A hyperbola passes through the foci of the ellipse
Question: A hyperbola passes through the foci of the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$ and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola is:(1) $\frac{x^{2}}{9}-\frac{y^{2}}{4}=1$(2) $\frac{x^{2}}{9}-\frac{y^{2}}{16}=1$(3) $x^{2}-y^{2}=9$(4) $\frac{x^{2}}{9}-\frac{y^{2}}{25}=1$Correct Option: , 2 Solution: $e_{1}=\sqrt{1-\frac{16}{25}}=\frac{3}{5} \qu...
Read More →An electromagnetic wave is represented by the electric
Question: An electromagnetic wave is represented by the electric field $\vec{E}=E_{0} \hat{n} \sin [\omega t+(6 y-8 z)]$. Taking unit vectors in $x$, $y$ and $z$ directions to be $\hat{i}, \hat{j}, \hat{k}$, the direction of propogation $\hat{s}$ is:(1) $\hat{s}=\frac{3 \hat{i}-4 \hat{j}}{5}$(2) $\hat{s}=\frac{-4 \hat{k}+3 \hat{j}}{5}$(3) $\hat{s}=\left(\frac{-3 \hat{j}+4 \hat{k}}{5}\right)$(4) $\hat{s}=\frac{3 \hat{j}-3 \hat{k}}{5}$Correct Option: , 3 Solution: (3) $\hat{S}=\frac{6 \hat{j}+8 \h...
Read More →Solve this
Question: $\sqrt{3} x^{2}+11 x+6 \sqrt{3}=0$ Solution: Given : $\sqrt{3} x^{2}+11 x+6 \sqrt{3}=0$ $\Rightarrow \sqrt{3} x^{2}+9 x+2 x+6 \sqrt{3}=0$ $\Rightarrow \sqrt{3} x(x+3 \sqrt{3})+2(x+3 \sqrt{3})=0$ $\Rightarrow(x+3 \sqrt{3})(\sqrt{3} x+2)=0$ $\Rightarrow x+3 \sqrt{3}=0$ or $\sqrt{3} x+2=0$ $\Rightarrow x=-3 \sqrt{3}$ or $x=\frac{-2}{\sqrt{3}}=\frac{-2 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}}=\frac{-2 \sqrt{3}}{3}$ Hence, the roots of the equation are $-3 \sqrt{3}$ and $\frac{-2 \sqrt{3}...
Read More →The locus of the point of intersection
Question: The locus of the point of intersection of the lines $(\sqrt{3}) \mathrm{kx}+\mathrm{ky}-4 \sqrt{3}=0$ and $\sqrt{3} \mathrm{x}-\mathrm{y}-4(\sqrt{3}) \mathrm{k}=0$ is a conic, whose eccentricity is Solution: $\sqrt{3} \mathrm{kx}+\mathrm{ky}=4 \sqrt{3}$ $\sqrt{3} \mathrm{kx}-\mathrm{ky}=4 \sqrt{3} \mathrm{k}^{2}$ Adding equation (1) $\backslash \(2) 2 \sqrt{3} \mathrm{kx}=4 \sqrt{3}\left(\mathrm{k}^{2}+1\right)$ $x=2\left(k+\frac{1}{k}\right)$ Substracting equation (1) $\backslash \(2)...
Read More →Match List - I with List - II :
Question: Match List - I with List - II : (a)-(i i),(b)-(i i i),(c)-(i v),(d)-(i)$(\mathrm{a})-(\mathrm{iv}),(\mathrm{b})-(\mathrm{iii}),(\mathrm{c})-(\mathrm{ii}),(\mathrm{d})-(\mathrm{i})$(a)-(iii), (b)-(iv), (c)-(i), (d)-(ii)$(\mathrm{a})-(\mathrm{ii}),(\mathrm{b})-(\mathrm{iv}),(\mathrm{c})-(\mathrm{i}),(\mathrm{d})-(\mathrm{iii})$Correct Option: , 3 Solution: (A) Antifertility drug $\rightarrow$ (iii) Nor ethindrone (B) Antibiotic $\rightarrow$ (iv) Salvarsan (C) Tranquilizer $\rightarrow$ ...
