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JEE Mains 2025 (29 Jan, Shift 1), 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
    Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : Choke coil is simply a coil having a large inductance but a small resistance. Choke coils are used with fluorescent mercury-tube fittings. If household electric power is directly connected to a mercury tube, the tube will be damaged. Reason (R): By using the choke coil, the voltage across the tube is reduced by a factor (R/R2+ω2L2)\left( R/\sqrt{R^{2}+\omega^{2}L^{2}} \right), where ω\omega is frequency of the supply across resistor RR and inductor LL. If the choke coil were not used, the voltage across the resistor would be the same as the applied voltage. In the light of the above statements, choose the most appropriate answer from the options given below:

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  2. 2
    Two projectiles are fired with same initial speed from same point on ground at angles of (45∘−α)\left( 45^{\circ}- \alpha \right) and (45∘+α)\left( 45^{\circ}+ \alpha \right), respectively, with the horizontal direction. The ratio of their maximum heights attained is :

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  3. 3
    An electric dipole of mass mm, charge qq, and length ll is placed in a uniform electric field E→=E0i^\overrightarrow{E}=E_{0}\widehat{i}. When the dipole is rotated slightly from its equilibrium position and released, the time period of its oscillations will be :

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  4. 4
    The pair of physical quantities not having same dimensions is :

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  5. 5
    Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : Time period of a simple pendulum is longer at the top of a mountain than that at the base of the mountain. Reason (R): Time period of a simple pendulum decreases with increasing value of acceleration due to gravity and vice-versa. In the light of the above statements, choose the most appropriate answer from the options given below:

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  6. 6
    The expression given below shows the variation of velocity (v) with time ( t ), v=At2+BtC+tv =At^{2}+\frac{Bt}{C + t}. The dimension of ABC is :

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  7. 7
    Consider I1I_{1} and I2I_{2} are the currents flowing simultaneously in two nearby coils 1&21\& 2, respectively. If L1=L_{1}= self inductance of coil 1 , M12=M_{12}= mutual inductance of coil 1 with respect to coil 2 , then the value of induced emf in coil 1 will be

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  8. 8
    At the interface between two materials having refractive indices n1n_{1} and n2n_{2}, the critical angle for reflection of an em wave is θ1c\theta_{1c}. The n2n_{2} material is replaced by another material having refractive index n3n_{3}, such that the critical angle at the interface between n1n_{1} and n3n_{3} materials is θ2c\theta_{2c}. If n3>n2>n1n_{3}>n_{2}>n_{1}; n2n3=25\frac{n_{2}}{n_{3}}=\frac{2}{5} and sin⁡θ2C−sin⁡θ1C=12\sin\theta_{2C}- \sin\theta_{1C}=\frac{1}{2}, then θ1C\theta_{1C} is

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  9. 9
    Consider a long straight wire of a circular cross-section (radius a) carrying a steady current I. The current is uniformly distributed across this cross-section. The distances from the centre of the wire's cross-section at which the magnetic field [inside the wire, outside the wire] is half of the maximum possible magnetic field, any where due to the wire, will be

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  10. 10
    As shown below, bob A of a pendulum having massless string of length ' R ' is released from 60∘60^{\circ} to the vertical. It hits another bob B of half the mass that is at rest on a friction less table in the centre. Assuming elastic collision, the magnitude of the velocity of bob A after the collision will be (take gg as acceleration due to gravity)

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  11. 11
    Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : Emission of electrons in photoelectric effect can be suppressed by applying a sufficiently negative electron potential to the photo emissive substance. Reason (R) : A negative electric potential, which stops the emission of electrons from the surface of a photo emissive substance, varies linearly with frequency of incident radiation. In the light of the above statements, choose the most appropriate answer from the options given below:

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  12. 12
    A coil of area A and N turns is rotating with angular velocity ω\omega in a uniform magnetic field B→\overrightarrow{B} about an axis perpendicular to B→\overrightarrow{B}. Magnetic flux φ\varphi and induced emf ε\varepsilon across it, at an instant when B→\overrightarrow{B} is parallel to the plane of coil, are :

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  13. 13
    The fractional compression (ΔVV)\left( \frac{\Delta V}{V} \right) of water at the depth of 2.5 km below the sea level is ____\_\_\_\_ %. Given, the Bulk modulus of water =2×109Nm−2= 2 \times10^{9}Nm^{- 2}, density of water =103 kg m−3=10^{3}\text{ }kg{\text{ }m}^{- 3}, acceleration due to gravity =g=10 ms−2= g = 10{\text{ }ms}^{- 2}.

