At a particular station, the TV transmission tower has a height of $100\,m$ . To triple its coverage range, height of
the tower should be increased to
a.
$200\,m$
b.
$300\,m$
c.
$600\,m$
d.
$900\,m$
Correct Answer
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Q2mcq
Two sources of equal emfs are connected in series. This combination is connected to an external resistance $R$ .
The internal resistances of the two sources are $r_{1}$ and $r_{2}(r_{1} >r_{2})$ . If the potential difference across the source
of internal resistance $r_{1}$ is zero then the value of $R$ will be
a.
$r_{1}- r_{2}$
Correct Answer
b.
$\frac{r_{1}r_{2}}{r_{1}+r_{2}}$
c.
$\frac{r_{1}+r_{2}}{2}$
d.
$r_{2}- r_{1}$
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Q3mcq
A beam of light travelling along $X$ -axis is described by the electric field $E_{y} = 900 sin\omega (t- \frac{x}{c})$ . The ratio of
electric force to magnetic force on a charge $q$ moving along $Y$ -axis with a speed of $3 \times 10^{7}\,m\,s^{-1}$ will be :
[Given speed of light $= 3 \times 10^{8}\,m\,s^{-1}$ ]
a.
$1 : 1$
b.
$1 : 10$
c.
$10 : 1$
Correct Answer
d.
$1 : 2$
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Q4nat
A ball of mass $m$ is thrown vertically upward. Another ball of mass $2\,m$ is thrown an angle $\theta$ with the vertical.
Both the balls stay in air for the same period of time. The ratio of the heights attained by the two balls
respectively is $\frac{1}{x}$ . The value of $x$ is _____ .
✓ Accepted Answer:1
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Q5nat
To light, a $50\,W, 100\,V$ lamp is connected, in series with a capacitor of capacitance $\frac{50}{\pi \sqrt{x}} \mu F$ , with
$200\,V, \,50\, Hz \, AC$ source. The value of $x$ will be _____ .
✓ Accepted Answer:3
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Q6mcq
A bag is gently dropped on a conveyor belt moving at a speed of $2\,m\,s^{-1}$ . The coefficient of friction between
the conveyor belt and bag is $0. 4$ Initially, the bag slips on the belt before it stops due to friction. The distance
travelled by the bag on the belt during slipping motion is : [Take $g= 10\,m\,s^{-2}$ ]
a.
$2\,m$
b.
$0. 5\,m$
Correct Answer
c.
$3. 2\,m$
d.
$0. 8\, ms$
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Q7mcq
What is the half-life period of a radioactive material if its activity drops to $\frac{1}{16^{th \,}}$ of its initial value of $30$ years ?
a.
$9. 5$ years
b.
$8. 5$ years
c.
$7. 5$ years
Correct Answer
d.
$10. 5$ years
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Q8mcq
Sand is being dropped from a stationary dropper at a rate of $0. 5\, kg \,s^{-1}$ on a conveyor belt moving with a
velocity of $5\,m\,s^{-1}$ . The power needed to keep belt moving with the same velocity will be
a.
$1. 25\,W$
b.
$2. 5\,W$
c.
$6. 25\,W$
d.
$12. 5\,W$
Correct Answer
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Q9mcq
An electron (mass $m$ ) with an initial velocity $\vec{v}=v_{0}ˆi(v_{0} > 0)$ is moving in an electric field
$\vec{E} = -E_{0}ˆi(E_{0} > 0)$ where $E_{0}$ is constant. If at $t= 0$ , de-Broglie wavelength is $\lambda _{0} = \frac{h}{mv_{0}}$ , then its de-Broglie
wavelength after time $t$ is given by
A bullet is shot vertically downwards with an initial velocity of $100\,m\,s^{-1}$ from a certain height. Within $10\,s$ ,
the bullet reaches the ground and instantaneously comes to rest due to the perfectly inelastic collision. The
velocity-time curve for total time $t = 20\,s$ will be : (Take $g= 10\,m\,s^{-2}$ )
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The one division of main scale of vernier callipers reads $1\, mm$ and $10$ divisions of Vernier scale is equal to the
$9$ divisions on main scale. When the two jaws of the instrument touch each other the zero of the Vernier lies to
the right of zero of the main scale and its fourth division coincides with a main scale division. When a
spherical bob is tightly placed between the two jaws, the zero of the Vernier scale lies in between $4. 1\, cm$ and
$4. 2\, cm$ and $6^{th \,}$ Vernier division coincides with a main scale division. The diameter of the bob will be _____
$10^{-2}\, cm$ .
