The major products of the following reaction are:
Question: The major products of the following reaction are: Correct Option: , 4 Solution:...
Read More →If p(x) be a polynomial of degree three that has
Question: If $p(x)$ be a polynomial of degree three that has a local maximum value 8 at $x=1$ and a local minimum value 4 at $x$ $=2 ;$ then $p(0)$ is equal to :(1) 6(2) $-12$(3) $-24$(4) 12Correct Option: , 2 Solution: Let $p^{\prime}(x)=\lambda(x-1)(x-2)$ where $\lambda0$ $p(x)=\lambda\left[\frac{x^{3}}{3}-\frac{3 x^{2}}{2}+2 x\right]+C$ Since $p(1)=8 \Rightarrow \lambda\left(\frac{1}{3}-\frac{3}{2}+2\right)+C=8$ $\Rightarrow \frac{5 \lambda}{6}+C=8$ ...........(1) Also, $p(2)=4 \Rightarrow \l...
Read More →A series LCR circuit is designed to resonate at an angular frequency
Question: A series LCR circuit is designed to resonate at an angular frequency $\omega_{0}=10^{5} \mathrm{rad} / \mathrm{s}$. The circuit draws $16 \mathrm{~W}$ power from $120 \mathrm{~V}$ source at resonance. The value of resistance ' $\mathrm{R}^{\prime}$ in the circuit is $\Omega$ Solution: (900) $P=\frac{V^{2}}{R}$ $16=\frac{120^{2}}{R} \Rightarrow R=\frac{14400}{16}$ $\Rightarrow R=900 \Omega$...
Read More →Figure shows a circuit that contains four identical resistors with resistance
Question: Figure shows a circuit that contains four identical resistors with resistance $\mathrm{R}=2.0 \Omega$. TWo identical inductors with inductance $\mathrm{L}=2.0 \mathrm{mH}$ and an ideal battery with emf $\mathrm{E}=9$. V. The current 'i' just after the switch 's' is closed will be : (1) $9 \mathrm{~A}$(2) $3.0 \mathrm{~A}$(3) $2.25 \mathrm{~A}$(4) $3.37 \mathrm{~A}$Correct Option: , 3 Solution: (3) When switch $S$ is closed- Given : $v=9 v$ From $V=I R$ $I=\frac{V}{R}$ $R_{\mathrm{eq} ....
Read More →If the tangent to the curve
Question: If the tangent to the curve $y=x+\sin y$ at a point $(a, b)$ is parallel to the line joining $\left(0, \frac{3}{2}\right)$ and $\left(\frac{1}{2}, 2\right)$, then :(1) $b=a$(2) $|b-a|=1$(3) $|a+b|=1$(4) $b=\frac{\pi}{2}+a$Correct Option: Solution: The given curve $y=x+\sin y$ $\because$ The point $(a, b)$ lie on the curve $\therefore b=a+\sin b$ $\Rightarrow \frac{d y}{d x}=1+\cos y \frac{d y}{d x} \Rightarrow(1-\cos y) \frac{d y}{d x}=1$ $\Rightarrow \frac{d y}{d x}=\frac{1}{1-\cos y}...
Read More →The major product of the following reaction is:
Question: The major product of the following reaction is: Correct Option: 1 Solution:...
Read More →Find the perimeter and area of a quadrilateral ABCD in which BC = 12 cm,
Question: Find the perimeter and area of a quadrilateralABCDin whichBC= 12 cm,CD= 9 cm,BD= 15 cm,DA= 17 cm and ABD= 90. Solution: We know that $\triangle A B D$ is a right-angled triangle. $\therefore A B^{2}=\sqrt{A D^{2}-D B^{2}}=\sqrt{17^{2}-15^{2}}=\sqrt{289-225}=\sqrt{64}=8 \mathrm{~cm}$ Now, Area of triangle $A B D=\frac{1}{2} \times$ Base $\times$ Height $=\frac{1}{2} \times A B \times B D$ $=\frac{1}{2} \times 8 \times 15$ $=60 \mathrm{~cm}^{2}$ Let : $a=9 \mathrm{~cm}, b=15 \mathrm{~cm}...
