If the line, 2 x-y+3=0 is at a distance
Question: If the line, $2 x-y+3=0$ is at a distance $\frac{1}{\sqrt{5}}$ and $\frac{2}{\sqrt{5}}$ from the lines $4 x-2 y+\alpha=0$ and $6 x-3 y+\beta=0$, respectively, then the sum of all possible value of $\alpha$ and $\beta$ is__________. Solution: $L_{1}: 2 x-y+3=0$ $L_{1}: 4 x-2 y+\alpha=0 \Rightarrow 2 x-y+\frac{\alpha}{2}=0$ $L_{1}: 6 x-3 y+\beta=0 \Rightarrow 2 x-y+\frac{\beta}{3}=0$ Distance between $L_{1}$ and $L_{2}$; $\left|\frac{\alpha-6}{2 \sqrt{5}}\right|=\frac{1}{\sqrt{5}} \Right...
Read More →Match List-I with List-II :
Question: Match List-I with List-II : Choose the correct answer from the ontions given below$(\mathrm{a})-(\mathrm{ii}),(\mathrm{b})-(\mathrm{iii}),(\mathrm{c})-(\mathrm{iv}),(\mathrm{d})-(\mathrm{i})$(a)-(iii), (b)-(iv), (c)-(i), (d)-(ii)(a)-(iii), (b)-(i), (c)-(iv), (d)-(ii)$(\mathrm{a})-(\mathrm{iv}),(\mathrm{b})-(\mathrm{i}),(\mathrm{c})-(\mathrm{ii}),(\mathrm{d})-(\mathrm{iii})$Correct Option: , 3 Solution: (a) Haber's process is used for $\mathrm{NH}_{3}$ synthesis. (b) Ostwald's process i...
Read More →If the perpendicular bisector of the line
Question: If the perpendicular bisector of the line segment joining the points $P(1,4)$ and $Q(k, 3)$ has $y$-intercept equal to $-4$, then a value of $k$ is :(1) $-2$(2) $-4$(3) $\sqrt{14}$(4) $\sqrt{15}$Correct Option: , 2 Solution: Mid point of line segment $P Q$ be $\left(\frac{k+1}{2}, \frac{7}{2}\right)$. $\therefore$ Slope of perpendicular line passing through $(0,-4)$ and $\left(\frac{k+1}{2}, \frac{7}{2}\right)=\frac{\frac{7}{2}+4}{\frac{k+1}{2}-0}=\frac{15}{k+1}$ Slope of $P Q=\frac{4-...
Read More →A small bob tied at one end of a thin string of length
Question: A small bob tied at one end of a thin string of length $1 \mathrm{~m}$ is describing a vertical circle so that the maximum and minimum tension in the string are in the rato $5: 1$. The velocity of the bob at the highest position is $\mathrm{m} / \mathrm{s}$. (take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$ ) Solution: (5) by conservation of energy, $v_{\min }^{2}=V^{2}-4 g \mid$ $\mathrm{T}_{\max }=\mathrm{mg}+\frac{\mathrm{mv}^{2}}{\mathrm{I}} \quad \ldots(2)$ $\mathrm{T}_{\min }=\f...
Read More →A triangle A B C lying in the first quadrant
Question: A triangle $A B C$ lying in the first quadrant has two vertices as $A(1,2)$ and $B(3,1)$. If $\angle B A C=90^{\circ}$, and ar $(\triangle A B C)=5 \sqrt{5}$ sq. units, then the abscissa of the vertex $C$ is:(1) $1+\sqrt{5}$(2) $1+2 \sqrt{5}$(3) $2+\sqrt{5}$(4) $2 \sqrt{5}-1$Correct Option: , 2 Solution: Let $\triangle A B C$ be in the first quadrant Slope of line $A B=-\frac{1}{2}$ Slope of line $A C=2$ Length of $A B=\sqrt{5}$ It is given that $\operatorname{ar}(\triangle A B C)=5 \s...
