The correct match between Item-I and Item-II is:
Question: The correct match between Item-I and Item-II is: $\mathrm{A} \rightarrow \mathrm{R} ; \mathrm{B} \rightarrow \mathrm{P} ; \mathrm{C} \rightarrow \mathrm{S} ; \mathrm{D} \rightarrow \mathrm{Q}$$\mathrm{A} \rightarrow \mathrm{Q} ; \mathrm{B} \rightarrow \mathrm{S} ; \mathrm{C} \rightarrow \mathrm{P} ; \mathrm{D} \rightarrow \mathrm{R}$$\mathrm{A} \rightarrow \mathrm{R} ; \mathrm{B} \rightarrow \mathrm{S} ; \mathrm{C} \rightarrow \mathrm{P} ; \mathrm{D} \rightarrow \mathrm{Q}$$\mathrm{A} ...
Read More →The equation of a tangent to the hyperbola
Question: The equation of a tangent to the hyperbola $4 x^{2}-5 y^{2}=20$ parallel to the line $x-y=2$ is:(1) $x-y+1=0$(2) $x-y+7=0$(3) $x-y+9=0$(4) $x-y-3=0$Correct Option: 1 Solution: (1) Given, the equation of line, $x-y=2 \Rightarrow y=x-2$ $\therefore$ its slope $=m=1$ Equation of hyperbola is: $\frac{x^{2}}{5}-\frac{y^{2}}{4}=1$ $\Rightarrow a^{2}=5, b^{2}=4$ The equation of tangent to the hyperbola is, $y=m x \pm \sqrt{a^{2} m^{2}-b^{2}}$\ $=x \pm \sqrt{5-4}$ $\Rightarrow y=x \pm 1$...
Read More →A particle of mass $m$ and charge q is in an electric and magnetic field given by
Question: A particle of mass $m$ and charge $q$ is in an electric and magnetic field given by $\overrightarrow{\mathrm{E}}=2 \hat{i}+3 \hat{j} ; \overrightarrow{\mathrm{B}}=4 \hat{j}+6 \hat{k}$ The charged particle is shifted from the origin to the point $\mathrm{P}(x=1 ; y=1)$ along a straight path. The magnitude of the total work done is :(1) $(0.35) \mathrm{q}$(2) $5 \mathrm{q}$(3) (2.5)q(4) $(0.15) \mathrm{q}$Correct Option: , 2 Solution: (2) Resultant force on the charged particle $=\overri...
Read More →Prove that
Question: $10 x-\frac{1}{3}=3$ Solution: Given: $10 x-\frac{1}{x}=3$ $\Rightarrow 10 x^{2}-1=3 x \quad[$ Multiplying both sides by $x]$ $\Rightarrow 10 x^{2}-3 x-1=0$ $\Rightarrow 10 x^{2}-(5 x-2 x)-1=0$ $\Rightarrow 10 x^{2}-5 x+2 x-1=0$ $\Rightarrow 5 x(2 x-1)+1(2 x-1)=0$ $\Rightarrow(2 x-1)(5 x+1)=0$ $\Rightarrow 2 x-1=0$ or $5 x+1=0$ $\Rightarrow x=\frac{1}{2}$ or $x=\frac{-1}{5}$ Hence, the roots of the equation are $\frac{1}{2}$ and $\frac{-1}{5}$....
Read More →A hyperbola has its centre at the origin,
Question: A hyperbola has its centre at the origin, passes through the point $(4,2)$ and has transverse axis of length 4 along the $x$-axis. Then the eccentricity of the hyperbola is :(1) $\frac{3}{2}$(2) $\sqrt{3}$(3) 2(4) $\frac{2}{\sqrt{3}}$Correct Option: , 4 Solution: Consider equation of hyperbola $\frac{x^{2}}{2^{2}}-\frac{y^{2}}{b^{2}}=1$ $\because(4,2)$ lies on hyperbola $\therefore \frac{16}{4}-\frac{4}{b^{2}}=1$ $\therefore b^{2}=\frac{4}{3}$Since, eccentricity $=\sqrt{1+\frac{b^{2}}{...
Read More →The compound used in the treatment of lead poisoning is :
Question: The compound used in the treatment of lead poisoning is :D-penicillaminedesferrioxime BCis-platinEDTACorrect Option: , 4 Solution: EDTA is used in the treatment of lead poisoning. Deferrioxime B is used in treatment of iron poisoning and D-penicillamine is used in treatment of heavy metal poisoning, while cis-platin is used for treating cancer....
