What will be the major product when
Question: What will be the major product when $\mathrm{m}$-cresol is reacted with propargyl bromide $\left(\mathrm{HC} \equiv \mathrm{C}-\mathrm{CH}_{2} \mathrm{Br}\right)$ in presence of $\mathrm{K}_{2} \mathrm{CO}_{3}$ in acetone?Correct Option: 1 Solution:...
Read More →Find the area of a trapezium whose parallel sides are 11 m and 25 m long,
Question: Find the area of a trapezium whose parallel sides are 11 m and 25 m long, and the nonparallel sides are 15 m and 13 m long. Solution: In the given figure,ABCDis the trapezium. Draw a lineBEparallel toAD.In ∆BCE,The sides of the triangle are of length 15 m, 13 m and 14 m. Semi-perimeter of the triangle is $s=\frac{15+13+14}{2}=\frac{42}{2}=21 \mathrm{~m}$ By Heron's formula, Area of $\Delta B C E=\sqrt{s(s-a)(s-b)(s-c)}$ $=\sqrt{21(21-15)(21-13)(21-14)}$ $=\sqrt{21(6)(8)(7)}$ $=84 \math...
Read More →A transmitting station releases waves of wavelength
Question: A transmitting station releases waves of wavelength $960 \mathrm{~m}$. A capacitor of $256 \mu \mathrm{F}$ is used in the resonant circuit. The self inductance of coil necessary for resonance is $\times 10^{-8} \mathrm{H}$ Solution: (10) Since resonance $\omega_{\mathrm{r}}=\frac{1}{\sqrt{\mathrm{LC}}}$ $\therefore 2 \pi f=\frac{1}{\sqrt{L C}}$ $\therefore 4 \pi^{2} \frac{C^{2}}{\lambda^{2}}=\frac{1}{L C}$ $\therefore \frac{4 \pi^{2} \times 9 \times 10^{8} \times 9 \times 10^{8}}{960 \...
Read More →The major products of the following reaction are :
Question: The major products of the following reaction are : Correct Option: , 4 Solution:...
Read More →The position of a moving car at time
Question: The position of a moving car at time $t$ is given by $f(t)=a t^{2}$ $+b t+c, t0$, where $a, b$ and $c$ are real numbers greater than 1 . Then the average speed of the car over the time interval $\left[t_{1}, t_{2}\right]$ is attained at the point :(1) $\left(t_{2}-t_{1}\right) / 2$(2) $\hat{a}\left(t_{2}-t_{1}\right)+b$(3) $\left(t_{1}+t_{2}\right) / 2$(4) $2 a\left(t_{1}+t_{2}\right)+b$Correct Option: , 3 Solution: Average speed $=f^{\prime}(t)=\frac{f\left(t_{2}\right)-f\left(t_{1}\r...
Read More →Let L1 be a tangent to the parabola
Question: Let $\mathrm{L}_{1}$ be a tangent to the parabola $y^{2}=4(x+1)$ and $\mathrm{L}_{2}$ be a tangent to the parabola $y^{2}=8(x+2)$ such that $\mathrm{L}_{1}$ and $\mathrm{L}_{2}$ intersect at right angles. Then $\mathrm{L}_{1}$ and $\mathrm{L}_{2}$ meet on the straight line:(1) $x+3=0$(2) $2 x+1=0$(3) $x+2=0$(4) $x+2 y=0$Correct Option: 1 Solution: $L_{1}: y=m_{1}(x+1)+\frac{1}{m_{1}} \quad$ [Tangent to $\left.y^{2}=4(x+1)\right]$ $L_{2}: y=m_{2}(x+2)+\frac{2}{m_{2}} \quad$ [Tangent to ...
Read More →The shape of the cross section of a canal is a trapezium.
Question: The shape of the cross section of a canal is a trapezium. If the canal is 10 m wide at the top, 6 m wide at the bottom and the area of its cross section is 640 m2, find the depth of the canal. Solution: The top and the bottom of the canal are parallel to each other.Let the height of the trapezium beh. Area of trapezium $=\frac{1}{2} \times$ sum of parallel sides $\times$ height $\Rightarrow 640=\frac{1}{2} \times(10+6) \times h$ $\Rightarrow 640=8 \times h$ $\Rightarrow h=\frac{640}{8}...
Read More →A rectangular lawn, 75 m by 60 m, has two roads, each road 4 m wide, running through the middle of the lawn,
Question: A rectangular lawn, 75 m by 60 m, has two roads, each road 4 mwide, running through the middle of the lawn, one parallel to length and the other parallel to breadth, as shown in the figure. Find the cost of gravelling the roads at Rs 50 per m2. Solution: Area of rectangleABCD= Length Breath= 75 4= 300 m2Area of rectanglePQRS= Length Breath= 60 4= 240 m2Area of squareEFGH= (side)2= (4)2= 16 m2 Area of the footpath = Area of rectangleABCD+ Area of rectanglePQRS Area of squareEFGH= 300 + ...