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A $1\,m$ long copper wire carries a current of $1\,A$ . If the cross section of the wire is $2. 0\, mm^{2}$ and the resistivity
of copper is $1. 7 \times 10^{-8}\,\Omega \,m$ . The force experienced by moving electron in the wire is _____ $\times 10^{-23}\,N$
(Charge of electron $= 1. 6 \times 10^{-19}C$ )
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A direct current of $4\,A$ and an alternating current of peak value $4\,A$ flow through resistance of $3\,\Omega$ and $2\,\Omega$
respectively. The ratio of heat produced in the two resistances in same interval of time will be :
a.
$3 : 2$
b.
$3 : 1$
c.
$3 : 4$
d.
$4 : 3$
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Two cylindrical vessels of equal cross-sectional area $16\, cm^{2}$ contain water upto heights $100\, cm$ and $150\, cm$
respectively. The vessels are interconnected so that the water levels in them become equal. The work done by
the force of gravity during the process, is [Take density of water $= 10^{3}\, kg \,m^{-3}$ and $g= 10\, ms^{-2}$ ]
a.
$0. 25\,J$
b.
$1\,J$
c.
$8\,J$
d.
$12\,J$
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Two beams of light having intensities $I$ and $4I$ interfere to produce a fringe pattern on a screen. The phase
difference between the two beams are $\frac{\pi}{2}$ and $\frac{\pi}{3}$ at points $A$ and $B$ respectively. The difference between the
resultant intensities at the two points is $xI$ . The value of $x$ will be _____ .
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A mass $0. 9\, kg$ , attached to a horizontal spring, executes SHM with an amplitude $A_{1}$ . When this mass passes
through its mean position, then a smaller mass of $124\,g$ is placed over it and both masses move together with
amplitude $A_{2}$ . If the ratio $\frac{A_{1}}{A_{2}}$ is $\frac{\alpha}{\alpha -1}$ , then the value of $\alpha$ will be _____ .
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A torque meter is calibrated to reference standards of mass, length and time each with $5%$ accuracy. After
calibration, the measured torque with this torque meter will have net accuracy of
a.
$15%$
b.
$25%$
c.
$75%$
d.
$5%$
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A long cylindrical volume contains a uniformly distributed charge of density $\rho \,C\,m^{-3}$ . The electric field inside
the cylindrical volume at a distance $x= \frac{2\epsilon _{0}}{\rho} \,m$ from its axis is _____ $V\,m^{-1}$ .
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A microscope was initially placed in air (refractive index $1$ ). It is then immersed in oil (refractive index $2$ ). For
a light whose wavelength in air is $\lambda$ , calculate the change of microscope's resolving power due to oil and
choose the correct option
a.
Resolving power will be $\frac{1}{4}$ in the oil than it was
in the air
b.
Resolving power will be twice in the oil than it
was in the air.
c.
Resolving power will be four times in the oil than
it was in the air.
d.
Resolving power will be $\frac{1}{2}$ in the oil than it was
in the air.
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In meter bridge experiment for measuring unknown resistance ' $S$ ', the null point is obtained at a distance
$30\, cm$ from the left side as shown at point $D$ . If $R$ is $5. 6\,k\Omega$ , then the value of unknown resistance ' $S$ ' will be
_____ $\Omega$ .
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Read the following statements :
$A$ . When small temperature difference between a liquid and its surrounding is doubled the rate of loss of heat of
the liquid becomes twice.
