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Recent questions tagged isi2015-dcg

2 votes
1 answer
31
ISI2015-DCG-31
Let $A$ be an $n \times n$ matrix such that $\mid A^{2} \mid\: =1$. Here $\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 $\mid A^{2} \mid\: =1$. Here $\mid A \mid $ stands for determinant of matrix $A$. Then$\mid A \mid =1$$\mid A \mid =0 \text{ o...
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0 votes
1 answer
32
ISI2015-DCG-32
The set of vectors constituting an orthogonal basis in $\mathbb{R} ^3$ is $\begin{Bmatrix} \begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix}, & \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}, & \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \end{Bmatrix}$ ... None of these
The set of vectors constituting an orthogonal basis in $\mathbb{R} ^3$ is$\begin{Bmatrix} \begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix}, & \begin{pmatrix} 1 \\ 1 \\ 0 \end{p...
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3 votes
1 answer
36
ISI2015-DCG-36
Suppose $X$ and $Y$ are finite sets, each with cardinality $n$. The number of bijective functions from $X$ to $Y$ is $n^n$ $n \log_2 n$ $n^2$ $n!$
Suppose $X$ and $Y$ are finite sets, each with cardinality $n$. The number of bijective functions from $X$ to $Y$ is$n^n$$n \log_2 n$$n^2$$n!$
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0 votes
1 answer
38
ISI2015-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$$18$
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1 votes
1 answer
39
ISI2015-DCG-39
The medians $AD$ and $BE$ of the triangle with vertices $A(0,b)$, $B(0,0)$ and $C(a,0)$ are mutually perpendicular if $b=\sqrt{2}a$ $b=\pm \sqrt{2}b$ $b= – \sqrt{2}a$ $b=a$
The medians $AD$ and $BE$ of the triangle with vertices $A(0,b)$, $B(0,0)$ and $C(a,0)$ are mutually perpendicular if$b=\sqrt{2}a$$b=\pm \sqrt{2}b$$b= – \sqrt{2}a$$b=a$...
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0 votes
0 answers
40
ISI2015-DCG-40
The equations $x=a \cos \theta + b \sin \theta$ and $y=a \sin \theta + b \cos \theta$, $( 0 \leq \theta \leq 2 \pi$ 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 $y=a \sin \theta + b \cos \theta$, $( 0 \leq \theta \leq 2 \pi$ and $a,b$ are arbitrary constants) representa circlea ...
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0 votes
0 answers
41
ISI2015-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)$$y=\pm (x...
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0 votes
1 answer
42
ISI2015-DCG-42
In an ellipse, the distance between its foci is $6$ and its minor axis is $8$. hen its eccentricity is $\frac{4}{5}$ $\frac{1}{\sqrt{52}}$ $\frac{3}{5}$ $\frac{1}{2}$
In an ellipse, the distance between its foci is $6$ and its minor axis is $8$. hen its eccentricity is$\frac{4}{5}$$\frac{1}{\sqrt{52}}$$\frac{3}{5}$$\frac{1}{2}$
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0 votes
0 answers
43
ISI2015-DCG-43
Four tangents are drawn to the ellipse $\displaystyle{}\frac{x^2}{9} + \frac{y^2}{5} =1$ 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 $\displaystyle{}\frac{x^2}{9} + \frac{y^2}{5} =1$ at the ends of its latera recta. The area of the quadrilateral so formed is$27$$\...
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0 votes
1 answer
44
ISI2015-DCG-44
If the distance between the foci of a hyperbola is $16$ and its eccentricity is $\sqrt{2}$, then the equation of the hyperbola is $y^2-x^2=32$ $x^2-y^2=16$ $y^2-x^2=16$ $x^2-y^2=32$
If the distance between the foci of a hyperbola is $16$ and its eccentricity is $\sqrt{2}$, then the equation of the hyperbola is$y^2-x^2=32$$x^2-y^2=16$$y^2-x^2=16$$x^2-...
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1 votes
2 answers
45
ISI2015-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
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1 votes
1 answer
46
ISI2015-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$$(\sin x + \cos x)^2+K$$- \fra...
