The minimum value of
Question: The minimum value of $f(x)=a^{a^{x}}+a^{1-a^{x}}$, where $a, x \in R$ and $a0$, is equal to:(1) $\mathrm{a}+\frac{1}{\mathrm{a}}$(2) $a+1$(3) $2 \mathrm{a}$(4) $2 \sqrt{a}$Correct Option: , 4 Solution: $\mathrm{AM} \geq \mathrm{GM}$ $\frac{\mathrm{a}^{\mathrm{ax}}+\frac{\mathrm{a}}{\mathrm{anx}}}{2} \geq\left(\mathrm{a}^{\mathrm{ax} \cdot} \frac{\mathrm{a}}{\mathrm{a}^{\mathrm{ax}}}\right)^{1 / 2} \Rightarrow \mathrm{a}^{\mathrm{ax}}+\mathrm{a}^{1-\mathrm{ax}} \geq 2 \sqrt{\mathrm{a}}$...
Read More →Find three numbers in AP whose sum is 15 and whose product is 105.
Question: Find three numbers in AP whose sum is 15 and whose product is 105. Solution: Let the firstthree numbers in an arithmetic progression bea d,a,a + d.The sum of the first three numbers in an arithmetic progression is 15.a d+a+a + d= 15⇒ 3a= 15⇒a= 5Their product is 105. $(a-d) \times a \times(a+d)=105$ $\Rightarrow\left(a^{2}-d^{2}\right) \times a=105$ $\Rightarrow\left((5)^{2}-d^{2}\right) \times 5=105$ $\Rightarrow(5)^{2}-d^{2}=21$ $\Rightarrow 25-d^{2}=21$ $\Rightarrow d^{2}=25-21$ $\Ri...
Read More →Given below are two statements :
Question: Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R. Assertion A: Hydrogen is the most abundant element in the Universe, but it is not the most abundant gas in the troposphere. Reason $R$ : Hydrogen is the lightest element. In the light of the above statements, choose the correct answer from the given below$\mathrm{A}$ is false but $\mathrm{R}$ is trueBoth $\mathrm{A}$ and $\mathrm{R}$ are true and $\mathrm{R}$ is the correct explanatio...
Read More →Prove the following
Question: Let $A_{1}, A_{2}, A_{3}, \ldots \ldots$ be squares such that for each $n \geq 1$, the length of the side of $A_{n}$ equals the length of diagonal of $\mathrm{A}_{n+1}$. If the length of $\mathrm{A}_{1}$ is $12 \mathrm{~cm}$, then the smallest value of $\mathrm{n}$ for which area of $\mathrm{A}_{n}$ is less than one, is Solution: $x=\frac{12}{\sqrt{2}} \quad y=\frac{12}{(\sqrt{2})^{2}}$ $\therefore$ Side lengths are in G.P. $\mathrm{T}_{\mathrm{n}}=\frac{12}{(\sqrt{2})^{\mathrm{n}-1}}$...
Read More →Show that
Question: Show that $(a-b)^{2},\left(a^{2}+b^{2}\right)$ and $(a+b)^{2}$ are in AP. Solution: The given numbers are $(a-b)^{2},\left(a^{2}+b^{2}\right)$ and $(a+b)^{2}$. Now, $\left(a^{2}+b^{2}\right)-(a-b)^{2}=a^{2}+b^{2}-\left(a^{2}-2 a b+b^{2}\right)=a^{2}+b^{2}-a^{2}+2 a b-b^{2}=2 a b$ $(a+b)^{2}-\left(a^{2}+b^{2}\right)=a^{2}+2 a b+b^{2}-a^{2}-b^{2}=2 a b$ So, $\left(a^{2}+b^{2}\right)-(a-b)^{2}=(a+b)^{2}-\left(a^{2}+b^{2}\right)=2 a b$ (Constant) Since each term differs from its preceding ...
Read More →A copper wire is stretched to make it 0.5 %$ longer.
Question: A copper wire is stretched to make it $0.5 \%$ longer. The percentage change in its electrical resistance if its volume remains unchanged is:(1) $2.0 \%$(2) $2.5 \%$(3) $1.0 \%$(4) $0.5 \%$Correct Option: , 3 Solution: (3) Resistance, $R=\frac{\rho \ell}{A}$ $\mathrm{R}=\rho \frac{\ell}{\mathrm{A}} \times \frac{\ell}{\ell}=\frac{\rho \ell^{2}}{\mathrm{~V}}[\because$ Volume $(\mathrm{V})=\mathrm{A} \ell .]$ Since resistivity and volume remains constant therefore $\%$ change in resistanc...