Read More →A person is swimming with a speed of 10 m/s
Question: A person is swimming with a speed of $10 \mathrm{~m} / \mathrm{s}$ at an angle of $120^{\circ}$ with the flow and reaches to a point directly opposite on the other side of the river. The speed of the flow is ' $x$ ' $m / s$. The value of ' $x$ ' to the nearest integer is Solution: $10 \sin 30^{\circ}=x$ $x=5 \mathrm{~m} / \mathrm{s}$...
Read More →Prove the following
Question: If a $\triangle A B C$ has vertices $A(-1,7), B(-7,1)$ and $C(5,-5)$, then its orthocentre has coordinates :(1) $\left(-\frac{3}{5}, \frac{3}{5}\right)$(2) $(-3,3)$(3) $\left(\frac{3}{5},-\frac{3}{5}\right)$(4) $(3,-3)$Correct Option: , 2 Solution: $m_{B C}=\frac{6}{-12}=-\frac{1}{2}$ $\therefore$ Equation of AS is $y-7=2(x+1)$ $y=2 x+9$ ....(i) $m_{A C}=\frac{12}{-6}=-2$ $\therefore$ Equation of $B P$ is $y-1=\frac{1}{2}(x+7)$ $y=\frac{x}{2}+\frac{9}{2}$ ...(ii) From equs. (i) and (ii...
Read More →A mosquito is moving with a velocity
Question: A mosquito is moving with a velocity $\overrightarrow{\mathrm{v}}=0.5 \mathrm{t}^{2} \hat{\mathrm{i}}+3 \mathrm{t} \hat{\mathrm{j}}+9 \hat{\mathrm{k}} \mathrm{m} / \mathrm{s}$ and accelerating in uniform conditions. What will be the direction of mosquito after $2 \mathrm{~s}$ ?(1) $\tan ^{-1}\left(\frac{2}{3}\right)$ from $x$ - axis(2) $\tan ^{-1}\left(\frac{\sqrt{85}}{6}\right)$ from $y$-axis(3) $\tan ^{-1}\left(\frac{5}{2}\right)$ from $y$-axis(4) $\tan ^{-1}\left(\frac{5}{2}\right)$...
Read More →The set of all possible values of
Question: The set of all possible values of $\theta$ in the interval $(0, \pi)$ for which the points $(1,2)$ and $(\sin \theta, \cos \theta)$ lie on the same side of the line $x+y=1$ is :(1) $\left(0, \frac{\pi}{2}\right)$(2) $\left(\frac{\pi}{4}, \frac{3 \pi}{4}\right)$(3) $\left(0, \frac{3 \pi}{4}\right)$(4) $\left(0, \frac{\pi}{4}\right)$Correct Option: 1 Solution: Let $f(x, y)=x+y-1$ Given $(1,2)$ and $(\sin \theta, \cos \theta)$ are lies on same side. $\therefore f(1,2) \cdot f(\sin \theta,...
Read More →In the following compounds, the decreasing order of basic strength will be:
Question: In the following compounds, the decreasing order of basic strength will be:$\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{NH}_{2}\mathrm{NH}_{3}\left(\mathrm{C}_{2} \mathrm{H}_{5}\right)_{2} \mathrm{NH}$$\left(\mathrm{C}_{2} \mathrm{H}_{5}\right)_{2} \mathrm{NH}\mathrm{NH}_{3}\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{NH}_{2}$$\left(\mathrm{C}_{2} \mathrm{H}_{5}\right)_{2} \mathrm{NH}\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{NH}_{2}\mathrm{NH}_{3}$$\mathrm{NH}_{3}\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{N...
Read More →The decreasing order of basicity of the following amines is:
Question: The decreasing order of basicity of the following amines is: $(\mathrm{A})(\mathrm{C})(\mathrm{D})(\mathrm{B})$$(\mathrm{C})(\mathrm{A})(\mathrm{B})(\mathrm{D})$$(\mathrm{B})(\mathrm{C})(\mathrm{D})(\mathrm{A})$$(\mathrm{C})(\mathrm{B})(\mathrm{A})(\mathrm{D})$Correct Option: , 4 Solution: Basic strength of amines depends upon availability of lone pair of electrons. Aliphatic amines are more basic than aromatic amines....
