Let L be a common tangent line to the curves

Question: Let $L$ be a common tangent line to the curves $4 x^{2}+9 y^{2}=36$ and $(2 x)^{2}+(2 y)^{2}=31$. Then the square of the slope of the line $L$ is Solution: $E: \frac{x^{2}}{9}+\frac{y^{2}}{4}=1 \quad C: x^{2}+y^{2}=\frac{31}{4}$ equation of tangent to ellipse is $y=m x \pm \sqrt{9 m^{2}+4}$ equation of tangent to circle is $y=m x \pm \sqrt{\frac{31}{4} m^{2}+\frac{31}{4}}$...(2) Comparing equation (i) $\backslash$ (ii) $9 m^{2}+4=\frac{31}{4} m^{2}+\frac{31}{4}$ $\Rightarrow 36 m^{2}+1...

Question: In the given circuit the cells have zero internal resistance. The currents (in Amperes) passing through resistance $R_{1}$ and $R_{2}$ respectively, are: (1) 1,2(2) 2,2(3) $0.5,0$(4) 0,1Correct Option: , 3 Solution: (3) Current passing through resistance $\mathrm{R}_{1}$, $\mathrm{i}_{1}=\frac{\mathrm{v}}{\mathrm{R}_{1}}=\frac{10}{20}=0.5 \mathrm{~A}$ and, $\mathrm{i}_{2}=0$...

Obtain all the zeros of the polynomial

Question: Obtain all the zeros of the polynomial $x^{4}+x^{3}-14 x^{2}-2 x+24$ if two of its zeros are $\sqrt{2}$ and $-\sqrt{2}$. Solution: Let $f(x)=x^{4}+x^{3}-14 x^{2}-2 x+24$ It is given that $\sqrt{2}$ and $-\sqrt{2}$ are two zeroes of $f(x)$ Thus, $f(x)$ is completely divisible by $(x+\sqrt{2})$ and $(x-\sqrt{2})$. Therefore, one factor of $f(x)$ is $\left(x^{2}-2\right)$. We get another factor of $f(x)$ by dividing it with $\left(x^{2}-2\right)$. On division, we get the quotient $x^{2}+x...

If the curve

Question: If the curve $x^{2}+2 y^{2}=2$ intersects the line $x+y=1$ at two points $P$ and $Q$, then the angle subtended by the line segment $\mathrm{PQ}$ at the origin is:(1) $\frac{\pi}{2}+\tan ^{-1}\left(\frac{1}{4}\right)$(2) $\frac{\pi}{2}-\tan ^{-1}\left(\frac{1}{4}\right)$(3) $\frac{\pi}{2}+\tan ^{-1}\left(\frac{1}{3}\right)$(4) $\frac{\pi}{2}-\tan ^{-1}\left(\frac{1}{3}\right)$Correct Option: 1, Solution: Ellipse $\frac{x^{2}}{2}+\frac{y^{2}}{1}=1$ Line $x+y=1$ Using homogenisation $x^{2...

A potentiometer wire A B having length L

Question: A potentiometer wire $A B$ having length $L$ and resistance $12 \mathrm{r}$ is joined to a cell $\mathrm{D}$ of emf $\varepsilon$ and internal resistance r. A cell C having emf $\varepsilon / 2$ and internal resistance $3 \mathrm{r}$ is connected. The length $\mathrm{AJ}$ at which the galvanometer as shown in fig. shows no deflection is: (1) $\frac{11}{12} \mathrm{~L}$(2) $\frac{11}{24} \mathrm{~L}$(3) $\frac{13}{24} \mathrm{~L}$(4) $\frac{5}{12} \mathrm{~L}$Correct Option: , 3 Solutio...

