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Recent questions tagged isi2016-dcg
1
votes
1
answer
31
ISI2016-DCG-31
Let $A$ be an $n\times n$ matrix such that $\mid\: A^{2}\mid=1.\:\: \mid A\:\mid$ stands for determinant of matrix $A.$ Then $\mid\:(A)\mid=1$ $\mid\:(A)\mid=0\:\text{or}\:1$ $\mid\:(A)\mid=-1,0\:\text{or}\:1$ $\mid\:(A)\mid=-1\:\text{or}\:1$
Let $A$ be an $n\times n$ matrix such that $\mid\: A^{2}\mid=1.\:\: \mid A\:\mid$ stands for determinant of matrix $A.$ Then$\mid\:(A)\mid=1$$\mid\:(A)\mid=0\:\text{or}\...
gatecse
372
views
gatecse
asked
Sep 18, 2019
Linear Algebra
isi2016-dcg
linear-algebra
matrix
determinant
+
–
0
votes
0
answers
32
ISI2016-DCG-32
The set of vectors constituting an orthogonal basis in $\mathbb{R}^{3}$ is $\begin{Bmatrix} \begin{pmatrix} 1\\ -1 \\0 \end{pmatrix}&,\begin{pmatrix} 1\\ 1 \\0 \end{pmatrix}&,\begin{pmatrix} 0\\ 0 \\ 1 \end{pmatrix} \end{Bmatrix}$ ... None of these
The set of vectors constituting an orthogonal basis in $\mathbb{R}^{3}$ is$\begin{Bmatrix} \begin{pmatrix} 1\\ -1 \\0 \end{pmatrix}&,\begin{pmatrix} 1\\ 1 \\0 \end{pmatri...
gatecse
301
views
gatecse
asked
Sep 18, 2019
Linear Algebra
isi2016-dcg
linear-algebra
matrix
orthogonal-matrix
eigen-vectors
+
–
0
votes
0
answers
33
ISI2016-DCG-33
Suppose $A$ and $B$ are orthogonal $n\times n$ matrices. Which of the following is also an orthogonal matrix? Assume that $O$ is the null matrix of order $n\times n$ and $I$ is the identity matrix of order $n.$ $AB-BA$ $\begin{pmatrix} A & O \\ O & B \end{pmatrix}$ $\begin{pmatrix} A & I \\ I & B \end{pmatrix}$ $A^{2}-B^{2}$
Suppose $A$ and $B$ are orthogonal $n\times n$ matrices. Which of the following is also an orthogonal matrix? Assume that $O$ is the null matrix of order $n\times n$ and ...
gatecse
249
views
gatecse
asked
Sep 18, 2019
Linear Algebra
isi2016-dcg
linear-algebra
matrix
orthogonal-matrix
+
–
0
votes
0
answers
34
ISI2016-DCG-34
Let $A_{ij}$ denote the minors of an $n\times n$ matrix $A.$ What is the relationship between $\mid A_{ij}\mid$ and $\mid A_{ji}\mid$? They are always equal. $\mid A_{ij}\mid=-\mid A_{ji}\mid$ if $i\neq j.$ They are equal if $A$ is a symmetric matrix. If $\mid A_{ij}\mid=0$ then $\mid A_{ji}\mid=0.$
Let $A_{ij}$ denote the minors of an $n\times n$ matrix $A.$ What is the relationship between $\mid A_{ij}\mid$ and $\mid A_{ji}\mid$?They are always equal.$\mid A_{ij}\...
gatecse
282
views
gatecse
asked
Sep 18, 2019
Linear Algebra
isi2016-dcg
linear-algebra
matrix
minors
+
–
0
votes
1
answer
35
ISI2016-DCG-35
Let $A,B$ and $C$ be three non empty sets. Consider the two relations given below: $A-(B-C)=(A-B)\cup C$ $A-(B\cup C)=(A-B)-C$ Both (1) and (2) are correct. (1) is correct but (2) is not. (2) is correct but (1) is not. Both (1) and (2) are incorrect.