Read More →Let a be an integer such that all the real roots
Question: Let a be an integer such that all the real roots of the polynomial $2 x^{5}+5 x^{4}+10 x^{3}+10 x^{2}+10 x+10$ lie in the interval $(a, a+1)$ Then, $|\mathbf{a}|$ is equal to Solution: Let, $f(x)=2 x^{5}+5 x^{4}+10 x^{3}+10 x^{2}+10 x+10$ $\Rightarrow f^{\prime}(x)=10\left(x^{4}+2 x^{3}+3 x^{2}+2 x+1\right)$ $=10\left(x^{2}+\frac{1}{x^{2}}+2\left(x+\frac{1}{x}\right)+3\right)$ $=10\left(\left(x+\frac{1}{z}\right)^{2}+2\left(x+\frac{1}{x}\right)+1\right)$ $=10\left(\left(x+\frac{1}{x}\r...
Read More →The increasing order of boiling points of the following compounds is :
Question: The increasing order of boiling points of the following compounds is : $\mathrm{I}\mathrm{III}\mathrm{IV}\mathrm{II}$$\mathrm{I}\mathrm{IV}\mathrm{III}\mathrm{II}$$\mathrm{IV}\mathrm{I}\mathrm{II}\mathrm{III}$$\mathrm{III}\mathrm{I}\mathrm{II}\mathrm{IV}$Correct Option: , 2 Solution: (II) and (III) compounds almost have same boiling point. In the given options, option (b) will be the correct answer....
Read More →The sides of a quadrilateral ABCD taken in order are 6 cm, 8 cm,
Question: The sides of a quadrilateral $A B C D$ taken in order are $6 \mathrm{~cm}, 8 \mathrm{~cm}, 12 \mathrm{~cm}$ and $14 \mathrm{~cm}$ respectively and the angle between the first two sides is a right angle. Find its area. (Given, $\sqrt{6}=2.45$ ). Solution: In the given figure,ABCDis a quadrilateral with sides of length 6 cm, 8 cm, 12 cm and 14 cm respectively and the angle between the first two sides is a right angle. JoinAC. In right angled ∆ABC, $A C^{2}=A B^{2}+B C^{2}$ (Pythagoras Th...
Read More →Let the normals at all the points on a given curve
Question: Let the normals at all the points on a given curve pass through a fixed point $(a, b)$. If the curve passes through $(3,-3)$ and $(4,-2 \sqrt{2})$, and given that $a-2 \sqrt{2} \mathrm{~b}=3$, then $\left(\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{ab}\right)$ is equal to Solution: Let the equation of normal is $Y-y=-\frac{1}{m}(X-x)$, where, $m=\frac{d y}{d x}$ As it passes through $(a, b)$ $b-y=-\frac{1}{m}(a-x)=-\frac{d x}{d y}(a-x)$ $\Rightarrow(b-y) d y=(x-a) d x$ by $-\frac{y^{2}}{2}=\...
Read More →Which of the following derivatives of alcohols is unstable
Question: Which of the following derivatives of alcohols is unstable in an aqueous base?Correct Option: 1 Solution: Esters are hydrolysed in basic medium (saponification), so it is unstable in aqueous base....
Read More →The triangle of maximum area that can be inscribed in a given circle of
Question: The triangle of maximum area that can be inscribed in a given circle of radius ' $r$ ' is:(1) A right angle triangle having two of its sides of length $2 r$ and $r$.(2) An equilateral triangle of height $\frac{2 r}{3}$.(3) An isosceles triangle with base equal to $2 r$.(4) An equilateral triangle having each of its side of length $\sqrt{3} \mathrm{r}$.Correct Option: 4, Solution: Triangle of maximum area that can be inscribed in a circle is an equilateral triangle. Let $\triangle \math...
