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JEE Mains 2025 (23 Jan, Shift 2), Paper 1 (B.E./B.Tech) question paper

75 questions · 300 marks · 180 minutes · +4 per correct answer, -1 per wrong one

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Physics

25 questions

  1. 1
    A ball having kinetic energy KE , is projected at an angle of 60∘60^{\circ} from the horizontal. What will be the kinetic energy of ball at the highest point of its flight?

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  2. 2
    Two charges 7μc7\mu c and −4μc- 4\mu c are placed at (−7 cm( - 7\text{ }cm, 0,0 ) and ( 7 cm,0,07\text{ }cm,0,0 ) respectively. Given, ϵ0=8.85×10−12C2 N−1 m−2\epsilon_{0}= 8.85 \times10^{- 12}C^{2}{\text{ }N}^{- 1}{\text{ }m}^{- 2}, the electrostatic potential energy of the charge configuration is :

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  3. 3
    The refractive index of the material of a glass prism is 3\sqrt{3}. The angle of minimum deviation is equal to the angle of the prism. What is the angle of the prism?

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  4. 4
    The equation of a transverse wave travelling along a string is y(x,t)=4.0sin⁡[20×10−3x+600t]mmy(x,t) = 4.0\sin\left\lbrack 20 \times10^{- 3}x + 600t \right\rbrack mm, where x is in the mm and t is in second. The velocity of the wave is :

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  5. 5
    The energy of a system is given as E(t)=α3e−βtE(t) =\alpha^{3}e^{- \beta t}, where tt is the time and β=0.3 s−1\beta = 0.3{\text{ }s}^{- 1}. The errors in the measurement of α\alpha and tt are 1.2%1.2\% and 1.6%1.6\%, respectively. At t=5 st = 5\text{ }s, maximum percentage error in the energy is :

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  6. 6
    In photoelectric effect an em-wave is incident on a metal surface and electrons are ejected from the surface. If the work function of the metal is 2.14 eV and stopping potential is 2 V , what is the wavelength of the em-wave? (Given hc =1242eVnm= 1242eVnm where h is the Planck's constant and c is the speed of light in vaccum.)

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  7. 7
    A circular disk of radius RR meter and mass M kgM\text{ }kg is rotating around the axis perpendicular to the disk. An external torque is applied to the disk such that θ(t)=5t2−8t\theta(t) = 5t^{2}- 8t, where θ(t)\theta(t) is the angular position of the rotating disc as a function of time tt. How much power is delivered by the applied torque, when t=2 st = 2\text{ }s ?

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  8. 8
    Water flows in a horizontal pipe whose one end is closed with a valve. The reading of the pressure gauge attached to the pipe is P1P_{1}. The reading of the pressure gauge falls to P2P_{2} when the valve is opened. The speed of water flowing in the pipe is proportional to

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  9. 9
    Match List-I with List-II. List-I : (A) Permeability of free space; (B) Magnetic field; (C) Magnetic moment; (D) Torsional constant. List-II : (I) [ML2 T−2]\left\lbrack ML^{2}{\text{ }T}^{- 2} \right\rbrack; (II) [MT−2 A−1]\left\lbrack MT^{- 2}{\text{ }A}^{- 1} \right\rbrack; (III) [MLT−2 A−2]\left\lbrack MLT^{- 2}{\text{ }A}^{- 2} \right\rbrack; (IV) [L2 A]\left\lbrack L^{2}\text{ }A \right\rbrack. Choose the correct answer from the options given below :

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  10. 10
    If a satellite orbiting the Earth is 9 times closer to the Earth than the Moon, what is the time period of rotation of the satellite? Given rotational time period of Moon =27= 27 days and gravitational attraction between the satellite and the moon is neglected.

