An inductance coil has a reactance
Question: An inductance coil has a reactance of $100 \Omega$. When an $\mathrm{AC}$ signal of frequency $1000 \mathrm{~Hz}$ is applied to the coil, the applied voltage leads the current by $45^{\circ}$. The self-inductance of the coil is:(1) $1.1 \times 10^{-2} \mathrm{H}$(2) $1.1 \times 10^{-1} \mathrm{H}$(3) $5.5 \times 10^{-5} \mathrm{H}$(4) $6.7 \times 10^{-7} \mathrm{H}$Correct Option: 1 Solution: (1) Given, Reactance of inductance coil, $Z=100 \Omega$ Frequency of $\mathrm{AC}$ signal, $v=...
Read More →Which is the most suitable reagent for the following transformation?
Question: Which is the most suitable reagent for the following transformation? Tollen's reagent$\mathrm{I}_{2} / \mathrm{NaOH}$$\mathrm{CrO}_{2} \mathrm{Cl}_{2} / \mathrm{CS}_{2}$alkaline $\mathrm{KMnO}_{4}$Correct Option: , 2 Solution: The most suitable reagent for the given reaction is $\mathrm{I}_{2} / \mathrm{NaOH}$ (Iodoform reaction)....
Read More →The lengths of three sides of a triangle are 20 cm, 16 cm and 12 cm.
Question: The lengths of three sides of a triangle are 20 cm, 16 cm and 12 cm. The area of the triangle is(a) 96 cm2(b) 120 cm2(c) 144 cm2(d) 160 cm2 Solution: (a) 96 cm2 Let: $a=20 \mathrm{~cm}, b=16 \mathrm{~cm}$ and $c=12 \mathrm{~cm}$ $s=\frac{a+b+c}{2}=\frac{20+16+12}{2}=24 \mathrm{~cm}$ By Heron's formula, we have : Area of triangle $=\sqrt{s(s-a)(s-b)(s-c)}$ $=\sqrt{24(24-20)(24-16)(24-12)}$ $=\sqrt{24 \times 4 \times 8 \times 12}$ $=\sqrt{6 \times 4 \times 4 \times 4 \times 4 \times 6}$ ...
Read More →Let f(x) be a polynomial of degree 5 such that
Question: Let $f(x)$ be a polynomial of degree 5 such that $x=\pm 1$ are its critical points. If $\lim _{x \rightarrow 0}\left(2+\frac{f(x)}{x^{3}}\right)=4$, then which one of the following is not true?(1) $f$ is an odd function.(2) $f(1)-4 f(-1)=4$.(3) $x=1$ is a point of maxima and $x=-1$ is a point of minima of $f$.(4) $x=1$ is a point of minima and $x=-1$ is a point of maxima of $f$.Correct Option: , 4 Solution: $f(x)=a x^{5}+b x^{4}+c x^{3}$ $\lim _{x \rightarrow 0}\left(2+\frac{a x^{5}+b ...
Read More →In a ∆ABC it is given that base = 12 cm and height = 5 cm. Its area is
Question: In a∆ABCit is given that base = 12 cm and height = 5 cm. Its area is (a) $60 \mathrm{~cm}^{2}$ (b) $30 \mathrm{~cm}^{2}$ (c) $15 \sqrt{3} \mathrm{~cm}^{2}$ (d) $45 \mathrm{~cm}^{2}$ Solution: (b) $30 \mathrm{~cm}^{2}$ Area of triangle $=\frac{1}{2} \times$ Base $\times$ Height Area of $\triangle A B C=\frac{1}{2} \times 12 \times 5=30 \mathrm{~cm}^{2}$...
Read More →Let f(x) be a polynomial of degree 5 such that
Question: Let $f(x)$ be a polynomial of degree 5 such that $x=\pm 1$ are its critical points. If $\lim _{x \rightarrow 0}\left(2+\frac{f(x)}{x^{3}}\right)=4$, then which one of the following is not true?(1) $f$ is an odd function.(2) $f(1)-4 f(-1)=4$.(3) $x=1$ is a point of maxima and $x=-1$ is a point of minima of $f$.(4) $x=1$ is a point of minima and $x=-1$ is a point of maxima of $f$.Correct Option: , 4 Solution: $f(x)=a x^{5}+b x^{4}+c x^{3}$...