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  14. 14
    If λ\lambda and K are de Broglie Wavelength and kinetic energy, respectively, of a particle with constant mass. The correct graphical representation for the particle will be :-

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  15. 15
    For the circuit shown above, equivalent GATE is :

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  16. 16
    A body of mass ' m ' connected to a massless and unstretchable string goes in a vertical circle of radius ' R ' under gravity g . The other end of the string is fixed at the center of circle. If velocity at top of circular path is ngRn\sqrt{gR}, where, n≥1n \geq 1, then ratio of kinetic energy of the body at bottom to that at top of the circle is

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  17. 17
    Let uu and vv be the distances of the object and the image from a lens of focal length ff. The correct graphical representation of uu and vv for a convex lens when ∣u∣>f|u| > f, is

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  18. 18
    Match List-I with List-II.
    List-IList-II
    (A) Electric field inside (distance r>0r > 0 from center) of a uniformly charged spherical shell with surface charge density σ\sigma, and radius RR.(I) σ/ε0\sigma/\varepsilon_{0}
    (B) Electric field at distance r>0r > 0 from a uniformly charged infinite plane sheet with surface charge density σ\sigma.(II) σ/2ε0\sigma/2\varepsilon_{0}
    (C) Electric field outside (distance r>0r > 0 from center) of a uniformly charged spherical shell with surface charge density σ\sigma, and radius RR.(III) 0
    (D) Electric field between 2 oppositely charged infinite plane parallel sheets with uniform surface charge density σ\sigma.(IV) σε0r2\frac{\sigma}{\varepsilon_{0}r^{2}}
    Choose the correct answer from the options given below :

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  19. 19
    The workdone in an adiabatic change in an ideal gas depends upon only:

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  20. 20
    Given below are two statements : one is labelled as Assertion (A) and other is labelled as Reason (R). Assertion (A) : Electromagnetic waves carry energy but not momentum. Reason (R): Mass of a photon is zero. In the light of the above statements, choose the most appropriate answer from the options given below :

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  21. 21
    The coordinates of a particle with respect to origin in a given reference frame is (1,1,1)(1,1,1) meters. If a force of F→=i^−j^+k^\overrightarrow{F}=\widehat{i}-\widehat{j}+\widehat{k} acts on the particle, then the magnitude of torque (with respect to origin) in zz-direction is _____\_\_\_\_\_
  22. 22
    A container of fixed volume contains a gas at 27∘C27^{\circ}C. To double the pressure of the gas, the temperature of gas should be raised to  ____\ \_\_\_\_  ∘C\ ^{\circ}C.
  23. 23
    Two light beams fall on a transparent material block at point 1 and 2 with angle θ1\theta_{1} and θ2\theta_{2}, respectively, as shown in figure. After refraction, the beams intersect at point 3 which is exactly on the interface at other end of the block. Given : the distance between 1 and 2,d=43 cm2,d = 4\sqrt{3}\text{ }cm and θ1=θ2=cos⁡−1(n22n1)\theta_{1}=\theta_{2}=\cos^{- 1}\left( \frac{n_{2}}{2n_{1}} \right), where refractive index of the block n2>n_{2}> refractive index of the outside medium n1n_{1}, then the thickness of the block is_____ cm .
  24. 24
    In a hydraulic lift, the surface area of the input piston is 6 cm26{\text{ }cm}^{2} and that of the output piston is 1500 cm21500{\text{ }cm}^{2}. If 100 N force is applied to the input piston to raise the output piston by 20 cm , then the work done is _____\_\_\_\_\_ kJ .
  25. 25
    The maximum speed of a boat in still water is 27 km/h27\ km/h. Now this boat is moving downstream in a river flowing at 9 km/h9\ km/h. A man in the boat throws a ball vertically upwards with speed of 10 m/s10\ m/s. Range of the ball as observed by an observer at rest on the riverbank is ____\_\_\_\_ cm. (Take g=10 m/s2g = 10\ m/s^{2})