Read More →Solve each of the following quadratic equations:
Question: Solve each of the following quadratic equations: $\sqrt{3} x^{2}+10 x-8 \sqrt{3}=0$ Solution: Consider $\sqrt{3} x^{2}+10 x-8 \sqrt{3}=0$ Factorising by splitting the middle term; $\sqrt{3} x^{2}+12 x-2 x-8 \sqrt{3}=0$ $\Rightarrow \sqrt{3} x(x+4 \sqrt{3})-2(x+4 \sqrt{3})=0$ $\Rightarrow(\sqrt{3} x-2)(x+4 \sqrt{3})=0$ $\Rightarrow \sqrt{3} x-2=0$ or $x+4 \sqrt{3}=0$ $\Rightarrow x=\frac{2}{\sqrt{3}}$ or $x=-4 \sqrt{3}$ Hence, the roots of the given equation are $\frac{2}{\sqrt{3}}$ and...
Read More →Solve each of the following quadratic equations:
Question: Solve each of the following quadratic equations: $x^{2}+2 \sqrt{2} x-6=0$ Solution: We write, $2 \sqrt{2} x=3 \sqrt{2} x-\sqrt{2} x$ as $x^{2} \times(-6)=-6 x^{2}=3 \sqrt{2} x \times(-\sqrt{2} x)$ $\therefore x^{2}+2 \sqrt{2} x-6=0$ $\Rightarrow x^{2}+3 \sqrt{2} x-\sqrt{2} x-6=0$ $\Rightarrow x(x+3 \sqrt{2})-\sqrt{2}(x+3 \sqrt{2})=0$ $\Rightarrow(x+3 \sqrt{2})(x-\sqrt{2})=0$ $\Rightarrow x+3 \sqrt{2}=0$ or $x-\sqrt{2}=0$ $\Rightarrow x=-3 \sqrt{2}$ or $x=\sqrt{2}$ Hence, the roots of t...
Read More →Match the list -I with list - II
Question: Match the list -I with list - II $(\mathrm{a})-(\mathrm{ii}),(\mathrm{b})-(\mathrm{iv}),(\mathrm{c})-(\mathrm{i}),(\mathrm{d})-(\mathrm{iii})$(a) $-($ iv $),(b)-(\mathrm{i}),(\mathrm{c})-(\mathrm{ii}),(\mathrm{d})-(\mathrm{iii})$$(\mathrm{a})-(\mathrm{iv}),(\mathrm{b})-(\mathrm{iii}),(\mathrm{c})-(\mathrm{i}),(\mathrm{d})-(\mathrm{ii})$$(\mathrm{a})-(\mathrm{ii}),(\mathrm{b})-(\mathrm{iv}),(\mathrm{c})-(\mathrm{iii}),(\mathrm{d})-(\mathrm{i})$Correct Option: 1 Solution: (a) Antacid : C...
Read More →Consider a hyperbola H:
Question: Consider a hyperbola $\mathrm{H}: \mathrm{x}^{2}-2 \mathrm{y}^{2}=4$. Let the tangent at a point $\mathrm{P}(4, \sqrt{6})$ meet the $\mathrm{x}$-axis at $\mathrm{Q}$ and latus rectum at $\mathrm{R}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right), \mathrm{x}_{1}0$. If $\mathrm{F}$ is a focus of $\mathrm{H}$ which is nearer to the point $\mathrm{P}$, then the area of $\Delta Q F R$ is equal to(1) $4 \sqrt{6}$(2) $\sqrt{6}-1$(3) $\frac{7}{\sqrt{6}}-2$(4) $4 \sqrt{6}-1$Correct Option: , 3 Solut...
Read More →Prove that
Question: $48 x^{2}-13 x-1=0$ Solution: Given: $48 x^{2}-13 x-1=0$ $\Rightarrow 48 x^{2}-(16 x-3 x)-1=0$ $\Rightarrow 48 x^{2}-16 x+3 x-1=0$ $\Rightarrow 16 x(3 x-1)+1(3 x-1)=0$ $\Rightarrow(16 x+1)(3 x-1)=0$ $\Rightarrow 16 x+1=0$ or $3 x-1=0$ $\Rightarrow x=\frac{-1}{16}$ or $x=\frac{1}{3}$ Hence, the roots of the equation are $\frac{-1}{16}$ and $\frac{1}{3}$....
Read More →Solve this
Question: $4-11 x=3 x^{2}$ Solution: Given: $4-11 x=3 x^{2}$ $\Rightarrow 3 x^{2}+11 x-4=0$ $\Rightarrow 3 x^{2}+12 x-x-4=0$ $\Rightarrow 3 x(x+4)-1(x+4)=0$ $\Rightarrow(x+4)(3 x-1)=0$ $\Rightarrow x+4=0$ or $3 x-1=0$ $\Rightarrow x=-4$ or $x=\frac{1}{3}$ Hence, the roots of the equation are $-4$ and $\frac{1}{3}$....