$B$ . Two bodies $P$ and $Q$ having equal surface areas are maintained at temperature $10^{\circ}C$ and $20^{\circ}C$ . The thermal
radiation emitted in a given time by $P$ and $Q$ are in the ratio $1 : 1. 15$
$C$ . A carnot Engine working between $100\,K$ and $400\,K$ has an efficiency of $75%$
$D$ . When small temperature difference between a liquid and its surrounding is quadrupled, the rate of loss of
heat of the liquid becomes twice.
Choose the correct answer from the options given below :
a.
$A, B, C$ only
b.
$A, B$ only
c.
$A, C$ only
d.
$B, C, D$ only
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Same gas is filled in two vessels of the same volume at the same temperature. If the ratio of the number of
molecules is $1 : 4$ , then
$A$ . The r.m.s. velocity of gas molecules in two vessels will be the same.
$B$ . The ratio of pressure in these vessels will be $1 : 4$ .
$C$ . The ratio of pressure will be $1 : 1$ .
$D$ . The r.m.s. velocity of gas molecules in two vessels will be in the ratio of $1 : 4$ .
a.
$A$ and $C$ only
b.
$B$ and $D$ only
c.
$A$ and $B$ only
d.
$C$ and $D$ only
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Two bar magnets oscillate in a horizontal plane in earth's magnetic field with time periods of $3\,s$ and $4\,s\,$
respectively. If their moments of inertia are in the ratio of $3 : 2$ then the ratio of their magnetic moments will
be
a.
$2 : 1$
b.
$8 : 3$
c.
$1 : 3$
d.
$27 : 16$
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Two identical positive charges $Q$ each are fixed at a distance of $2a$ apart from each other. Another point charge
$q_{0}$ with mass $m$ is placed at midpoint between two fixed charges. For a small displacement along the line
joining the fixed charges, the charge $q_{0}$ executes SHM. The time period of oscillation of charge $q_{0}$ will be
A square aluminium (shear modulus is $25 \times 10^{9} Nm^{-2}$ ) slab of side $60\, cm$ and thickness $15\, cm$ is subjected
to a shearing force (on its narrow face) of $18. 0 \times 10^{4}\,N$ . The lower edge is riveted to the floor. The
displacement of the upper edge is _____ $\mu m$ .
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If $K_{1}$ and $K_{2}$ are the thermal conductivities $L_{1}$ and $L_{2}$ are the lengths and $A_{1}$ and $A_{2}$ are the cross sectional
areas of steel and copper rods respectively such that $\frac{K_{2}}{K_{1}} = 9, \frac{A_{1}}{A_{2}} = 2, \frac{L_{1}}{L_{2}} = 2$ . Then, for the arrangement as
shown in the figure. The value of temperature $T$ of the steel - copper junction in the steady state will be
a.
$18^{\circ}C$
b.
$14^{\circ}C$
c.
$45^{\circ}C$
d.
$150^{\circ}C$
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Two satellites $A$ and $B$ having masses in the ratio $4 : 3$ are revolving in circular orbits of radii $3r$ and $4r$
respectively around the earth. The ratio of total mechanical energy of $A$ to $B$ is
a.
$9 : 16$
b.
$16 : 9$
c.
$1 : 1$
d.
$4 : 3$
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A pulley of radius $1. 5\,m$ is rotated about its axis by a force $F =(12t-3t^{2})\,N$ applied tangentially (while $t$ is
measured in seconds). If moment of inertia of the pulley about its axis of rotation is $4. 5\, kg \,m^{2}$ , the number of
rotations made by the pulley before its direction of motion is reversed, will be $\frac{K}{\pi}$ . The value of $K$ is _____ .