Chemistry

25 questions

  1. 26
    Total number of nucleophiles from the following is :-
    NH3,PhSH, (H3C)2 S,H2C=CH2,O⊖H,H3O⊕,(CH3)2CO, >C=NCH3\begin{matrix} & NH_{3},PhSH,\ \left( H_{3}C \right)_{2}\text{ }S,H_{2}C =CH_{2},\overset{\ominus}{O}H,H_{3}O^{\oplus}, \\ & \left( CH_{3} \right)_{2}CO,\ >C=NCH_{3} \end{matrix}

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  2. 27
    The standard reduction potential values of some of the p-block ions are given below. Predict the one with the strongest oxidising capacity.

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  3. 28
    The molar conductivity of a weak electrolyte when plotted against the square root of its concentration, which of the following is expected to be observed?

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  4. 29
    At temperature T , compound AB2( g)AB_{2(\text{ }g)} dissociates as AB2( g)⇌AB(g)+12 B2( g)AB_{2(\text{ }g)}\rightleftharpoons AB_{(g)}+\frac{1}{2}{\text{ }B}_{2(\text{ }g)} having degree of dissociation x (small compared to unity). The correct expression for x in terms of KpK_{p} and p is

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  5. 30
    Match List-I with List-II.
    List-I (Structure)List-II (IUPAC Name)
    (A)H3C−CH2−CH(C2H5)−CH2−CH(CH3)−C2H5\mathrm{H_{3}C-CH_{2}-CH(C_{2}H_{5})-CH_{2}-CH(CH_{3})-C_{2}H_{5}}(I)4-Methylpent-1-ene
    (B)
    (CH3)2C(C3H7)2\left( CH_{3} \right)_{2}C\left( C_{3}H_{7} \right)_{2}
    		(II)3-Ethyl-5-methylheptane
    (C)CH2=C(CH3)−CH=CH−CH3\mathrm{CH_{2}=C(CH_{3})-CH=CH-CH_{3}}(III)4,4-Dimethylheptane
    (D)CH2=CH−CH2−CH(CH3)2\mathrm{CH_{2}=CH-CH_{2}-CH(CH_{3})_{2}}(IV)2-Methyl-1,3-pentadiene
    Choose the correct answer from the options given below:

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  6. 31
    Choose the correct statements. (A) Weight of a substance is the amount of matter present in it. (B) Mass is the force exerted by gravity on an object. (C) Volume is the amount of space occupied by a substance. (D) Temperatures below 0∘C0^{\circ}C are possible in Celsius scale, but in Kelvin scale negative temperature is not possible. (E) Precision refers to the closeness of various measurements for the same quantity.

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  7. 32
    The correct increasing order of stability of the complexes based on Δ∘\Delta_{\circ} value is : (I) [Mn(CN)6]3−\left\lbrack Mn(CN)_{6} \right\rbrack^{3 -} (II) [Co(CN)6]4−\left\lbrack Co(CN)_{6} \right\rbrack^{4 -} (III) [Fe(CN)6]4−\left\lbrack Fe(CN)_{6} \right\rbrack^{4 -} (IV) [Fe(CN)6]3−\left\lbrack Fe(CN)_{6} \right\rbrack^{3 -}

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  8. 33
    Match List-I with List-II.
    List-I (Complex)List-II (Hybridisation and magnetic character)
    (A) [MnBr4]2−[MnBr_{4}]^{2-}(I) d2sp3d^{2}sp^{3}, diamagnetic
    (B) [FeF6]3−[FeF_{6}]^{3-}(II) sp3d2sp^{3}d^{2}, paramagnetic
    (C) [Co(C2O4)3]3−[Co(C_{2}O_{4})_{3}]^{3-}(III) sp3sp^{3}, diamagnetic
    (D) [Ni(CO)4][Ni(CO)_{4}](IV) sp3sp^{3}, paramagnetic
    Choose the correct answer from the options given below :

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  9. 34
    In the following substitution reaction : ⟶C2H5ONaC2H5OHP\underset{C_{2}H_{5}OH}{\overset{C_{2}H_{5}ONa}{\longrightarrow}} P (product). The product P formed is:

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  10. 35
    For a Mg∣Mg2+Mg \mid Mg^{2 +} (aq) ∥Ag+(aq)∣Ag\parallel Ag^{+}(aq) \mid Ag the correct Nernst Equation is :

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  11. 36
    The correct option with order of melting points of the pairs (Mn,Fe),(Tc,Ru)(Mn,Fe),(Tc,Ru) and (Re,Os)(Re,Os) is :

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  12. 37
    1.24 g of AX2AX_{2} (molar mass 124 g mol−1124\text{ }g{\text{ }mol}^{- 1} ) is dissolved in 1 kg of water to form a solution with boiling point of 100.0156∘C100.0156^{\circ}C, while 25.4 g of AY2AY_{2} (molar mass 250 g mol−1250\text{ }g{\text{ }mol}^{- 1} ) in 2 kg of water constitutes a solution with a boiling point of 100.0260∘C100.0260^{\circ}C.Kb(H2O)=0.52 K kg mol−1K_{b}\left( H_{2}O \right)= 0.52\text{ }K\text{ }kg{\text{ }mol}^{- 1} Which of the following is correct?

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  13. 38
    500 J of energy is transferred as heat to 0.5 mol of Argon gas at 298 K and 1.00 atm . The final temperature and the change in internal energy respectively are : Given : R=8.3 J K−1 mol−1R = 8.3\text{ }J{\text{ }K}^{- 1}{\text{ }mol}^{- 1}

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  14. 39
    The reaction A2+B2→2ABA_{2}+B_{2}\rightarrow 2AB follows the mechanism
    A2⇌k1k−1 A+A( fast )A+B2→K2AB+B (slow) A+B→AB (fast) \begin{matrix} & A_{2}\underset{k_{- 1}}{\overset{k_{1}}{\rightleftharpoons}}\text{ }A + A(\text{~fast~}) \\ & A +B_{2}\overset{K_{2}}{\rightarrow}AB + B\text{~(slow)~} \\ & A + B \rightarrow AB\text{~(fast)~} \end{matrix}
    The overall order of the reaction is :

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  15. 40
    If a0a_{0} is denoted as the Bohr radius of hydrogen atom, then what is the de-Broglie wavelength ( λ\lambda ) of the electron present in the second orbit of hydrogen atom? [ n : any integer]

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  16. 41
    The product (P)(P) formed in the following reaction is →Zn−HgHCl  Product (p)\underset{HCl}{\overset{Zn - Hg}{\rightarrow}}\ \ Product\ (p)

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  17. 42
    An element ' EE ' has the ionisation enthalpy value of 374kJmol−1374kJmol^{- 1}. ' E ' reacts with elements A,B,CA,B,C and D with electron gain enthalpy values of −328,−349- 328, - 349, -325 and −295 kJ mol−1- 295\text{ }kJ{\text{ }mol}^{- 1}, respectively. The correct order of the products EA, EB, EC and ED in terms of ionic character is :

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  18. 43
    Match List-I with List-II.
    List-I (Carbohydrate)List-II (Linkage, Source)
    (A)Amylose(I)β−C1−C4\beta-C_{1}-C_{4}, plant
    (B)Cellulose(II)α−C1−C4\alpha-C_{1}-C_{4}, animal
    (C)Glycogen(III)α−C1−C4\alpha-C_{1}-C_{4}, α−C1−C6\alpha-C_{1}-C_{6}, plant
    (D)Amylopectin(IV)α−C1−C4\alpha-C_{1}-C_{4}, plant
    Choose the correct answer from the options given below:

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  19. 44
    The steam volatile compounds among the following are : (A) (B) (C) (D) Choose the correct answer from the options given below :

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  20. 45
    Given below are two statements : Statement (I): The radii of isoelectronic species increases in the order.
    Mg2+<Na+<F−<O2−Mg^{2 +}<Na^{+}<F^{-}<O^{2 -}
    Statement (II) : The magnitude of electron gain enthalpy of halogen decreases in the order.
    Cl>F>Br>ICl > F > Br > I
    In the light of the above statements, choose the most appropriate answer from the options given below :

