A spaceship in space sweeps stationary interplanetary dust.

Question: A spaceship in space sweeps stationary interplanetary dust. As a result, its mass increases at a rate $\frac{d M(t)}{d t}=b v^{2}(t)$, where $v(t)$ is its instantaneous velocity. The instantaneous acceleration of the satellite is :(1) $-b v^{3}(t)$$-\frac{b v^{3}}{M(t)}$(3) $-\frac{2 b v^{3}}{M(t)}$(4) $-\frac{b v^{3}}{2 M(t)}$Correct Option: , 2 Solution: (2) From the Newton's second law, $F=\frac{d p}{d t}=\frac{d(m v)}{d t}=v\left(\frac{d m}{d t}\right)$ ...(1) We have given, $\frac...

A block starts moving up an inclined plane of inclination

Question: A block starts moving up an inclined plane of inclination $30^{\circ}$ with an initial velocity of $v_{0}$. It comes back to its initial position with velocity $\frac{v_{0}}{2}$. The value of the coefficient of kinetic friction between the block and the inclined plane is close to $\frac{I}{1000}$. The nearest integer to $I$ is Solution: (346) Acceleration of block while moving up an inclined plane, $a_{1}=g \sin \theta+\mu g \cos \theta$ $\Rightarrow a_{1}=g \sin 30^{\circ}+\mu g \cos ...

An inclined plane making an angle

Question: An inclined plane making an angle of $30^{\circ}$ with horizontal is placed in a uniform horizontal electric field $200 \frac{N}{C}$ as shown in the figure. A body of mass $1 \mathrm{~kg}$ and charge $5 \mathrm{mC}$ is allowed to slide down from rest at a height of $1 \mathrm{~m}$. If the coefficient of frication is $0.2$, find the time taken by the body to reach the bottom. $\left[g=9.8 \mathrm{~m} / \mathrm{s}^{2}, \sin 30^{\circ}=\frac{1}{2} ; \cos 30^{\circ}=\frac{\sqrt{3}}{2}\righ...

Objective Questions (MCQ)

Question: Objective Questions (MCQ) In the equation $a x^{2}+b x+c=0$, it is given that $D=\left(b^{2}-4 a c\right)0$. Then, the roots of the equation are (a) real and equal (b) real and unequal(c) imaginary (d) none of these Solution: We know that when discriminant, $D0$, the roots of the given quadratic equation are real and unequal. Hence, the correct answer is option B....

The roots of

Question: The roots of $a x^{2}+b x+c=0, a \neq 0$ are real and unequal, if $\left(b^{2}-4 a c\right)$ (a) 0(b) = 0(c) 0(d) none of these Solution: (a) 0 The roots of the equation are real and unequal when $\left(b^{2}-4 a c\right)0$....

Some identical balls are arranged in rows to form an equilateral

Question: Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If 99 more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square whose each side contains exactly 2 balls less than the number of balls each side of the triangle contains. Then the number of balls used to form the equilateral triangle is:(...

If the equation

Question: If the equation $4 x^{2}-3 k x+1=0$ has equal roots, then $k=$ ? (a) $\pm \frac{2}{3}$ (b) $\pm \frac{1}{3}$ (c) $\pm \frac{3}{4}$ (d) $\pm \frac{4}{3}$ Solution: (d) $\pm \frac{4}{3}$ It is given that the roots of the equation $\left(4 x^{2}-3 k x+1=0\right)$ are equal. $\therefore\left(b^{2}-4 a c\right)=0$ $\Rightarrow(3 k)^{2}-4 \times 4 \times 1=0$ $\Rightarrow 9 k^{2}=16$ $\Rightarrow k^{2}=\frac{16}{9}$ $\Rightarrow k=\pm \frac{4}{3}$...

As shown in the figure, a block of mass

Question: As shown in the figure, a block of mass $\sqrt{3} \mathrm{~kg}$ is kept on a horizontal rough surface of coefficient of friction $\frac{1}{3 \sqrt{3}}$. The critical force to be applied on the vertical surface as shown at an angle $60^{\circ}$ with horizontal such that it does not move, will be $3 \mathrm{x}$. The value of $\mathrm{x}$ will be $\left[\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2} ; \sin 60^{\circ}=\frac{\sqrt{3}}{2} ; \cos 60^{\circ}=\frac{1}{2}\right]$ (round off to neare...

