How many two-digit numbers are divisible by 6?
Question: How many two-digit numbers are divisible by 6? Solution: The two-digit numbers divisible by 6 are 12, 18, 24, ..., 96.Clearly, these number are in AP.Here,a= 12 andd= 18 12 = 6Let this AP containsnterms. Then, $a_{n}=96$ $\Rightarrow 12+(n-1) \times 6=96 \quad\left[a_{n}=a+(n-1) d\right]$ $\Rightarrow 6 n+6=96$ $\Rightarrow 6 n=96-6=90$ $\Rightarrow n=15$ Hence, there are 15 two-digit numbers divisible by 6....
Read More →Consider the following reactions:
Question: Consider the following reactions: Which of these reaction(s) will not produce Saytzeff product?(a), (c) and (d)(d) only(c) only(b) and (d)Correct Option: , 3 Solution:...
Read More →The first and last terms of an AP are a and l, respectively.
Question: The first and last terms of an AP areaandl, respectively. Show that the sum of thenth term from the beginning and thenth term from the end is (a+ 1). Solution: In the given AP, first term =aand last term =l.Let thecommon difference bed. Then, nthterm from the beginning is given byTn=a+ (n- 1)d ...(1)Similarly, nthterm from the end is given byTn= {l- (n- 1)d} ...(2)Adding (1) and (2), we get:a+ (n- 1)d+{l- (n- 1)d}=a+ (n- 1)d+l- (n- 1)d= a+l Hence, thesum of thenthterm from the beginn...
Read More →When the temperature of a metal wire is increased from
Question: When the temperature of a metal wire is increased from $0^{\circ} \mathrm{C}$ to $10^{\circ} \mathrm{C}$, its length increased by $0.02 \%$. The percentage change in its mass density will be closest to :(1) $0.06$(2) $2.3$(3) $0.008$(4) $0.8$Correct Option: 1 Solution: (1) Change in length of the metal wire $(\Delta l)$ when its temperature is changed by $\Delta T$ is given by $\Delta l=l \alpha \Delta T$ Here, $\alpha=$ Coefficient of linear expansion Here, $\Delta l=0.02 \%, \Delta T...
Read More →Consider the following reactions:
Question: Consider the following reactions: Which of these reaction(s) will not produce Saytzeff product?(a), (c) and (d)(d) only(c) only(b) and (d)Correct Option: , 3 Solution:...
Read More →If the pth term of an AP is q and its qth term is p then show that its (p + q)th term is zero.
Question: If thepth term of an AP isqand itsqth term ispthen show that its (p+q)th term is zero. Solution: In the given AP, let the first term beaand the common difference bed.Then Tn=a+ (n- 1)d⇒ Tp=a+ (p- 1)d=q ...(i) ⇒Tq=a+ (q- 1)d=p ...(ii) On subtracting (i) from (ii), we get:(q-p)d= (p-q)⇒d= -1Puttingd= -1 in (i), we get:a= (p+q- 1)Thus,a=(p+q- 1) andd= -1Now, Tp+q=a+ (p+q- 1)d = (p+q- 1)+ (p+q- 1)(-1) = (p+q- 1) - (p+q- 1)=0 Hence, the (p+q)th term is 0 (zero)....
Read More →Consider the following reactions:
Question: Consider the following reactions: Which of these reaction(s) will not produce Saytzeff product?(a), (c) and (d) (d) only(c) only(b) and (d)Correct Option: , 3 Solution:...
Read More →The 19th term of an AP is equal to 3 times its 6th term. If its 9th term is 19, find the AP.
Question: The 19th term of an AP is equal to 3 times its 6th term. If its 9th term is 19, find the AP. Solution: Letabe the first term anddbe the common difference of the AP. Then, $a_{19}=3 a_{6}$ (Given) $\Rightarrow a+18 d=3(a+5 d) \quad\left[a_{n}=a+(n-1) d\right]$ $\Rightarrow a+18 d=3 a+15 d$ $\Rightarrow 3 a-a=18 d-15 d$ $\Rightarrow 2 a=3 d$ $\ldots \ldots(1)$ Also, $a_{9}=19$ (Given) $\Rightarrow a+8 d=19 \quad \ldots \ldots(2)$ From (1) and (2), we get $\frac{3 d}{2}+8 d=19$ $\Rightarr...
