, y_2, y_3$ and $y_4$ be fixed real numbers, not all of them equal to zero. Define a $4 \times 4$ matrix $\textbf{A}$ ... $(\textbf{A})$ equals $1$ or $2$ $0$ $4$ $2$ or $3$
asked
Sep 23, 2019
in
Linear Algebra

37
views
isi2015mma
linearalgebra
matrices
rankofmatrix
0
votes
1
answer
23
ISI2015MMA41
Let $k$ and $n$ be integers greater than $1$. Then $(kn)!$ is not necessarily divisible by $(n!)^k$ $(k!)^n$ $n! \cdot k! \cdot$ $2^{kn}$
asked
Sep 23, 2019
in
Numerical Ability

21
views
isi2015mma
numericalability
numbersystem
remaindertheorem
0
votes
0
answers
24
ISI2015MMA42
Let $\lambda_1, \lambda_2, \lambda_3$ denote the eigenvalues of the matrix $A \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos t & \sin t \\ 0 &  \sin t & \cos t \end{pmatrix}.$ If $\lambda_1+\lambda_2+\lambda_3 = \sqrt{2}+1$ ... $\{  \frac{\pi}{4}, \frac{\pi}{4} \}$ $\{  \frac{\pi}{3}, \frac{\pi}{3} \}$
asked
Sep 23, 2019
in
Linear Algebra

42
views
isi2015mma
linearalgebra
matrices
eigenvalue
0
votes
1
answer
25
ISI2015MMA43
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, 2019
in
Linear Algebra

27
views
isi2015mma
linearalgebra
systemofequations
0
votes
1
answer
26
ISI2015MMA44
Let $P_1$, $P_2$ and $P_3$ denote, respectively, the planes defined by $\begin{array} {} a_1x +b_1y+c_1z=\alpha _1 \\ a_2x +b_2y+c_2z=\alpha _2 \\ a_3x +b_3y+c_3z=\alpha _3 \end{array}$ It is given that $P_1$, $P_2$ and $P_3$ ... then the planes do not have any common point of intersection intersect at a unique point intersect along a straight line intersect along a plane
asked
Sep 23, 2019
in
Linear Algebra

30
views
isi2015mma
linearalgebra
systemofequations
0
votes
0
answers
27
ISI2015MMA45
Angles between any pair of $4$ main diagonals of a cube are $\cos^{1} 1/\sqrt{3}, \pi – \cos ^{1} 1/\sqrt{3}$ $\cos^{1} 1/3, \pi – \cos ^{1} 1/3$ $\pi/2$ none of the above
asked
Sep 23, 2019
in
Geometry

16
views
isi2015mma
cubes
nongate
0
votes
0
answers
28
ISI2015MMA46
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

13
views
isi2015mma
lines
nongate
0
votes
0
answers
29
ISI2015MMA47
Consider the family $\mathcal{F}$ of curves in the plane given by $x=cy^2$, where $c$ is a real parameter. Let $\mathcal{G}$ be the family of curves having the following property: every member of $\mathcal{G}$ intersect each member of $\mathcal{F}$ orthogonally. Then $\mathcal{G}$ is given by $xy=k$ $x^2+y^2=k^2$ $y^2+2x^2=k^2$ $x^2y^2+2yk=k^2$
asked
Sep 23, 2019
in
Geometry

14
views
isi2015mma
curves
0
votes
0
answers
30
ISI2015MMA48
Suppose the circle with equation $x^2+y^2+2fx+2gy+c=0$ cuts the parabola $y^2=4ax, \: (a>0)$ at four distinct points. If $d$ denotes the sum of the ordinates of these four points, then the set of possible values of $d$ is $\{0\}$ $(4a,4a)$ $(a,a)$ $( \infty, \infty)$
asked
Sep 23, 2019
in
Geometry

14
views
isi2015mma
circle
parabola
nongate
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