Let $A,B$ and $C$ be three non empty sets. Consider the two relations given below:$A-(B-C)=(A-B)\cup C$$A-(B\cup C)=(A-B)-C$Both (1) and (2) are correct.(1) is correct bu...
gatecse
353
views
gatecse
asked
Sep 18, 2019
Set Theory & Algebra
isi2016-dcg
set-theory
+
–
0
votes
1
answer
36
ISI2016-DCG-36
Suppose $X$ and $Y$ are finite sets, each with cardinality $n$.. The number of bijective functions from $X$ to $Y$ is $n^{n}$ $n\log_{2}n$ $n^{2}$ $n!$
Suppose $X$ and $Y$ are finite sets, each with cardinality $n$.. The number of bijective functions from $X$ to $Y$ is$n^{n}$$n\log_{2}n$$n^{2}$$n!$
gatecse
335
views
gatecse
asked
Sep 18, 2019
Set Theory & Algebra
isi2016-dcg
set-theory
functions
+
–
0
votes
0
answers
37
ISI2016-DCG-37
Suppose $f_{\alpha}\::\:[0,1]\rightarrow[0,1],\:-1<\alpha<\infty$ is given by $f_{\alpha}(x)=\dfrac{(\alpha+1)x}{\alpha x+1}.$ Then $f_{\alpha}$ is A bijective (one-one and onto) function. A surjective (onto) function. An injective (one-one) function. We can not conclude about the type.
Suppose $f_{\alpha}\::\:[0,1]\rightarrow[0,1],\:-1<\alpha<\infty$ is given by$$f_{\alpha}(x)=\dfrac{(\alpha+1)x}{\alpha x+1}.$$Then $f_{\alpha}$ isA bijective (one-one an...
gatecse
291
views
gatecse
asked
Sep 18, 2019
Set Theory & Algebra
isi2016-dcg
set-theory
functions
+
–
0
votes
0
answers
38
ISI2016-DCG-38
The length of the chord on the straight line $3x-4y+5=0$ intercepted by the circle passing through the points $(1,2),(3,-4)$ and $(5,6)$ is $12$ $14$ $16$ $18$
The length of the chord on the straight line $3x-4y+5=0$ intercepted by the circle passing through the points $(1,2),(3,-4)$ and $(5,6)$ is$12$$14$$16$$18$
gatecse
189
views
gatecse
asked
Sep 18, 2019
Geometry
isi2016-dcg
lines
non-gate
+
–
0
votes
1
answer
39
ISI2016-DCG-39
The medians $AD$ and $BE$ of the triangle with vertices $A(0,b),B(0,0)$ and $C(a,0)$ are mutually perpendicular if $b=\sqrt{2}a$ $a=\pm\sqrt{2}b$ $b=-\sqrt{2}a$ $b=a$
The medians $AD$ and $BE$ of the triangle with vertices $A(0,b),B(0,0)$ and $C(a,0)$ are mutually perpendicular if$b=\sqrt{2}a$$a=\pm\sqrt{2}b$$b=-\sqrt{2}a$$b=a$
gatecse
302
views
gatecse
asked
Sep 18, 2019
Geometry
isi2016-dcg
triangles
non-gate
+
–
0
votes
0
answers
40
ISI2016-DCG-40
The equations $x=a\cos\theta+b\sin\theta$ and $y=a\sin\theta+b\cos\theta,(0\leq\theta\leq2\pi$ and $a,b$ are arbitrary constants$)$ represent a circle a parabola an ellipse a hyperbola
The equations $x=a\cos\theta+b\sin\theta$ and $y=a\sin\theta+b\cos\theta,(0\leq\theta\leq2\pi$ and $a,b$ are arbitrary constants$)$ representa circlea parabolaan ellipsea...
gatecse
272
views
gatecse
asked
Sep 18, 2019
Geometry
isi2016-dcg
trigonometry
curves
non-gate
+
–
0
votes
0
answers
41
ISI2016-DCG-41
A straight line touches the circle $x^{2}+y^{2}=2a^{2}$ and also the parabola $y^{2}=8ax.$ Then the equation of the straight line is $y=\pm x$ $y=\pm(x+a)$ $y=\pm(x+2a)$ $y=\pm(x-21)$
A straight line touches the circle $x^{2}+y^{2}=2a^{2}$ and also the parabola $y^{2}=8ax.$ Then the equation of the straight line is$y=\pm x$$y=\pm(x+a)$$y=\pm(x+2a)$$y=...
