Ship A is sailing towards north-east with velocity
Question: Ship $A$ is sailing towards north-east with velocity $\vec{v}=30 \hat{i}+50 \hat{j} \mathrm{~km} / \mathrm{hr}$ where $\hat{i}$ points east and $\hat{j}$, north. Ship B is at a distance of $80 \mathrm{~km}$ east and $150 \mathrm{~km}$ north of Ship A and is sailing towards west at $10 \mathrm{~km} / \mathrm{hr}$. A will be at minimum distance from $B$ in:(1) $4.2 \mathrm{hrs}$(2) $2.6 \mathrm{hrs}$(3) $3.2 \mathrm{hrs}$(4) $2.2 \mathrm{hrs}$Correct Option: , 2 Solution: $\vec{v}_{A}=30...
Read More →The number of integral values of m so
Question: The number of integral values of $m$ so that the abscissa of point of intersection of lines $3 x+4 y=9$ and $y=m x+1$ is also an integer, is :(1) 1(2) 2(3) 3(4) 0Correct Option: , 2 Solution: $3 x+4 y=9$ $y=m x+1$ $\Rightarrow 3 x+4 m x+4=9$ $\Rightarrow(3+4 \mathrm{~m}) \mathrm{x}=5$ $\Rightarrow \mathrm{x}$ will be an integer when $3+4 m=5,-5,1,-1$ $\Rightarrow \mathrm{m}=\frac{1}{2},-2,-\frac{1}{2},-1$ so, number of integral values of $m$ is 2...
Read More →In the following reaction sequence,
Question: In the following reaction sequence, the major product B is:Correct Option: 1 Solution:...
Read More →Prove the following
Question: Let $\tan \alpha, \tan \beta$ and $\tan \gamma ; \alpha, \beta, \gamma \neq \frac{(2 n-1) \pi}{2}$ $\mathrm{n} \in \mathrm{N}$ be the slopes of three line segments $\mathrm{OA} \mathrm{OB}$ and $\mathrm{OC}$, respectively, where $\mathrm{O}$ is origin.If circumcentre of $\Delta \mathrm{ABC}$ coincides with origin and its orthocentre lies on $y$-axis, then the value of $\left(\frac{\cos 3 \alpha+\cos 3 \beta+\cos 3 \gamma}{\cos \alpha \cos \beta \cos \gamma}\right)^{2}$ is equal to : So...
Read More →A particle starts from the origin at t=0
Question: A particle starts from the origin at $t=0$ with an initial velocity of $3.0 \hat{i} \mathrm{~m} / \mathrm{s}$ and moves in the $x-y$ plane with a constant acceleration $(6.0 \hat{i}+4.0 \hat{j}) \mathrm{m} / \mathrm{s}^{2}$. The $x$-coordinate of the particle at the instant when its $y$-coordinate is $32 \mathrm{~m}$ is $\mathrm{D}$ meters. The value of $\mathrm{D}$ is:(1) 32(2) 50(3) 60(4) 40Correct Option: , 3 Solution: (3) Using $S=u t+\frac{1}{2} a t^{2}$ $y=u_{y} t+\frac{1}{2} a_{...
Read More →Let the tangent to the circle
Question: Let the tangent to the circle $x^{2}+y^{2}=25$ at the point $\mathrm{R}(3,4)$ meet $\mathrm{x}$-axis and $\mathrm{y}$-axis at point $\mathrm{P}$ and $\mathrm{Q}$, respectively. If $\mathrm{r}$ is the radius of the circle passing through the origin $\mathrm{O}$ and having centre at the incentre of the triangle $\mathrm{OPQ}$, then $\mathrm{r}^{2}$ is equal to(1) $\frac{529}{64}$(2) $\frac{125}{72}$ (3) $\frac{625}{72}$(4) $\frac{585}{66}$Correct Option: , 3 Solution: Tangent to circle $...
Read More →The distance x covered by a particle in one dimensional motion varies
Question: The distance $x$ covered by a particle in one dimensional motion varies with time $t$ as $x^{2}=a t^{2}+2 b t+c$. If the acceleration of the particle depends on $x$ as $x^{-} n$, where $n$ is an integer, the value of $n$ is_______ Solution: (3) Distance $\mathrm{X}$ varies with time $\mathrm{t}$ as $x^{2}=a t^{2}+2 b t+c$ $\Rightarrow 2 x \frac{d x}{d t}=2 a t+2 b$ $\Rightarrow x \frac{d x}{d t}=a t+b \Rightarrow \frac{d x}{d t}=\frac{(a t+b)}{x}$ $\Rightarrow x \frac{d^{2} x}{d t^{2}}...
Read More →Consider the following reaction:
Question: Consider the following reaction: The product ' $X$ ' is used:in protein estimation as an alternative to ninhydrinin acid base titration as an indicatoras food grade colourantin laboratory test for phenolsCorrect Option: , 2 Solution: Methyl orange is used as an indicator in acid base titrations....