Read More →What is the product of following reaction?
Question: What is the product of following reaction? Hex-3-ynal (i) $\stackrel{\mathrm{NaBH}_{4}}{\longrightarrow}$ ? (ii) $\mathrm{PBr}_{3}$ (iii) $\mathrm{Mg}$ /ether (iv) $\mathrm{CO}_{2} / \mathrm{H}_{3} \mathrm{O}^{+}$Correct Option: , 4 Solution:...
Read More →A wire
Question: A wire of $1 \Omega$ has a length of $1 \mathrm{~m}$. It is stretched till its length increases by $25 \%$. The percentage change in a resistance to the nearest integer is:(1) $25 \%$(2) $12.5 \%$(3) $76 \%$(4) $56 \%$Correct Option: , 4 Solution: (4) For stretched or compressed wire $\mathrm{R} \propto \mathrm{I}^{2}$ $\frac{R_{1}}{R_{2}}=\frac{l_{1}^{2}}{l_{2}^{2}}$ $\Rightarrow \frac{\mathrm{R}}{\mathrm{R}_{2}}=\frac{\mathrm{R}^{2}}{(1.251)^{2}}$ $\Rightarrow \mathrm{R}_{2}=1.5625 \...
Read More →The 24th term of an AP is twice its 10th term. Show that its 72nd term is 4 times its 15th term.
Question: The 24th term of an AP is twice its 10th term. Show that its 72nd term is 4 times its 15th term. Solution: Letabe the first term anddbe the common difference of the AP. Then, $a_{24}=2 a_{10}$ (Given) $\Rightarrow a+23 d=2(a+9 d) \quad\left[a_{n}=a+(n-1) d\right]$ $\Rightarrow a+23 d=2 a+18 d$ $\Rightarrow 2 a-a=23 d-18 d$ $\Rightarrow a=5 d$ $\ldots \ldots(1)$ Now, $\frac{a_{72}}{a_{15}}=\frac{a+71 d}{a+14 d}$ $\Rightarrow \frac{a_{72}}{a_{15}}=\frac{5 d+71 d}{5 d+14 d} \quad[$ From $...
Read More →if
Question: If $\bar{P} \times \bar{Q}=\vec{Q} \times \vec{P}$, the angle between $\bar{P}$ and $\bar{Q}$ is $\theta\left(0^{\circ}\theta360^{\circ}\right)$. The value of ' $\theta$ ' will be Solution: (180) If $\vec{P} \times \vec{Q}=\vec{Q} \times \bar{P}$ Only if $\vec{P}=0$ Or $\overrightarrow{\mathrm{Q}}=0$ The angle b/w $\overrightarrow{\mathrm{P}} \ \overrightarrow{\mathrm{Q}}$ is $\theta\left(0^{\circ}\theta360^{\circ}\right)$ So $\theta=180^{\circ}$...
Read More →The major product obtained from the following reaction is:
Question: The major product obtained from the following reaction is: Correct Option: 1 Solution:...
Read More →The percentage increase in the speed of transverse waves produced
Question: The percentage increase in the speed of transverse waves produced in a stretched string if the tension is increased by $4 \%$ will be_____ $\%$ Solution: (2) Speed of transverse wave is $V=\sqrt{\frac{T}{\mu}}$ $\frac{\Delta n}{v}=\frac{1}{2} \ell n T-\frac{1}{2} \ell n \mu$ $\frac{\Delta v}{v}=\frac{1}{2} \frac{\Delta T}{T}$ $=\frac{1}{2} \times 4$ $\frac{\Delta v}{v}=2 \%$...
