The length of the hypotenuse of a right-angled triangle exceeds the length of the base by 2 cm and exceeds twice the length of the altitude by 1 cm.
Question: The length of the hypotenuse of a right-angled triangle exceeds the length of the base by 2 cm and exceeds twice the length of the altitude by 1 cm. Find the length of each side of the triangle. Solution: Let the base and altitude of the right-angled triangle be $x$ and $y \mathrm{~cm}$, respectively. Therefore, the hypotenuse will be $(x+2) \mathrm{cm}$. $\therefore(x+2)^{2}=y^{2}+x^{2} \quad \ldots$ (i) Again, the hypotenuse exceeds twice the length of the altitude by $1 \mathrm{~cm}...
Read More →The missing value in the following figure is
Question: The missing value in the following figure is Use the logic which gives answer in single digit Solution: $x=(2-1)^{1 !}=1$ $w=(12-8)^{4 !}=4^{24}$ $z=(7-4)^{3 !}=3^{6}$ hence $y=(5-3)^{2 !}=2^{2}$...
Read More →Solve this
Question: A $25 \times 10^{-3} \mathrm{~m}^{3}$ volume cylinder is filled with $1 \mathrm{~mol}$ of $\mathrm{O}_{2}$ gas at room temperature $(300 \mathrm{~K})$. The molecular diameter of $\mathrm{O}_{2}$, and its root mean square speed, are found to be $0.3 \mathrm{~nm}$ and $200 \mathrm{~m} / \mathrm{s}$, respectively. What is the average(1) $\sim 10^{12}$(2) $\sim 10^{11}$(3) $\sim 10^{10}$(4) $\sim 10^{13}$Correct Option: , 3 Solution: (3) $\mathrm{V}=25 \times 10^{-3} \mathrm{~m}^{3}, \math...
Read More →Which amongst the following is the strongest acid?
Question: Which amongst the following is the strongest acid?$\mathrm{CHBr}_{3}$$\mathrm{CH}(\mathrm{CN})_{3}$$\mathrm{CHI}_{3}$$\mathrm{CHCl}_{3}$Correct Option: , 3 Solution: Due to the resonance stabilisation of the conjugate base, $\mathrm{CH}(\mathrm{CN})_{3}$ is the strongest acid amongst the given compounds. The conjugate bases of $\mathrm{CHBr}_{3}$ and $\mathrm{CHl}_{3}$ are stabilised by inductive effect of halogens. This is why, they are less stable. Also, the conjugate base of $\mathr...
Read More →The hypotenuse of a right-angled triangle is 20 metres.
Question: The hypotenuse of a right-angled triangle is 20 metres. If the difference between the length of the other sides be 4 metres, find the other sides. Solution: Let one side of the right-angled triangle be $x \mathrm{~m}$ and the other side be $(x+4) \mathrm{m}$. On applying Pythagoras theorem, we have: $20^{2}=(x+4)^{2}+x^{2}$ $\Rightarrow 400=x^{2}+8 x+16+x^{2}$ $\Rightarrow 2 x^{2}+8 x-384=0$ $\Rightarrow x^{2}+4 x-192=0$ $\Rightarrow x^{2}+(16-12) x-192=0$ $\Rightarrow x^{2}+16 x-12 x-...
Read More →The area of a right-angled triangle is 165 sq metres.
Question: The area of a right-angled triangle is 165 sq metres. Determine its base and altitude if the latter exceeds the former by 7 metres. Solution: Let the base be $x \mathrm{~m}$. Therefore, the altitude will be $(x+7) \mathrm{m}$. Area of a triangle $=\frac{1}{2} \times$ Base $\times$ Altitude $\therefore \frac{1}{2} \times x \times(x+7)=165$ $\Rightarrow x^{2}+7 x=330$ $\Rightarrow x^{2}+7 x-330=0$ $\Rightarrow x^{2}+(22-15) x-330=0$ $\Rightarrow x^{2}+22 x-15 x-330=0$ $\Rightarrow x(x+22...
Read More →n moles of an ideal gas with constant volume heat capacity
Question: $\mathrm{n}$ moles of an ideal gas with constant volume heat capacity $\mathrm{C}_{\mathrm{V}}$ undergo an isobaric expansion by certain volume. The ratio of the work done in the process, to the heat supplied is:(1) $\frac{\mathrm{nR}}{\mathrm{C}_{\mathrm{V}}+\mathrm{nR}}$(2) $\frac{\mathrm{nR}}{\mathrm{C}_{\mathrm{V}}-\mathrm{n} \mathrm{R}}$(3) $\frac{4 n R}{C_{V}-n R}$(4) $\frac{4 n R}{C_{\mathrm{V}}+\mathrm{n} \mathrm{R}}$Correct Option: 1 Solution: (1) At constant volume Work done ...
Read More →The number of times the digit 3
Question: The number of times the digit 3 will be written when listing the integers from 1 to 1000 is Solution: $3_{-}-=10 \times 10=100$ $-^{3}-=10 \times 10=100$ $--3=10 \times 10=\frac{100}{300}$...