Read More →The angular frequency of alternating current in a L-C-R circuit
Question: The angular frequency of alterlating current in a L-C-R circuit is $100 \mathrm{rad} / \mathrm{s}$. The components connected are shown in the figure. Find the value of inductance of the coil and capacity of condenser. (1) $0.8 \mathrm{H}$ and $250 \mu \mathrm{F}$(2) $0.8 \mathrm{H}$ and $150 \mu \mathrm{F}$(3) $1.33 \mathrm{H}$ and $250 \mu \mathrm{F}$(4) $1.33 \mathrm{H}$ and $150 \mu \mathrm{F}$Correct Option: 1 Solution: (1) Since key is open, circuit is series $15=\mathrm{i}_{\math...
Read More →The major product of the following reaction is:
Question: The major product of the following reaction is: Correct Option: , 2 Solution:...
Read More →Which of the following points lies on the tangent to the curve
Question: Which of the following points lies on the tangent to the curve $x^{4} e^{y}+2 \sqrt{y+1}=3$ at the point $(1,0)$ ?(1) $(2,2)$(2) $(2,6)$(3) $(-2,6)$(4) $(-2,4)$Correct Option: , 3 Solution: The given curve is, $x^{4} \cdot e^{y}+2 \sqrt{y+1}=3$ Differentiating w.r.t. $x$, we get $\left(4 x^{3}+x^{4} \cdot y^{\prime}\right) e^{y}+\frac{y^{\prime}}{\sqrt{1+y}}=0$ $\Rightarrow\left(\frac{d y}{d x}\right)=\frac{-4 x^{3} e^{y}}{\left(\frac{1}{\sqrt{y+1}}+e^{y} x^{4}\right)}$ $\Rightarrow\le...
Read More →In the given figure, ABCD is a square with diagonal 44 cm.
Question: In the given figure,ABCDis a square with diagonal 44 cm. How much paper of each shade is needed to make a kite given in the figure? Solution: In the given figure,ABCDis a square with diagonal 44 cm.AB=BC=CD=DA. ....(1)In right angled ∆ABC, $A C^{2}=A B^{2}+B C^{2}$ (Pythagoras Theorem) $\Rightarrow 44^{2}=2 A B^{2}$ $\Rightarrow 1936=2 A B^{2}$ $\Rightarrow A B^{2}=\frac{1936}{2}$ $\Rightarrow A B^{2}=968$ $\Rightarrow A B=22 \sqrt{2} \mathrm{~cm}$ ...(2) $\therefore$ Sides of square $...
Read More →If x=1 is a critical point of the function
Question: If $x=1$ is a critical point of the function $f(x)=\left(3 x^{2}+a x-2-a\right) e^{x}$, then : (1) $x=1$ and $x=-\frac{2}{3}$ are local minima of $f$.(2) $x=1$ and $x=-\frac{2}{3}$ are local maxima of $f$.(3) $x=1$ is a local maxima and $x=-\frac{2}{3}$ is a local minima of $f$.(4) $x=1$ is a local minima and $x=-\frac{2}{3}$ is a local maxima of $f$.Correct Option: , 4 Solution: The given function $f(x)=\left(3 x^{2}+a x-2-a\right) e^{x}$ $f^{\prime}(x)=(6 x+a) e^{x}+\left(3 x^{2}+a x...
Read More →The major product obtained in the given reaction is :
Question: The major product obtained in the given reaction is : W" Correct Option: , 3 Solution:...
Read More →If the minimum and the maximum values of the function
Question: If the minimum and the maximum values of the function $f:\left[\frac{\pi}{4}, \frac{\pi}{2}\right] \rightarrow \mathrm{R}$, defined by $f(\theta)=\left|\begin{array}{ccc}-\sin ^{2} \theta -1-\sin ^{2} \theta 1 \\ -\cos ^{2} \theta -1-\cos ^{2} \theta 1 \\ 12 10 -2\end{array}\right|$ are $m$ and $M$ respectively, then the ordered pair $(m, M)$ is equal to : (1) $(0,2 \sqrt{2})$(2) $(-4,0)$(3) $(-4,4)$(4) $(0,4)$Correct Option: , 2 Solution: Applying $C_{2} \rightarrow C_{2}-C_{1}$ $f(\t...
Read More →The major product of the following reaction is:
Question: The major product of the following reaction is: Correct Option: , 3 Solution:...
Read More →An umbrella is made by stitching 12 triangular pieces of cloth, each measuring
Question: An umbrella is made by stitching 12 triangular pieces of cloth, each measuring (50 cm 20 cm 50 cm). Find the area of the cloth used in it. Solution: We know that the triangle is an isosceles triangle.Thus, we can find out the area of one triangular piece of cloth. Area of isosceles triangle $=\frac{b}{4} \sqrt{4 a^{2}-b^{2}}$ $=\frac{20}{4} \times \sqrt{4(50)^{2}-20^{2}} \quad(a=50 \mathrm{~cm}$ and $b=20 \mathrm{~cm})$ $=5 \times \sqrt{10000-400}$ $=5 \times \sqrt{9600}$ $=5 \times 40...