gatecse
178
views
gatecse
asked
Sep 18, 2019
Geometry
isi2016-dcg
lines
parabola
non-gate
+
–
0
votes
0
answers
42
ISI2016-DCG-42
In an ellipse, the distance between its foci is $6$ and its minor axis is $8.$ Then its eccentricity is $\frac{4}{5}$ $\frac{1}{\sqrt{52}}$ $\frac{3}{5}$ $\frac{1}{2}$
In an ellipse, the distance between its foci is $6$ and its minor axis is $8.$ Then its eccentricity is$\frac{4}{5}$$\frac{1}{\sqrt{52}}$$\frac{3}{5}$$\frac{1}{2}$
gatecse
162
views
gatecse
asked
Sep 18, 2019
Geometry
isi2016-dcg
ellipse
non-gate
+
–
0
votes
1
answer
43
ISI2016-DCG-43
Four tangents are drawn to the ellipse $\dfrac{x^{2}}{9}+\dfrac{y^{2}}{5}=1$ at the ends of its latera recta. The area of the quadrilateral so formed is $27$ $\frac{13}{2}$ $\frac{15}{4}$ $45$
Four tangents are drawn to the ellipse $\dfrac{x^{2}}{9}+\dfrac{y^{2}}{5}=1$ at the ends of its latera recta. The area of the quadrilateral so formed is$27$$\frac{13}{2}$...
gatecse
331
views
gatecse
asked
Sep 18, 2019
Geometry
isi2016-dcg
ellipse
quadrilateral
area
non-gate
+
–
0
votes
1
answer
44
ISI2016-DCG-44
If the distance between the foci of a hyperbola is $16$ and its eccentricity is $\sqrt{2},$ then the equation of the hyperbola is $y^{2}-x^{2}=32$ $x^{2}-y^{2}=16$ $y^{2}-x^{2}=16$ $x^{2}-y^{2}=32$
If the distance between the foci of a hyperbola is $16$ and its eccentricity is $\sqrt{2},$ then the equation of the hyperbola is$y^{2}-x^{2}=32$$x^{2}-y^{2}=16$$y^{2}-x^...
gatecse
243
views
gatecse
asked
Sep 18, 2019
Geometry
isi2016-dcg
hyperbola
curves
non-gate
+
–
2
votes
2
answers
45
ISI2016-DCG-45
The value of $\underset{x \to 0}{\lim} \dfrac{\tan^{2}\:x-x\:\tan\:x}{\sin\:x}$ is $\frac{\sqrt{3}}{2}$ $\frac{1}{2}$ $0$ None of these
The value of $\underset{x \to 0}{\lim} \dfrac{\tan^{2}\:x-x\:\tan\:x}{\sin\:x}$ is$\frac{\sqrt{3}}{2}$$\frac{1}{2}$$0$None of these
gatecse
439
views
gatecse
asked
Sep 18, 2019
Calculus
isi2016-dcg
limits
+
–
1
votes
1
answer
46
ISI2016-DCG-46
Let $I=\int(\sin\:x-\cos\:x)(\sin\:x+\cos\:x)^{3}dx$ and $K$ be a constant of integration. Then the value of $I$ is $(\sin\:x+\cos\:x)^{4}+K$ $(\sin\:x+\cos\:x)^{2}+K$ $-\frac{1}{4}(\sin\:x+\cos\:x)^{4}+K$ None of these
Let $I=\int(\sin\:x-\cos\:x)(\sin\:x+\cos\:x)^{3}dx$ and $K$ be a constant of integration. Then the value of $I$ is$(\sin\:x+\cos\:x)^{4}+K$$(\sin\:x+\cos\:x)^{2}+K$$-\fr...
gatecse
389
views
gatecse
asked
Sep 18, 2019
Calculus
isi2016-dcg
calculus
integration
non-gate
+
–
0
votes
1
answer
47
ISI2016-DCG-47
The Taylor series expansion of $f(x)=\ln(1+x^{2})$ about $x=0$ is $\sum_{n=1}^{\infty}(-1)^{n}\frac{x^{n}}{n}$ $\sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^{2n}}{n}$ $\sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^{2n+1}}{n+1}$ $\sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^{n+1}}{n+1}$
The Taylor series expansion of $f(x)=\ln(1+x^{2})$ about $x=0$ is$\sum_{n=1}^{\infty}(-1)^{n}\frac{x^{n}}{n}$$\sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^{2n}}{n}$$\sum_{n=1}^{\...