If the equation

Question: If the equation $x^{2}+2(k+2) x+9 k=0$ has equal rots, then $k=?$ (a) 1 or 4(b) 1 or 4(c) 1 or 4(d) 1 or 4 Solution: (a) 1 or 4 It is given that the roots of the equation $\left(x^{2}+2(k+2) x+9 k=0\right)$ are equal. $\therefore\left(b^{2}-4 a c\right)=0$ $\Rightarrow\{2(k+2)\}^{2}-4 \times 1 \times 9 k=0$ $\Rightarrow 4\left(k^{2}+4 k+4\right)-36 k=0$ $\Rightarrow 4 k^{2}+16 k+16-36 k=0$ $\Rightarrow 4 k^{2}-20 k+16=0$ $\Rightarrow k^{2}-5 k+4=0$ $\Rightarrow k^{2}-4 k-k+4=0$ $\Right...

If the equation

Question: If the equation $9 x^{2}+6 k x+4=0$ has equal roots then $k=?$ (a) 2 or 0(b) 2 or 0(c) 2 or 2(d) 0 only Solution: (c) 2 or 2 It is given that the roots of the equation $\left(9 x^{2}+6 k x+4=0\right)$ are equal. $\therefore\left(b^{2}-4 a c\right)=0$ $\Rightarrow(6 k)^{2}-4 \times 9 \times 4=0$ $\Rightarrow 36 k^{2}=144$ $\Rightarrow k^{2}=4$ $\Rightarrow k=\pm 2$...

If the roots of the equation

Question: If the roots of the equation $a x^{2}+b x+c=0$ are equal, then then $c=?$ (a) $\frac{-b}{2 a}$ (b) $\frac{b}{2 a}$ (c) $\frac{-b^{2}}{4 a}$ (d) $\frac{b^{2}}{4 a}$ Solution: (d) $\frac{b^{2}}{4 a}$ It is given that the roots of the equation $\left(a x^{2}+b x+c=0\right)$ are equal. $\therefore\left(b^{2}-4 a c\right)=0$ $\Rightarrow b^{2}=4 a c$ $\Rightarrow c=\frac{b^{2}}{4 a}$...

An Ellingham diagram provides information about:

Question: An Ellingham diagram provides information about:the conditions of $\mathrm{pH}$ and potential under which a species is thermodynamically stable.the temperature dependence of the standard Gibbs energies of formation of some metal oxides.the pressure dependence of the standard electrode potentials of reduction reactions involved in the extraction of metals.the kinetics of the reduction process.Correct Option: , 2 Solution: Ellingham diagram is the graph of $\Delta \mathrm{G}^{0} \mathrm{...

A committee of 11 members is to be formed

Question: A committee of 11 members is to be formed from 8 males and 5 females. If $m$ is the number of ways the committee is formed with at least 6 males and $n$ is the number of ways the committee is formed with at least 3 females, then:(1) $m+n=68$(2) $m=n=78$(3) $\mathrm{n}=\mathrm{m}-8$(4) $\mathrm{m}=\mathrm{n}=68$Correct Option: , 2 Solution: Since, $m=$ number of ways the committee is formed with at least 6 males $={ }^{8} \mathrm{C}_{6} \cdot{ }^{5} \mathrm{C}_{5}+{ }^{8} \mathrm{C}_{7}...

The roots of the equation

Question: The roots of the equation $a x^{2}+b x+c=0$ will be reciprocal of each other if (a)a=b(b)b=c(c)c=a(d) none of these Solution: (c)c = a Let the roots of the equation $\left(a x^{2}+b x+c=0\right)$ be $\alpha$ and $\frac{1}{\alpha}$. $\therefore$ Product of the roots $=\alpha \times \frac{1}{\alpha}=1$ $\Rightarrow \frac{c}{a}=1$ $\Rightarrow c=a$...

The number of four-digit numbers strictly

Question: The number of four-digit numbers strictly greater than 4321 that can be formed using the digits $0,1,2,3,4,5$ (repetition of digits is allowed) is:(1) 288(2) 360(3) 306(4) 310Correct Option: , 4 Solution:...

The processes of calcination and roasting in metallurgical industries,

Question: The processes of calcination and roasting in metallurgical industries, respectively, can lead to :Global warming and photochemical smogGlobal warming and acid rainPhotochemical smog and ozone layer depletionPhotochemical smog and global warmingCorrect Option: , 2 Solution:...

Objective Questions (MCQ)

Question: Objective Questions (MCQ) If $\alpha$ and $\beta$ are the roots of the equation $3 x^{2}+8 x+2=0$ then $\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)=$ ? (a) $\frac{-3}{8}$ (b) $\frac{2}{3}$ (c) $-4$ (d) 4 Solution: It is given that $\alpha$ and $\beta$ are the roots of the equation $3 x^{2}+8 x+2=0$. $\therefore \alpha+\beta=-\frac{8}{3}$ and $\alpha \beta=\frac{2}{3}$ $\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha \beta}=\frac{-\frac{8}{3}}{\frac{2}{3}}=-4$ Hence, the c...

