If the line $x-2 y=12 is tangent to the ellipse
Question: If the line $x-2 y=12$ is tangent to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ at the point $\left(3, \frac{-9}{2}\right)$, then the length of the latus rectum of the ellipse is : (1) 9(2) $12 \sqrt{2}$(3) 5(4) $8 \sqrt{3}$Correct Option: 1 Solution: Equation of tangent to $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ at $\left(3,-\frac{9}{2}\right)$ is, $\frac{3 x}{a^{2}}-\frac{9 y}{2 b^{2}}=1$ But given equation of tangent is, $x-2 y=12$ $\therefore \frac{3}{a^{2}}=\frac{...
Read More →If α and β are the zeroes of a polynomial
Question: If $\alpha$ and $\beta$ are the zeroes of a polynomial $2 x^{2}+7 x+5$, write the value of $\alpha+\beta+\alpha \beta$. Solution: By using the relationship between the zeros of the quadratic ploynomial.We have, Sum of zeroes $=\frac{-(\text { coefficient of } x)}{\text { coefficent of } x^{2}}$ and Product of zeroes $=\frac{\text { constant term }}{\text { coefficent of } x^{2}}$ $\therefore \alpha+\beta=\frac{-7}{2}$ and $\alpha \beta=\frac{5}{2}$ Now, $\alpha+\beta+\alpha \beta=\frac...
Read More →In an ellipse, with centre at the origin,
Question: In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at $(0,5 \sqrt{3})$, then the length of its latus rectum is:(1) 10(2) 5(3) 8(4) 6Correct Option: , 2 Solution: Given that focus is $(0,5 \sqrt{3}) \Rightarrow|b||a|$ Let $ba0$ and foci is $(0, \pm b e)$ $\because a^{2}=b^{2}-b^{2} e^{2} \Rightarrow b^{2} e^{2}=b^{2}-a^{2}$ $b e=\sqrt{b^{2}-a^{2}} \Rightarrow b^{2}-a^{2}=75$.....(i) $\because 2 b-2 a=10 \R...
Read More →In a meter bridge, the wire of length 1m has a non-uniform
Question: In a meter bridge, the wire of length $1 \mathrm{~m}$ has a non-uniform cross-section such that, the variation $\frac{\mathrm{dR}}{\mathrm{dl}}$ of its resistance $\mathrm{R}$ with length $l$ is $\frac{\mathrm{dR}}{\mathrm{dl}} \propto \frac{1}{\sqrt{l}}$. Two equal resistances are connected as shown in the figure. The galvanometer has zero deflection when the jockey is at point P. What is the length AP? (1) $0.2 \mathrm{~m}$(2) $0.3 \mathrm{~m}$(3) $0.25 \mathrm{~m}$(4) $0.35 \mathrm{...
Read More →If x3 + x2 − ax + b is divisible by
Question: If $x^{3}+x^{2}-a x+b$ is divisible by $\left(x^{2}-x\right)$, write the values of $a$ and $b$. Solution: Equating $x^{2}-x$ to 0 to find the zeros, we will get $x(x-1)=0$ $\Rightarrow x=0$ or $x-1=0$ $\Rightarrow x=0$ or $x=1$ Since, $x^{3}+x^{2}-a x+b$ is divisible by $x^{2}-x$ Hence, the zeros of $x^{2}-x$ will satisfy $x^{3}+x^{2}-a x+b$ $\therefore(0)^{3}+0^{2}-a(0)+b=0$ $\Rightarrow b=0$ and $(1)^{3}+1^{2}-a(1)+0=0 \quad[\because b=0]$ $\Rightarrow a=2$...
Read More →If (a − b), a and (a + b) are zeros of the polynomial
Question: If $(a-b), a$ and $(a+b)$ are zeros of the polynomial $2 x^{3}-6 x^{2}+5 x-7$, write the value of $a$. Solution: By using the relationship between the zeroes of the cubic ploynomial.We have Sum of zeroes $=\frac{-\left(\text { coefficient of } x^{2}\right)}{\text { coefficent of } x^{3}}$ $\Rightarrow a-b+a+a+b=\frac{-(-6)}{2}$ $\Rightarrow 3 a=3$ $\Rightarrow a=1$...