Read More →A 27 mW laser beam has a cross-sectional area
Question: A $27 \mathrm{~mW}$ laser beam has a cross-sectional area of $10 \mathrm{~mm}^{2}$. The magnitude of the maximum electric field in this electromagnetic wave is given by: [Given permittivity of space $\epsilon_{0}=9 \times 10^{-12}$ SI units, Speed of light $\mathrm{c}=3 \times 10^{8} \mathrm{~m} / \mathrm{s}$ ](1) $2 \mathrm{kV} / \mathrm{m}$(2) $0.7 \mathrm{kV} / \mathrm{m}$(3) $1 \mathrm{kV} / \mathrm{m}$(4) $1.4 \mathrm{kV} / \mathrm{m}$Correct Option: , 4 Solution: (4) EM wave inte...
Read More →Solve each of the following quadratic equations:
Question: Solve each of the following quadratic equations: $2 x^{2}-x+\frac{1}{8}=0$ Solution: We write, $-x=-\frac{x}{2}-\frac{x}{2}$ as $2 x^{2} \times \frac{1}{8}=\frac{x^{2}}{4}=\left(-\frac{x}{2}\right) \times\left(-\frac{x}{2}\right)$ $\therefore 2 x^{2}-x+\frac{1}{8}=0$ $\Rightarrow 2 x^{2}-\frac{x}{2}-\frac{x}{2}+\frac{1}{8}=0$ $\Rightarrow 2 x\left(x-\frac{1}{4}\right)-\frac{1}{2}\left(x-\frac{1}{4}\right)=0$ $\Rightarrow\left(x-\frac{1}{4}\right)\left(2 x-\frac{1}{2}\right)=0$ $\Righta...
Read More →Let 0 <
Question: Let $0\theta\frac{\pi}{2}$. If the eccentricity of the hyperbola $\frac{x^{2}}{\cos ^{2} \theta}-\frac{y^{2}}{\sin ^{2} \theta}=1$ is greater than 2 , then the length of its latus rectum lies in the interval: (1) $(3, \infty)$(2) $(3 / 2,2]$(3) $(2,3]$(4) $(1,3 / 2]$Correct Option: 1 Solution: $\because a^{2}=\cos ^{2} \theta, b^{2}=\sin ^{2} \theta$ and $e2 \Rightarrow e^{2}4 \Rightarrow 1+b^{2} / a^{2}4$ $\Rightarrow \quad 1+\tan ^{2} \theta4$ $\Rightarrow \sec ^{2} \theta4 \Rightarr...
Read More →Which of the following statements is not true about RNA?
Question: Which of the following statements is not true about RNA?It controls the synthesis of protein.It has always double stranded $\alpha$-helix structure.It usually does not replicate.It is present in the nucleus of the cell.Correct Option: , 2 Solution: RNA has a single helix structure, whereas, DNA has a double helix structure....
Read More →Solve each of the following quadratic equations:
Question: Solve each of the following quadratic equations: $100 x^{2}-20 x+1=0$ Solution: We write, $-20 x=-10 x-10 x$ as $100 x^{2} \times 1=100 x^{2}=(-10 x) \times(-10 x)$ $\therefore 100 x^{2}-20 x+1=0$ $\Rightarrow 100 x^{2}-10 x-10 x+1=0$ $\Rightarrow 10 x(10 x-1)-1(10 x-1)=0$ $\Rightarrow(10 x-1)(10 x-1)=0$ $\Rightarrow(10 x-1)^{2}=0$ $\Rightarrow 10 x-1=0$ $\Rightarrow x=\frac{1}{10}$ Hence, $\frac{1}{10}$ is the repreated root of the given equation....
Read More →Noradrenaline is a/an:
Question: Noradrenaline is a/an:AntacidNeurotransmitterAntidepressantAntihistamineCorrect Option: , 2 Solution: Noradrenaline, also called Norepinephrine, is neurotransmitter....
Read More →An electromagnetic wave of intensity
Question: An electromagnetic wave of intensity $50 \mathrm{Wm}^{-2}$ enters in a medium of refractive index ' $\mathrm{n}$ ' without any loss. The ratio of the magnitudes of electric fields, and the ratio of the magnitudes of magnetic fields of the wave before and after entering into the medium are respectively, given by:(1) $\left(\frac{1}{\sqrt{\mathrm{n}}}, \frac{1}{\sqrt{\mathrm{n}}}\right)$(2) $(\sqrt{\mathrm{n}}, \sqrt{\mathrm{n}})$(3) $\left(\sqrt{n}, \frac{1}{\sqrt{n}}\right)$(4) $\left(...