Read More →Prove that
Question: $15 x^{2}-28=x$ Solution: Given: $15 x^{2}-28=x$ $\Rightarrow 15 x^{2}-x-28=0$ $\Rightarrow 15 x^{2}-(21 x-20 x)-28=0$ $\Rightarrow 15 x^{2}-21 x+20 x-28=0$ $\Rightarrow 3 x(5 x-7)+4(5 x-7)=0$ $\Rightarrow(3 x+4)(5 x-7)=0$ $\Rightarrow 3 x+4=0$ or $5 x-7=0$ $\Rightarrow x=\frac{-4}{3}$ or $x=\frac{7}{5}$ Hence, the roots of the equation are $\frac{-4}{3}$ and $\frac{7}{5}$....
Read More →A non-reducing sugar "A" hydrolyses to give two reducing mono saccharides.
Question: A non-reducing sugar "A" hydrolyses to give two reducing mono saccharides. Sugar A is-FructoseGalaetoseGlucoseSucroseCorrect Option: , 4 Solution:...
Read More →A square ABCD has
Question: A square $\mathrm{ABCD}$ has all its vertices on the curve $\mathrm{x}^{2} \mathrm{y}^{2}=1$. The midpoints of its sides also lie on the same curve. Then, the square of area of $\mathrm{ABCD}$ is Solution: $x y=1,-1$ $\frac{\mathrm{t}_{1}+\mathrm{t}_{2}}{2} \cdot \frac{\frac{1}{\mathrm{t}_{1}}-\frac{1}{\mathrm{t}_{2}}}{2}=1$ $\Rightarrow \mathrm{t}_{1}^{2}-\mathrm{t}_{2}^{2}=4 \mathrm{t}_{1} \mathrm{t}_{2}$ $\frac{1}{t_{1}^{2}} \times\left(-\frac{1}{t_{2}^{2}}\right)=-1 \Rightarrow t_{...
Read More →The electric field of a plane electromagnetic wave is given
Question: The electric field of a plane electromagnetic wave is given by $\overrightarrow{\mathrm{E}}=\mathrm{E}_{0} \hat{\mathrm{i}} \cos (\mathrm{kz}) \cos (\omega \mathrm{t})$ The corresponding magnetic field $\vec{B}$ is then given by :(1) $\vec{B}=\frac{E_{0}}{C} \hat{j} \sin (k z) \sin (\omega t)$(2) $\vec{B}=\frac{E_{0}}{C} \hat{j} \sin (k z) \cos (\omega t)$(3) $\overrightarrow{\mathrm{B}}=\frac{\mathrm{E}_{0}}{\mathrm{C}} \hat{\mathrm{j}} \cos (\mathrm{kz}) \sin (\omega \mathrm{t})$(4) ...
Read More →Solve this
Question: $4 x^{2}-9 x=100$ Solution: Given: $4 x^{2}-9 x=100$ $\Rightarrow 4 x^{2}-9 x-100=0$ $\Rightarrow 4 x^{2}-(25 x-16 x)-100=0$ $\Rightarrow 4 x^{2}-25 x+16 x-100=0$ $\Rightarrow x(4 x-25)+4(4 x-25)=0$ $\Rightarrow(4 x-25)(x+4)=0$ $\Rightarrow 4 x-25=0$ or $x+4=0$ $\Rightarrow x=\frac{25}{4}$ or $x=-4$ Hence, the roots of the equation are $\frac{25}{4}$ and $-4$....
Read More →The locus of the midpoints of the chord of the circle,
Question: The locus of the midpoints of the chord of the circle, $x^{2}+y^{2}=25$ which is tangent to the hyperbola, $\frac{x^{2}}{9}-\frac{y^{2}}{16}=1$ is :(1) $\left(x^{2}+y^{2}\right)^{2}-16 x^{2}+9 y^{2}=0$(2) $\left(x^{2}+y^{2}\right)^{2}-9 x^{2}+144 y^{2}=0$(3) $\left(x^{2}+y^{2}\right)^{2}-9 x^{2}-16 y^{2}=0$(4) $\left(x^{2}+y^{2}\right)^{2}-9 x^{2}+16 y^{2}=0$Correct Option: , 4 Solution: Equation of chord $y-k=-\frac{h}{k}(x-h)$ $k y-k^{2}=-h x+h^{2}$ $h x+k y=h^{2}+k^{2}$ $y=-\frac{h ...