gatecse
397
views
gatecse
asked
Sep 18, 2019
Calculus
isi2016-dcg
calculus
taylor-series
non-gate
+
–
0
votes
1
answer
48
ISI2016-DCG-48
The piecewise linear function for the following graph is $f(x)=\begin{cases} = x,x\leq-2 \\ =4,-2<x<3 \\ = x+1,x\geq 3\end{cases}$ $f(x)=\begin{cases} = x-2,x\leq-2 \\ =4,-2<x<3 \\ = x-1,x\geq 3\end{cases}$ $f(x)=\begin{cases} = 2x,x\leq-2 \\ =x,-2<x<3 \\ = x+1,x\geq 3\end{cases}$ $f(x)=\begin{cases} = 2-x,x\leq-2 \\ =4,-2<x<3 \\ = x+1,x\geq 3\end{cases}$
The piecewise linear function for the following graph is$f(x)=\begin{cases} = x,x\leq-2 \\ =4,-2<x<3 \\ = x+1,x\geq 3\end{cases}$$f(x)=\begin{cases} = x-2,x\leq-2 \\ =4,...
gatecse
417
views
gatecse
asked
Sep 18, 2019
Calculus
isi2016-dcg
calculus
functions
curves
non-gate
+
–
0
votes
1
answer
49
ISI2016-DCG-49
$\underset{x\rightarrow 1}{\lim}\dfrac{x^{\frac{1}{3}}-1}{x^{\frac{1}{4}}-1}$ equals $\frac{4}{3}$ $\frac{3}{4}$ $1$ None of these
$\underset{x\rightarrow 1}{\lim}\dfrac{x^{\frac{1}{3}}-1}{x^{\frac{1}{4}}-1}$ equals$\frac{4}{3}$$\frac{3}{4}$$1$None of these
gatecse
243
views
gatecse
asked
Sep 18, 2019
Calculus
isi2016-dcg
calculus
limits
+
–
1
votes
1
answer
50
ISI2016-DCG-50
The domain of the function $\ln(3x^{2}-4x+5)$ is set of positive real numbers set of real numbers set of negative real numbers set of real numbers larger than $5$
The domain of the function $\ln(3x^{2}-4x+5)$ isset of positive real numbersset of real numbersset of negative real numbersset of real numbers larger than $5$
gatecse
378
views
gatecse
asked
Sep 18, 2019
Set Theory & Algebra
isi2016-dcg
functions
+
–
0
votes
0
answers
51
ISI2016-DCG-51
Four squares of sides $x\: cm$ each are cut off from the four corners of a square metal sheet having side $10\: cm.$ The residual sheet is then folded into an open box which is then filled with a liquid costing Rs. $x^{2}$ per $cm^{3}.$ The value of $x$ for which the cost of filling the box completely with the liquid is maximized, is $100$ $50$ $30$ $10$
Four squares of sides $x\: cm$ each are cut off from the four corners of a square metal sheet having side $10\: cm.$ The residual sheet is then folded into an open box w...