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  11. 11
    Two point charges −4μc- 4\mu c and 4μc4\mu c, constituting an electric dipole, are placed at (−9,0,0)cm( - 9,0,0)cm and (9,0,0)cm(9,0,0)cm in a uniform electric field of strength 104NC−110^{4}NC^{- 1}. The work done on the dipole in rotating it from the equilibrium through 180∘180^{\circ} is :

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  12. 12
    A galvanometer having a coil of resistance 30Ω30\Omega need 20 mA of current for full-scale deflection. If a maximum current of 3 A is to be measured using this galvanometer, the resistance of the shunt to be added to the galvanometer should be 30XΩ\frac{30}{X}\Omega, where X is

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  13. 13
    The width of one of the two slits in Young's double slit experiment is dd while that of the other slit is xd . If the ratio of the maximum to the minimum intensity in the interference pattern on the screen is 9:49:4 then what is the value of xx ? (Assume that the field strength varies according to the slit width.)

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  14. 14
    Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : The binding energy per nucleon is found to be practically independent of the atomic number A , for nuclei with mass numbers between 30 and 170. Reason (R): Nuclear force is long range. In the light of the above statements, choose the correct answer from the options given below :

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  15. 15
    Water of mass mm gram is slowly heated to increase the temperature from T1T_{1} to T2T_{2}. The change in entropy of the water, given specific heat of water is 1Jkg−1 K−11Jkg^{- 1}{\text{ }K}^{- 1}, is :

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  16. 16
    What is the current through the battery in the circuit shown below?

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  17. 17
    A plane electromagnetic wave of frequency 20 MHz travels in free space along the +x direction. At a particular point in space and time, the electric field vector of the wave is Ey=9.3 Vm−1E_{y}= 9.3\ Vm^{-1}. Then, the magnetic field vector of the wave at that point is-

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  18. 18
    Using the given P-V diagram, the work done by an ideal gas along the path ABCD is-

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  19. 19
    A concave mirror of focal length ff in air is dipped in a liquid of refractive index μ\mu. Its focal length in the liquid will be :

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  20. 20
    A massless spring gets elongated by amount X1X_{1} under a tension of 5 N . Its elongation is x2x_{2} under the tension of 7 N . For the elongation of (5x1−2x2)\left( 5x_{1}- 2x_{2} \right), the tension in the spring will be,

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  21. 21
    An air bubble of radius 1.0 mm is observed at a depth of 20 cm below the free surface of a liquid having surface tension 0.095 J/m20.095\text{ }J/m^{2} and density 103 kg/m310^{3}\text{ }kg/m^{3}. The difference between pressure inside the bubble and atmospheric pressure ____N/m2\_\_\_\_N/m^{2}. (Take g=10 m/s2g = 10\text{ }m/s^{2} )
  22. 22
    A satellite of mass M2\frac{M}{2} is revolving around earth in a circular orbit at a height of R3\frac{R}{3} from earth surface. The angular momentum of the satellite is MGMRxM\sqrt{\frac{GMR}{x}}. The value of xx is ____\_\_\_\_ , where M and RR are the mass and radius of earth, respectively. ( G is the gravitational constant)
  23. 23
    At steady state the charge on the capacitor, as shown in the circuit below, is ___\_\_\_ μC\mu C.
  24. 24
    A time varying potential difference is applied between the plates of a parallel plate capacitor of capacitance 2.5μ F2.5\mu\text{ }F. The dielectric constant of the medium between the capacitor plates is 1 . It produces an instantaneous displacement current of 0.25 mA in the intervening space between the capacitor plates, the magnitude of the rate of change of the potential difference will be ____\_\_\_\_ Vs−1Vs^{- 1}.
  25. 25
    In a series LCR circuit, a resistor of 300Ω300\Omega, a capacitor of 25 nF and an inductor of 100 mH are used. For maximum current in the circuit, the angular frequency of the ac source is _____\_\_\_\_\_ ×104\times10^{4} radians s  −1\ ^{- 1}.