Read More →Let P be the point of intersection
Question: Let $\mathrm{P}$ be the point of intersection of the common tangents to the parabola $y^{2}=12 x$ and hyperbola $8 x^{2}-y^{2}=8$. If $\mathrm{S}$ and $\mathrm{S}^{\prime}$ denote the foci of the hyperbola where $S$ lies on the positive $x$-axis then P divides SS ' in a ratio :(1) $13: 11$(2) $14: 13$(3) $5: 4$(4) $2: 1$Correct Option: , 3 Solution: Equation of tangent to $y^{2}=12 x$ is $y=m x+\frac{3}{m}$ Equation of tangent to $\frac{x^{2}}{1}-\frac{y^{2}}{8}=1$ is $y=m x \pm \sqrt{...
Read More →Solve this
Question: $9 x^{2}+6 x+1=0$ Solution: Given : $9 x^{2}+6 x+1=0$ $\Rightarrow 9 x^{2}+3 x+3 x+1=0$ $\Rightarrow 3 x(3 x+1)+1(3 x+1)=0$ $\Rightarrow(3 x+1)(3 x+1)=0$ $\Rightarrow 3 x+1=0$ or $3 x+1=0$ $\Rightarrow x=\frac{-1}{3}$ or $x=\frac{-1}{3}$ Hence, $\frac{-1}{3}$ is the root of the equation $9 x^{2}+6 x+1=0$....
Read More →A chemist has 4 samples of artificial sweetener A, B, C and D.
Question: A chemist has 4 samples of artificial sweetener A, B, C and D. To identify these samples, he performed certain experiments and noted the following observations: (I) A and D both form blue-biolet colour with ninhydrin. (II) Lassaigne extract of $\mathrm{C}$ gives positive $\mathrm{AgNO}_{3}$ test and negative $\mathrm{Fe}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]_{3}$ test. (III)Lassaigne extract of B and D gives positive sodium nitroprusside test. Based on these observations which o...
Read More →Solve each of the following quadratic equations:
Question: Solve each of the following quadratic equations: $x^{2}-(1+\sqrt{2}) x+\sqrt{2}=0$ Solution: $x^{2}-(1+\sqrt{2}) x+\sqrt{2}=0$ $x^{2}-x-\sqrt{2} x+\sqrt{2}=0$ $x(x-1)-\sqrt{2}(x-1)=0$ $(x-\sqrt{2})(x-1)=0$ $\Rightarrow x-\sqrt{2}=0$ and $x-1=0$ $\Rightarrow x=\sqrt{2}$ and $x=1$...
Read More →Solve each of the following quadratic equations:
Question: Solve each of the following quadratic equations: $x^{2}-(1+\sqrt{2}) x+\sqrt{2}=0$ Solution: $x^{2}-(1+\sqrt{2}) x+\sqrt{2}=0$ $x^{2}-x-\sqrt{2} x+\sqrt{2}=0$ $x(x-1)-\sqrt{2}(x-1)=0$ $(x-\sqrt{2})(x-1)=0$ $\Rightarrow x-\sqrt{2}=0$ and $x-1=0$ $\Rightarrow x=\sqrt{2}$ and $x=1$...
Read More →Solve each of the following quadratic equations:
Question: Solve each of the following quadratic equations: $5 x^{2}+13 x+8=0$ Solution: We write, $13 x=5 x+8 x$ as $5 x^{2} \times 8=40 x^{2}=5 x \times 8 x$ $\therefore 5 x^{2}+13 x+8=0$ $\Rightarrow 5 x^{2}+5 x+8 x+8=0$ $\Rightarrow 5 x(x+1)+8(x+1)=0$ $\Rightarrow(x+1)(5 x+8)=0$ $\Rightarrow x+1=0$ or $5 x+8=0$ $\Rightarrow x=-1$ or $x=-\frac{8}{5}$ Hence, $-1$ and $-\frac{8}{5}$ are the roots of the given equation....
Read More →If 5 x+9=0 is the directrix of the hyperbola
Question: If $5 x+9=0$ is the directrix of the hyperbola $16 x^{2}-9 y^{2}=144$, then its corresponding focus is :(1) $(5,0)$(2) $\left(-\frac{5}{3}, 0\right)$(3) $\left(\frac{5}{3}, 0\right)$(4) $(-5,0)$Correct Option: , 4 Solution: $16 x^{2}-9 y^{2}=144 \Rightarrow \frac{x^{2}}{9}-\frac{y^{2}}{16}=1$ Then focus is S' $(-a e, 0)$ $x=\frac{-9}{5}$ $\mathrm{a}=3, \mathrm{~b}=4 \Rightarrow e^{2}=1+\frac{16}{9}=\frac{25}{9} \quad\left[\because e=\sqrt{1+\frac{b^{2}}{a^{2}}}\right]$ $\therefore$ the...
