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Recent questions tagged isi2015-dcg

2 votes
4 answers
1
ISI2015-DCG-1
The sequence $\dfrac{1}{\log_3 2}, \: \dfrac{1}{\log_6 2}, \: \dfrac{1}{\log_{12} 2}, \: \dfrac{1}{\log_{24} 2} \dots $ is in Arithmetic progression (AP) Geometric progression ( GP) Harmonic progression (HP) None of these
The sequence $\dfrac{1}{\log_3 2}, \: \dfrac{1}{\log_6 2}, \: \dfrac{1}{\log_{12} 2}, \: \dfrac{1}{\log_{24} 2} \dots $ is inArithmetic progression (AP)Geometric progress...
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1 votes
4 answers
2
ISI2015-DCG-2
Let $S=\{6, 10, 7, 13, 5, 12, 8, 11, 9\}$ and $a=\underset{x \in S}{\Sigma} (x-9)^2$ & $b = \underset{x \in S}{\Sigma} (x-10)^2$. Then $a <b$ $a>b$ $a=b$ None of these
Let $S=\{6, 10, 7, 13, 5, 12, 8, 11, 9\}$ and $a=\underset{x \in S}{\Sigma} (x-9)^2$ & $b = \underset{x \in S}{\Sigma} (x-10)^2$. Then$a <b$$a>b$$a=b$None of these
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2 answers
4
ISI2015-DCG-4
If $\tan x=p+1$ and $\tan y=p-1$, then the value of $2 \cot (x-y)$ is $2p$ $p^2$ $(p+1)(p-1)$ $\frac{2p}{p^2-1}$
If $\tan x=p+1$ and $\tan y=p-1$, then the value of $2 \cot (x-y)$ is$2p$$p^2$$(p+1)(p-1)$$\frac{2p}{p^2-1}$
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0 votes
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5
ISI2015-DCG-5
If $f(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}$ then the value of $\big(f(x)\big)^2$ is $f(x)$ $f(2x)$ $2f(x)$ None of these
If $$f(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ then the value of $\big(f(x)\big)^2$ is$f(x)$$f(2x)$$2f(x)$None of th...
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1 votes
1 answer
6
ISI2015-DCG-6
The coefficient of $x^2$ in the product $(1+x)(1+2x)(1+3x) \dots (1+10x)$ is $1320$ $1420$ $1120$ None of these
The coefficient of $x^2$ in the product $(1+x)(1+2x)(1+3x) \dots (1+10x)$ is$1320$$1420$$1120$None of these
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0 votes
1 answer
7
ISI2015-DCG-7
Let $x^2-2(4k-1)x+15k^2-2k-7>0$ for any real value of $x$. Then the integer value of $k$ is $2$ $4$ $3$ $1$
Let $x^2-2(4k-1)x+15k^2-2k-7>0$ for any real value of $x$. Then the integer value of $k$ is$2$$4$$3$$1$
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1 votes
2 answers
8
ISI2015-DCG-8
Let $S=\{0, 1, 2, \cdots 25\}$ and $T=\{n \in S: n^2+3n+2$ is divisible by $6\}$. Then the number of elements in the set $T$ is $16$ $17$ $18$ $10$
Let $S=\{0, 1, 2, \cdots 25\}$ and $T=\{n \in S: n^2+3n+2$ is divisible by $6\}$. Then the number of elements in the set $T$ is$16$$17$$18$$10$
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2 votes
1 answer
9
ISI2015-DCG-9
Let $a$ be the $81$ – digit number of which all the digits are equal to $1$. Then the number $a$ is, divisible by $9$ but not divisible by $27$ divisible by $27$ but not divisible by $81$ divisible by $81$ None of the above
Let $a$ be the $81$ – digit number of which all the digits are equal to $1$. Then the number $a$ is,divisible by $9$ but not divisible by $27$divisible by $27$ but not ...