Chemistry

25 questions

  1. 26
    The effect of temperature on spontaneity of reactions are represented as:
    ΔH\Delta H
    ΔS\Delta S
    		TemperatureSpontaneity
    (A)+-any TNon spontaneous
    (B)++low Tspontaneous
    (C)--low TNon spontaneous
    (D)-+any Tspontaneous

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  2. 27
    Standard electrode potentials for a few half cells are mentioned below:
    ECu2+/Cu∘=0.34 V,EZn2+/Zn∘=−0.76 VEAg+/Ag∘=0.80 V,EMg2+/Mg∘=−2.37 V\begin{matrix} & E_{Cu^{2 +}/Cu}^{\circ}= 0.34\text{ }V,E_{Zn^{2 +}/Zn}^{\circ}= - 0.76\text{ }V \\ & E_{Ag^{+}/Ag}^{\circ}= 0.80\text{ }V,E_{Mg^{2 +}/Mg}^{\circ}= - 2.37\text{ }V \end{matrix}
    Which one of the following cells gives the most negative value of ΔG∘\Delta G^{\circ} ?

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  3. 28
    The α\alpha - Helix and β\beta - Pleated sheet structures of protein are associated with its:

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  4. 29
    Given below are two statements: Consider the following reaction Statement (I): In the case of formaldehyde ,K,K is about 2280 , due to small substituents, hydration is faster. Statement (II) : In the case of trichloro acetaldehyde K is about 2000 due to -I effect of -Cl . In the light of the above statements, choose the correct answer form the options given below;

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  5. 30
    Consider the reaction
    X2Y( g)⇌X2( g)+12Y2( g)X_{2}Y(\text{ }g) \rightleftharpoons X_{2}(\text{ }g) +\frac{1}{2}Y_{2}(\text{ }g)
    The equation representing correct relationship between the degree of dissociation ( x ) of X2Y(g)X_{2}Y(g) with its equilibrium constant Kp is _____\_\_\_\_\_ . Assume x to be very very small.

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  6. 31
    Identify A,BA,B and C in the given below reaction sequence

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  7. 32
    Given below are two statements: Statement (I). The boiling points of alcohols and phenols increase with increase in the number of C -atoms. Statement (II): The boiling points of alcohols and phenols are higher in comparison to other class of compounds such as ethers, haloalkanes. In the light of the above statements, choose the correct answer from the options given below:

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  8. 33
    When a non-volatile solute is added to the solvent, the vapour pressure of the solvent decreases by 10 mm of Hg . The mole fraction of the solute in the solution is 0.2 . What would be the mole fraction of the solvent if decrease in vapour pressure is 20 mm of Hg ?

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  9. 34
    Given below are two statements: Statement (I) : For a given shell, the total number of allowed orbitals is given by n2n^{2}. Statement (II) : For any subshell, the spatial orientation of the orbitals is given by −l-l to +l+l values including zero. In the light of the above statements, choose the correct answer from the options given below:

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  10. 35
    The ascending order of relative rate of solvolysis of following compounds is (A) (B) (C) (D)

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  11. 36
    Match List - I (Isomers of C10H14C_{10}H_{14}) with List - II (Ozonolysis product). The figure shows the table row by row: isomer (A) then product (I), isomer (B) then product (II), isomer (C) then product (III), isomer (D) then product (IV). Choose the correct answer from the options given below:

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  12. 37
    Which of the following graphs most appropriately represents a zero- order reaction?

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  13. 38
    Match List - I with List - II.
    List - IList - II
    (A)Bronze(I)
    Cu,NiCu,Ni
    (B)Brass(II)
    Fe,Cr,Ni,CFe,Cr,Ni,C
    (C)UK silver coin(III)
    Cu,ZnCu,Zn
    (D)Stainless Steel(IV)
    Cu,SnCu,Sn
    Choose the correct answer from the options given below :