Read More →When a car si at rest, its driver sees raindrops falling on it vertically.
Question: When a car si at rest, its driver sees raindrops falling on it vertically. When driving the car with speed $v$, he sees that raindrops are coming at an angle $60^{\circ}$ from the horizontal. On furter increasing the speed of the car to $(1+\beta) v$, this angle changes to $45^{\circ}$. The value of $\beta$ is close to:(1) $0.50$(2) $0.41$(3) $0.37$(4) $0.73$Correct Option: , 4 Solution: (4) The given situation is shown in the diagram. Here $v_{r}$ be the velocity of rain drop. When ca...
Read More →In a triangle
Question: In a triangle $\mathrm{PQR}$, the co-ordinates of the points $\mathrm{P}$ and $\mathrm{Q}$ are $(-2,4)$ and $(4,-2)$ respectively. If the equation of the perpendicular bisector of $\mathrm{PR}$ is $2 x-y+2=0$, then the centre of the circumcircle of the $\triangle \mathrm{PQR}$ is :(1) $(-1,0)$(2) $(-2,-2)$(3) $(0.2)$(4) $(1,4)$Correct Option: , 2 Solution: Equation of perpendicular bisector of $P R$ is $y=x$ Solving with $2 \mathrm{x}-\mathrm{y}+2=0$ will give $(-2,2)$...
Read More →The increasing order of
Question: The increasing order of $\mathrm{p} K_{\mathrm{b}}$ for the following compounds will be: $(\mathrm{B})(\mathrm{C})(\mathrm{A})$$(\mathrm{A})(\mathrm{B})(\mathrm{C})$$(\mathrm{C})(\mathrm{A})(\mathrm{B})$$(\mathrm{B})(\mathrm{A})(\mathrm{C})$Correct Option: , 4 Solution: Conjugate acid of guanadine(B) is resonance stabilised and have 2 resonance structure. Similarly conjugate acid of (A) is also resonance stabilised and have one resonance structure. (C) does not exhibit resonance struct...
Read More →Prove the following
Question: Let $\mathrm{A}(-1,1), \mathrm{B}(3,4)$ and $\mathrm{C}(2,0)$ be given three points. A line $y=m x, m0$, intersects lines $\mathrm{AC}$ and $\mathrm{BC}$ at point $\mathrm{P}$ and $\mathrm{Q}$ respectively. Let $\mathrm{A}_{1}$ and $\mathrm{A}_{2}$ be the areas of $\Delta \mathrm{ABC}$ and $\Delta \mathrm{PQC}$ respectively, such that $\mathrm{A}_{1}=3 \mathrm{~A}_{2}$, then the value of $\mathrm{m}$ is equal to :(1) $\frac{4}{15}$(2) 1(3) 2(4) 3Correct Option: , 2 Solution: $\mathrm{P...
Read More →The velocity v and time t graph of a body in a straight line motion is shown in the figure
Question: The velocity $(v)$ and time $(t)$ graph of a body in a straight line motion is shown in the figure. The point $S$ is at $4.333$ seconds. The total distance covered by the body in $6 \mathrm{~s}$ is: (1) $\frac{37}{3} \mathrm{~m}$(2) $12 \mathrm{~m}$(3) $11 \mathrm{~m}$(4) $\frac{49}{4} \mathrm{~m}$Correct Option: 1 Solution: $O S=4+\frac{1}{3}=\frac{13}{3}$ $S D=2-\frac{1}{3}=\frac{5}{3}$ Distance covered by the body $=$ area of $v-t$ graph $=\operatorname{ar}(O A B S)+\operatorname{ar...
Read More →Which of the following compounds can be prepared in good yield by Gabriel phthalimide synthesis?
Question: Which of the following compounds can be prepared in good yield by Gabriel phthalimide synthesis?Correct Option: 1 Solution: Gabriel phthalimide synthesis gives 1 amine in good yield....
Read More →The increasing order of
Question: The increasing order of $\mathrm{p} K_{\mathrm{b}}$ values of the following compounds is : $\mathrm{II}\mathrm{IV}\mathrm{III}\mathrm{I}$$\mathrm{I}\mathrm{II}\mathrm{IV}\mathrm{III}$$\mathrm{II}\mathrm{I}\mathrm{III}\mathrm{IV}$$\mathrm{I}\mathrm{II}\mathrm{III}\mathrm{IV}$Correct Option: , 2 Solution:...
Read More →The most appropriate reagent for conversion of
Question: The most appropriate reagent for conversion of $\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{CN}$ into $\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{NH}_{2}$ is :$\mathrm{NaBH}_{4}$$\mathrm{CaH}_{2}$$\mathrm{LiAlH}_{4}$$\mathrm{Na}(\mathrm{CN}) \mathrm{BH}_{3}$Correct Option: , 3 Solution:...
Read More →A helicopter rises from rest on the ground vertically upwards with a constant acceleration g.