Find all the zeros of

Question: Find all the zeros of $\left(x^{4}+x^{3}-23 x^{2}-3 x+60\right)$, if it is given that two of its zeros are $\sqrt{3}$ and $-\sqrt{3}$. Solution: Let $f(x)=x^{4}+x^{3}-23 x^{2}-3 x+60$ Since $\sqrt{3}$ and $-\sqrt{3}$ are the zeroes of $f(x)$, it follows that each one of $(x-\sqrt{3})$ and $(x+\sqrt{3})$ is a factor of $f(x)$. Consequently, $(x-\sqrt{3})(x+\sqrt{3})=\left(x^{2}-3\right)$ is a factor of $f(x)$. On dividing $f(x)$ by $\left(x^{2}-3\right)$, we get: $\therefore f(x)=0$ $=\...

A 2 W carbon resistor is color coded with green, black, red and brown respectively.

Question: A 2 W carbon resistor is color coded with green, black, red and brown respectively. The maximum current which can be passed through this resistor is:(1) $20 \mathrm{~mA}$(2) $100 \mathrm{~mA}$(3) $0.4 \mathrm{~mA}$(4) $63 \mathrm{~mA}$Correct Option: 1 Solution: (1) Colour code for carbon resistor $\mathrm{Bl}, \mathrm{Br}, \mathrm{R}, \mathrm{O}, \mathrm{Y}, \mathrm{G}$, Blue, $\mathrm{V}, \mathrm{Gr}, \mathrm{W}$ $\begin{array}{llllllllll}0 1 2 3 4 5 6 7 8 9\end{array}$ Resistance, $...

If 2 and –2 are two zeros of the polynomial

Question: If 2 and $-2$ are two zeros of the polynomial $2 x^{4}-5 x^{3}-11 x^{2}+20 x+12$, find all the zeros of the given polynomial. Solution: Let $f(x)=2 x^{4}-5 x^{3}-11 x^{2}+20 x+12$ It is given that 2 and $-2$ are two zeroes of $f(x)$ Thus, $f(x)$ is completely divisible by $(x+2)$ and $(x-2)$. Therefore, one factor of $f(x)$ is $\left(x^{2}-4\right)$. We get another factor of $f(x)$ by dividing it with $\left(x^{2}-4\right)$. On division, we get the quotient $2 x^{2}-5 x-3$ $\Rightarrow...

Let L be a tangent line to the parabola

Question: Let $L$ be a tangent line to the parabola $y^{2}=4 x-20$ at $(6,2)$. If $L$ is also a tangent to the ellipse $\frac{x^{2}}{2}+\frac{y^{2}}{b}=1$, then the value of $b$ is equal to :(1) 11(2) 14(3) 16(4) 20Correct Option: , 2 Solution: Tangent to parabola $2 y=2(x+6)-20$ $\Rightarrow y=x-4$ Condition of tangency for ellipse. $16=2(1)^{2}+b$ $\Rightarrow b=14$...

If the point of intersections of the ellipse

Question: If the point of intersections of the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{b^{2}}=1$ and the circle $\mathrm{x}^{2}+\mathrm{y}^{2}=4 \mathrm{~b}, \mathrm{~b}4$ lie on the curve $y^{2}=3 x^{2}$, then $b$ is equal to:(1) 12(2) 5(3) 6(4) 10Correct Option: 1 Solution: $y^{2}=3 x^{2}$ and $x^{2}+y^{2}=4 b$ Solve both we get so $x^{2}=b$ $\frac{x^{2}}{16}+\frac{3 x^{2}}{b^{2}}=1$ $\frac{b}{16}+\frac{3}{b}=1$ $b^{2}-16 b+48=0$ $(b-12)(b-4)=0$ $b=12, b4$...

A uniform metallic wire has a resistance

Question: A uniform metallic wire has a resistance of $18 \Omega$ and is bent into an equilateral triangle. Then, the resistance between any two vertices of the triangle is:(1) $4 \Omega$(2) $8 \Omega$(3) $12 \mathrm{~W}$(4) $2 \mathrm{~W}$Correct Option: 1 Solution: Resistance, $R \propto l$ so resistance of each side of the equilateral triangle $=6 \Omega$ Resistance $R_{e q}$ between any two vertices $\frac{1}{\mathrm{R}_{\mathrm{eq}}}=\frac{1}{12}+\frac{1}{6} \Rightarrow \mathrm{R}_{\text {e...