Read More →The products formed in the reaction of cumene with
Question: The products formed in the reaction of cumene with $\mathrm{O}_{2}$ followed by treatment with dil. HCI are:Correct Option: , 3 Solution: Reaction involved:...
Read More →Find the peak current and resonant frequency of the following circuit (as shown in figure)
Question: Find the peak current and resonant frequency of the following circuit (as shown in figure) (1) $0.2 \mathrm{~A}$ and $100 \mathrm{~Hz}$(2) $2 \mathrm{~A}$ and $50 \mathrm{~Hz}$(3) $2 \mathrm{~A}$ and $100 \mathrm{~Hz}$(4) $0.2 \mathrm{~A}$ and $50 \mathrm{~Hz}$Correct Option: , 4 Solution: (4) Peak current in series LCR CKT $i=\frac{v_{0}}{z} \Rightarrow \frac{30}{\sqrt{\left(x_{L}-x_{C}\right)^{2}+R^{2}}}$ $i=\frac{30}{\sqrt{(10-100)^{2}+(120)^{2}}}$ $i \Rightarrow \frac{30}{150} \Rig...
Read More →The area of a rhombus is 480 cm2, and one of its diagonals measures 48 cm.
Question: The area of a rhombus is 480 cm2, and one of its diagonals measures 48 cm. Find (i) the length of the other diagonal, (ii) the length of each of its sides, and (iii) its perimeter. Solution: It is given that,Area of rhombus = 480 cm2.One of the diagonal = 48 cm. (i) Area of the rhombus $=\frac{1}{2} \times d_{1} \times d_{2}$ $\Rightarrow 480=\frac{1}{2} \times 48 \times d_{2}$ $\Rightarrow 480=24 \times d_{2}$ $\Rightarrow d_{2}=\frac{480}{24}$ $\Rightarrow d_{2}=20 \mathrm{~cm}$ $\Ri...
Read More →The value of c in the Lagrange's mean value theorem for the function
Question: The value of $c$ in the Lagrange's mean value theorem for the function $f(x)=x^{3}-4 x^{2}+8 x+11$, when $x \in[0,1]$ is: (1) $\frac{4-\sqrt{5}}{3}$(2) $\frac{4-\sqrt{7}}{3}$(3) $\frac{2}{3}$(4) $\frac{\sqrt{7}-2}{3}$Correct Option: 2, Solution: Since, $f(x)$ is a polynomial function. $\therefore \quad$ It is continuous and differentiable in $[0,1]$ Here, $f(0)=11, f(1)=1-4+8+11=16$ $f^{\prime}(x)=3 x^{2}-8 x+8$ $\therefore \quad f^{\prime}(c)=\frac{f(1)-f(0)}{1-0}=\frac{16-11}{1}$ $=3...
Read More →The major product obtained in the following reaction is:
Question: The major product obtained in the following reaction is: Correct Option: , 4 Solution: Reaction involved:...
Read More →Find the area of a rhombus one side
Question: Find the area of a rhombus one side of which measures 20 cm and one of whose diagonals is 24 cm. Solution: It is given that,The sides of rhombus = 20 cm.One of the diagonal = 24 cm. In ∆ABC,The sides of the triangle are of length 20 cm, 20 cm and 24 cm. Semi-perimeter of the triangle is $s=\frac{20+20+24}{2}=\frac{64}{2}=32 \mathrm{~cm}$ By Heron's formula, Area of $\Delta A B C=\sqrt{s(s-a)(s-b)(s-c)}$ $=\sqrt{32(32-20)(32-20)(32-24)}$ $=\sqrt{32(12)(12)(8)}$ $=192 \mathrm{~cm}^{2} \q...