Read More →The area (in sq. units) of the largest rectangle
Question: The area (in sq. units) of the largest rectangle $A B C D$ whose vertices $A$ and $B$ lie on the $x$-axis and vertices $C$ and $D$ lie on the parabola, $y=x^{2}-1$ below the $x$-axis, is :(1) $\frac{2}{3 \sqrt{3}}$(2) $\frac{1}{3 \sqrt{3}}$(3) $\frac{4}{3}$(4) $\frac{4}{3 \sqrt{3}}$Correct Option: Solution: Area of rectangle $A B C D$ $A=2 x \cdot\left(x^{2}-1\right)=2 x^{3}-2 x$ $\therefore \quad \frac{d A}{d x}=6 x^{2}-2$ For maximum area $\frac{d A}{d x}=0 \Rightarrow x=\pm \frac{1}...
Read More →The organic compound that gives following qualitative analysis is:
Question: The organic compound that gives following qualitative analysis is: Correct Option: 1 Solution:...
Read More →The current (i) at time $t=0$ and $t=infty$ respectively for the given circuit is :
Question: The current (i) at time $t=0$ and $t=\infty$ respectively for the given circuit is : $\mathrm{R}_{\mathrm{eq}}=\frac{1 \times 4}{1+4}+\frac{5 \times 5}{5+5}$ (1) $\frac{18 E}{55}, \frac{5 E}{18}$(2) $\frac{5 E}{18}, \frac{18 E}{55}$(3) $\frac{5 \mathrm{E}}{18}, \frac{10 \mathrm{E}}{33}$(4) $\frac{10 \mathrm{E}}{33}, \frac{5 \mathrm{E}}{18}$Correct Option: , 3 Solution: (3) at $t=0$, inductor is removed, so circuit will look like this at $t=0$ $\mathrm{R}_{\mathrm{eq}}=\frac{6 \times 9}...
Read More →A floral design on a floor is made up of 16 tiles, each triangular in shape having sides 16 cm,
Question: A floral design on a floor is made up of 16 tiles, each triangular in shape having sides 16 cm, 12 cm and 20 cm. Find the cost of polishing the tiles at Re 1 per sq cm. Solution: Area of one triangular-shaped tile can be found in the following manner: Let : $a=16 \mathrm{~cm}, b=12 \mathrm{~cm}$ and $c=20 \mathrm{~cm}$ $s=\frac{a+b+c}{2}=\frac{16+12+20}{2}=24 \mathrm{~cm}$ By Heron's formula, we have: Area of triangle $=\sqrt{s(s-a)(s-b)(s-c)}$ $=\sqrt{24(24-16)(24-12)(24-20)}$ $=\sqrt...
Read More →Let f be a twice differentiable function on
Question: Let $f$ be a twice differentiable function on $(1,6)$. If $f(2)=8$, $f^{\prime}(2)=5, f^{\prime}(x) \geq 1$ and $f^{\prime \prime}(x) \geq 4$, for all $x \in(1,6)$, then :(1) $f(5)+f^{\prime}(5) \leq 26$(2) $f(5)+f^{\prime}(5) \geq 28$(3) $f^{\prime}(5)+f^{\prime \prime}(5) \leq 20$(4) $f(5) \leq 10$Correct Option: , 2 Solution: Let $f$ be twice differentiable function $\because f^{\prime}(x) \geq 1$ $\Rightarrow \frac{f(5)-f(2)}{3} \geq 1$ $\Rightarrow f(5) \geq 3+f(2)$ $\Rightarrow f...
Read More →The difference between the semiperimeter and the sides of a ∆ABC are 8 cm,
Question: The difference between the semiperimeter and the sides of a ∆ABCare 8 cm, 7 cm and 5 cm respectively. Find the area of the triangle. Solution: Let the semi-perimeter of the triangle bes.Let the sides of the triangle bea,bandc.Given:sa= 8,sb= 7 andsc= 5 ....(1)Adding all three equations, we get3s (a + b + c)= 8 + 7 + 5⇒ 3s (a + b + c)= 20 ⇒ 3s 2s= 20 $\left(\because s=\frac{a+b+c}{2}\right)$ ⇒s= 20 cm ...(2) By Heron's formula, Area of $\Delta=\sqrt{s(s-a)(s-b)(s-c)}$ $=\sqrt{20(8)(7)(5...
Read More →If the tangent to the curve,
Question: If the tangent to the curve, $y=e^{x}$ at a point $\left(c, e^{c}\right)$ and the normal to the parabola, $y^{2}=4 x$ at the point $(1,2)$ intersect at the same point on the $x$-axis, then the value of $c$ is Solution: For $(1,2)$ of $y^{2}=4 x \Rightarrow t=1, a=1$ Equation of normal to the parabola $\Rightarrow t x+y=2 a t+a t^{3}$ $\Rightarrow x+y=3$ intersect $x$-axis at $(3,0)$ $y=e^{x} \Rightarrow \frac{d y}{d x}=e^{x}$ Equation of tangent to the curve $\Rightarrow y-e^{c}=e^{c}(...
Read More →The major product of the following reaction is :
Question: The major product of the following reaction is : Correct Option: , 2 Solution:...
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