Read More →In basic medium,
Question: In basic medium, $\mathrm{H}_{2} \mathrm{O}_{2}$ exhibits which of the following reactions? (A) $\mathrm{Mn}^{2+} \rightarrow \mathrm{Mn}^{4+}$ (B) $\mathrm{I}_{2} \rightarrow \mathrm{I}^{-}$ (C) $\mathrm{PbS} \rightarrow \mathrm{PbSO}_{4}$ Choose the most appropriate answer from the options given below:$(\mathrm{A}),(\mathrm{C})$ only(A) only(B) only(A), (B) onlyCorrect Option: , 4 Solution: In basic medium, oxidising action of $\mathrm{H}_{2} \mathrm{O}_{2}$ $\mathrm{Mn}^{2+}+\mathrm...
Read More →Prove the following
Question: If $0\theta, \phi\frac{\pi}{2}, \mathrm{x}=\sum_{\mathrm{n}=0}^{\infty} \cos ^{2 \mathrm{n}} \theta, \mathrm{y}=\sum_{\mathrm{n}=0}^{\infty} \sin ^{2 \mathrm{n}} \phi$ and $\mathrm{z}=\sum_{\mathrm{n}=0}^{\infty} \cos ^{2 \mathrm{n}} \theta \cdot \sin ^{2 \mathrm{n}} \phi$ then(1) $x y z=4$(2) $x y-z=(x+y) z$(3) $x y+y z+z x=z$(4) $x y+z=(x+y) z$Correct Option: , 4 Solution: $x=1+\cos ^{2} \theta+\ldots \ldots \ldots \infty$ $x=\frac{1}{1-\cos ^{2} \theta}=\frac{1}{\sin ^{2} \theta}$ ....
Read More →Write the value of x for which
Question: Write the value ofxfor which (x+ 2), 2x, (2x+ 3) are three consecutive terms of an AP? Solution: Since (x+ 2), 2xand (2x+ 3) are in AP, we have: $2 x-(x+2)=(2 x+3)-2 x$ $\Rightarrow x-2=3$ $\Rightarrow x=5$ $\therefore x=5$...
Read More →The position vector of a particle changes
Question: The position vector of a particle changes with time according to the relation $\vec{r}(\mathrm{t})=15 \mathrm{t}^{2} \hat{i}+\left(4-20 \mathrm{t}^{2}\right) \hat{j}$. What is the magnitude of the acceleration at $t=1$ ?(1) 40(2) 25(3) 100(4) 50Correct Option: , 4 Solution: (4) $\vec{r}=15 t^{2} \hat{i}+\left(4-20 t^{2}\right) \hat{j}$ $\vec{v}=\frac{d \vec{r}}{d t}=30 t \hat{i}-40 \hat{t j}$ Acceleration, $\vec{a}=\frac{d \vec{v}}{d t}=30 \hat{i}-40 \hat{j}$ $\therefore a=\sqrt{30^{2}...
Read More →The INCORRECT statement(s) about heavy water is
Question: The INCORRECT statement(s) about heavy water is (are) (a) used as a moderator in nuclear reactor (b) obtained as a by-product in fertilizer industry. (c) used for the study of reaction mechanism (d) has a higher dielectric constant than water Choose the correct answer from the options given below:(B) only(C) only(D) only(B) and (D) onlyCorrect Option: , 3 Solution: The dielectric constant of $\mathrm{H}_{2} \mathrm{O}$ is greater than heavy water....
Read More →The sum of first four terms of a geometric
Question: The sum of first four terms of a geometric progression (G.P.) is $\frac{65}{12}$ and the sum of their respective reciprocals is $\frac{65}{18} .$ If the product of first three terms of the G.P. is 1, and the third term is $\alpha$, then $2 \alpha$ is Solution: $a, a r, a r^{2}, a r^{3}$ $a+a r+a r^{2}+a r^{3}=\frac{65}{12} \ldots(1)$ $\frac{1}{a}+\frac{1}{a r}+\frac{1}{a r^{2}}+\frac{1}{a r^{3}}=\frac{65}{18}$ $\frac{1}{a}\left(\frac{r^{3}+r^{2}+r+1}{r^{3}}\right)=\frac{65}{18} \ldots(...
Read More →If (3y − 1), (3y + 5) and (5y + 1) are three consecutive terms of an AP then find the value of y.