Read More →If the locus of the mid-point of the line segment
Question: If the locus of the mid-point of the line segment from the point $(3,2)$ to a point on the circle, $x^{2}+y^{2}=1$ is a circle of the radius $r$, then $r$ is equal to:(1) $\frac{1}{4}$(2) $\frac{1}{2}$(3) 1(4) $\frac{1}{3}$Correct Option: , 2 Solution: $\therefore P \equiv(2 h-3,2 k-2) \rightarrow$ on circle $\therefore\left(h-\frac{3}{2}\right)^{2}+(k-1)^{2}=\frac{1}{4}$ $\Rightarrow$ radius $=\frac{1}{2}$...
Read More →The position of a particle as a function of time $t$, is given by
Question: The position of a particle as a function of time $t$, is given by $x(t)=a t+b t^{2}-c t^{3}$ where, $a, b$ and $c$ are constants. When the particle attains zero acceleration, then its velocity will be:(1) $a+\frac{b^{2}}{4 c}$(2) $a+\frac{b^{2}}{3 c}$(3) $a+\frac{b^{2}}{c}$(4) $a+\frac{b^{2}}{2 c}$Correct Option: , 2 Solution: (2) $x=a t+b t^{2}-c t^{3}$ Velocity, $v=\frac{d x}{d t}=\frac{d}{d t}\left(a t+b t^{2}+c t^{3}\right)$ $=a+2 b t-3 c t^{2}$ Acceleration, $\frac{d v}{d t}=\frac...
Read More →The intersection of three lines
Question: The intersection of three lines $x-y=0, x+2 y=3$ and $2 x+y=6$ is a:(1) Equilateral triangle(2) None of the above(3) Isosceles triangle(4) Right angled triangleCorrect Option: , 3 Solution:...
Read More →Consider the following reactions,
Question: Consider the following reactions, The compound [P] is:Correct Option: , 3 Solution:...
Read More →A ball is thrown vertically up (taken as + z-axis) from the ground.
Question: A ball is thrown vertically up (taken as $+$ z-axis) from the ground. The correct momentum-height ( $\mathrm{p}-\mathrm{h}$ ) diagram is:Correct Option: , 4 Solution: (4) $v^{2}=u^{2}-2 g h$ Therefore graph between $\mathrm{p}$ and $\mathrm{h}$ cannot have straight line. (2) and (3) are not possible. During upward journey as $\mathrm{h}$ increases, $\mathrm{p}$ decreases and in downward journey as $\mathrm{h}$ decreases $\mathrm{p}$ increases. Therefore 4 is the correct option....
Read More →The image of the point
Question: The image of the point $(3,5)$ in the line $x-y+1=0$, lies on :(1) $(x-2)^{2}+(y-4)^{2}=4$(2) $(x-4)^{2}+(y+2)^{2}=16$(3) $(x-4)^{2}+(y-4)^{2}=8$(4) $(x-2)^{2}+(y-2)^{2}=12$Correct Option: 1 Solution: Image of $\mathrm{P}(3,5)$ on the line $\mathrm{x}-\mathrm{y}+1=0$ is $\frac{x-3}{1}=\frac{y-5}{-1}=\frac{-2(3-5+1)}{2}=1$ $x=4, y=4$ $\therefore$ Image is $(4,4)$ Which lies on $(x-2)^{2}+(y-4)^{2}=4$...
Read More →Let a point P be such
Question: Let a point $\mathrm{P}$ be such that its distance from the point $(5,0)$ is thrice the distance of $\mathrm{P}$ from the point $(-5,0)$. If the locus of the point $P$ is a circle of radius $r$, then $4 r^{2}$ is equal to (Round off to the nearest integer) Solution: Let $P(h, k)$ Given $P A=3 P B$ $P A^{2}=9 P B^{2}$ $\Rightarrow(h-5)^{2}+k^{2}=9\left[(h+5)^{2}+k^{2}\right]$ $\Rightarrow 8 h^{2}+8 k^{2}+100 h+200=0$ $\therefore$ Locus $x^{2}+y^{2}+\left(\frac{25}{2}\right) x+25=0$ $\th...