An inclined plane is bent in such a way that the vertical cross-section is given by

Question: An inclined plane is bent in such a way that the vertical cross-section is given by $y=\frac{x^{2}}{4}$ where $y$ is in vertical and $x$ in horizontal direction. If the upper surface of this curved plane is rough with coefficient of friction $\mu=0.5$, the maximum height in $\mathrm{cm}$ at which a stationary block will not slip downward is $\mathrm{cm}$. Solution: given $y=\frac{x^{2}}{4}$ $\mu=0.5$ condition for block will not slip downward $\mathrm{mg} \sin \theta=\mu \mathrm{mg} \c...

The sum of all natural numbers

Question: The sum of all natural numbers ' $n$ ' such that $100n200$ and H.C.F. $(91, n)1$ is :(1) 3203(2) 3303(3) 3221(4) 3121Correct Option: , 4 Solution: $\because 91=13 \times 7$ Then, the required numbers are either divisible by 7 or 13 . $\therefore$ Sum of such numbers $=$ Sum of no. divisible by $7+$ sum of the no. divisible by 13 - Sum of the numbers divisible by 91 $=(105+112+\ldots+196)+(104+117+\ldots+195)-182$ $=2107+1196-182$ $=3121$...

Among statements (1) - (4), the correct ones are :

Question: Among statements (1) - (4), the correct ones are : (a) Lime stone is decomposed to $\mathrm{CaO}$ during the extraction of iron from its oxides. (b) In the extraction of silver, silver is extracted as an anionic complex. (c) Nickel is purified by Mond's process. (d) Zr and Ti are purified by Van Arkel method.(a), (b), (c) and (d)(a), (c) and (d) only(b), (c) and (d) only(c) and (d) onlyCorrect Option: 1 Solution:...

If the sum of the roots of a quadratic equation is 6 and their product is 6, the equation is

Question: If the sum of the roots of a quadratic equation is 6 and their product is 6, the equation is (a) $x^{2}-6 x+6=0$ (b) $x^{2}+6 x-6=0$ (c) $x^{2}-6 x-6=0$ (d) $x^{2}+6 x+6=0$ Solution: (a) $x^{2}-6 x+6=0$ Given : Sum of roots $=6$ Product of roots $=6$ Thus, the equation is: $x^{2}-6 x+6=0$...

All possible numbers are formed using the digits

Question: All possible numbers are formed using the digits $1,1,2,2$, $2,2,3,4,4$ taken all at a time. The number of such numbers in which the odd digits occupy even places is :(1) 180(2) 175(3) 160(4) 162Correct Option: 1 Solution: $\because$ There are total 9 digits and out of which only 3 digits are odd. $\therefore$ Number of ways to arrange odd digits first $={ }^{4} C_{3} \cdot \frac{3 !}{2 !}$ Hence, total number of 9 digit numbers $=\left({ }^{4} C_{3} \cdot \frac{3 !}{2 !}\right) \cdot ...

Question: The coefficient of static friction between a wooden block of mass $0.5 \mathrm{~kg}$ and a vertical rough wall is $0.2$. The magnitude of horizontal force that should be applied on the block to keep it adhere to the wall will be $\mathrm{N}\left[\mathrm{g}=10 \mathrm{~ms}^{-2}\right]$ Solution: Given : $\mu_{\mathrm{s}}=0.2$ $m=0.5 \mathrm{~kg}$ $\mathrm{~g}=10 \mathrm{~m} / \mathrm{s}^{2}$ we know that $f_{s}=\mu N$ and $\ldots(1)$ To keep the block adhere to the wall here $\mathrm{N}...

Cast iron is used for the manufacture of:

Question: Cast iron is used for the manufacture of:wrought iron and pig ironpig iron, scrap iron and steelwrought iron, pig iron and steelwrought iron and steelCorrect Option: , 4 Solution: Cast iron is made from pig iron which is used for production of wroght iron and steel....

The root of a quadratic equation are 5 and −2. Then, the equation is

Question: The root of a quadratic equation are 5 and 2. Then, the equation is (a) $x^{2}-3 x+10=0$ (b) $x^{2}-3 x-10=0$ (c) $x^{2}+3 x-10=0$ (d) $x^{2}+3 x+10=0$ Solution: (b) $x^{2}-3 x-10=0$ It is given that the roots of the quadratic equation are 5 and $-2$. Then, the equation is: $x^{2}-(5-2) x+5 \times(-2)=0$ $\Rightarrow x^{2}-3 x-10=0$...