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0 votes
1 answer
10
ISI2015-DCG-10
The $5000$th term of the sequence $1,2,2, 3,3,3,4,4,4,4, \cdots$ is $98$ $99$ $100$ $101$
The $5000$th term of the sequence $1,2,2, 3,3,3,4,4,4,4, \cdots$ is$98$$99$$100$$101$
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1 votes
1 answer
11
ISI2015-DCG-11
Let two systems of linear equations be defined as follows: $\begin{array} {} & x+y & =1 \\ P: & 3x+3y & =3 \\ & 5x+5y & =5 \end{array}$ ... $P$ and $Q$ are inconsistent $P$ and $Q$ are consistent $P$ is consistent but $Q$ is inconsistent None of the above
Let two systems of linear equations be defined as follows:$\begin{array} {} & x+y & =1 \\ P: & 3x+3y & =3 \\ & 5x+5y & =5 \end{array}$ and $\begin{array} {} & x+y & =3 \\...
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1 votes
1 answer
12
ISI2015-DCG-12
The highest power of $3$ contained in $1000!$ is $198$ $891$ $498$ $292$
The highest power of $3$ contained in $1000!$ is$198$$891$$498$$292$
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1 answer
13
ISI2015-DCG-13
For all the natural number $n \geq 3, \: n^2+1$ is divisible by $3$ not divisible by $3$ divisible by $9$ None of these
For all the natural number $n \geq 3, \: n^2+1$ isdivisible by $3$not divisible by $3$divisible by $9$None of these
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0 votes
1 answer
14
ISI2015-DCG-14
For natural numbers $n$, the inequality $2^n >2n+1$ is valid when $n \geq 3$ $n < 3$ $n=3$ None of these
For natural numbers $n$, the inequality $2^n >2n+1$ is valid when$n \geq 3$$n < 3$$n=3$None of these
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1 answer
15
ISI2015-DCG-15
The smallest integer $n$ for which $1+2+2^2+2^3+2^4+ \cdots +2^n$ exceeds $9999$, given that $\log_{10} 2=0.30103$, is $12$ $13$ $14$ None of these
The smallest integer $n$ for which $1+2+2^2+2^3+2^4+ \cdots +2^n$ exceeds $9999$, given that $\log_{10} 2=0.30103$, is$12$$13$$14$None of these
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0 votes
2 answers
16
ISI2015-DCG-16
The shaded region in the following diagram represents the relation $y \leq x$ $\mid y \mid \leq \mid x \mid$ $y \leq \mid x \mid$ $\mid y \mid \leq x$
The shaded region in the following diagram represents the relation$y \leq x$$\mid y \mid \leq \mid x \mid$$y \leq \mid x \mid$$\mid y \mid \leq x$
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17
ISI2015-DCG-17
The set $\{(x,y): \mid x \mid + \mid y \mid \leq 1\}$ is represented by the shaded region in
The set $\{(x,y): \mid x \mid + \mid y \mid \leq 1\}$ is represented by the shaded region in
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18
ISI2015-DCG-18
The value of $(1.1)^{10}$ correct to $4$ decimal places is $2.4512$ $1.9547$ $2.5937$ $1.4512$
The value of $(1.1)^{10}$ correct to $4$ decimal places is$2.4512$$1.9547$$2.5937$$1.4512$
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1 answer
19
ISI2015-DCG-19
The expression $3^{2n+1} + 2^{n+2}$ is divisible by $7$ for all positive integer values of $n$ all non-negative integer values of $n$ only even integer values of $n$ only odd integer values of $n$
The expression $3^{2n+1} + 2^{n+2}$ is divisible by $7$ forall positive integer values of $n$all non-negative integer values of $n$only even integer values of $n$only odd...
