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Recent questions tagged isi2016-dcg

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ISI2016-DCG-1
The sequence $\dfrac{1}{\log_{3} 2},\dfrac{1}{\log_{6} 2},\dfrac{1}{\log_{12} 2},\dfrac{1}{\log_{24} 2}\cdots$ is inArithmetic progression (AP)Geometric progression (GP)H...
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699
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2 answers
1 votes
ISI2016-DCG-2
Let $S=\{6,10,7,13,5,12,8,11,9\},$ and $a=\sum_{x\in S}(x-9)^{2}\:\&\: b=\sum_{x\in S}(x-10)^{2}.$ Then$a<b$$a>b$$a=b$None of these
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445
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1 answers
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ISI2016-DCG-3
The value of $\begin{vmatrix} 1+a& 1& 1& 1\\ 1&1+b &1 &1 \\ 1&1 &1+c &1 \\ 1&1 &1 &1+d \end{vmatrix}$ is$abcd(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d})$$abcd(\f...
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354
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1 answers
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ISI2016-DCG-4
If $f(x)=\begin{bmatrix}\cos\:x & -\sin\:x & 0 \\ \sin\:x & \cos\:x & 0 \\ 0 & 0 & 1 \end{bmatrix}$ then the value of $\big(f(x)\big)^2$ is$f(x)$$f(2x)$$2f(x)$None of t...
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481
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2 answers
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ISI2016-DCG-5
If $\tan\: x=p+1$ and $\tan\; y=p-1,$ then the value of $2\:\cot\:(x-y)$ is$2p$$p^{2}$$(p+1)(p-1)$$\frac{2p}{p^{2}-1}$
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391
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1 answers
1 votes
ISI2016-DCG-6
The coefficient of $x^{2}$ in the product $(1+x)(1+2x)(1+3x)\cdots (1+10x)$ is$1320$$1420$$1120$None of these
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280
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1 answers
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ISI2016-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$$1$
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260
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1 answers
1 votes
ISI2016-DCG-8
Let $S=\{0,1,2,\cdots,25\}$ and $T=\{n\in S\: : \: n^{2}+3n+2\: \text{is divisible by}\: 6\}$. Then the number of elements in the set $T$ is$16$$17$$18$$10$
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383
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1 answers
1 votes
ISI2016-DCG-9
The $5000$th term of the sequence $1,2,2,3,3,3,4,4,4,4,\cdots$ is$98$$99$$100$$101$
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288
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1 answers
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ISI2016-DCG-10
Let $a$ be the $81$-digit number of which all the digits are equal to $1.$ Then the number $a$ is ,divisible by $9$ but not divisible by $27$ divisible by $27$ but not di...
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496
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1 answers
1 votes
ISI2016-DCG-11
Let two systems of linear equations be defined as follows:$\begin{array}{lll} & x+y & =1 \\ P: & 3x+3y & =3 \\ & 5x+5y & =5 \end{array}$ and $\begin{array}{lll} & x+y...
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284
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1 answers
1 votes
ISI2016-DCG-12
The highest power of $3$ contained in $1000!$ is$198$$891$$498$$292$
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350
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1 answers
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ISI2016-DCG-13
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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390
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1 answers
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ISI2016-DCG-14
For natural numbers $n,$ the inequality $2^{n}>2n+1$ is valid when$n\geq 3$$n<3$$n=3$None of these
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402
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ISI2016-DCG-15
The shaded region in the following diagram represents the relation$y\:\leq\: x$$\mid \:y\mid \:\leq\: \mid x\:\mid $$y\:\leq\: \mid x\:\mid$$\mid \:y\mid\: \leq\: x$
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298
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ISI2016-DCG-16
The set $\{(x,y)\: :\: \mid x\mid+\mid y\mid\:\leq\:1\}$ is represented by the shaded region in
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297
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1 answers
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ISI2016-DCG-17
The smallest integer $n$ for which $1+2+2^{2}+2^{3}+2^{4}+\cdots+2^{n}$ exceeds $9999$, given that $\log_{10}2=0.30103$, is$12$$13$$14$None of these
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349
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0 answers
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ISI2016-DCG-18
The value of $(1.1)^{10}$ correct to $4$ decimal places is$2.4512$$1.9547$$2.5937$$1.4512$
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469
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2 answers
1 votes
ISI2016-DCG-19
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 i...
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338
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1 answers
1 votes
ISI2016-DCG-20
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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496
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1 answers
2 votes
ISI2016-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
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378
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1 answers
1 votes
ISI2016-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
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434
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1 answers
1 votes
ISI2016-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
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407
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1 answers
2 votes
ISI2016-DCG-24
If the letters of the word $\text{COMPUTER}$ be arranged in random order, the number of arrangements in which the three vowels $O, U$ and $E$ occur together is$8!$$6!$$3!...
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341
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1 answers
1 votes
ISI2016-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$$x^{2}-63...
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454
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2 answers
1 votes
ISI2016-DCG-26
If $r$ be the ratio of the roots of the equation $ax^{2}+bx+c=0,$ then $\frac{r}{b}=\frac{r+1}{ac}$$\frac{r+1}{b}=\frac{r}{ac}$$\frac{(r+1)^{2}}{r}=\frac{b^{2}}{ac}$$\lef...
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332
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1 answers
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ISI2016-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 t...
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685
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1 answers
2 votes
ISI2016-DCG-28
If one root of a quadratic equation $ax^{2}+bx+c=0$ be equal to the n th power of the other, then$(ac)^{\frac{n}{n+1}}+b=0$$(ac)^{\frac{n+1}{n}}+b=0$$(ac^{n})^{\frac{1}{n...
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237
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0 answers
1 votes
ISI2016-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}-...
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405
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ISI2016-DCG-30
Let $p,q,r,s$ be real numbers such that $pr=2(q+s).$ Consider the equations $x^{2}+px+q=0$ and $x^{2}+rx+s=0.$ Thenat least one of the equations has real roots.both these...
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