Read More →Prove the following
Question: If $\alpha$ and $\beta$ be the roots of the equation $x^{2}-2 x+2=0$, then the least value of $n$ for which $\left(\frac{\alpha}{\beta}\right)^{n}=1$ is :(1) 2(2) 5(3) 4(4) 3Correct Option: 3, Solution: The given quadratic equation is $x^{2}-2 x+2=0$ Then, the roots of the this equation are $\frac{2 \pm \sqrt{-4}}{2}=1 \pm i$ Now, $\frac{\alpha}{\beta}=\frac{1-i}{1+i}=\frac{(1-i)^{2}}{1-i^{2}}=i$ or $\frac{\alpha}{\beta}=\frac{1-i}{1+i}=\frac{(1-i)^{2}}{1-i^{2}}=i$ So, $\frac{\alpha}{\...
Read More →Prove the following
Question: If $\alpha$ and $\beta$ be the roots of the equation $x^{2}-2 x+2=0$, then the least value of $n$ for which $\left(\frac{\alpha}{\beta}\right)^{n}=1$ is :(1) 9(2) 12(3) 4(4) 10Correct Option: , 3 Solution: The given quadratic equation is $x^{2}-2 x+2=0$ Then, the roots of the this equation are $\frac{2 \pm \sqrt{-4}}{2}=1 \pm i$ Now, $\frac{\alpha}{\beta}=\frac{1-i}{1+i}=\frac{(1-i)^{2}}{1-i^{2}}=i$ or $\frac{\alpha}{\beta}=\frac{1-i}{1+i}=\frac{(1-i)^{2}}{1-i^{2}}=i$ So, $\frac{\alpha...
Read More →The pitch of the screw guage is 1 mm and there are 100 divisions on the circular scale.
Question: The pitch of the screw guage is $1 \mathrm{~mm}$ and there are 100 divisions on the circular scale. When nothing is put in between the jaws, the zero of the circular scale lines 8 divisions below the reference line. When a wire is placed between the jaws, the first linear scale division is clearly visible while $72^{\text {nd }}$ division on circular scale coincides with the reference line. The radius of the wire is :(1) $1.64 \mathrm{~mm}$(2) $1.80 \mathrm{~mm}$(3) $0.82 \mathrm{~mm}$...
Read More →The 17th term of an AP is 5 more than twice its 8th term.
Question: The 17th term of an AP is 5 more than twice its 8th term. If the 11th term of the AP is 43, find itsnth term. Solution: Letabe the first term anddbe the common difference of the AP. Then, $a_{17}=2 a_{8}+5$ (Given) $\therefore a+16 d=2(a+7 d)+5 \quad\left[a_{n}=a+(n-1) d\right]$ $\Rightarrow a+16 d=2 a+14 d+5$ $\Rightarrow a-2 d=-5 \quad \ldots$ (1) Also, $a_{11}=43$ (Given) $\Rightarrow a+10 d=43 \quad \ldots(2)$ From (1) and (2), we get $-5+2 d+10 d=43$ $\Rightarrow 12 d=43+5=48$ $\R...
Read More →The major product formed in the following reaction is :
Question: The major product formed in the following reaction is : $\mathrm{CH}_{3} \mathrm{CH}=\mathrm{CHCH}\left(\mathrm{CH}_{3}\right)_{2} \stackrel{\mathrm{HBr}}{\longrightarrow}$$\mathrm{CH}_{3} \mathrm{CH}(\mathrm{Br}) \mathrm{CH}_{2} \mathrm{CH}\left(\mathrm{CH}_{3}\right)_{2}$$\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CH}(\mathrm{Br}) \mathrm{CH}\left(\mathrm{CH}_{3}\right)_{2}$$\mathrm{Br}\left(\mathrm{CH}_{2}\right)_{3} \mathrm{CH}\left(\mathrm{CH}_{3}\right)_{2}$$\mathrm{CH}_{3} \mathrm{...