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2 votes
2 answers
20
ISI2015-DCG-20
The total number of factors of $3528$ greater than $1$ but less than $3528$ is $35$ $36$ $34$ None of these
The total number of factors of $3528$ greater than $1$ but less than $3528$ is$35$$36$$34$None of these
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0 votes
1 answer
21
ISI2015-DCG-21
The value of the term independent of $x$ in the expansion of $(1-x)^2(x+\frac{1}{x})^7$ is $-70$ $70$ $35$ None of these
The value of the term independent of $x$ in the expansion of $(1-x)^2(x+\frac{1}{x})^7$ is$-70$$70$$35$None of these
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1 votes
3 answers
22
ISI2015-DCG-22
The value of $\begin{vmatrix} 1 & \log _x y & \log_x z \\ \log _y x & 1 & \log_y z \\ \log _z x & \log _z y & 1 \end{vmatrix}$ is $0$ $1$ $-1$ None of these
The value of $$\begin{vmatrix} 1 & \log _x y & \log_x z \\ \log _y x & 1 & \log_y z \\ \log _z x & \log _z y & 1 \end{vmatrix}$$ is$0$$1$$-1$None of these
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1 votes
2 answers
23
ISI2015-DCG-23
The value of $\log _2 e – \log _4 e + \log _8 e – \log _{16} e + \log_{32} e – \cdots$ is $-1$ $0$ $1$ None of these
The value of $\log _2 e – \log _4 e + \log _8 e – \log _{16} e + \log_{32} e – \cdots$ is$-1$$0$$1$None of these
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2 votes
1 answer
24
ISI2015-DCG-24
If the letters of the word $\textbf{COMPUTER}$ be arranged in random order, the number of arrangements in which the three vowels $O, U$ and $E$ occur together is $8!$ $6!$ $3!6!$ None of these
If the letters of the word $\textbf{COMPUTER}$ be arranged in random order, the number of arrangements in which the three vowels $O, U$ and $E$ occur together is$8!$$6!$$...
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1 votes
1 answer
25
ISI2015-DCG-25
If $\alpha$ and $\beta$ be the roots of the equation $x^2+3x+4=0$, then the equation with roots $(\alpha + \beta)^2$ and $(\alpha – \beta)^2$ is $x^2+2x+63=0$ $x^2-63x+2=0$ $x^2-2x-63=0$ None of the above
If $\alpha$ and $\beta$ be the roots of the equation $x^2+3x+4=0$, then the equation with roots $(\alpha + \beta)^2$ and $(\alpha – \beta)^2$ is$x^2+2x+63=0$$x^2-63x+2=...
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0 votes
1 answer
26
ISI2015-DCG-26
If $r$ be the ratio of the roots of the equation $ax^{2}+bx+c=0,$ then $\frac{r}{b}=\frac{r+1}{ac}$ $\frac{r+1}{b}=\frac{r}{ac}$ $\frac{(r+1)^{2}}{r}=\frac{b^{2}}{ac}$ $\left(\frac{r}{b}\right)^{2}=\frac{r+1}{ac}$
If $r$ be the ratio of the roots of the equation $ax^{2}+bx+c=0,$ then $\frac{r}{b}=\frac{r+1}{ac}$$\frac{r+1}{b}=\frac{r}{ac}$$\frac{(r+1)^{2}}{r}=\frac{b^{2}}{ac}$$\lef...
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1 votes
1 answer
27
ISI2015-DCG-27
If $A$ be the set of triangles in a plane and $R^{+}$ be the set of all positive real numbers, then the function $f\::\:A\rightarrow R^{+},$ defined by $f(x)=$ area of triangle $x,$ is one-one and into one-one and onto many-one and onto many-one and into
If $A$ be the set of triangles in a plane and $R^{+}$ be the set of all positive real numbers, then the function $f\::\:A\rightarrow R^{+},$ defined by $f(x)=$ area of t...
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3 votes
1 answer
29
ISI2015-DCG-29
The condition that ensures that the roots of the equation $x^3-px^2+qx-r=0$ are in $H.P.$ is $r^2-9pqr+q^3=0$ $27r^2-9pqr+3q^3=0$ $3r^3-27pqr-9q^3=0$ $27r^2-9pqr+2q^3=0$
The condition that ensures that the roots of the equation $x^3-px^2+qx-r=0$ are in $H.P.$ is$r^2-9pqr+q^3=0$$27r^2-9pqr+3q^3=0$$3r^3-27pqr-9q^3=0$$27r^2-9pqr+2q^3=0$
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