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  21. 46
    Given below are some nitrogen containing compounds. Each of them is treated with HCl separately. 1.0 g of the most basic compound will consume _____\_\_\_\_\_ mg of HCl . (Given molar mass in gmol−1C:12,H:1,O:16gmol^{- 1}C:12,H:1,O:16, Cl:35.5)Cl:35.5)
  22. 47
    The molar mass of the water insoluble product formed from the fusion of chromite ore (FeCr2O4)\left( FeCr_{2}O_{4} \right) with Na2CO3Na_{2}CO_{3} in presence of O2O_{2} is ______\_\_\_\_\_\_ gmol−1gmol^{- 1}.
  23. 48
    The sum of sigma (σ)(\sigma) and pi(π)pi(\pi) bonds in Hex-1,3-dien-5-yne is ____\_\_\_\_ .
  24. 49
    If A2 BA_{2}\text{ }B is 30%30\% ionised in an aqueous solution, then the value of van't Hoff factor (i) is _____\_\_\_\_\_ ×10−1\times10^{- 1}.
  25. 50
    0.1 mole of compound ' SS ' will weigh _____\_\_\_\_\_ g. (Given molar mass in gmol−1C:12,H:1,O:16gmol^{- 1}C:12,H:1,O:16 )

Mathematics

25 questions

  1. 51
    Let the line x+y=1x + y = 1 meet the circle x2+y2=4x^{2}+y^{2}= 4 at the points AA and BB. If the line perpendicular to ABAB and passing through the mid point of the chord ABAB intersects the circle at C and D , then the area of the quadrilateral ADBC is equal to

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  2. 52
    Let M and m respectively be the maximum and the minimum values of
    f(x)=∣1+sin⁡2xcos⁡2x4sin⁡4xsin⁡2x1+cos⁡2x4sin⁡4xsin⁡2xcos⁡2x1+4sin⁡4x∣,x∈Rf(x) =\left| \begin{matrix} 1 +\sin^{2}x & \cos^{2}x & 4\sin4x \\ \sin^{2}x & 1 +\cos^{2}x & 4\sin4x \\ \sin^{2}x & \cos^{2}x & 1 + 4\sin4x \end{matrix} \right|,x \in R
    Then M4−m4M^{4}-m^{4} is equal to :

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  3. 53
    Two parabolas have the same focus (4,3)(4,3) and their directrices are the xx-axis and the yy-axis, respectively. If these parabolas intersects at the points AA and BB, then (AB)2(AB)^{2} is equal to

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  4. 54
    Let ABCABC be a triangle formed by the lines 7x−6y+3=0,x+2y−31=07x- 6y+ 3 = 0,x+ 2y- 31 = 0 and 9x−2y−19=09x- 2y- 19 = 0, Let the point (h,k)(h,k) be the image of the centroid of ΔABC\Delta ABC in the line 3x+6y−53=03x + 6y - 53 = 0. Then h2+k2+hkh^{2}+k^{2}+ hk is equal to

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  5. 55
    Let a→=2i^−j^+3k^,b→=3i^−5j^+k^\overrightarrow{a}= 2\widehat{i}-\widehat{j}+ 3\widehat{k},\overrightarrow{b}= 3\widehat{i}- 5\widehat{j}+\widehat{k} and c→\overrightarrow{c} be a vector such that a→×c→=c→×b→\overrightarrow{a}\times\overrightarrow{c}=\overrightarrow{c}\times\overrightarrow{b} and (a→+c→)⋅(b→+c→)=168(\overrightarrow{a}+\overrightarrow{c}) \cdot (\overrightarrow{b}+\overrightarrow{c}) = 168. Then the maximum value of ∣c→∣2|\overrightarrow{c}|^{2} is :

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  6. 56
    Let PP be the set of seven -digit numbers with sum of their digits equal to 11 . If the numbers in P are formed by using the digits 1,2 and 3 only, then the number of elements in the set PP is :

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  7. 57
    Let the area of the region {(x,y):2y≤x2+3\left\{ (x,y):2y\leq x^{2}+ 3 \right., y+∣x∣≤3,y≥∣x−1∣}y+ |x| \leq 3,y\geq |x- 1|\} be AA. Then 6A6A is equal to:

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  8. 58
    The least value of n for which the number of integral terms in the Binomial expansion of (73+1112)n(\sqrt[3]{7}+\sqrt[12]{11})^{n} is 183, is :

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  9. 59
    The number of solutions of the equation (9x−9x+2)(2x−7x+3)=0\left( \frac{9}{x}-\frac{9}{\sqrt{x}}+ 2 \right)\left( \frac{2}{x}-\frac{7}{\sqrt{x}}+ 3 \right)= 0 is :

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  10. 60
    Let y=y(x)y = y(x) be the solution of the differential equation cos⁡x(log⁡e(cos⁡x))2dy+(sin⁡x−3ysinxlog⁡e(cos⁡x))dx=0\cos x\left( \log_{e}(\cos x) \right)^{2}dy +\left( \sin x - 3ysinx\log_{e}(\cos x) \right)dx = 0, x∈(0,π2)x \in\left( 0,\frac{\pi}{2} \right). If y(π4)=−1log⁡e2y\left( \frac{\pi}{4} \right)=\frac{- 1}{\log_{e}2}, then y(π6)y\left( \frac{\pi}{6} \right) is :

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  11. 61
    Define a relation R on the interval [0,π2)\left\lbrack 0,\frac{\pi}{2} \right) by xR y if and only if sec⁡2x−tan⁡2y=1\sec^{2}x-\tan^{2}y= 1. Then RR is :

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  12. 62
    Let the ellipse, E1:x2a2+y2 b2=1,a>bE_{1}:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{{\text{ }b}^{2}}= 1,a > b and E2:x2 A2+y2 B2=1, A<BE_{2}:\frac{x^{2}}{{\text{ }A}^{2}}+\frac{y^{2}}{{\text{ }B}^{2}}= 1,\text{ }A < B have same eccentricity 13\frac{1}{\sqrt{3}}. Let the product of their lengths of latus rectums be 323\frac{32}{\sqrt{3}}, and the distance between the foci of E1E_{1} be 4. If E1E_{1} and E2E_{2} meet at A,B,CA,B,C and DD, then the area of the quadrilateral ABCD equals:

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  13. 63
    Consider an A.P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800 . Then its 11th 11^{\text{th~}} term is :

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  14. 64
    Let a→=i^+2j^+k^\overrightarrow{a}=\widehat{i}+ 2\widehat{j}+\widehat{k} and b→=2i^+7j^+3k^\overrightarrow{b}= 2\widehat{i}+ 7\widehat{j}+ 3\widehat{k}. Let L1:r→=(−i^+2j^+k^)+λa→,λ∈RL_{1}:\overrightarrow{r}= ( -\widehat{i}+ 2\widehat{j}+\widehat{k}) + \lambda\overrightarrow{a},\lambda \in R and L2:r→=(j^+k^)+μb→,μ∈RL_{2}:\overrightarrow{r}= (\widehat{j}+\widehat{k}) + \mu\overrightarrow{b},\mu \in R be two lines. If the line L3L_{3} passes through the point of intersection of L1L_{1} and L2L_{2}, and is parallel to a→+b→\overrightarrow{a}+\overrightarrow{b}, then L3L_{3} passes through the point:

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  15. 65
    The value of lim⁡n→∞(∑K=1nk3+6k2+11k+5(k+3)!)\lim_{n \rightarrow \infty} \left( \sum_{K = 1}^{n} \frac{k^{3}+ 6k^{2}+ 11k + 5}{(k + 3)!} \right) is:

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  16. 66
    The integral 80∫0π4(sin⁡θ+cos⁡θ9+16sin⁡2θ)dθ80\int_{0}^{\frac{\pi}{4}} \left( \frac{\sin\theta + \cos\theta}{9 + 16\sin2\theta} \right)d\theta is equal to :

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  17. 67
    Let L1:x−11=y−2−1=z−12L_{1}:\frac{x - 1}{1}=\frac{y - 2}{- 1}=\frac{z - 1}{2} and L2:x+1−1=y−22=z1L_{2}:\frac{x + 1}{- 1}=\frac{y - 2}{2}=\frac{z}{1} be two lines. Let L3L_{3} be a line passing through the point (α,β,γ)(\alpha,\beta,\gamma) and be perpendicular to both L1L_{1} and L2L_{2}. If L3L_{3} intersects L1L_{1}, then ∣5α−11β−8γ∣|5\alpha - 11\beta - 8\gamma| equals :