Question: If (3y 1), (3y+ 5) and (5y+ 1) are three consecutive terms of an AP then find the value ofy. Solution: It is given that (3y 1), (3y+ 5) and (5y+ 1) are three consecutive terms of an AP. $\therefore(3 y+5)-(3 y-1)=(5 y+1)-(3 y+5)$ $\Rightarrow 3 y+5-3 y+1=5 y+1-3 y-5$ $\Rightarrow 6=2 y-4$ $\Rightarrow 2 y=6+4=10$ $\Rightarrow y=5$ Hence, the value ofyis 5....
Read More →The area of a square
Question: The area of a square is $5.29 \mathrm{~cm}^{2}$. The area of 7 such squares taking into account the significant figures is:(1) $37 \mathrm{~cm}^{2}$(2) $37.030 \mathrm{~cm}^{2}$(3) $37.03 \mathrm{~cm}^{2}$(4) $37.0 \mathrm{~cm}^{2}$Correct Option: , 4 Solution: (4) $A=7 \times 5.29=37.03 \mathrm{~cm}^{2}$ The result should have three significant figures, so $A=37.0 \mathrm{~cm}^{2}$...
Read More →The correct statements about
Question: The correct statements about $\mathrm{H}_{2} \mathrm{O}_{2}$ are: (A) used in the treatment of effluents. (B) used as both oxidising and reducing agents. (C) the two hydroxyl groups lie in the same plane. (D) miscible with water. Choose the correct answer from the options given below:$(\mathrm{A}),(\mathrm{B}),(\mathrm{C})$ and $(\mathrm{D})$(A), (B) and (D) only$(\mathrm{B}),(\mathrm{C})$ and $(\mathrm{D})$ only$(\mathrm{A}),(\mathrm{C})$ and $(\mathrm{D})$ onlyCorrect Option: , 2 Sol...
Read More →Prove the following
Question: Let $A=\{n \in N: n$ is a 3 -digit number $\}$ $B=\{9 k+2: k \in N\}$ and $\mathrm{C}:\{9 \mathrm{k}+\ell: \mathrm{k} \in \mathrm{N}\}$ for some $\ell(0\ell9)$ If the sum of all the elements of the set $A \cap(B \cup C)$ is $274 \times 400$, then $\ell$ is equal to Solution: 3 digit number of the form $9 \mathrm{~K}+2$ are $\{101,109, \ldots \ldots \ldots . .992\}$ $\Rightarrow$ Sum equal to $\frac{100}{2}(1093)=\mathrm{s}_{1}=54650$ $274 \times 400=s_{1}+s_{2}$ $274 \times 400=\frac{1...
Read More →Solve this
Question: Let $\left|\overrightarrow{\mathrm{A}_{1}}\right|=3,\left|\overrightarrow{\mathrm{A}_{2}}\right|=5$ and $\left|\overrightarrow{\mathrm{A}_{1}}+\overrightarrow{\mathrm{A}_{2}}\right|=5$. The value of $\left(2 \overrightarrow{\mathrm{A}_{1}}+3 \overrightarrow{\mathrm{A}_{2}}\right) \cdot\left(3 \overrightarrow{\mathrm{A}_{1}}-2 \overrightarrow{\mathrm{A}_{2}}\right)$ is :(1) $-106.5$(2) $-99.5$(3) $-112.5$(4) $-118.5$Correct Option: , 4 Solution: (4) Using, $R^{2}=A_{1}^{2}+A_{2}^{2}+2 A...
Read More →Prove the following
Question: If $\sum_{r=1}^{10} r !\left(r^{3}+6 r^{2}+2 r+5\right)=\alpha(11 !)$, then the value of $\alpha$ is equal to____________. Solution: $\sum_{r=1}^{10} \mathrm{r} !\{(\mathrm{r}+1)(\mathrm{r}+2)(\mathrm{r}+3)-9(\mathrm{r}+1)+8\}$ $=\sum_{\mathrm{r}=1}^{10}[\{(\mathrm{r}+3) !-(\mathrm{r}+1) !\}-8\{(\mathrm{r}+1) !-\mathrm{r} !\}]$ $=(13 !+12 !-2 !-3 !)-8(11 !-1)$ $=(12.13+12-8) \cdot 11 !-8+8$ $=(160)(11) !$ Hence $\alpha=160$...
Read More →Given below are two statements:
Question: Given below are two statements: Statement $\mathrm{I}: \mathrm{H}_{2} \mathrm{O}_{2}$ can act as both oxidising and reducing agent in basic medium. Statement II : In the hydrogen economy, the energy is transmitted in the form of dihydrogen. In the light of the above statements, choose the correct answer from the options given below:Both statement I and statement II are falseBoth statement I and statement II are trueStatement I is true but statement II is falseStatement I is false but s...
Read More →For the four sets of three measured physical quantities as given below. Which of the following options is correct?