Read More →A resonance circuit having inductance and resistance $2 imes 10^{-4} mathrm{H}$ and $6.28 Omega$
Question: A resonance circuit having inductance and resistance $2 \times 10^{-4} \mathrm{H}$ and $6.28 \Omega$ respectively oscillates at $10 \mathrm{MHz}$ frequency. The value of quality factor of this resonator is $[\pi=3.14]$ Solution: $(2000)$ Given : $\mathrm{R}=6.28 \Omega$ $\mathrm{L}=2 \times 10^{-4}$ Henry we know that quality factor $Q$ is given by $\Rightarrow Q=\frac{X_{L}}{R}=\frac{\omega L}{R}$ also, $\omega=2 \pi f$, so $\Rightarrow Q=\frac{2 \pi f L}{R}$ $\Rightarrow Q=\frac{2 \p...
Read More →The major product [B] in the following reactions is :
Question: The major product [B] in the following reactions is : Correct Option: , 2 Solution:...
Read More →The maximum slope of the curve
Question: The maximum slope of the curve $y=\frac{1}{2} x^{4}-5 x^{3}+18 x^{2}-19 x$ occurs at the point:(1) $(2,9)$ (2) $(2,2)$(3) $\left(3, \frac{21}{2}\right)$(4) $(0,0)$Correct Option: , 2 Solution: $\frac{d y}{d x}=2 x^{3}-15 x^{2}+36 x-19$ Let $f(x)=2 x^{3}-15 x^{2}+36 x-19$ $f^{\prime}(x)=6 x^{2}-30 x+36=0$ $x^{2}-5 x+6=0$ $x=2,3$ $f "(x)=12 x-30$ $f "(x)0$ for $x=2$ At $x=2$ $y=8-40+72-38$ $y=72-70=2$ $\Rightarrow(2,2)$...
Read More →When neopentyl alcohol is heated with an acid,
Question: When neopentyl alcohol is heated with an acid, it slowly converted into an $85: 15$ mixture of alkenes $\mathrm{A}$ and $\mathrm{B}$, respectively. What are these alkenes?Correct Option: , 2 Solution:...
Read More →In a series LCR circuit, the inductive reactance $left(X_{L}ight)$ is $10 Omega$ and the
Question: In a series LCR circuit, the inductive reactance $\left(X_{L}\right)$ is $10 \Omega$ and the capacitive reactance $\left(\mathrm{X}_{\mathrm{C}}\right)$ is $4 \Omega$. The resistance $(\mathrm{R})$ in the circuit is $6 \Omega$. The power factor of the circuit is:(1) $\frac{1}{2}$(2) $\frac{1}{2 \sqrt{2}}$(3) $\frac{1}{\sqrt{2}}$(4) $\frac{\sqrt{3}}{2}$Correct Option: 3, Solution: (3) We know that power factor is $\cos \phi$, $\cos \phi=\frac{\mathrm{R}}{\mathrm{Z}} \ldots(1)$ $\mathrm{...
Read More →If the curves
Question: If the curves $x=y^{4}$ and $x y=k$ cut at right angles, then $(4 k)^{6}$ is equal to Solution: $4 y^{3} \frac{d y}{d x}=1 \quad \ \quad x \frac{d y}{d x}+y=0$ $m_{1}=\frac{1}{4 y^{3}} \frac{d y}{d x}=\frac{-y}{x}=m_{2}$ $m_{1} m_{2}=-1$ $\frac{1}{4 \cdot y^{3}} \times \frac{-y}{x}=-1 \because x=y^{4}$ $\frac{1}{4 \cdot y^{6}}=1$ and $x y=k$ $y^{6}=\frac{1}{4}$ $\Rightarrow k^{6}=y^{30}$ $\Rightarrow k^{6}=\left(\frac{1}{4}\right)^{5}$ $\therefore(4 k)^{6}=4^{6} \times k^{6}=4$...