Read More →If the tangents on the ellipse
Question: If the tangents on the ellipse $4 x^{2}+y^{2}=8$ at the points $(1,2)$ and $(a, b)$ are perpendicular to each other, then $a^{2}$ is equal(1) $\frac{128}{17}$(2) $\frac{64}{17}$(3) $\frac{4}{17}$(4) $\frac{2}{17}$Correct Option: , 4 Solution: Since $(a, b)$ touches the given ellipse $4 x^{2}+y^{2}=8$ $\therefore 4 a^{2}+b^{2}=8$ $\ldots(1)$ Equation of tangent on the ellipse at the point $A(1,2)$ is: $4 x+2 y=8 \Rightarrow 2 x+y=4 \Rightarrow y=-2 x+4$ But, also equation of tangent at ...
Read More →If (x + a) is a factor of
Question: If $(x+a)$ is a factor of $\left(2 x^{2}+2 a x+5 x+10\right)$, find the value of $a .$ Solution: Given: (x+a) is a factor of 2x2+ 2ax+ 5x+ 10We have $x+a=0$ $\Rightarrow x=-a$ Since, $(x+a)$ is a factor of $2 x^{2}+2 a x+5 x+10$ Hence, It will satisfy the above polynomial $\therefore 2(-a)^{2}+2 a(-a)+5(-a)+10=0$ $\Rightarrow-5 a+10=0$ $\Rightarrow a=2$...
Read More →The galvanometer deflection,
Question: The galvanometer deflection, when key $\mathrm{K}_{1}$ is closed but $\mathrm{K}_{2}$ is open, equals $\theta_{0}$ (see figure). On closing $\mathrm{K}_{2}$ also and adjusting $R_{2}$ to $5 \Omega$, the deflection in galvanometer becomes $\frac{\theta_{0}}{5}$. The resistance of the galvanometer is, then, given by [Neglect the internal resistance of battery]: (1) $5 \Omega$(2) $22 \Omega$(3) $25 \Omega$(4) $12 \Omega$Correct Option: , 2 Solution: (2) When key $\mathrm{K}_{1}$ is closed...
Read More →The length of the minor axis (along y-axis) of an ellipse in
Question: The length of the minor axis (along y-axis) of an ellipse in the standard form is $\frac{4}{\sqrt{3}}$. If this ellipse touches the line, $x+6 y=8$; then its eccentricity is:(1) $\frac{1}{2} \sqrt{\frac{11}{3}}$(2) $\sqrt{\frac{5}{6}}$(3) $\frac{1}{2} \sqrt{\frac{5}{3}}$(4) $\frac{1}{3} \sqrt{\frac{11}{3}}$Correct Option: 1 Solution: Let $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 ; ab$ $2 b=\frac{4}{\sqrt{3}} \Rightarrow b=\frac{2}{\sqrt{3}}$ Equation of tangent $\equiv y=m x \pm \sqrt...
Read More →If the product of the zeros of the quadratic polynomial
Question: If the product of the zeros of the quadratic polynomial $x^{2}-4 x+k$ is 3 then write the value of $k$. Solution: By using the relationship between the zeros of the quadratic ploynomial.We have Product of zeroes $=\frac{\text { constant term }}{\text { coefficent of } x^{2}}$ $\Rightarrow 3=\frac{k}{1}$ $\Rightarrow k=3$...
Read More →If the sum of the zeros of the quadratic polynomial
Question: If the sum of the zeros of the quadratic polynomial $k x^{2}-3 x+5$ is 1 , write the value of $k$. Solution: By using the relationship between the zeros of the quadratic ploynomial.We have Sum of zeroes $=\frac{-(\text { coefficient of } x)}{\text { coefficent of } x^{2}}$ $\Rightarrow 1=\frac{-(-3)}{k}$ $\Rightarrow k=3$...