Read More →If the tangent to the curve,
Question: If the tangent to the curve, $y=f(x)=x \log _{e} x,(x0)$ at a point $(c, f(c))$ is parallel to the line segement joining the points $(1,0)$ and $(e, e)$, then $c$ is equal to:(1) $\frac{e-1}{e}$(2) $\mathrm{e}^{\left(\frac{1}{\mathrm{e}-1}\right)}$(3) $\mathrm{e}^{\left(\frac{1}{1-\mathrm{e}}\right)}$(4) $\frac{e}{e-1}$Correct Option: , 2 Solution: The given tangent to the curve is, $y=x \log _{e} x \quad(x0)$ $\Rightarrow \frac{d y}{d x}=1+\log _{e} x$ $\left.\Rightarrow \frac{d y}{d ...
Read More →The major product of the following reaction is:
Question: The major product of the following reaction is: Correct Option: 1 Solution: Mechanism involved for the given reaction is?...
Read More →A parallelogram and a square have the same area.
Question: A parallelogram and a square have the same area. If the sides of the square measure 40 m and altitude of the parallelogram measures 25 m, find the length of the corresponding base of the parallelogram. Solution: It is given that,Sides of the square = 40 mAltitude of the parallelogram = 25 m Now,Area of the parallelogram = Area of the square $\Rightarrow$ Base $\times$ Height $=(\text { side })^{2}$ $\Rightarrow$ Base $\times 25=(40)^{2}$ $\Rightarrow$ Base $\times 25=1600$ $\Rightarrow...
Read More →In a series LCR resonant circuit, the quality factor is measured as 100 . If
Question: In a series LCR resonant circuit, the quality factor is measured as 100 . If the inductance is increased by two fold and resistance is decreased by two fold, then the quality factor after this change will be (round off to nearest integer) Solution: (283) Quality factor $=\frac{X_{\mathrm{L}}}{\mathrm{R}}=\frac{\omega \mathrm{L}}{\mathrm{R}}$ $\mathrm{Q}=\frac{1}{\sqrt{\mathrm{LC}}} \frac{\mathrm{L}}{\mathrm{R}}$ $\mathrm{Q}=\left(\frac{1}{\sqrt{\mathrm{C}}}\right) \frac{\sqrt{\mathrm{L...
Read More →A parallelogram and a rhombus are equal in area
Question: A parallelogram and a rhombus are equal in area. The diagonals of the rhombus measure 120 m and 44 m. If one of the sides of the parallelogram measures 66 m, find its corresponding altitude. Solution: Diagonals $d_{1}$ and $d_{2}$ of the rhombus measure $120 \mathrm{~m}$ and $44 \mathrm{~m}$, respectively. Base of the parallelogram = 66 m Now,Area of the rhombus = Area of the parallelogram $\Rightarrow \frac{1}{2} \times d_{1} \times d_{2}=$ Base $\times$ Height $\Rightarrow \frac{1}{2...
Read More →For all twice differentiable functios
Question: For all twice differentiable functios $f: \mathrm{R} \rightarrow \mathrm{R}$, with $f(0)=f(1)=f^{\prime}(0)=0$(1) $f^{\prime \prime}(x) \neq 0$ at every point $x \in(0,1)$(2) $f^{\prime \prime}(x)=0$, for some $x \in(0,1)$(3) $f^{\prime \prime}(0)=0$(4) $f^{\prime \prime}(x)=0$, at every point $x \in(0,1)$Correct Option: , 2 Solution: Let $f: \mathrm{R} \rightarrow \mathrm{R}$, with $f(0)=f(1)=0$ and $f^{\prime}(0)=0$ $\because f(x)$ is differentiable and continuous and $f(0)=f(1)=0$ T...
Read More →Consider the following reaction :
Question: Consider the following reaction : $\mathrm{CH} \equiv \mathrm{CH}$$\mathrm{CH}_{3}-\mathrm{C} \equiv \mathrm{C}-\mathrm{CH}_{3}$$\mathrm{CH}_{3}-\mathrm{C} \equiv \mathrm{CH}$$\mathrm{CH}_{2}=\mathrm{CH}_{2}$Correct Option: , 3 Solution:...