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  18. 68
    Let x1,x2,……x10x_{1},x_{2},\ldots\ldots x_{10} be ten observations such that ∑i=110(xi−2)=30,∑i=110(xi−β)2=98,β>2\sum_{i = 1}^{10} \left( x_{i}- 2 \right)= 30,\sum_{i = 1}^{10} \left( x_{i}- \beta \right)^{2}= 98,\beta > 2 and their variance is 45\frac{4}{5}. If μ\mu and σ2\sigma^{2} are respectively the mean and the variance of 2(x1−1)+4β,2(x2−1)+2\left( x_{1}- 1 \right)+ 4\beta,2\left( x_{2}- 1 \right)+ 4β,….,2(x10−1)+4β4\beta,\ldots.,2\left( x_{10}- 1 \right)+ 4\beta, then βμσ2\frac{\beta\mu}{\sigma^{2}} is equal to :

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  19. 69
    Let ∣z1−8−2i∣≤1\left| z_{1}- 8 - 2i \right|\leq 1 and ∣z2−2+6i∣≤2\left| z_{2}- 2 + 6i \right|\leq 2, z1,z2∈Cz_{1},z_{2}\in C. Then the minimum value of ∣z1−z2∣\left| z_{1}-z_{2} \right| is :

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  20. 70
    Let A=[aij]=[log⁡5128log⁡45log⁡58log⁡425]A =\left\lbrack a_{ij} \right\rbrack=\begin{bmatrix} \log_{5}128 & \log_{4}5 \\ \log_{5}8 & \log_{4}25 \end{bmatrix}. If AijA_{ij} is the cofactor of aij,Cij=∑k=12aikAjk,1≤ia_{ij},C_{ij}=\sum_{k = 1}^{2} a_{ik}A_{jk},1 \leq i, j≤2j \leq 2, and C=[Cij]C =\left\lbrack C_{ij} \right\rbrack, then 8∣C∣8|C| is equal to :

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  21. 71
    Let f:(0,∞)→Rf:(0,\infty) \rightarrow R be a twice differentiable function. If for some a≠0,∫01f(λx)dλ=af(x)a \neq 0,\int_{0}^{1} f(\lambda x)d\lambda = af(x), f(1)=1f(1) = 1 and f(16)=18f(16) =\frac{1}{8}, then 16−f′(116)16 -f^{'}\left( \frac{1}{16} \right) is equal to ____\_\_\_\_ ,
  22. 72
    Let S={m∈Z:Am2+Am=3I−A−6}S =\left\{ m \in Z:A^{m^{2}}+A^{m}= 3I -A^{- 6} \right\}, where A=[2−110]A =\begin{bmatrix} 2 & - 1 \\ 1 & 0 \end{bmatrix}. Then n(S)n(S) is equal to
  23. 73
    Let [t]\lbrack t\rbrack be the greatest integer less than or equal to tt. Then the least value of p∈Np \in N for which
    lim⁡x→0+(x([1x]+[2x]+…..+[px])−x2([1x2]+[22x2]+….+[92x2]))≥1\underset{x \rightarrow0^{+}}{\lim} \left( x\left( \left\lbrack \frac{1}{x} \right\rbrack+\left\lbrack \frac{2}{x} \right\rbrack+ \ldots.. +\left\lbrack \frac{p}{x} \right\rbrack \right)-x^{2}\left( \left\lbrack \frac{1}{x^{2}} \right\rbrack+\left\lbrack \frac{2^{2}}{x^{2}} \right\rbrack+ \ldots. +\left\lbrack \frac{9^{2}}{x^{2}} \right\rbrack \right) \right)\geq 1
    is equal to
  24. 74
    The number of 6 -letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is 4  ______\ \_\_\_\_\_\_
  25. 75
    Let S={x:cos⁡−1x=π+sin⁡−1x+sin⁡−1(2x+1)}S=\{x:\cos^{-1}x=\pi+\sin^{-1}x+\sin^{-1}(2x+1)\}. Then ∑x∈S(2x−1)2\sum_{x\in S}(2x-1)^{2} is equal to

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