Read More →In the following skew conformation of ethane,
Question: In the following skew conformation of ethane, $\mathrm{H}^{\prime}-\mathrm{C}-\mathrm{C}-\mathrm{H}^{\prime \prime}$ dihedral angle is : $58^{\circ}$$149^{\circ}$$151^{\circ}$$120^{\circ}$Correct Option: , 2 Solution: $\therefore$ Angle between $\mathrm{H}^{\prime}$ and $\mathrm{H}^{\prime \prime}=120^{\circ}+29^{\circ}=149^{\circ}$...
Read More →The sum of all the 4 -digit distinct numbers
Question: The sum of all the 4 -digit distinct numbers that can be formed with the digits $1,2,2$ and 3 is:(1) 26664(2) 122664(3) 122234(4) 22264Correct Option: 1 Solution: Digits are $1,2,2,3$ total distinct numbers $\frac{4 !}{2 !}=12$. total numbers when 1 at unit place is 3 . 2 at unit place is 63 at unit place is 3 . So, sum $=(3+12+9)\left(10^{3}+10^{2}+10+1\right)$ $=(1111) \times 24$ $=26664$...
Read More →The area of a right-angled triangle is 96 sq metres
Question: The area of a right-angled triangle is 96 sq metres. If the base is three times the altitude, find the base. Solution: Let the altitude of the triangle bexm.Therefore, the base will be3xm. Area of a triangle $=\frac{1}{2} \times$ Base $\times$ Altitude $\therefore \frac{1}{2} \times 3 x \times x=96(\because$ Area $=96$ sq $\mathrm{m})$ $\Rightarrow \frac{x^{2}}{2}=32$ $\Rightarrow x^{2}=64$ $\Rightarrow x=\pm 8$ The value of $x$ cannot be negative. Therefore, the altitude and base of t...
Read More →A cylinder with fixed capacity of 67.2 lit contains helium gas at STP.
Question: A cylinder with fixed capacity of $67.2$ lit contains helium gas at STP. The amount of heat needed to raise the temperature of the gas by $20^{\circ} \mathrm{C}$ is : [Given that $\mathrm{R}=8.31 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$ ](1) $350 \mathrm{~J}$(2) $374 \mathrm{~J}$(3) $748 \mathrm{~J}$(4) $700 \mathrm{~J}$Correct Option: , 3 Solution: (3) As the process is isochoric so, $\mathrm{Q}=\mathrm{nc}_{\mathrm{v}} \Delta \mathrm{T}=\frac{67.2}{22.4} \times \frac{3 \mathr...
Read More →If the sides AB, BC and
Question: If the sides $\mathrm{AB}, \mathrm{BC}$ and $\mathrm{CA}$ of a triangle $\mathrm{ABC}$ have 3,5 and 6 interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to :(1) 364(2) 240(3) 333(4) 360Correct Option: , 3 Solution: Total Number of triangles formed $={ }^{14} \mathrm{C}_{3}-{ }^{3} \mathrm{C}_{3}-{ }^{5} \mathrm{C}_{3}-{ }^{6} \mathrm{C}_{3}$ $=333$...
Read More →The IUPAC name for the following compound is :
Question: The IUPAC name for the following compound is : 3-methyl-4-(3-methylprop-l-enyl)-l-heptyne3,5-dimethyl-4-propylhept-6-en-l-yne3-methyl-4-(1-methylprop-2-ynyl)-l-heptene3,5-dimethyl-4-propylhept-1-en-6-yneCorrect Option: , 4 Solution:...
Read More →The specific heats,
Question: The specific heats, $\mathrm{C}_{p}$ and $\mathrm{C}_{v}$ of a gas of diatomic molecules, $\mathrm{A}$, are given (in units of $\mathrm{J} \mathrm{mol}^{-1} \mathrm{k}^{-1}$ ) by 29 and 22, respectively. Another gas of diatomic molecules, B, has the corresponding values 30 and 21 . If they are treated as ideal gases, then:(1) A is rigid but B has a vibrational mode.(2) A has a vibrational mode but B has none.(3) A has one vibrational mode and B has two.(4) Both A and B have a vibration...
Read More →Team 'A' consists of 7 boys and n girls and
Question: Team 'A' consists of 7 boys and $\mathrm{n}$ girls and Team 'B' has 4 boys and 6 girls. If a total of 52 single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl, then $\mathrm{n}$ is equal to :(1) 5(2) 2(3) 4(4) 6Correct Option: , 3 Solution: Total matches between boys of both team $={ }^{7} \mathrm{C}_{1} \times{ }^{4} \mathrm{C}_{1}=28$ Total matches between girls of both team $={ }^{n} C_{1}{ }^{6} C_{1}=6 n$ Now, $28+6 n...
Read More →Which of these factors does not govern the stability of a conformation in acyclic compounds ?
Question: Which of these factors does not govern the stability of a conformation in acyclic compounds ?Steric interactionsTorsional strainElectrostatic forces of interactionAngle strainCorrect Option: , 4 Solution: Angle strain is present in cyclic compounds....