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At $310\,K$ , the solubility of $CaF_{2}$ in water is $2. 34 \times 10^{-3}\,g/100\, mL$ . The solubility product of $CaF_{2}$ is ----
$\times 10^{-8}(mol /L)^{3}$ (nearest integer). (Given molar mass : $CaF_{2} = 78\,g\, mol^{-1}$ )
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$20\, mL$ of $0. 02 MK_{2}Cr_{2}O_{7}$ solution is used for the titration of $10\, mL$ of $Fe^{2+}$ solution in the acidic medium.
The molarity of $Fe^{2+}$ solution is____ $\times 10^{-2}M$
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Match List I with List II
List-I List-II
Pollutant Disease /sickness
A Sulphate $(> 500 ppm)$ I Methemoglobinemia
B Nitrate $(> 50 ppm)$ II Brown mottling of teeth
C Lead $(> 50 ppb)$ III Laxative effect
D Fluoride $(> 2 ppm)$ IV Kidney damage
Choose the correct answer from the options given below
a.
$A -IV, B -I, C -II, D -III$
b.
$A -III, B -I, \,C -IV, D -II$
c.
$A -II, B -IV, C -I, D -III$
d.
$A -II, B -IV, C -III, D -I$
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Given below are two statements.
Statement I : $O_{2}, Cu^{2+}$ and $Fe^{3+}$ are weakly attracted by magnetic field and are magnetized in the same
direction as magnetic field.
Statement II : $NaCl$ and $H_{2}O$ are weakly magnetized in opposite direction to magnetic field.
In the light of the above statements, choose the most appropriate answer form the options given below
a.
Both Statement I and Statement II are correct.
b.
Both Statement I and Statement II are incorrect.
c.
Statement I is correct but Statement II is
incorrect.
d.
Statement I is incorrect but Statement II is
correct.
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Given below are two statements:
Statement I : The chlorides of $Be$ and $Al$ have Cl-bridged structure. Both are soluble in organic solvents and
act as Lewis bases.
Statement II: Hydroxides of $Be$ and $Al$ dissolve in excess alkali to give beryllate and aluminate ions. In the
light of the above statements. Choose the correct answer from the options given below.
a.
Both statement I and Statement II are true
b.
Both statement I and Statement II are false
c.
Statement I is true but Statement II is false
d.
Statement I is false but Statement II is true
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Given below are two statements. One is labelled as Assertion $A$ and the other is labelled as Reason $R$ .
Assertion $A$ : Activated charcoal adsorbs $SO_{2}$ more efficiently than $CH_{4}$ .
Reason $R$ : Gases with lower critical temperatures are readily adsorbed by activated charcoal. In the light of
the above statements, choose the correct answer from the options given below.
a.
Both $A$ and $R$ are correct and $R$ is the correct
explanation of $A$ .
b.
Both $A$ and $R$ are correct but $R$ is NOT the
correct explanation of $A$ .
c.
$A$ is correct but $R$ is not correct.
d.
$A$ is not correct but $R$ is correct.
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Which of the following methods are not used to refine any metal?
(A) Liquation
(B) Calcination
(C) Electrolysis
(D) Leaching
(E) Distillation
Choose the correct answer from the options given below
a.
$B$ and $D$ only
b.
$A, B, D$ and $E$ only
c.
$B, D$ and $E$ only
d.
$A, C$ and $E$ only
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Given below are two statements:
Statement I : Hydrogen peroxide can act as an oxidizing agent in both acidic and basic conditions.
Statement II: Density of hydrogen peroxide at $298\,K$ is lower than that of $D_{2}O$ .
In the light of the above statements. Choose the correct answer from the options.
a.
Both statement I and Statement II are ture
b.
Both statement I and Statement II are false
c.
Statement I is true but Statement II is false
d.