Read More →The increasing order of basic it for the following intermediates is
Question: The increasing order of basic it for the following intermediates is (from weak to strong) (iii) $$ (i) $$ (ii) $$ (iv) $$ (v)$(\mathrm{v})(\mathrm{i})(\mathrm{iv})(\mathrm{ii})(\mathrm{iii})$(v) $$ (iii) $$ (ii) $$ (iv) $$ (i)(iii) $$ (iv) $$ (ii) $$ (i) $$ (v)Correct Option: , 3 Solution: Basicity order can be determined by the cummulative effect of the factors on the electron density of concerned atom....
Read More →A man is walking on a straight line.
Question: A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the coordinate axes is $\frac{1}{4}$. Three stones $A, B$ and $C$ are placed at the points $(1,1),(2,2)$ and $(4,4)$ respectively. Then which of these stones is/are on the path of the man?(1) B only(2) A only(3) All the three(4) $\mathrm{C}$ onlyCorrect Option: 1 Solution: $\frac{x}{a}+\frac{y}{b}=1$ $\frac{h}{a}+\frac{k}{b}=1 \ldots$ (1) $\frac{\frac{1}{a}+\frac{1}{b}}{2}=\fra...
Read More →The major product Z obtained in the following reaction scheme is:
Question: The major product $\mathrm{Z}$ obtained in the following reaction scheme is: Correct Option: , 2 Solution:...
Read More →A particle starts from origin O from rest and moves with a uniform acceleration along the positive x-axis.
Question: A particle starts from origin $\mathrm{O}$ from rest and moves with a uniform acceleration along the positive $x$-axis. Identify all figures that correctly represents the motion qualitatively ( $a=$ acceleration, $v=$ velocity,$x=$ displacement $t=$ time $)$ (1) (B), (C)(2) (A)(3) (A), (B), (C)(4) (A), (B), (D)Correct Option: , 4 Solution: (4) For constant acceleration, there is straight line parallel to t-axis on $\vec{a}-t$. Inclined straight line on $\vec{v}-t$, and parabola on $\ve...
Read More →Let the centroid of an equilateral triangle
Question: Let the centroid of an equilateral triangle $\mathrm{ABC}$ be at the origin. Let one of the sides of the equilateral triangle be along the straight line $x+y=3$. If $R$ and $r$ be the radius of circumcircle and incircle respectively of $\Delta \mathrm{ABC}$, then $(\mathrm{R}+\mathrm{r})$ is equal to:(1) $\frac{9}{\sqrt{2}}$(2) $7 \sqrt{2}$(3) $2 \sqrt{2}$(4) $3 \sqrt{2}$Correct Option: 1 Solution: $r=O M=\frac{3}{\sqrt{2}}$ $\ \sin 30^{\circ}=\frac{1}{2}=\frac{r}{R} \Rightarrow R=\fra...
Read More →The equation of one of the
Question: The equation of one of the straight lines which passes through the point $(1,3)$ and makes an angles $\tan ^{-1}(\sqrt{2})$ with the straight line, $y+1=3 \sqrt{2} x$ is(1) $4 \sqrt{2} \mathrm{x}+5 \mathrm{y}-(15+4 \sqrt{2})=0$(2) $5 \sqrt{2} \mathrm{x}+4 \mathrm{y}-(15+4 \sqrt{2})=0$(3) $4 \sqrt{2} x+5 y-4 \sqrt{2}=0$(4) $4 \sqrt{2} \mathrm{x}-5 \mathrm{y}-(5+4 \sqrt{2})=0$Correct Option: 1 Solution: $y=m x+c$ $3=m+c$ $\sqrt{2}=\left|\frac{m-3 \sqrt{2}}{1+3 \sqrt{2} m}\right|$ $=6 m+\...
Read More →The major products $mathrm{A}$ and $mathrm{B}$ in the following reactions are:
Question: The major products $\mathrm{A}$ and $\mathrm{B}$ in the following reactions are: Correct Option: , 4 Solution:...
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