Read More →An HCl molecule has rotational,
Question: An $\mathrm{HCl}$ molecule has rotational, translational and vibrational motions. If the rms velocity of $\mathrm{HCl}$ molecules in its gaseous phase is $\bar{v}, \mathrm{~m}$ is its mass and $\mathrm{k}_{\mathrm{B}}$ is Boltzmann constant, then its temperature will be:(1) $\frac{m \bar{v}^{2}}{6 k_{B}}$(2) $\frac{m \bar{v}^{2}}{3 k_{B}}$(3) $\frac{m \bar{v}^{2}}{7 k_{B}}$(4) $\frac{m \bar{v}^{2}}{5 k_{B}}$Correct Option: 1 Solution: (1) In this case the total degree of freedom is 6 ....
Read More →Number of stereo centers present in linear and cyclic structures of glucose are respectively:
Question: Number of stereo centers present in linear and cyclic structures of glucose are respectively:$5 \ 4$$4 \ 4$$5 \ 5$$4 \ 5$Correct Option: , 4 Solution: Linear structure of glucose,...
Read More →For a given gas at 1 atm pressure,
Question: For a given gas at $1 \mathrm{~atm}$ pressure, rms speed of the molecules is $200 \mathrm{~m} / \mathrm{s}$ at $127^{\circ} \mathrm{C}$. At 2 atm pressure and at $227^{\circ} \mathrm{C}$, the rms speed of the molecules will be:(1) $100 \mathrm{~m} / \mathrm{s}$(2) $80 \sqrt{5} \mathrm{~m} / \mathrm{s}$(3) $100 \sqrt{5} \mathrm{~m} / \mathrm{s}$(4) $80 \mathrm{~m} / \mathrm{s}$Correct Option: , 3 Solution: (3) $V_{r m s}=\sqrt{\frac{3 R T}{M}}$ $\frac{v_{1}}{v_{2}}=\sqrt{\frac{T_{1}}{T_...
Read More →The temperature, at which the root mean square velocity of hydrogen molecules
Question: The temperature, at which the root mean square velocity of hydrogen molecules equals their escape velocity from the earth, is closest to : [Boltzmann Constant $k_{\mathrm{B}}=1.38 \times 10^{-23} \mathrm{~J} / \mathrm{K}$ Avogadro Number $\mathrm{N}_{\mathrm{A}}=6.02 \times 10^{26} / \mathrm{kg}$ Radius of Earth : $6.4 \times 10^{6} \mathrm{~m}$ Gravitational acceleration on Earth $=10 \mathrm{~ms}^{-2}$ ](1) $800 \mathrm{~K}$(2) $3 \times 10^{5} \mathrm{~K}$(3) $10^{4} \mathrm{~K}$(4)...
Read More →The correct IUPAC name of the following compound is:
Question: The correct IUPAC name of the following compound is: 5-chloro-4-methyl-1-nitrobenzene2 -chloro-1-methyl-4-nitrobenzene3 -chloro-4-methyl-1-nitrobenzene2 -methyl-5-nitro-1-chlorobenzeneCorrect Option: Solution: 2-Chloro-1-methyl-4-nitrobenzene...
Read More →if
Question: If $10^{22}$ gas molecules each of mass $10^{-26}$ $\mathrm{kg}$ collide with a surface (perpendicular to it) elastically per second over an area $1 \mathrm{~m}^{2}$ with a speed $10^{4}$ $\mathrm{m} / \mathrm{s}$, the pressure exerted by the gas molecules will be of the order of :(1) $2 \mathrm{~N} / \mathrm{m}^{2}$(2) $4 \mathrm{~N} / \mathrm{m}^{2}$(3) $8 \mathrm{~N} / \mathrm{m}^{2}$(4) $16 \mathrm{~N} / \mathrm{m}^{2}$Correct Option: 1 Solution: (1) Rate of change of momentum duri...
Read More →The area of a right-triangle is 600 cm2. If the base of the triangle exceeds the altitude by 10 cm,
Question: The area of a right-triangle is 600 cm2. If the base of the triangle exceeds the altitude by 10 cm, find the dimensions of the triangle. Solution: Let the altitude of the triangle bexcm.Therefore, the base of the triangle will be (x+ 10) cm. Area of triangle $=\frac{1}{2} x(x+10)=600$ $\Rightarrow x(x+10)=1200$ $\Rightarrow x^{2}+10 x-1200=0$ $\Rightarrow x^{2}+(40-30) x-1200=0$ $\Rightarrow x^{2}+40 x-30 x-1200=0$ $\Rightarrow x(x+40)-30(x+40)=0$ $\Rightarrow(x+40)(x-30)=0$ $\Rightarr...
Read More →Consider a rectangle ABCD having
Question: Consider a rectangle $\mathrm{ABCD}$ having $5,7,6,9$ points in the interior of the line segments $\mathrm{AB}, \mathrm{CD}, \mathrm{BC}, \mathrm{DA}$ respectively. Let $\alpha$ be the numberof triangles having these points from different sides as vertices and $\beta$ be the number of quadrilaterals having these points from different sides as vertices. Then $(\beta-\alpha)$ is equal to :(1) 795(2) 1173(3) 1890(4) 717Correct Option: , 4 Solution: $\alpha=$ Number of triangles $\alpha=5 ...
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