Statement I is false but Statement II is true
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In the following reaction
The $%$ yield for reaction $I$ is $60%$ and that of reaction II is $50%$ . The overall yield of the complete reaction is $%$
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$2NO + 2H_{2} \rightarrow N\rightarrow_{2}+ 2H_{2}O$
The above reaction has been studied at $800\,^{\circ}C$ . The related data are given in the table below
Initial
Reaction serial Initial pressure of Initial rate
Pressure
number $H_{2}/ kPa ( \frac{-dp}{dt})/(kPa /s)$
of $NO / kPa$
$1 65. 6 40. 0 0. 135$
$2 65. 6 20. 1 0. 033$
$3 38. 6 65. 6 0. 214$
$4 19. 2 65. 6 0. 106$
The order of the reaction with respect to NO is____
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In Carius method of estimation of halogen. $0. 45\,g$ of an organic compound gave $0. 36\,g$ of $AgBr$ . Find out the
percentage of bromine in the compound.
(Molar masses : $AgBr = 188\,g\, mol^{-1} : Br = 80\,g\, mol^{-1}$ )
a.
$34. 04%$
b.
$40. 04%$
c.
$36. 03%$
d.
$38. 04%$
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$250\,g$ solution of $D$ -glucose in water contains $10. 8%$ of carbon by weight. The molality of the solution is
nearest to (Given: Atomic Weights are $H = 1u; \,C = 12\,u; \,O = 16\,u$ )
a.
$1. 03$
b.
$2. 06$
c.
$3. 09$
d.
$5. 40$
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Given below are two statements. One is labelled as Assertion $A$ and the other is labelled as Reason $R$ .
Assertion $A : [6]$ Annulene. $[8]$ Annulene and cis $-[10]$ Annulene, are respectively aromatic, not-aromatic and
aromatic.
Reason $R$ : Planarity is one of the requirements of aromatic systems.
In the light of the above statements, choose the most appropriate answer from the options given below.
a.
Both $A$ and $R$ are correct and $R$ is the correct
explanation of $A$ .
b.
Both $A$ and $R$ are correct but $R$ is NOT the
correct explanation of $A$ .
c.
$A$ is correct but $R$ is not correct.
d.
$A$ is not correct but $R$ is correct.
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Boiling point of a $2%$ aqueous solution of a nonvolatile solute $A$ is equal to the boiling point of $8%$ aqueous
solution of a non-volatile solute $B$ . The relation between molecular weights of $A$ and $B$ is.
a.
$M_{A} = 4M_{B}$
b.
$M_{B} = 4M_{A}$
c.
$M_{A} = 8M_{B}$
d.
$M_{B} = 8M_{A}$
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The molar heat capacity for an ideal gas at constant pressure is $20. 785\,J\,K^{-1}\, mol^{-1}$ . The change in internal
energy is $5000\,J$ upon heating it from $300\,K$ to $500\,K$ . The number of moles of the gas at constant volume
is____
(Given: $R = 8. 314\,J\,K^{-1}\, mol^{-1}$ )
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Match List-I with List-II
List-I List-II
A Benzenesulphonyl chloride I Test for primary amines
B Hoffmann bromamide reaction II Anti Saytzeff
C Carbylamine reaction III Hinsberg reagent
D Hoffmann orientation IV Known reaction of Isocyanates.
Choose the correct answer from the options given below
a.
$A -IV, B -III, C -II, D -I$
b.
$A -IV, B -II, C -I, D -III$
c.
$A -III, B -IV, C -I, D -II$
d.
$A -IV, B -III, C -I, D -II$
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Given below are two statements. One is labelled as Assertion $A$ and the other is labelled as Reason $R$ .
Assertion $A$ : Energy of $2\,s$ orbital of hydrogen atom is greater than that of $2\,s$ orbital of lithium.
Reason $R$ : Energies of the orbitals in the same subshell decrease with increase in the atomic number.
In the light of the above statements, choose the correct answer from the options given below.
a.
Both $A$ and $R$ are true and $R$ is the correct
explanation of $A$ .
b.