Question: A helicopter rises from rest on the ground vertically upwards with a constant acceleration $g$. A food packet is dropped from the helicopter when it is at a height $h$. The time taken by the packet to reach the ground is close to [ $g$ is the accelertion due to gravity] :(1) $t=\frac{2}{3} \sqrt{\left(\frac{h}{g}\right)}$(2) $t=1.8 \sqrt{\frac{h}{g}}$(3) $t=3.4 \sqrt{\left(\frac{h}{g}\right)}$(4) $t=\sqrt{\frac{2 h}{3 g}}$Correct Option: , 3 Solution: (3) For upward motion of helicopte...
Read More →In the following reaction sequence, [C] is :
Question: In the following reaction sequence, $[\mathrm{C}]$ is : Correct Option: 1 Solution: b...
Read More →The speed verses time graph for a particle is shown in the figure.
Question: The speed verses time graph for a particle is shown in the figure. The distance travelled (in $\mathrm{m}$ ) by the particle during the time interval $t=0$ to $t=5 \mathrm{~s}$ will be_____________ Solution: (20) Distance travelled = Area of speed-time graph $=\frac{1}{2} \times 5 \times 8=20 \mathrm{~m}$...
Read More →Let the observations
Question: Let the observations $x_{i}(1 \leq i \leq 10)$ satisfy the equations, $\sum_{i=1}^{10}\left(x_{i}-5\right)=10$ and $\sum_{i=1}^{10}\left(x_{i}-5\right)^{2}=40$. If $\mu$ and $\lambda$ are the mean and the variance of the observations, $x_{1}-3, x_{2}-3, \ldots$ $x_{10}-3$, then the ordered pair $(\mu, \lambda)$ is equal to:(1) $(3,3)$(2) $(6,3)$(3) $(6,6)$(4) $(3,6)$Correct Option: 1 Solution: Mean of the observation $\left(x_{i}-5\right)=\frac{\Sigma\left(x_{i}-5\right)}{10}=1$ $\ther...
Read More →Solve the following
Question: Three isomers $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ (mol. formula $\mathrm{C}_{8} \mathrm{H}_{11} \mathrm{~N}$ ) give the following results : Correct Option: , 2 Solution:...
Read More →A small ball of mass $m$ is thrown upward with velocity u from the ground.
Question: A small ball of mass $m$ is thrown upward with velocity $u$ from the ground. The ball experiences a resistive force $m k v^{2}$ where $v$ is its speed. The maximum height attained by the ball is :(1) $\frac{1}{2 k} \tan ^{-1} \frac{k u^{2}}{g}$(2) $\frac{1}{k} \ln \left(1+\frac{k u^{2}}{2 g}\right)$(3) $\frac{1}{k} \tan ^{-1} \frac{k u^{2}}{2 g}$(4) $\frac{1}{2 k} \ln \left(1+\frac{k u^{2}}{g}\right)$Correct Option: , 4 Solution: $\vec{F}=m k v^{2}-m g$ $\left(\because m g\right.$ and ...
Read More →The mean and variance of 20 observations are
Question: The mean and variance of 20 observations are found to be 10 and 4 , respectively. On rechecking, it was found that an observation 9 was incorrect and the correct observation was 11 . Then the correct variance is:(1) $3.99$(2) $4.01$(3) $4.02$(4) $3.98$Correct Option: 1 Solution: Let $x_{1}, x_{2}, \ldots . ., x_{20}$ be 20 observations, then Mean $=\frac{x_{1}+x_{2}+\ldots . .+x_{20}}{20}=10$ $\Rightarrow \frac{\sum_{i=1}^{20} x_{i}}{20}=10$...(i) Variance $=\frac{\Sigma x_{i}^{2}}{n}-...
Read More →Starting from the origin at time $t=0$, with initial velocity
Question: Starting from the origin at time $t=0$, with initial velocity $5 \hat{j} \mathrm{~ms}^{-1}$, a particle moves in the $x-y$ plane with a constant acceleration of $(10 \hat{i}+4 \hat{j}) \mathrm{ms}^{-2}$. At time $t$, its coordiantes are $\left(20 \mathrm{~m}, y_{0} \mathrm{~m}\right)$. The values of $t$ and $y_{0}$ are, respectively:(1) $2 \mathrm{~s}$ and $18 \mathrm{~m}$(2) $4 \mathrm{~s}$ and $52 \mathrm{~m}$(3) $2 \mathrm{~s}$ and $24 \mathrm{~m}$(4) $5 \mathrm{~s}$ and $25 \mathrm...
Read More →The mean and the standard deviation
Question: The mean and the standard deviation (s.d.) of 10 observations are 20 and 2 respectively. Each of these 10 observations is multiplied by $p$ and then reduced by $q$, where $p \neq 0$ and $q \neq 0$. If the new mean and new s.d. become half of their original values, then $q$ is equal to:(1) $-5$(2) 10(3) $-20$(4) $-10$Correct Option: , 3 Solution: Let $\bar{x}$ and $\sigma$ be the mean and standard deviations of given observations. If each observation is multiplied with $p$ and then $q$ ...
Read More →