Question: For the four sets of three measured physical quantities as given below. Which of the following options is correct? (A) $\mathrm{A}_{1}=24.36, \mathrm{~B}_{1}=0.0724, \mathrm{C}_{1}=256.2$ (B) $A_{2}=24.44, B_{2}=16.082, C_{2}=240.2$ (C) $\mathrm{A}_{3}=25.2, \mathrm{~B}_{3}=19.2812, \mathrm{C}_{3}=236.183$ (D) $\mathrm{A}_{4}=25, \mathrm{~B}_{4}=236.191, \mathrm{C}_{4}=19.5$(1) $\begin{aligned} A_{4}+B_{4}+C_{4}A_{1}+B_{1}+C_{1}A_{3}+B_{3}+C_{3}A_{2} +B_{2} \\ +C_{2} \end{aligned}$(2) ...
Read More →Let S1 be the sum of first 2 n terms
Question: Let $S_{1}$ be the sum of first $2 \mathrm{n}$ terms of an arithmetic progression. Let $S_{2}$ be the sum of first $4 \mathrm{n}$ terms of the same arithmetic progression. If $\left(S_{2}-S_{1}\right)$ is 1000 , then the sum of the first 6 n terms of the arithmetic progression is equal to:(1) 1000(2) 7000(3) 5000(4) 3000Correct Option: , 4 Solution: $S_{2 n}=\frac{2 n}{2}[2 a+(2 n-1) d], S_{4 n}=\frac{4 n}{2}[2 a+(4 n-$1)d] $\Rightarrow \mathrm{S}_{2}-\mathrm{S}_{1}=\frac{4 \mathrm{n}}...
Read More →Find the value of x for which the numbers (5x + 2),
Question: Find the value ofxfor which the numbers (5x+ 2), (4x 1) and (x+ 2) are in AP. Solution: It is given that (5x+ 2), (4x 1) and (x+ 2) are in AP. $\therefore(4 x-1)-(5 x+2)=(x+2)-(4 x-1)$ $\Rightarrow 4 x-1-5 x-2=x+2-4 x+1$ $\Rightarrow-x-3=-3 x+3$ $\Rightarrow 3 x-x=3+3$ $\Rightarrow 2 x=6$ $\Rightarrow x=3$ Hence, the value ofxis 3....
Read More →Prove the following
Question: $\frac{1}{3^{2}-1}+\frac{1}{5^{2}-1}+\frac{1}{7^{2}-1}+\ldots+\frac{1}{(201)^{2}-1}$ is equal to(1) $\frac{101}{404}$(2) $\frac{25}{101}$(3) $\frac{101}{408}$(4) $\frac{99}{400}$Correct Option: , 2 Solution: $\mathrm{T}_{\mathrm{n}}=\frac{1}{(2 \mathrm{n}+1)^{2}-1} \frac{1}{(2 \mathrm{n}+2) 2 \mathrm{n}}=\frac{1}{4(\mathrm{n})(\mathrm{n}+1)}$ $=\frac{(\mathrm{n}+1)-\mathrm{n}}{4 \mathrm{n}(\mathrm{n}+1)}=\frac{1}{4}\left(\frac{1}{\mathrm{n}}-\frac{1}{\mathrm{n}+1}\right)$ $\mathrm{S}=\...
Read More →Determine k so that (3k − 2), (4k − 6) and (k + 2) are three consecutive terms of an AP.
Question: Determinekso that (3k 2),(4k 6) and(k+ 2) are three consecutive terms of an AP. Solution: It is given that (3k 2),(4k 6) and(k+ 2) are three consecutive terms of an AP.(4k 6)(3k 2) =(k+ 2)(4k 6)⇒ 4k 6 3k+ 2 =k+ 2 4k+ 6⇒k 4 =3k+ 8⇒k+ 3k=8 + 4⇒ 4k=12⇒k=3Hence, the value ofkis 3....
Read More →A particle moves such that its position vector
Question: A particle moves such that its position vector $\vec{r}(t)$ $=\cos \omega \mathrm{t} \hat{i}+\sin \omega \mathrm{t} \hat{j}$ where $\omega$ is a constant and $t$ is time. Then which of the following statements is true for the velocity $\vec{v}(t)$ and acceleration $\vec{a}(t)$ of the particle:(1) $\vec{v}$ is perpendicular to $\vec{r}$ and $\vec{a}$ is directed away from the origin(2) $\vec{v}$ and $\vec{a}$ both are perpendicular to $\vec{r}$(3) $\vec{v}$ and $\vec{a}$ both are parall...
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