Recent questions tagged 2005

Let $f$ be differentiable for all $x$. If $f(1) = -2$ and $f'(x) \geq 2$ for $x \in [1,6]$, then :

asked Nov 5, 2018 by pady_1

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Suppose $f(x)$ is differentiable at $x=1$ and $ \displaystyle \lim_{h \to 0} \frac{1}{h} f(1+h) =5 $, then $f'(1)$ equals :

asked Nov 5, 2018 by pady_1

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If the roots of the equation $x^2 - bx + c = 0$ be two consecutive integers, then $b^2 - 4c$ equals :

asked Nov 5, 2018 by pady_1

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The value of $a$ for which the sum of the squares of the roots of the equation $x^2 - (a-2)x -a - 1 = 0$ assume the least value is :

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The system of equations $\alpha x + y + z = \alpha -1$ $x + \alpha y + z = \alpha -1 $ $x + y + \alpha z = \alpha - 1$ has no solution, if $\alpha$ is :

asked Nov 5, 2018 by pady_1

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If $a^2+b^2+c^2 = -2$ and $f(x) = \begin{vmatrix} 1+a^2x & (1+b^2)x & (1+c^2) x \\ (1+a^2)x & 1+b^2x & (1+c^2)x \\ (1+a^2)x & (1+b^2)x & 1+c^2x \end{vmatrix}$, then $f(x)$ is a polynomial of degree :

asked Nov 5, 2018 by pady_1

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If $w = \frac{z}{z - \frac{1}{3} i } $ and $|w| = 1$, then $z$ lies on :

asked Nov 5, 2018 by pady_1

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If $z_1$ and $z_2$ are two non-zero complex numbers such that $|z_1+z_2| = |z_1| + |z_2|$, then arg $z_1$ - arg $z_2$ is equal to :

asked Nov 5, 2018 by pady_1

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Let $ f : (-1, 1) \to B$ be a function defined by$f(x) = \tan^{-1} \frac{2x}{1-x^2}$, then $f$ is both one-one and onto when $B$ is the interval :

asked Nov 5, 2018 by pady_1

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If the coefficient of $x^7$ in $\begin{bmatrix} ax^2 + \begin{pmatrix} \frac{1}{bx} \end{pmatrix} \end{bmatrix}^{11} $ equals the coefficient of $x^{-7}$ in $\begin{bmatrix} ax^2 - \begin{pmatrix} \frac{1}{bx} \end{pmatrix} \end{bmatrix}^{11} $, then $a$ and $b$ satisfy the relation :

asked Nov 5, 2018 by pady_1

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If $A = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$ and $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, then which one of the following holds for all $n \geq 1$, by the principle of mathematical induction ?

asked Nov 5, 2018 by pady_1

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The value of $^{50}C_4 $ $\displaystyle\sum_{r = 1}^{6}$ $^{56-r}C_3$ is :

asked Nov 5, 2018 by pady_1

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If the letters of the word SACHIN are arranged in all possible ways and these words are written out as in dictionary, then the word SACHIN appears at serial number :

asked Nov 5, 2018 by pady_1

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In a triangle PQR, $\angle{R} = \frac{\pi}{2}$. If $\tan (\frac{P}{2})$ and $\tan (\frac{Q}{2})$ are the roots of $ax^2 + bx + c = 0, a \neq 0$, then :

asked Nov 5, 2018 by pady_1

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If the coefficients of $r^{th}$, $(r+1)^{th}$ and $(r+2)^{th}$ terms in the binomial expansion of $(1 + y)^m$ are in AP, then $m$ and $r$ satisfy the equation :

asked Nov 5, 2018 by pady_1

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ABC is a triangle. Forces $\overrightarrow{P}, \overrightarrow{Q}, \overrightarrow{R}$ acting along $IA, IB$ and $IC$ respectively are in equilibrium, where $I$ is the incentre of $\Delta ABC$. Then $\overrightarrow{P} : \overrightarrow{Q} : \overrightarrow{R}$ is :

asked Nov 5, 2018 by pady_1

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The differential equation representing the family of curves $y^2 = 2c (x + \sqrt{c})$, where $c > 0$, is a parameter, is of order and degree as follows:

asked Nov 5, 2018 by pady_1

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Area of the greatest rectangle that can be inscribed in the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $ is :

asked Nov 5, 2018 by pady_1

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$ \displaystyle{\lim_{x \to \infty} [\frac{1}{n^2} \sec^2 \frac{1}{n^2} + \frac{2}{n^2} \sec^2 \frac{4}{n^2} + . . . . . . + \frac{n}{n^2} \sec^2 1]}$