Read More →If e1 and e2 are the eccentricities of the ellipse
Question: If $e_{1}$ and $e_{2}$ are the eccentricities of the ellipse, $\frac{x^{2}}{18}+\frac{y^{2}}{4}=1$ and the hyperbola, $\frac{x^{2}}{9}-\frac{y^{2}}{4}=1$ respectively and $\left(e_{1}, e_{2}\right)$ is a point on the ellipse, $15 x^{2}+3 y^{2}=k$, then $k$ is equal to(1) 16(2) 17(3) 15(4) 14Correct Option: 1 Solution: Eccentricity of ellipse $e_{1}=\sqrt{1-\frac{4}{18}}=\sqrt{\frac{7}{9}}=\frac{\sqrt{7}}{3}$ Eccentricity of hyperbola $e_{2}=\sqrt{1+\frac{4}{9}}=\sqrt{\frac{13}{9}}=\fra...
Read More →Write the zeros of the polynomial
Question: Write the zeros of the polynomial $x^{2}-x-6$ Solution: $f(x)=x^{2}-x-6$ $=x^{2}-3 x+2 x-6$ $=x(x-3)+2(x-3)$ $=(x-3)(x+2)$ $f(x)=0 \Rightarrow(x-3)(x+2)=0$ $\Rightarrow(x-3)=0$ or $(x+2)=0$ $\Rightarrow x=3$ or $x=-2$ So, the zeros off(x) are 3 and 2....
Read More →Two electric bulbs, rated at (25 W, 220 V) and (100 W, 220 V),
Question: Two electric bulbs, rated at $(25 \mathrm{~W}, 220 \mathrm{~V})$ and $(100 \mathrm{~W}, 220 \mathrm{~V})$, are connected in series across a $220 \mathrm{~V}$ voltage source. If the $25 \mathrm{~W}$ and $100 \mathrm{~W}$ bulbs draw powers $\mathrm{P}_{1}$ and $\mathrm{P}_{2}$ respectively, then:(1) $P_{1}=16 W, P_{2}=4 W$(2) $\quad P_{1}=16 \mathrm{~W}, P_{2}=9 \mathrm{~W}$(3) $\mathrm{P}_{1}=9 \mathrm{~W}, \mathrm{P}_{2}=16 \mathrm{~W}$(4) $\mathrm{P}_{1}=4 \mathrm{~W}, \mathrm{P}_{2}=...
Read More →If −2 is a zero of the polynomial
Question: If $-2$ is a zero of the polynomial $3 x^{2}+4 x+2 k$ then find the value of $k$. Solution: Given:x= 2 is one zero of the polynomial 3x2+ 4x+ 2kTherefore, It will satisfy the above polynomial.Now, we have $3(-2)^{2}+4(-2)+2 k=0$ $\Rightarrow 12-8+2 k=0$ $\Rightarrow k=-2$...
Read More →Let the line y=m x and the ellipse
Question: Let the line $y=m x$ and the ellipse $2 x^{2}+y^{2}=1$ intersect at a point $P$ in the first quadrant. If the normal to this ellipse at $P$ meets the co-ordinate axes at $\left(-\frac{1}{3 \sqrt{2}}, 0\right)$ and $(0, \beta)$, then $\beta$ is equal to:(1) $\frac{2 \sqrt{2}}{3}$(2) $\frac{2}{\sqrt{3}}$(3) $\frac{2}{3}$(4) $\frac{\sqrt{2}}{3}$Correct Option: , 4 Solution: Let $P$ be $\left(x_{1}, y_{1}\right)$ So, equation of normal at $P$ is $\frac{x}{2 x_{1}}-\frac{y}{y_{1}}=-\frac{1}...
Read More →If 1 is a zero of the polynomial
Question: If 1 is a zero of the polynomial $a x^{2}-3(a-1) x-1$, then find the value of $a$. Solution: Given:x= 1 is one zero of the polynomialax2 3(a 1)x 1Therefore, It will satisfy the above polynomial.Now, we have $a(1)^{2}-3(a-1) 1-1=0$ $\Rightarrow a-3 a+3-1=0$ $\Rightarrow-2 a=-2$ $\Rightarrow a=1$...
Read More →If 3 x+4 y=12 is a tangent
Question: If $3 x+4 y=12 \sqrt{2}$ is $a$ tangent to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{9}=1$ for some $a \in R$, then the distance between the foci of the ellipse is:(1) $2 \sqrt{7}$(2) 4(3) $2 \sqrt{5}$(4) $2 \sqrt{2}$Correct Option: 1 Solution: $3 x+4 y=12 \sqrt{2}$ $\Rightarrow \quad 4 y=-3 x+12 \sqrt{2}$ $\Rightarrow \quad y=-\frac{3}{4} x+3 \sqrt{2}$ Now, condition of tangency, $c^{2}=a^{2} m^{2}+b^{2}$ $\therefore \quad 18=a^{2} \cdot \frac{9}{16}+9 \quad \Rightarrow \quad a^{2...