Both $A$ and $R$ are true but $R$ is NOT the correct
explanation of $A$ .
c.
$A$ is true but $R$ is false.
d.
$A$ is false but $R$ is true.
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Given below are two statements:
Statement I : Iron (III) catalyst, acidified $K_{2}Cr_{2}O_{7}$ and neutral $KMnO_{4}$ have the ability to oxidise $I^{-}$ to $I_{2}$
independently.
Statement II: Manganate ion is paramagnetic in nature and involves $p$ $\pi$ $-p$ $\pi$bonding.
In the light of the above statements, choose the correct answer from the options.
a.
Both statement I and Statement II are ture
b.
Both statement I and Statement II are false
c.
Statement I is true but Statement II is false
d.
Statement I is false but Statement II is true
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Match List-I with List-II
List-I List-II
A I Anti-depressant
B II $550$ times sweeter than cane sugar
C III Narcotic analgesic
D IV Antiseptic
Choose the correct answer from the options given below
a.
$A -IV, B -III, C -II, D -I$
b.
$A -III, B -I, C -II, D -IV$
c.
$A -III, B -IV, C -I, D -II$
d.
$A -III, B -I, C -IV, D -II$
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Match List I with List II
List-I List-II
Polymers Commenrcial names
A Phenol-formaldehyde resin I Glyptal
B Copolymer of $1, 3$ -butadiene and styrene II Novolac
C Polyester of glycol and phthalic acid III Buna- $s$
D Polyester of glycol and terephthalic acid IV Dacron
Choose the correct answer from the options given below
a.
$A -II, B -III, C -IV, D -I$
b.
$A -II, B -III, C -I, D -IV$
c.
$A -II, B -I, C -III, D -IV$
d.
$A -III, B -II, C -IV, D -I$
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The conductivity of a solution of complex with formula $CoCl_{3}(NH_{3})_{4}$ corresponds to $1 : 1$ electrolyte, then
the primary valency of central metal ion is
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A sugar ' $X$ ' dehydrates very slowly under acidic condition to give furfural which on further reaction with
resorcinol gives the coloured product after sometime. Sugar ' $X$ ' is
a.
Aldopentose
b.
Aldotetrose
c.
Oxalic acid
d.
Ketotetrose
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Optical activity of an enantiomeric mixture is $+12. 6^{\circ}$ and the specific rotation of $(+)$ isomer is $+30^{\circ}$ . The
optical purity is____ $%$
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Let $\vec{a}=\alpha ˆi+ ˆj+\beta ˆk$ and $\vec{b}= 3ˆi-5ˆj+ 4ˆk$ be two vectors, such that $\vec{a}\times \vec{b}= -ˆi+ 9ˆi+ 12ˆk$ . Then the
projection of $\vec{b}-2\vec{a}$ on $\vec{b}+ \vec{a}$ is equal to
a.
$2$
b.
$\frac{39}{5}$
c.
$9$
d.
$\frac{46}{5}$
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Let $y=y(x)$ be the solution curve of the differential equation
$sin(2x^{2})log_{e}(tanx^{2})dy+(4xy-4\sqrt{2}xsin(x^{2} - \frac{\pi}{4}))dx= 0, \,0 <x<\sqrt{\frac{\pi}{2}}$
, which passes through the point $(\sqrt{\frac{\pi}{6}}, 1)$ . Then $y(\sqrt{\frac{\pi}{3}})$ is equal to _______.
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If the circle $x^{2} +y^{2} -2gx+ 6y-19c= 0, \,g,c\in R$ passes through the point $(6, 1)$ and its centre lies on the
line $x-2cy= 8$ , then the length of intercept made by the circle on $x$ -axis is
a.
$\sqrt{11}$
b.
$4$
c.
$3$
d.