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  14. 39
    Identify the coordination complexes in which the central metal ion has d4d^{4} configuration. (A) [FeO4]2−\left\lbrack FeO_{4} \right\rbrack^{2 -} (B) [Mn(CN)6]3−\left\lbrack Mn(CN)_{6} \right\rbrack^{3 -} (C) [Fe(CN)6]3−\left\lbrack Fe(CN)_{6} \right\rbrack^{3 -} (D) (E) [NiF6]2−\left\lbrack NiF_{6} \right\rbrack^{2 -} Choose the correct answer from the options given below :

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  15. 40
    Given below are the atomic numbers of some group 14 elements. The atomic number of the element with lowest melting point is :

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  16. 41
    pH of water is 7 at 25∘C25^{\circ}C. If water is heated to 80∘C80^{\circ}C, it's pH will :

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  17. 42
    Identify the products [A]\lbrack A\rbrack and [B]\lbrack B\rbrack, respectively in the following reaction : chlorobenzene →(ii) H+(i) NaOH, 623 K, 300 atm[A]→Na2Cr2O7H2SO4[B]\xrightarrow[(ii)\ H^{+}]{(i)\ NaOH,\ 623\ K,\ 300\ atm}\left\lbrack A \right\rbrack\underset{H_{2}SO_{4}}{\overset{Na_{2}Cr_{2}O_{7}}{\rightarrow}}\lbrack B\rbrack

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  18. 43
    Consider a binary solution of two volatile liquid components 1 and 2. x1x_{1} and y1y_{1} are the mole fractions of component 1 in liquid and vapour phase, respectively. The slope and intercept of the linear plot of 1x1\frac{1}{x_{1}} vs 1y1\frac{1}{y_{1}} are given respectively as :

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  19. 44
    Given below are two statements about X-ray spectra of elements : Statement (I) : A plot of v(v=\sqrt{v}(v = frequency of X-rays emitted) vs atomic mass is a straight line. Statement (II) : A plot of v(v=v(v = frequency of X-rays emitted) vs atomic number is a straight line. In the light of the above statements choose the correct answer from the options given below :

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  20. 45
    Consider the following reactions
    K2Cr2O7→KOH−H2O[ A]→H2SO4−H2O[ B]+K2SO4K_{2}Cr_{2}O_{7}\underset{-H_{2}O}{\overset{KOH}{\rightarrow}}\lbrack\text{ }A\rbrack\underset{-H_{2}O}{\overset{H_{2}SO_{4}}{\rightarrow}}\lbrack\text{ }B\rbrack +K_{2}SO_{4}
    The products [A]\lbrack A\rbrack and [B]\lbrack B\rbrack, respectively are :