Read More →The difference between the lengths of the parallel sides of a trapezium is 8 cm,
Question: The difference between the lengths of the parallel sides of a trapezium is 8 cm, the perpendicular distance between these sides is 24 cm and the area of the trapezium is 312 cm2. Find the length of each of the parallel sides. Solution: Let the length of the parallel sides bexandx 8.The height of the trapezium = 24 cm Area of trapezium $=\frac{1}{2} \times$ sum of parallel sides $\times$ height $\Rightarrow 312=\frac{1}{2} \times(x+x-8) \times 24$ ⇒ 312 = 12(2x 8) $\Rightarrow 2 x-8=\fr...
Read More →An alternating current is given by the equation
Question: An alternating current is given by the equation $\mathrm{i}=\mathrm{i}_{1} \sin \omega \mathrm{t}+\mathrm{i}_{2} \operatorname{coscot}$. The rms current will be :(1) $\frac{1}{2}\left(\mathrm{i}_{1}^{2}+\mathrm{i}_{2}^{2}\right)^{\frac{1}{2}}$(2) $\frac{1}{\sqrt{2}}\left(\mathrm{i}_{1}^{2}+\mathrm{i}_{2}^{2}\right)^{\frac{1}{2}}$(3) $\frac{1}{\sqrt{2}}\left(\mathrm{i}_{1}+\mathrm{i}_{2}\right)^{2}$(4) $\frac{1}{\sqrt{2}}\left(\mathrm{i}_{1}+\mathrm{i}_{2}\right)$Correct Option: , 2 Sol...
Read More →The set of all real values of
Question: The set of all real values of $\lambda$ for which the function $f(x)=\left(1-\cos ^{2} x\right) \cdot(\lambda+\sin x), x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, has exactly one maxima and exactly minima, is: (1) $\left(-\frac{1}{2}, \frac{1}{2}\right)-\{0\}$(2) $\left(-\frac{3}{2}, \frac{3}{2}\right)$(3) $\left(-\frac{1}{2}, \frac{1}{2}\right)$(4) $\left(-\frac{3}{2}, \frac{3}{2}\right)-\{0\}$Correct Option: , 4 Solution: $f(x)=\left(1-\cos ^{2} x\right)(\lambda+\sin x)=\sin ^{...
Read More →Heating of 2-chloro-1-phenylbutane with EtOK/EtOH gives X as the major product.
Question: Heating of 2-chloro-1-phenylbutane with EtOK/EtOH gives $X$ as the major product. Reaction of $X$ with $\mathrm{Hg}(\mathrm{OAc})_{2} / \mathrm{H}_{2} \mathrm{O}$ followed by $\mathrm{NaBH}_{4}$ gives $\mathrm{Y}$ as the major product. $Y$ is :Correct Option: , 3 Solution:...
Read More →An LCR circuit contains resistance
Question: An LCR circuit contains resistance of $110 \Omega$ and a supply of $220 \mathrm{~V}$ at $300 \mathrm{rad} / \mathrm{s}$ angular frequency. If only capacitance is removed from the circuit, current lags behind the voltage by $45^{\circ}$. If on the other hand, only inductor is removed the current leads by $45^{\circ}$ with the applied voltage. The rms current flowing in the circuit will be:(1) $2.5 \mathrm{~A}$(2) $2 \mathrm{~A}$(3) $1 \mathrm{~A}$(4) $1.5 \mathrm{~A}$Correct Option: , 2...
Read More →be respectively the minimum and maximum values of
Question: Let $m$ and $M$ be respectively the minimum and maximum values of $\left|\begin{array}{ccc}\cos ^{2} x 1+\sin ^{2} x \sin 2 x \\ 1+\cos ^{2} x \sin ^{2} x \sin 2 x \\ \cos ^{2} x \sin ^{2} x 1+\sin 2 x\end{array}\right|$ Then the ordered pair $(m, M)$ is equal to : (1) $(-3,3)$(2) $(-3,-1)(3)(-4,-1)$(3) $(-4,-1)$(4) $(1,3)$Correct Option: , 2 Solution: $C_{1} \rightarrow C_{1}+C_{2}$ Let $f(x)=\left|\begin{array}{ccc}2 1+\sin ^{2} x \sin 2 x \\ 2 \sin ^{2} x \sin 2 x \\ 1 \sin ^{2} x 1...
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