Read More →Solve each of the following quadratic equations:
Question: Solve each of the following quadratic equations: $3 x^{2}-2 x-1=0$ Solution: We write, $-2 x=-3 x+x$ as $3 x^{2} \times(-1)=-3 x^{2}=(-3 x) \times x$ $\therefore 3 x^{2}-2 x-1=0$ $\Rightarrow 3 x^{2}-3 x+x-1=0$ $\Rightarrow 3 x(x-1)+1(x-1)=0$ $\Rightarrow(x-1)(3 x+1)=0$ $\Rightarrow x-1=0$ or $3 x+1=0$ $\Rightarrow x=1$ or $x=-\frac{1}{3}$ Hence, the roots of the given equation are 1 and $-\frac{1}{3}$....
Read More →Match List-I with List-II :
Question: Match List-I with List-II : Choose the correct match :$-(\mathrm{i} \mathrm{v}),(\mathrm{b})-(\mathrm{i}$ i 1$),(\mathrm{c})-(\mathrm{i} \mathrm{i}),(\mathrm{d})-(\mathrm{i}) \mathrm{S}$(a)-(ii), (b)-(i), (c)-(iv), (d)-(iii)(a)-(iii), (b)-(ii), (c)-(iv), (d)-(i)(a)-(i), (b)-(ii), (c)-(iv), (d)-(iii)Correct Option: , 2 Solution: Artificial sweetner : Sucralose Antiseptic: Bithional Preservative : Sodium Benzoate Glyceryl ester of stearic acid : Sodium steasate...
Read More →Solve this
Question: $6 x^{2}+x-12=0$ Solution: Given: $6 x^{2}+x-12=0$ $\Rightarrow 6 x^{2}+9 x-8 x-12=0$ $\Rightarrow 3 x(2 x+3)-4(2 x+3)=0$ $\Rightarrow(3 x-4)(2 x+3)=0$ $\Rightarrow 3 x-4=0$ or $2 x+3=0$ $\Rightarrow x=\frac{4}{3}$ or $x=\frac{-3}{2}$ Hence, $\frac{4}{3}$ and $\frac{-3}{2}$ are the roots of the equation $6 x^{2}+x-12=0$....
Read More →Solve this
Question: $50 \mathrm{~W} / \mathrm{m}^{2}$ energy density of sunlight is normally incident on the surface of a solar panel. Some part of incident energy $(25 \%)$ is reflected from the surface and the rest is absorbed. The force exerted on $1 \mathrm{~m}^{2}$ surface area will be close to $\left(\mathrm{c}=3 \times 10^{8} \mathrm{~m} / \mathrm{s}\right)$(1) $15 \times 10^{-8} \mathrm{~N}$(2) $20 \times 10^{-8} \mathrm{~N}$(3) $10 \times 10^{-8} \mathrm{~N}$(4) $35 \times 10^{-8} \mathrm{~N}$Cor...
Read More →Solve this
Question: $6 x^{2}+11 x+3=0$ Solution: Given : $6 x^{2}+11 x+3=0$ $\Rightarrow 6 x^{2}+9 x+2 x+3=0$ $\Rightarrow 3 x(2 x+3)+1(2 x+3)=0$ $\Rightarrow(3 x+1)(2 x+3)=0$ $\Rightarrow 3 x+1=0$ or $2 x+3=0$ $\Rightarrow x=\frac{-1}{3}$ or $x=\frac{-3}{2}$ Hence, $\frac{-1}{3}$ and $\frac{-3}{2}$ are the roots of the equation $6 x^{2}+11 x+3=0$....
Read More →Which of the following is correct structure of tyrosine?
Question: Which of the following is correct structure of tyrosine?Correct Option: , 4 Solution: The structure of Tyrosine amino acid is...
Read More →Which of the following is correct structure of tyrosine?
Question: Which of the following is correct structure of tyrosine?Correct Option: Solution: The structure of Tyrosine amino acid is...
Read More →Solve this
Question: $x^{2}=18 x-77$ Solution: Given : $x^{2}=18 x-77$ $\Rightarrow x^{2}-18 x+77=0$ $\Rightarrow x^{2}-(11 x+7 x)+77=0$ $\Rightarrow x^{2}-11 x-7 x+77=0$ $\Rightarrow x(x-11)-7(x-11)=0$ $\Rightarrow(x-7)(x-11)=0$ $\Rightarrow x-7=0$ or $x-11=0$ $\Rightarrow x=7$ or $x=11$ Hence, 7 and 11 are the roots of the equation $x^{2}=18 x-77$....
Read More →