Read More →If −4 is a zero of the quadratic polynomial
Question: If $-4$ is a zero of the quadratic polynomial $x^{2}-x-(2 k+2)$ then find the value of $k$ Solution: Given:x= 4 is one zero of the polynomialx2x(2k+ 2)Therefore, It will satisfy the above polynomial.Now, we have $(-4)^{2}-(-4)-(2 k+2)=0$ $\Rightarrow 16+4-2 k-2=0$ $\Rightarrow-2 k=-18$ $\Rightarrow k=9$...
Read More →The charge on a capacitor plate in a circuit, as a function of time, is shown in the figure:
Question: The charge on a capacitor plate in a circuit, as a function of time, is shown in the figure: What is the value of current at $\mathrm{t}=4 \mathrm{~s}$ ?(1) Zero(2) $3 \mu \mathrm{A}$(3) $2 \mu \mathrm{A}$(4) $1.5 \mu \mathrm{A}$Correct Option: 1 Solution: (1) Clearly, from graph Current, I $=\frac{\mathrm{dq}}{\mathrm{dt}}=0$ at $\mathrm{t}=4 \mathrm{~s}$ [Since $\mathrm{q}$ is constant]...
Read More →If 3 is a zero of the polynomial
Question: If 3 is a zero of the polynomial $2 x^{2}+x+k$, find the value of $k$. Solution: Given:x= 3 is one zero of the polynomial 2x2+x+kTherefore, It will satisfy the above polynomial.Now, we have $2(3)^{2}+3+k=0$ $\Rightarrow 21+k=0$ $\Rightarrow k=-21$...
Read More →If the distance between the foci of an
Question: If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12 , then the length of its latus rectum is:(1) $\sqrt{3}$(2) $3 \sqrt{2}$(3) $\frac{3}{\sqrt{2}}$(4) $2 \sqrt{3}$Correct Option: , 2 Solution: $2 a e=6$ and $\frac{2 a}{e}=12$ $\Rightarrow a e=3$ $\ldots$ (i) and $\frac{a}{e}=6 \Rightarrow e=\frac{a}{6}$..(ii) $\Rightarrow a^{2}=18 \quad$ [From (i) and (ii)] $\Rightarrow \quad b^{2}=a^{2}-a^{2} e^{2}=18-9=9$ $\therefore \quad$ Latus rectum ...
Read More →In the given circuit diagram, the currents,
Question: In the given circuit diagram, the currents, $\mathrm{I}_{1}=-0.3 \mathrm{~A}, \mathrm{I}_{4}=0.8 \mathrm{~A}$ and $\mathrm{I}_{5}=0.4 \mathrm{~A}$, are flowing as shown. The currents $\mathrm{I}_{2}, \mathrm{I}_{3}$ and $\mathrm{I}_{6}$, respectively, are : (1) $1.1 \mathrm{~A},-0.4 \mathrm{~A}, 0.4 \mathrm{~A}$(2) $1.1 \mathrm{~A}, 0.4 \mathrm{~A}, 0.4 \mathrm{~A}$(3) $0.4 \mathrm{~A}, 1.1 \mathrm{~A}, 0.4 \mathrm{~A}$(4) $-0.4 \mathrm{~A}, 0.4 \mathrm{~A}, 1.1 \mathrm{~A}$Correct Opt...
Read More →If one zero of the quadratic polynomial
Question: If one zero of the quadratic polynomial $k x^{2}+3 x+k$ is 2 then find the value of $k$. Solution: Given:x= 2 is one zero of the quadratic polynomialkx2+ 3x+kTherefore, It will satisfy the above polynomial.Now, we have $k(2)^{2}+3(2)+k=0$ $\Rightarrow 4 k+6+k=0$ $\Rightarrow 5 k+6=0$ $\Rightarrow k=-\frac{6}{5}$...
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