Read More →Prove the following
Question: Let $a, b \in \mathrm{R}, a \neq 0$ be such that the equation, $a x^{2}-2 b x+5=0$ has a repeated root $\alpha$, which is also a root of the equation, $x^{2}-2 b x-10=0$. If $\beta$ is the other root of this equation, then $\alpha^{2}+\beta^{2}$ is equal to :(1) 25(2) 26(3) 28(4) 24Correct Option: 1 Solution: $a x^{2}-2 b x+5=0$ If $\alpha$ and $\alpha$ are roots of equations, then sum of roots $2 \alpha=\frac{2 b}{a} \Rightarrow \alpha=\frac{b}{a}$ and product of roots $=\alpha^{2}=\f...
Read More →In an octagon ABCDEFGH of equal side, what is the sum of
Question: In an octagon $\mathrm{ABCDEFGH}$ of equal side, what is the sum of $\overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{AC}}+\overrightarrow{\mathrm{AD}}+\overrightarrow{\mathrm{AE}}+\overrightarrow{\mathrm{AF}}+\overrightarrow{\mathrm{AG}}+\overrightarrow{\mathrm{AH}}$ If, $\overrightarrow{\mathrm{AO}}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-4 \hat{\mathrm{k}}$ (1) $16 \hat{\mathrm{i}}+24 \hat{\mathrm{j}}-32 \hat{\mathrm{k}}$(2) $-16 \hat{\mathrm{i}}-24 \hat{\mathrm{j}}-32 \hat{\mathr...
Read More →The equation that represents the water-gas shift reaction is:
Question: The equation that represents the water-gas shift reaction is: Correct Option: , 4 Solution: This reaction is called water gas shift reaction....
Read More →For what value of n, the nth terms of the arithmetic progressions 63, 65, 67,... and 3, 10, 17,... are equal?
Question: For what value ofn, thenth terms of the arithmetic progressions 63, 65, 67,... and 3, 10, 17,... are equal? Solution: Let thenthterm of the given progressions be tnand Tn, respectively.The first AP is 63, 65, 67,...Let its first term beaand common difference bed.Thena= 63 andd= (65 - 63) = 2So, itsnthterm is given bytn=a+ (n- 1)d⇒63+(n- 1) ⨯2⇒61+2nThe second AP is 3, 10, 17,...Let its first term be A and common difference be D.Then A = 3 and D = (10 - 3) = 7So, its nthterm is given byT...
Read More →The number of real roots of the equation,
Question: The number of real roots of the equation, $e^{4 x}+e^{3 x}-4 e^{2 x}+e^{x}+1=0$ is:(1) 1(2) 3(3) 2(4) 4Correct Option: 1 Solution: Let $e^{x}=t \in(0, \infty)$ Given equation $t^{4}+t^{3}-4 t^{2}+t+1=0$ $\Rightarrow t^{2}+t-4+\frac{1}{t}+\frac{1}{t^{2}}=0$ $\Rightarrow\left(t^{2}+\frac{1}{t^{2}}\right)+\left(t+\frac{1}{t}\right)-4=0$ Let $t+\frac{1}{t}=y$ $\left(y^{2}-2\right)+y-4=0 \Rightarrow y^{2}+y-6=0$ $y^{2}+y-6=0 \Rightarrow y=-3,2$ $\Rightarrow y=2 \quad \Rightarrow \quad t+\fr...
Read More →The sum of the 2nd and the 7th terms of an AP is 30. If its 15th term is 1 less than twice its 8th term, find the AP.
Question: The sum of the 2nd and the 7th terms of an AP is 30. If its 15th term is 1 less than twice its 8th term, find the AP. Solution: Letabe the first term anddbe the common difference of the AP. Then, $a_{2}+a_{7}=30$ (Given) $\therefore(a+d)+(a+6 d)=30 \quad\left[a_{n}=a+(n-1) d\right]$ $\Rightarrow 2 a+7 d=30 \quad \ldots$ (1) Also, $a_{15}=2 a_{8}-1$ (Given) $\Rightarrow a+14 d=2(a+7 d)-1$ $\Rightarrow a+14 d=2 a+14 d-1$ $\Rightarrow-a=-1$ $\Rightarrow a=1$ Puttinga= 1 in (1), we get $2 ...
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