If 3 and −3 are two zeros of the polynomial

Question: If 3 and $-3$ are two zeros of the polynomial $\left(x^{4}+x^{3}-11 x^{2}-9 x+18\right)$, find all the zeros of the given polynomial. Solution: Let $f(x)=x^{4}+x^{3}-11 x^{2}-9 x+18$ Since 3 and $-3$ are the zeroes of $f(x)$, it follows that each one of $(x+3)$ and $(x-3)$ is a factor of $f(x)$ Consequently, $(x-3)(x+3)=\left(x^{2}-9\right)$ is a factor of $f(x) .$ On dividing $f(x)$ by $\left(x^{2}-9\right)$, we get: $f(x)=0=\left(x^{2}+x-2\right)\left(x^{2}-9\right)=0$ $=\left(x^{2}+...

If 1 and −2 are two zeros of the polynomial

Question: If 1 and $-2$ are two zeros of the polynomial $\left(x^{3}-4 x^{2}-7 x+10\right)$, find its third zero. Solution: Let $f(x)=x^{3}-4 x^{2}-7 x+10$ Since 1 and $-2$ are the zeroes of $f(x)$, it follows that each one of $(x-1)$ and $(x+2)$ is a factor of $f(x)$ Consequently, $(x-1)(x+2)=\left(x^{2}+x-2\right)$ is a factor of $f(x)$. On dividing $f(x)$ by $\left(x^{2}+x-2\right)$, we get: $f(x)=0=\left(x^{2}+x-2\right)(x-5)=0$ $=(x-1)(x+2)(x-5)=0$ $\Rightarrow x=1$ or $x=-2$ or $x=5$ Hence...

In the given circuit the internal resistance of the 18 V cell is negligible.

Question: In the given circuit the internal resistance of the $18 \mathrm{~V}$ cell is negligible. If $\mathrm{R}_{1}=400 \Omega, \mathrm{R}_{3}=100 \Omega$ and $\mathrm{R}_{4}=500 \Omega$ and the reading of an ideal voltmeter across $R_{4}$ is $5 V$, then the value of $R_{2}$ will be: (1) $300 \Omega$(2) $450 \Omega$(3) $550 \Omega$(4) $230 \Omega$Correct Option: 1 Solution: (1) Across $\mathrm{R}_{4}$ reading of voltmeter, $\mathrm{V}_{4}=5 \mathrm{~V}$ Current, $\mathrm{i}_{4}=\frac{\mathrm{V...

It is given that −1 is one of the zeros of the polynomial

Question: It is given that $-1$ is one of the zeros of the polynomial $x^{3}+2 x^{2}-11 x-12$. Find all the given zeros of the given polynomial. Solution: Let $f(x)=x^{3}+2 x^{2}-11 x-12$ Since $-1$ is a zero of $f(x),(x+1)$ is a factor of $f(x)$. On dividing $f(x)$ by $(x+1)$, we get: $f(x)=x^{3}+2 x^{2}-11 x-12$ $=(x+1)\left(x^{2}+x-12\right)$ $=(x+1)\left\{x^{2}+4 x-3 x-12\right\}$ $=(x+1)\{x(x+4)-3(x+4)\}$ $=(x+1)(x-3)(x+4)$ $\therefore f(x)=0=(x+1)(x-3)(x+4)=0$ $=(x+1)=0$ or $(x-3)=0$ or $(...