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  21. 46
    0.01 mole of an organic compound (X) containing 10%10\% hydrogen, on complete combustion produced 0.9 gH2O0.9\text{ }gH_{2}O. Molar mass of (X)(X) is ___\_\_\_ gmol−1gmol^{- 1}.
  22. 47
    Consider the following sequence of reactions. →(i)NaNO2,HCl0−5∘C(A)\overset{\begin{array}{r} \left( i \right)NaNO_{2},HCl \\ \mathbf{0 - 5{^\circ}C} \end{array}}{\rightarrow}(A) Total number of sp3sp^{3} hybridised carbon atoms in the major product C formed is ____\_\_\_\_
  23. 48
    When 81.0 g of aluminium is allowed to react with 128.0 g of oxygen gas, the mass of aluminium oxide produced in grams is ____\_\_\_\_ . (Nearest integer) Given : Molar mass of Al is 27.0 g mol−127.0\text{ }g{\text{ }mol}^{- 1} Molar mass of O is 16.0 g mol−116.0\text{ }g{\text{ }mol}^{- 1}
  24. 49
    The bond dissociation enthalpy of X2X_{2}, ΔHbond∘\Delta H_{\text{bond}}^{\circ}, calculated from the given data is _____\_\_\_\_\_ kJmol−1kJmol^{- 1}. (Nearest integer)
    M+X−(s)→M+(g)+X−(g)ΔHlattice ∘=800 kJ mol−1M( s)→M( g)ΔHsub ∘=100 kJ mol−1M( g)→M+(g)+e−(g)ΔH∘ i=500 kJ mol−1X( g)+e−(g)→X−(g)ΔH∘ eg=−300 kJ mol−1M( s)+12X2( g)→M+X−(s)ΔHf∘=−400 kJ mol−1\begin{matrix} & M^{+}X^{-}(s) \rightarrow M^{+}(g) +X^{-}(g)\Delta H_{\text{lattice~}}^{\circ}= 800\text{ }kJ{\text{ }mol}^{- 1} \\ & M(\text{ }s) \rightarrow M(\text{ }g)\Delta H_{\text{sub~}}^{\circ}= 100\text{ }kJ{\text{ }mol}^{- 1} \\ & M(\text{ }g) \rightarrow M^{+}(g)+e^{-}(g)\Delta H^{\circ}\ _{i}= 500\text{ }kJ{\text{ }mol}^{- 1} \\ & X(\text{ }g) +e^{-}(g) \rightarrow X^{-}(g)\Delta H^{\circ}\ _{eg}= - 300\text{ }kJ{\text{ }mol}^{- 1} \\ & M(\text{ }s) +\frac{1}{2}X_{2}(\text{ }g) \rightarrow M^{+}X^{-}(s)\Delta H_{f}^{\circ}= - 400\text{ }kJ{\text{ }mol}^{- 1} \end{matrix}
    [Given : M+X−M^{+}X^{-}is a pure ionic compound and X forms a diatomic molecule X2X_{2} in gaseous state]
  25. 50
    A compound ' X ' absorbs 2 moles of hydrogen and ' X ' upon oxidation with KMnO4/H+KMnO_{4}/H^{+} gives the three products shown (propanone, ethanoic acid and 4-oxopentanoic acid). The total number of σ\sigma bonds present in the compound ' X ' is ______\_\_\_\_\_\_ .

Mathematics

25 questions

  1. 51
    If in the expansion of (1+x)p(1−x)q(1 + x)^{p}(1 - x)^{q}, the coefficients of xx and x2x^{2} are 1 and -2 , respectively, then p2+q2p^{2}+q^{2} is equal to :

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  2. 52
    Let A={(x,y)∈R×R:∣x+y∣≥3}\mathbf{A = \{(x,y) \in R \times R:|x + y| \geq 3\}} and
    B={(x,y)∈R×R:∣x∣+∣y∣≤3}.\mathbf{B = \{(x,y) \in R \times R:|x| + |y| \leq 3\}.}
    If C={(x,y)∈A∩B:x=0\mathbf{C = \{(x,y) \in A \cap B:x = 0} or y=0}y = 0\}, then ∑(x,y)∈C∣x+y∣\sum_{(x,y) \in C} |x + y| is :

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  3. 53
    The system of equations
    x+y+z=6x+2y+5z=9x+5y+λz=μ\begin{matrix} & x + y + z = 6 \\ & x + 2y + 5z = 9 \\ & x + 5y + \lambda z = \mu \end{matrix}
    has no solution if

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  4. 54
    Let ∫x3sin⁡x dx=g(x)+C\int x^{3}\sin x\,dx = g(x)+C, where CC is the constant of integration. If 8(g(π2)+g′(π2))=απ3+βπ2+γ, α,β,γ∈Z8\left(g\left(\frac{\pi}{2}\right)+g'\left(\frac{\pi}{2}\right)\right)=\alpha\pi^{3}+\beta\pi^{2}+\gamma,\ \alpha,\beta,\gamma\in Z, then α+β−γ\alpha+\beta-\gamma equals:

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  5. 55
    A rod of length eight units moves such that its ends AA and BB always lie on the lines x−y+2=0x - y + 2 = 0 and y+2=0y + 2 = 0, respectively. If the locus of the point PP, that divides the rod ABAB internally in the ratio 2:12:1 is 9(x2+αy2+βxy+γx+28y)−76=09\left( x^{2}+ \alpha y^{2}+ \beta xy + \gamma x + 28y \right)- 76 = 0, then α−β−γ\alpha - \beta - \gamma is equal to :

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  6. 56
    The distance of the line x−22=y−63=z−34\frac{x - 2}{2}=\frac{y - 6}{3}=\frac{z - 3}{4} from the point (1,4,0)(1,4,0) along the line x1=y−22=z+33\frac{x}{1}=\frac{y - 2}{2}=\frac{z + 3}{3} is :

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  7. 57
    Let the point A divide the line segment joining the points P(−1,−1,2)P( - 1, - 1,2) and Q(5,5,10)Q(5,5,10) internally in the ratio r:1(r>0)r:1(r > 0). If OO is the origin and (OQ→⋅OA→)−15∣OP→×OA→∣2=10(\overrightarrow{OQ}\cdot\overrightarrow{OA}) -\frac{1}{5}|\overrightarrow{OP}\times\overrightarrow{OA}|^{2}= 10, then the value of r is :

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  8. 58
    If the area of the region {(x,y):−1≤x≤1,0≤y≤a+e∣x∣−e−x,a>0}\left\{ (x,y): - 1 \leq x \leq 1,0 \leq y \leq a +e^{|x|}-e^{- x},a > 0 \right\} is e2+8e+1e\frac{e^{2}+ 8e + 1}{e}, then the value of a is :

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  9. 59
    A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm , the ice-cream melts at the rate of 81 cm3/min81{\text{ }cm}^{3}/min and the thickness of the ice-cream layer decreases at the rate of 14π cm/min\frac{1}{4\pi}\text{ }cm/min. The surface area (in cm2cm^{2} ) of the chocolate ball (without the ice-cream layer) is :

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  10. 60
    A board has 16 squares as shown in the figure : Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is :

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  11. 61
    Let x=x(y)x = x(y) be the solution of the differential equation
    y=(x−ydxdy)sin⁡(xy),y>0 and x(1)=π2y =\left( x - y\frac{dx}{dy} \right)\sin\left( \frac{x}{y} \right),y > 0\text{~and~}x(1) =\frac{\pi}{2}
    Then cos⁡(x(2))\cos(x(2)) is equal to :

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  12. 62
    Let the range of the function f(x)=6+16cos⁡x⋅cos⁡(π3−x)⋅cos⁡(π3+x)⋅sin⁡3x⋅cos⁡6x, x∈Rf(x) = 6 + 16\cos x \cdot \cos\left( \frac{\pi}{3}- x \right)\cdot \cos\left( \frac{\pi}{3}+ x \right) \cdot \sin3x \cdot \cos6x,\ x \in R be [α,β]\lbrack\alpha,\beta\rbrack. Then the distance of the point (α,β)(\alpha,\beta) from the line 3x+4y+12=03x + 4y + 12 = 0 is :

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  13. 63
    Let the shortest distance from (a,0),a>0(a,0),a > 0, to the parabola y2=4xy^{2}= 4x be 4 . Then the equation of the circle passing through the point (a,0)(a,0) and the focus of the parabola, and having its centre on the axis of the parabola is:

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  14. 64
    Let X=R×RX = R \times R. Define a relation R on X as:
    (a1,b1)R(a2,b2)⇔b1=b2.\left( a_{1},b_{1} \right)R\left( a_{2},b_{2} \right)\Leftrightarrow b_{1}=b_{2}.
    Statement-I: R is an equivalence relation. Statement-II: For some (a,b)∈X(a,b) \in X, the set S={(x,y)∈X:(x,y)R(a,b)}\mathbf{S = \{(x,y) \in X:(x,y)R(a,b)\}} represents a line parallel to y=xy = x. In the light of the above statements, choose the correct answer from the options given below:

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  15. 65
    The length of the chord of the ellipse x24+y22=1\frac{x^{2}}{4}+\frac{y^{2}}{2}= 1 whose mid-point is (1,12)\left( 1,\frac{1}{2} \right), is:

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  16. 66
    Let A=[aij]A =\left\lbrack a_{ij} \right\rbrack be a 3×33 \times 3 matrix such that A[010]=[001],A[413]=[010]A\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}=\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix},A\begin{bmatrix} 4 \\ 1 \\ 3 \end{bmatrix}=\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} and A[212]=[100]A\begin{bmatrix} 2 \\ 1 \\ 2 \end{bmatrix}=\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, then a23a_{23} equals:

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  17. 67
    The number of complex numbers zz, satisfying ∣z∣=1|z| = 1 and ∣zzˉ+zˉz∣=1\left| \frac{z}{\bar{z}}+\frac{\bar{z}}{z} \right|= 1, is :

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  18. 68
    If the square of the shortest distance between the lines x−21=y−12=z+3−3\frac{x - 2}{1}=\frac{y - 1}{2}=\frac{z + 3}{- 3} and x+12=y+34=z+5−5\frac{x + 1}{2}=\frac{y + 3}{4}=\frac{z + 5}{- 5} is mn\frac{m}{n}, where m,nm,n are coprime numbers, then m+nm + n is equal to:

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  19. 69
    If I=∫0π2sin⁡32xsin⁡32x+cos⁡32xdxI =\int_{0}^{\frac{\pi}{2}} \frac{\sin^{\frac{3}{2}}x}{\sin^{\frac{3}{2}}x +\cos^{\frac{3}{2}}x}dx, then ∫02Ixsin⁡xcos⁡xsin⁡4x+cos⁡4xdx\int_{0}^{2I} \frac{x\sin x\cos x}{\sin^{4}x +\cos^{4}x}dx equals:

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  20. 70
    lim⁡x→∞(2x2−3x+5)(3x−1)x2(3x2+5x+4)(3x+2)x\lim_{x \rightarrow \infty} \frac{\left( 2x^{2}- 3x + 5 \right)(3x - 1)^{\frac{x}{2}}}{\left( 3x^{2}+ 5x + 4 \right)\sqrt{(3x + 2)^{x}}} is equal to:

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  21. 71
    The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is
  22. 72
    Let α,β\alpha,\beta be the roots of the equation x2−ax−b=0x^{2}- ax - b = 0 with Im(α)<Im(β)Im(\alpha) < Im(\beta). Let Pn=αn−βnP_{n}=\alpha^{n}-\beta^{n}. If P3=−57i, P4=−37i, P5=117iP_{3}= - 5\sqrt{7}i,\ P_{4}= - 3\sqrt{7}i,\ P_{5}= 11\sqrt{7}i and P6=457iP_{6}= 45\sqrt{7}i, then ∣α4+β4∣\left| \alpha^{4}+\beta^{4} \right| is equal to  ___\ \_\_\_ .
  23. 73
    The focus of the parabola y2=4x+16y^{2}= 4x + 16 is the centre of the circle C of radius 5 . If the values of λ\lambda, for 3x−y=03x - y = 0 and x+λy=4x + \lambda y = 4, are λ1\lambda_{1} and λ2,λ1<λ2\lambda_{2},\lambda_{1}<\lambda_{2}, then 12λ1+29λ212\lambda_{1}+ 29\lambda_{2} is equal to _____\_\_\_\_\_ .
  24. 74
    The variance of the numbers 8,21,34,47,…,3208,21,34,47,\ldots,320, is ____\_\_\_\_
  25. 75
    The roots of the quadratic equation 3x2−px+q=03x^{2}-px+q=0 are the 10th10^{\text{th}} and 11th11^{\text{th}} terms of an arithmetic progression with common difference 32\frac{3}{2}. If the sum of the first 11 terms of this arithmetic progression is 88, then q−2pq-2p is equal to

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