asked Nov 5, 2018 by pady_1

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If the cube roots of unity are 1, $\omega, \omega^2$, then the roots of the equation $(x-1)^3 + 8 = 0$, are :

asked Nov 5, 2018 by pady_1

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If $A^2 - A + I = 0$, then the inverse of A is :

asked Nov 5, 2018 by pady_1

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Let $R = \{ (3,3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6) \}$ be a relation on the set $A = \{ 3, 6, 9, 12\}$. The relation is :

asked Nov 5, 2018 by pady_1

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If in a frequency distribution, the Mean and Medium are 21 and 22 respectively, then its Mode is approximately :

asked Nov 5, 2018 by pady_1

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Let P be the point (1, 0) and Q a point on the locus $y^2=8x$. The locus of mid point of PQ is :

asked Nov 5, 2018 by pady_1

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If $C$ is the mid point of AB and P is any point outside AB, then :

asked Nov 5, 2018 by pady_1

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$t_{1/4}$ can be taken as the time taken for the concentration of a reactant to drop to $\frac{3}{4}$ of its initial value. If the rate constant for a first order reaction is $k$, the $t_{1/4}$ can be written as :

asked Nov 5, 2018 by pady_1

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An organic compound having molecular mass 60 is found to contain $C=20 \%, H=6.67 \%$ and $N = 46.67 \%$ while rest is oxygen. On heating it gives $NH_3$ along with a solid residue. The solid residue give violet colour with alkaline copper sulphate solution. The compound is :

asked Nov 5, 2018 by pady_1

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The amount of solid $NH_4HS$ is placed in a flask already containing ammonia gas at a certain temperature and 0.50 atm pressure. Ammonium hydrogen sulphide decomposes to yield $NH_3$ and $H_2S$ gases in the flask. When the decomposition reaction reaches equilibrium, the total pressure in the flask rises to 0.84 atm ? The equilibrium constant for $NH_2HS$ decomposition at this temperature is :

asked Nov 5, 2018 by pady_1

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If the bond dissociation energies of $XY, X_2$ and $Y_2$ (all diatomic molecules) are in the ratio of $1 : 1 : 0.5$ and $\Delta H_f$ for the formation of $XY$ is $-200 \; kJ \; mol^{-1}$. The bond dissociation energy of $X_2$ will be :

asked Nov 5, 2018 by pady_1

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p-cresol reacts with chloroform in alkaline medium to give the compound A which adds hydrogen cyanide to form, the compound B. The latter on acidic hydrolysis gives chiral carboxylic acid. The structure of the carboxylic acid is :

asked Nov 5, 2018 by pady_1

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Reaction of cyclohexanone with dimethlamine in the presence of catalic amount of an acid forms a compound. Water during the reaction is continuously removed. The compound formed is generally known as :

asked Nov 5, 2018 by pady_1

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The value of the 'spin only' magnetic moment for one of the following configurations is 2.84 BM. The correct one is :

asked Nov 5, 2018 by pady_1

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Which of the following factors may be regarded as the main cause of lanthanide contraction ?

asked Nov 5, 2018 by pady_1

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Which of the following statements in relation to the hydrogen atom is correct ?

asked Nov 5, 2018 by pady_1

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The structure of diborane $(B_2H_6)$ contains :

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The reaction is fastest when $X$ is :

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Equimolar solutions in the same solvent have :

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Elimination of bromine from 2-bromobutane results in the formation of :

asked Nov 5, 2018 by pady_1

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Which of the following is fully fluorinated polymer ?

asked Nov 5, 2018 by pady_1

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Acid catalyzed hydration of alkenes except ethene leads to the formation of :

asked Nov 5, 2018 by pady_1

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Amongst the following the most basic compound is :

asked Nov 5, 2018 by pady_1

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Which types of isomerism is shown by 2, 3-dichlorobutane ?

asked Nov 5, 2018 by pady_1

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Which one of the following methods is neither meant for the synthesis nor for separation of amines ?

asked Nov 5, 2018 by pady_1

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Alkyl halides react with dialkyl copper reagents to give :

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Of the five isomeric hexanes, the isomer which can give two monochlorinated compound is :

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Among the following acids which has the lowest $pK_a$ value ?

asked Nov 5, 2018 by pady_1

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In both DNA and RNA, heterocylic base and phosphate ester linkages are at :

asked Nov 5, 2018 by pady_1

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Tertiary alkyl halides are practically inert to substitution by $S_N 2$ mechanism because of :

asked Nov 5, 2018 by pady_1

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The decreasing order of nucleophilicity among the nucleophiles :

asked Nov 5, 2018 by pady_1

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Reaction of one molecule of HBr with one molecule of 1, 3-butadiene at $40^{\circ} C$ gives predominantly :

asked Nov 5, 2018 by pady_1

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