gatecse
316
views
gatecse
asked
Sep 18, 2019
Quantitative Aptitude
isi2016-dcg
quantitative-aptitude
geometry
+
–
0
votes
1
answer
52
ISI2016-DCG-52
The area bounded by $y=x^{2}-4,y=0$ and $x=4$ is $\frac{64}{3}$ $6$ $\frac{16}{3}$ $\frac{32}{3}$
The area bounded by $y=x^{2}-4,y=0$ and $x=4$ is$\frac{64}{3}$$6$$\frac{16}{3}$$\frac{32}{3}$
gatecse
324
views
gatecse
asked
Sep 18, 2019
Geometry
isi2016-dcg
curves
area
non-gate
+
–
0
votes
1
answer
53
ISI2016-DCG-53
$\underset{x\rightarrow-1}{\lim}\dfrac{1+\sqrt[3]{x}}{1+\sqrt[5]{x}}$ equals $\frac{3}{5}$ $\frac{5}{3}$ $1$ $\infty$
$\underset{x\rightarrow-1}{\lim}\dfrac{1+\sqrt[3]{x}}{1+\sqrt[5]{x}}$ equals$\frac{3}{5}$$\frac{5}{3}$$1$$\infty$
gatecse
219
views
gatecse
asked
Sep 18, 2019
Calculus
isi2016-dcg
calculus
limits
+
–
0
votes
0
answers
54
ISI2016-DCG-54
$\underset{x\rightarrow 0}{\lim}x\sin\left(\dfrac{1}{x}\right)$ equals $-1$ $0$ $1$ Does not exist
$\underset{x\rightarrow 0}{\lim}x\sin\left(\dfrac{1}{x}\right)$ equals$-1$$0$$1$Does not exist
gatecse
225
views
gatecse
asked
Sep 18, 2019
Calculus
isi2016-dcg
calculus
limits
+
–
0
votes
0
answers
55
ISI2016-DCG-55
$\underset{x\rightarrow 0}{\lim}\sin\left(\dfrac{1}{x}\right)$ equals $-1$ $0$ $1$ Does not exist
$\underset{x\rightarrow 0}{\lim}\sin\left(\dfrac{1}{x}\right)$ equals$-1$$0$$1$Does not exist
gatecse
238
views
gatecse
asked
Sep 18, 2019
Calculus
isi2016-dcg
calculus
limits
+
–
0
votes
1
answer
56
ISI2016-DCG-56
$\underset{x\rightarrow \infty}{\lim} \left(1+\dfrac{1}{x^{2}}\right)^{x}$ equals $-1$ $0$ $1$ Does not exist
$\underset{x\rightarrow \infty}{\lim} \left(1+\dfrac{1}{x^{2}}\right)^{x}$ equals$-1$$0$$1$Does not exist
gatecse
313
views
gatecse
asked
Sep 18, 2019
Calculus
isi2016-dcg
calculus
limits
+
–
0
votes
0
answers
57
ISI2016-DCG-57
$\underset{x\rightarrow 1}{\lim} \dfrac{x^{16}-1}{\mid x-1\mid}$ equals $-1$ $0$ $1$ Does not exist
$\underset{x\rightarrow 1}{\lim} \dfrac{x^{16}-1}{\mid x-1\mid}$ equals$-1$$0$$1$Does not exist
gatecse
518
views
gatecse
asked
Sep 18, 2019
Calculus
isi2016-dcg
calculus
limits
+
–
0
votes
0
answers
58
ISI2016-DCG-58
Let $y=\left \lfloor x \right \rfloor$ where $\left \lfloor x \right \rfloor$ is greatest integer less than or equal to $x$. Then $y$ is continuous and many-one. $y$ is not differentiable and many-one. $y$ is not differentiable. $y$ is differentiable and many-one.
Let $y=\left \lfloor x \right \rfloor$ where $\left \lfloor x \right \rfloor$ is greatest integer less than or equal to $x$. Then$y$ is continuous and many-one.$y$ is not...
gatecse
365
views
gatecse
asked
Sep 18, 2019
Calculus
isi2016-dcg
calculus
continuity
differentiation
functions
+
–
0
votes
0
answers
59
ISI2016-DCG-59
If in a $\triangle ABC,\angle B=\dfrac{2\pi}{3},$ then $\cos A+\cos C$ lies in $\left[\:-\sqrt{3},\sqrt{3}\:\right]$ $\left(\:-\sqrt{3},\sqrt{3}\:\right]$ $\left(\:\frac{3}{2},\sqrt{3}\:\right)$ $\left(\:\frac{3}{2},\sqrt{3}\:\right]$
If in a $\triangle ABC,\angle B=\dfrac{2\pi}{3},$ then $\cos A+\cos C$ lies in$\left[\:-\sqrt{3},\sqrt{3}\:\right]$$\left(\:-\sqrt{3},\sqrt{3}\:\right]$$\left(\:\frac{3}{...
gatecse
339
views
gatecse
asked
Sep 18, 2019
Geometry
isi2016-dcg
geometry
triangles
trigonometry
non-gate
+
–
0
votes
0
answers
60
ISI2016-DCG-60
Which of the following relations is true for the following figure? $b^{2}=c(c+a)$ $c^{2}=a(a+b)$ $a^{2}=b(b+c)$ All of these
Which of the following relations is true for the following figure?$b^{2}=c(c+a)$$c^{2}=a(a+b)$$a^{2}=b(b+c)$All of these
gatecse
352
views
gatecse
asked
Sep 18, 2019
Geometry
isi2016-dcg
triangles
non-gate
+
–
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