$2\sqrt{23}$
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Let $M$ and $N$ be the number of points on the curve $y^{5} -9xy+ 2x= 0$ , where the tangents to the curve are
parallel to $x$ -axis and $y$ -axis, respectively. Then the value of $M +N$ equals _______.
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Let $A(1, 1), \,B(-4, 3),C(-2, -5)$ be vertices of a triangle $ABC, \,P$ be a point on side $BC$ , and $\Delta _{1}$ and $\Delta _{2}$ be
the areas of triangle $APB$ and $ABC$ .
Respectively.
If $\Delta _{1} :\Delta _{2} = 4 : 7$ , then the area enclosed by the lines $AP,AC$ and the $x$ -axis is
a.
$\frac{1}{4}$
b.
$\frac{3}{4}$
c.
$\frac{1}{2}$
d.
$1$
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Suppose $a_{1},a_{2}, … ,a_{n}, …$ be an arithmetic progression of natural numbers. If the ratio of the sum of the first
five terms to the sum of first nine terms of the progression is $5 : 17$ and $110 <a_{15} < 120$ , then the sum of the
first ten terms of the progression is equal to
a.
$290$
b.
$380$
c.
$460$
d.
$510$
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For $k\in R$ , let the solutions of the equation $cos(sin^{-1}(xcot(tan^{-1}(cos(sin^{-1} x)))))=k, 0 < |x| < \frac{1}{\sqrt{2}}$ be $\alpha$
and $\beta$ , where the inverse trigonometric functions take only principal values. If the solutions of the equation
$x^{2} - bx-5 = 0$ are $\frac{1}{\alpha ^{2}} + \frac{1}{\beta ^{2}}$ and $\frac{\alpha}{\beta}$ , then $\frac{b}{k^{2}}$ is equal to ______.
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Let $\vec{a}= 2ˆi-ˆj+ 5ˆk$ and $\vec{b}=\alpha ˆi+\beta ˆj+ 2ˆk$ . If $((\vec{a}\times \vec{b})\times ˆi)\cdot ˆk= \frac{23}{2}$ , then $\vec{b}\times 2ˆj$ is equal to
a.
$4$
b.
$5$
c.
$\sqrt{21}$
d.
$\sqrt{17}$
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If the length of the latus rectum of the ellipse $x^{2} + 4y^{2} + 2x+ 8y- \lambda = 0$ is $4$ , and $l$ is the length of its major
axis, then $\lambda +l$ is equal to _____.
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Let a vertical tower $AB$ of height $2h$ stands on a horizontal ground. Let from a point $P$ on the ground a man
can see upto height $h$ of the tower with an angle of elevation $2\alpha$ . When from $P$ , he moves a distance $d$ in the
$-$
direction of $A\vec{P}$ , he can see the top $B$ of the tower with an angle of elevation $\alpha$ . If $d= \sqrt{7}h$ , then $tan\alpha$ is
equal to
a.
$\sqrt{5}-2$
b.
$\sqrt{3}-1$
c.
$\sqrt{7}-2$
d.
$\sqrt{7}- \sqrt{3}$
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Let $y=y_{1}(x)$ and $y=y_{2}(x)$ be two distinct solutions of the differential equation $\frac{dy}{dx} =x+y$ , with $y_{1}(0) = 0$
and $y_{2}(0) = 1$ respectively. Then, the number of points of intersection of $y=y_{1}(x)$ and $y=y_{2}(x)$ is
a.
$0$
b.
$1$
c.
$2$
d.
$3$
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$||$
If the plane $P$ passes through the intersection of two mutually perpendicular planes $2x+ky-5z= 1$ and
$3kx- ky+z= 5,k< 3$ and intercepts a unit length on positive $x$ -axis, then the intercept made by the plane
$P$ on the $y$ -axis is
a.
$\frac{1}{11}$
b.
$\frac{5}{11}$
c.
$6$
d.