For x>1,

Question: For $x1$, if $(2 x)^{2 y}=4 e^{2 x-2 y}$, then $\left(1+\log _{e} 2 x\right)^{2} \frac{d y}{d x}$ is equal to :(1) $\frac{x \log _{e} 2 x-\log _{e} 2}{x}$(2) $\log _{\mathrm{e}} 2 \mathrm{x}$(3) $\frac{x \log _{e} 2 x+\log _{e} 2}{x}$(4) $x \log _{e} 2 x$Correct Option: 1 Solution: Consider the equation, $(2 x)^{2 y}=4 e^{2 x-2 y}$ Taking log on both sides $2 y \ln (2 x)=\ln 4+(2 x-2 y)$ $\ldots(1)$ Differentiating both sides w.r.t. $x$, $2 y \frac{1}{2 x} 2+2 \ln (2 x) \frac{d y}{d x}...

Verify division algorithm for the polynomials

Question: Verify division algorithm for the polynomials $f(x)=8+20 x+x^{2}-6 x^{3}$ and $g(x)=2+5 x-3 x^{2}$ Solution: We can write $f(x)$ as $-6 x^{3}+x^{2}+20 x+8$ and $g(x)$ as $-3 x^{2}+5 x+2$ Quotient $=2 x+3$ Remainder $=x+2$ By using division rule, we haveDivided = Quotient Divisor + Remainder $\therefore-6 x^{3}+x^{2}+20 x+8=\left(-3 x^{2}+5 x+2\right)(2 x+3)+x+2$ $\Rightarrow-6 x^{3}+x^{2}+20 x+8=-6 x^{3}+10 x^{2}+4 x-9 x^{2}+15 x+6+x+2$ $\Rightarrow-6 x^{3}+x^{2}+20 x+8=-6 x^{3}+x^{2}+...

A carbon resistance has following colour code. What is the value of the resistance?

Question: A carbon resistance has following colour code. What is the value of the resistance? (1) $530 \mathrm{k} \Omega \pm 5 \%$(2) $5.3 \mathrm{k} \Omega \pm 5 \%$(3) $6.4 \mathrm{M} \Omega \pm 5 \%$(4) $64 \mathrm{M} \Omega \pm 10 \%$Correct Option: 1 Solution: (1) Colour code for carbon resistor Bl, Br, R, O, Y, G, Blue, V, Gr, W $\begin{array}{llllllllll}0 1 2 3 4 5 6 7 8 9\end{array}$ Resistance, $\mathrm{R}=\mathrm{AB} \times \mathrm{C} \pm \mathrm{D}$ Bands $\mathrm{A}$ and $\mathrm{B}$...

On dividing,

Question: On dividing, $3 x^{3}+x^{2}+2 x+5$ by a polynomial $g(x)$, the quotient and remainder are $3 x-5$ and $9 x+10$ respectively. Find $g(x)$ Solution: By using division rule, we haveDivided = Quotient Divisor + Remainder $\therefore 3 x^{3}+x^{2}+2 x+5=(3 x-5) g(x)+9 x+10$ $\Rightarrow 3 x^{3}+x^{2}+2 x+5-9 x-10=(3 x-5) g(x)$ $\Rightarrow 3 x^{3}+x^{2}-7 x-5=(3 x-5) g(x)$ $\Rightarrow g(x)=\frac{3 x^{3}+x^{2}-7 x-5}{3 x-5}$ $\therefore g(x)=x^{2}+2 x+1$...

if xloge

Question: If $x \log _{e}\left(\log _{e} x\right)-x^{2}+y^{2}=4(y0)$, then $\frac{d y}{d x}$ at $\mathrm{x}=\mathrm{e}$ is equal to :(1) $\frac{(1+2 e)}{2 \sqrt{4+e^{2}}}$(2) $\frac{(2 e-1)}{2 \sqrt{4+e^{2}}}$(3) $\frac{(1+2 e)}{\sqrt{4+e^{2}}}$(4) $\frac{e}{\sqrt{4+e^{2}}}$Correct Option: , 2 Solution: Consider the equation, $x \log _{e}\left(\log _{e} x\right)-x^{2}+y^{2}=4$ Differentiate both sides w.r.t. $x$, $\log _{e}\left(\log _{e} x\right)+x \cdot \frac{1}{x \cdot \log _{e} x}-2 x+2 y \f...