Read More →An organic compound (A)
Question: An organic compound (A) (molecular formula $\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{2}$ ) was hydrolysed with dil. $\mathrm{H}_{2} \mathrm{SO}_{4}$ to give a carboxylic acid (B) and an alochol (C). 'C' gives white turbidity immediately when treated with anhydrous $\mathrm{ZnCl}_{2}$ and conc. $\mathrm{HCl}$. The organic compound (A) is :Correct Option: 1 Solution:...
Read More →Find the area of the shaded region in the figure given below.
Question: Find the area of the shaded region in the figure given below. Solution: In right angled ∆ABD, $A B^{2}=A D^{2}+D B^{2}$ (Pythagoras Theorem) $\Rightarrow A B^{2}=12^{2}+16^{2}$ $\Rightarrow A B^{2}=144+256$ $\Rightarrow A B^{2}=400$ $\Rightarrow A B=20 \mathrm{~cm}$ Area of $\triangle A D B=\frac{1}{2} \times D B \times A D$ $=\frac{1}{2} \times 16 \times 12$ = 96 cm2 ....(1) In ∆ACB,The sides of the triangle are of length 20 cm, 52 cm and 48 cm. Semi-perimeter of the triangle is $s=\f...
Read More →The shortest distance between the line
Question: The shortest distance between the line $x-y=1$ and the curve $x^{2}=2 y$ is:(1) $\frac{1}{2}$ (2) 0(3) $\frac{1}{2 \sqrt{2}}$(4) $\frac{1}{\sqrt{2}}$Correct Option: Solution: Shortest distance must be along common normal $m_{1}($ slope of line $x-y=1)=1 \Rightarrow$ slope of perpendicular line $=-1 m_{2}=\frac{2 x}{2}=x \Rightarrow m_{2}=h \Rightarrow$ slope of normal $-\frac{1}{h}$ $-\frac{1}{h}=-1 \Rightarrow h=1$ sopoint is $\left(1, \frac{1}{2}\right)$ $\mathrm{D}=\left|\frac{1-\fr...
Read More →Consider the following reaction :
Question: Consider the following reaction : The product 'P' gives positive ceric ammonium nitrate test. This is because of the presence of which of these $-\mathrm{OH} \operatorname{group}(\mathrm{s})$ ?(ii) only(iii) and (iv)(iv) only(ii) and (iv)Correct Option: 1 Solution: Generally CAN test is done for alcohols which give pink or red colour. But for phenols and phenolic compounds it gives brown or black colour. So, this test helps to diffirentiate phenols from alcohols....
Read More →Two compounds A and A with same molecular formula
Question: Two compounds $A$ and $B$ with same molecular formula $\left(\mathrm{C}_{3} \mathrm{H}_{6} \mathrm{O}\right)$ undergo Grignard's reaction with methylmagnesium bromide to give products $\mathrm{C}$ and $\mathrm{D}$. Products $\mathrm{C}$ and D show following chemical tests. $\bar{C}$ and $D$ respectively are :Correct Option: 1 Solution:...
Read More →Let $f(x)$ be a polynomial of degree 6 in
Question: Let $f(x)$ be a polynomial of degree 6 in $x$, in which the coefficient of $x^{6}$ is unity and it has extrema at $x=-1$ and $x=1$. If $\lim _{x \rightarrow 0} \frac{f(x)}{x^{3}}=1$, then 5. $f(2)$ is equal to Solution: $f(x)=x^{6}+a x^{5}+b x^{4}+x^{3}$ $\therefore f^{\prime}(x)=6 x^{5}+5 a x^{4}+4 b x^{3}+3 x^{2}$ Roots $1 \-1$ $\therefore 6+5 a+4 b+3=0 \-6+5 a-4 b+3=0$ solving $a=-\frac{3}{5} \quad b=-\frac{3}{2}$ $\therefore f(x)=x^{6}-\frac{3}{5} x^{5}-\frac{3}{2} x^{4}+x^{3}$ $\t...
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