Read More →Solve each of the following quadratic equations:
Question: Solve each of the following quadratic equations: $\sqrt{2} x^{2}+7 x+5 \sqrt{2}=0$ Solution: We write, $7 x=5 x+2 x$ as $\sqrt{2} x^{2} \times 5 \sqrt{2}=10 x^{2}=5 x \times 2 x$ $\therefore \sqrt{2} x^{2}+7 x+5 \sqrt{2}=0$ $\Rightarrow \sqrt{2} x^{2}+5 x+2 x+5 \sqrt{2}=0$ $\Rightarrow x(\sqrt{2} x+5)+\sqrt{2}(\sqrt{2} x+5)=0$ $\Rightarrow(\sqrt{2} x+5)(x+\sqrt{2})=0$ $\Rightarrow x+\sqrt{2}=0$ or $\sqrt{2} x+5=0$ $\Rightarrow x=-\sqrt{2}$ or $x=-\frac{5}{\sqrt{2}}=-\frac{5 \sqrt{2}}{2...
Read More →If a directrix of a hyperbola centred at the origin
Question: If a directrix of a hyperbola centred at the origin and passing through the point $(4,-2 \sqrt{3})$ is $5 x=4 \sqrt{5}$ and its eccentricity is e, then :(1) $4 \mathrm{e}^{4}-24 \mathrm{e}^{2}+27=0$(2) $4 e^{4}-12 e^{2}-27=0$(3) $4 \mathrm{e}^{4}-24 \mathrm{e}^{2}+35=0$(4) $4 \mathrm{e}^{4}+8 \mathrm{e}^{2}-35=0$Correct Option: , 3 Solution: $\because$ directrix of a hyperbola is, $5 x=4 \sqrt{5} \Rightarrow x=\frac{4}{\sqrt{5}} \Rightarrow \frac{a}{e}=\frac{4}{\sqrt{5}}$ Now, hyperbol...
Read More →The electric field of a plane polarized electromagnetic wave in free space at time t=0 is given by an expression
Question: The electric field of a plane polarized electromagnetic wave in free space at time $t=0$ is given by an expression $\overrightarrow{\mathrm{E}}(x, y)=10 \hat{\mathrm{j}} \cos [(6 x+8 z)]$ The magnetic field $\overrightarrow{\mathrm{B}}(x, z, t)$ is given by: ( $\mathrm{c}$ is the velocity of light)(1) $\frac{1}{\mathrm{c}}(6 \hat{\mathrm{k}}+8 \hat{\mathrm{i}}) \cos [(6 x-8 z+10 \mathrm{c} t)]$(2) $\frac{1}{\mathrm{c}}(6 \hat{\mathrm{k}}-8 \hat{\mathrm{i}}) \cos [(6 x+8 z-10 \mathrm{c}...
Read More →Match the following:
Question: Match the following: (i) $-(\mathrm{a}),(\mathrm{ii})-(\mathrm{d}),(\mathrm{iii})-(\mathrm{c}),(\mathrm{iv})-(\mathrm{b})$(i) $-$ (c), (ii) $-$ (d), (iii) $-$ (a), (iv) $-$ (b)(i) $-(\mathrm{c})$, (ii) $-(\mathrm{a}),(\mathrm{iii})-(\mathrm{d}),(\mathrm{iv})-(\mathrm{b})$(i) $-(\mathrm{d}),($ ii $)-(\mathrm{b})$, (iii) $-(\mathrm{a})$, (iv) $-(\mathrm{c})$Correct Option: , 3 Solution:...
Read More →Solve each of the following quadratic equations:
Question: Solve each of the following quadratic equations: $x^{2}+3 \sqrt{3} x-30=0$ Solution: We write, $3 \sqrt{3} x=5 \sqrt{3} x-2 \sqrt{3} x$ as $x^{2} \times(-30)=-30 x^{2}=5 \sqrt{3} x \times(-2 \sqrt{3} x)$ $\therefore x^{2}+3 \sqrt{3} x-30=0$ $\Rightarrow x^{2}+5 \sqrt{3} x-2 \sqrt{3} x-30=0$ $\Rightarrow x(x+5 \sqrt{3})-2 \sqrt{3}(x+5 \sqrt{3})=0$ $\Rightarrow(x+5 \sqrt{3})(x-2 \sqrt{3})=0$ $\Rightarrow x+5 \sqrt{3}=0$ or $x-2 \sqrt{3}=0$ $\Rightarrow x=-5 \sqrt{3}$ or $x=2 \sqrt{3}$ He...
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