Drift speed of electrons,

Question: Drift speed of electrons, when $1.5 \mathrm{~A}$ of current flows in a copper wire of cross section $5 \mathrm{~mm}^{2}$, is $v$. If the electron density in copper is $9 \times 10^{28} / \mathrm{m}^{3}$ the value of $v$ in $\mathrm{mm} / \mathrm{s}$ close to (Take charge of electron to be $=1.6 \times 10^{-19} \mathrm{C}$ )(1) $0.02$(2) 3(3) 2(4) $0.2$Correct Option: 1 Solution: (1) Using, $\mathrm{I}=\mathrm{neAv}_{\mathrm{d}}$ $\therefore$ Drift speed $\mathrm{v}_{\mathrm{d}}=\frac{1...

If the polynomial

Question: If the polynomial $\left(x^{4}+2 x^{3}+8 x^{2}+12 x+18\right)$ is divided by another polynomial $\left(x^{2}+5\right)$, the remainder comes out to be $(p x+q)$. Find the values of $p$ and $q$. Solution: Let $f(x)=x^{4}+2 x^{3}+8 x^{2}+12 x+18$ It is given that when $f(x)$ is divisible by $x^{2}+5$, the remainder comes out to be $p x+q$. On division, we get the quotient $x^{2}+2 x+3$ and the remainder $2 x+3$ Since,the remainder comes out to bepx+q.Therefore,p= 2 andq= 3.Hence, the valu...

By actual division, show that

Question: By actual division, show that $x^{2}-3$ is a factor of $2 x^{4}+3 x^{3}-2 x^{2}-9 x-12$ Solution: Let $f(x)=2 x^{4}+3 x^{3}-2 x^{2}-9 x-12$ and $g(x)=x^{2}-3$ Quotient $q(x)=2 x^{2}+3 x+4$ Remainder $r(x)=0$ Since, the remainder is 0 . Hence, $x^{2}-3$ is a factor of $2 x^{4}+3 x^{3}-2 x^{2}-9 x-12$...

A resistance is shown in the figure. Its value and tolerance are given respectively by:

Question: A resistance is shown in the figure. Its value and tolerance are given respectively by: (1) $270 \Omega, 10 \%$(2) $27 \mathrm{k} \Omega, 10 \%$(3) $27 \mathrm{k} \Omega, 20 \%$(4) $270 \Omega, 5 \%$Correct Option: , 2 Solution: (2) Color code: BI, Br, R, O, Y, G, B, V, Gr, W $0,1,2,3,4,5,6,7,89$ $\mathrm{R}=\mathrm{AB} \times \mathrm{C} \pm \mathrm{D} \%$ where $\mathrm{D}=$ tolerance $\mathrm{D}_{\text {gold }}=\pm 5 \%, \mathrm{D}_{\text {silver }}=\pm 10 \% ; \mathrm{D}_{\text {no ...

Let f

Question: Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a function such that $f(x)=x^{3}+x^{2} f^{\prime}(1)+x f^{\prime \prime}(2)+f^{\prime \prime \prime}(3), x \in \mathbf{R}$. Then $f(2)$ equals:(1) $-4$(2) 30(3) $-2$(4) 8Correct Option: 3, Solution: Let $f(x)=x^{3}+a x^{2}+b x+c$ $f^{\prime}(x)=3 x^{2}+2 a x+b \Rightarrow f^{\prime}(1)=3+2 a+b$ $f^{\prime \prime}(x)=6 x+2 a \Rightarrow f^{\prime \prime}(2)=12+2 a$ $f^{\prime \prime \prime}(x)=6 \Rightarrow f^{\prime \prime \prime}(3)=6$ $\b...