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0 votes
2 answers
47
ISI2015-DCG-47
The Taylor series expansion of $f(x)= \text{ln}(1+x^2)$ about $x=0$ is $\sum _{n=1}^{\infty} (-1)^n \frac{x^n}{n}$ $\sum _{n=1}^{\infty} (-1)^{n+1} \frac{x^{2n}}{n}$ $\sum _{n=1}^{\infty} (-1)^{n+1} \frac{x^{2n+1}}{n+1}$ $\sum _{n=0}^{\infty} (-1)^{n+1} \frac{x^{n+1}}{n+1}$
The Taylor series expansion of $f(x)= \text{ln}(1+x^2)$ 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 _...
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1 votes
1 answer
48
ISI2015-DCG-48
$\underset{x \to 1}{\lim} \dfrac{x^{\frac{1}{3}}-1}{x^{\frac{1}{4}}-1}$ equals $\frac{4}{3}$ $\frac{3}{4}$ $1$ None of these
$\underset{x \to 1}{\lim} \dfrac{x^{\frac{1}{3}}-1}{x^{\frac{1}{4}}-1}$ equals$\frac{4}{3}$$\frac{3}{4}$$1$None of these
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1 votes
2 answers
49
ISI2015-DCG-49
The domain of the function $\text{ln}(3x^2-4x+5)$ 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 $\text{ln}(3x^2-4x+5)$ isset of positive real numbersset of real numbersset of negative real numbersset of real numbers larger than $5$
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1 votes
1 answer
51
ISI2015-DCG-51
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}$$\frac{32}{3}$
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1 votes
1 answer
52
ISI2015-DCG-52
$\underset{x \to -1}{\lim} \dfrac{1+\sqrt[3]{x}}{1+\sqrt[5]{x}}$ equals $\frac{3}{5}$ $\frac{5}{3}$ $1$ $\infty$
$\underset{x \to -1}{\lim} \dfrac{1+\sqrt[3]{x}}{1+\sqrt[5]{x}}$ equals$\frac{3}{5}$$\frac{5}{3}$$1$$\infty$
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0 votes
1 answer
54
ISI2015-DCG-54
$\underset{x \to 0}{\lim} x \sin \left( \frac{1}{x} \right)$ equals $-1$ $0$ $1$ Does not exist
$\underset{x \to 0}{\lim} x \sin \left( \frac{1}{x} \right)$ equals$-1$$0$$1$Does not exist
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1 votes
0 answers
55
ISI2015-DCG-55
$\underset{x \to 0}{\lim} \sin \bigg( \dfrac{1}{x} \bigg)$ equals $-1$ $0$ $1$ Does not exist
$\underset{x \to 0}{\lim} \sin \bigg( \dfrac{1}{x} \bigg)$ equals$-1$$0$$1$Does not exist
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1 votes
1 answer
56
ISI2015-DCG-56
$\underset{x \to \infty}{\lim} \left( 1 + \dfrac{1}{x^2} \right) ^x$ equals $-1$ $0$ $1$ Does not exist
$\underset{x \to \infty}{\lim} \left( 1 + \dfrac{1}{x^2} \right) ^x$ equals$-1$$0$$1$Does not exist
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0 votes
2 answers
57
ISI2015-DCG-57
Let $y=\lfloor x \rfloor$, where $\lfloor x \rfloor$ 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 $y=\lfloor x \rfloor$, where $\lfloor x \rfloor$ is greatest integer less than or equal to $x$. Then$y$ is continuous and many-one$y$ is not differentiable and many-o...
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0 votes
1 answer
58
ISI2015-DCG-58
$\underset{x \to 1}{\lim} \dfrac{x^{16}-1}{\mid x-1 \mid}$ equals $-1$ $0$ $1$ Does not exist
$\underset{x \to 1}{\lim} \dfrac{x^{16}-1}{\mid x-1 \mid}$ equals$-1$$0$$1$Does not exist
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0 votes
1 answer
59
ISI2015-DCG-59
If in a $\Delta ABC$, $\angle B = \frac{2 \pi}{3}$, then $\cos A + \cos C$ lies in $[\:- \sqrt{3}, \sqrt{3}\:]$ $(\: – \sqrt{3}, \sqrt{3}\:]$ $(\:\frac{3}{2}, \sqrt{3}\:)$ $(\:\frac{3}{2}, \sqrt{3}\:]$
If in a $\Delta ABC$, $\angle B = \frac{2 \pi}{3}$, then $\cos A + \cos C$ lies in$[\:- \sqrt{3}, \sqrt{3}\:]$$(\: – \sqrt{3}, \sqrt{3}\:]$$(\:\frac{3}{2}, \sqrt{3}\:)$...
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0 votes
0 answers
60
ISI2015-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
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