$7$
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$||$
Let the line $\frac{x-3}{7} = \frac{y-2}{-1} = \frac{z-3}{-4}$ intersect the plane containing the lines $\frac{x-4}{1} = \frac{y+1}{-2} = \frac{z}{1}$ and
$4ax- y+ 5z-7a= 0 = 2x -5y -z -3,a\in R$ at the point $P(\alpha ,\beta ,\gamma )$ . Then the value of $\alpha +\beta +\gamma$ equals
______.
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An ellipse $E : \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ passes through the vertices of the hyperbola $H : \frac{x^{2}}{49} - \frac{y^{2}}{64} = -1$ . Let the major
and minor axes of the ellipse $E$ coincide with the transverse and conjugate axes of the hyperbola $H$ . Let the
product of the eccentricities of $E$ and $H$ be $\frac{1}{2}$ . If $l$ is the length of the latus rectum of the ellipse $E$ , then the
value of $113l$ is equal to _______.
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Let $f,g:N-{1} N$ be functions defined by $f(a) =\alpha$ , where $\alpha$ is the maximum of the powers of those
primes $p$ such that $\vec{p}^{\alpha }$ divides $a$ , and $g(a) =a+ 1$ , for all $a\in N-{1}$ . Then, the function $f+g$ is
a.
one-one but not onto
b.
onto but not one-one
c.
both one-one and onto
d.
neither one-one nor onto
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Let $f :R\rightarrow \rightarrow R$ be a function defined as $f(x) =asin( \frac{\pi [x]}{2} )+ [2 - x],a\in R$ , where $[t]$ is the greatest integer
less than or equal to $t$ . If $_{x}lim_{\rightarrow -1\rightarrow } f(x)$ exists, then the value of $\int _{0}^{4} f(x)dx$ is equal to
a.
$-1$
b.
$-2$
c.
$1$
d.
$2$
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Let a function $f :R\rightarrow \rightarrow R$ be defined as:
$\int ^{x}(5 -|t-3|)dt, x> 4$
$f(x) ={^{0}$
$x^{2} +bx, x\leq 4$
where $b\in R$ . If $f$ is continuous at $x= 4$ , then which of the following statements is NOT true?
a.
$f$ is not differentiable at $x= 4$
b.
$f'(3) +f'(5) = \frac{35}{4}$
c.
$f$ is increasing in $(-\infty , \frac{1}{8} )\cup (8, \infty )$
d.
$f$ has a local minima at $x= \frac{1}{8}$
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Let $S$ be the sample space of all five digit numbers. If $p$ is the probability that a randomly selected number
from $S$ , is a multiple of $7$ but not divisible by $5$ , then $9p$ is equal to
a.
$1. 0146$
b.
$1. 2085$
c.
$1. 0285$
d.
$1. 1521$
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Let the minimum value $v_{0}$ of $v= |z|^{2} + |z-3|^{2} + |z-6i|^{2}$ , $z\in C$ is attained at $z=z_{0}$ . Then
$2z^{2} - z^{3} + 3^{2}+v^{2}$ is equal to
$-_{0}_{0}_{0}$
a.
$1000$
b.
$1024$
c.
$1105$
d.
$1196$
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Let $S$ be the set containing all $3 \times 3$ matrices with entries from ${-1, 0, 1}$ . The total number of matrices
$A\in S$ such that the sum of all the diagonal elements of $A^{T}A$ is $6$ is ______.
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Let $P(a,b)$ be a point on the parabola $y^{2} = 8x$ such that the tangent at $P$ passes through the centre of the circle
$x^{2} +y^{2} -10x-14y+ 65 = 0$ . Let $A$ be the product of all possible values of $a$ and $B$ be the product of all
possible values of $b$ . Then the value of $A+B$ is equal to
a.
$0$
b.
$25$
c.
$40$
d.
$65$
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The mean and variance of $10$ observations were calculated as $15$ and $15$ respectively by a student who took by
mistake $25$ instead of $15$ for one observation. Then, the correct standard deviation is _______.
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