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JEE Mains 2026 (4 Apr, 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
    In a screw gauge when the circular scale is given five complete rotations it moves linearly by 2.5 mm . If the circular scale has 100 divisions, the least count of screw gauge is ____\_\_\_\_ mm.

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  2. 2
    The increase in the pressure required to decrease the volume ( ΔV\Delta V ) of water is 6.3×107 N/m26.3 \times10^{7}\text{ }N/m^{2}. The percentage decrease in the volume is ____\_\_\_\_ . (Bulk modulus of water =2.1×109 N/m2= 2.1 \times10^{9}\text{ }N/m^{2})

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  3. 3
    The time taken by a block of mass mm to slide down from the highest point to the lowest point on a rough inclined plane is 50%50\% more compared to the time taken by the same block on identical inclined smooth plane. Both inclined planes are at 45∘45^{\circ} with the horizontal. The coefficient of kinetic friction between the rough inclined surface and block is ____\_\_\_\_ .

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  4. 4
    Two nuclei of mass number 3 combine with another nucleus of mass number 4 to yield a nucleus of mass number 10. If the binding energy per nucleon for the mass number 3, 4 and 10 are 5.6MeV,7.4MeV5.6MeV,7.4MeV and 6.1 MeV , respectively, then in the process, ΔMc2=\Delta Mc^{2}= ____\_\_\_\_ MeV .

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  5. 5
    A solid sphere of mass M and radius R is divided into two unequal parts. The smaller part having mass M/8 is converted into a sphere of radius r and the larger part is converted into a circular disc of thickness tt and radius 2R2R. If I1I_{1} is moment of inertia of a sphere having radius r about an axis through its centre and I2I_{2} is the moment of inertia of a disc about its diameter, the ratio of their moment of inertia I2/I1=I_{2}/I_{1}= ____\_\_\_\_ .

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  6. 6
    The two projectiles are projected with the same initial velocities at the 15∘15^{\circ} and 30∘30^{\circ} with respect to the horizontal. The ratio of their ranges is 1:x1:x. The value of xx is

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  7. 7
    The graph shows variation of stopping potential VoV_{o} with the frequency vv of the incident radiation for three photosensitive metals X1,X2X_{1},X_{2} and X3X_{3}. Which metal will give out electrons with greater kinetic energy, for the same wavelength of incident radiation?

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  8. 8
    A slit of width a is illuminated by light of wavelength λ\lambda. The linear separation between 1st 1^{\text{st~}} and 3rd 3^{\text{rd~}} minima in the diffraction pattern produced on a screen placed at a distance D from the slit system is ____\_\_\_\_ .

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  9. 9
    A string A of length 0.314 m of Young's modulus 2×1010 N/m22 \times10^{10}\text{ }N/m^{2} is connected to another string B of length and Young's modulus both twice of those of A. This series combination of strings is then suspended from a rigid support and its free end is fixed to a load of mass 0.8 kg . The net change in length of the combination is ____\_\_\_\_ mm . (radius of both the strings is 0.2 mm and acceleration due to gravity =10 m/s2= 10\text{ }m/s^{2} ) (Mass of both strings is to be neglected as compared to the mass of load)

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  10. 10
    One gas of n1n_{1} mole of molecules at temperature T1T_{1}, volume V1V_{1}, and pressure P1P_{1}, and another gas of n2n_{2} mole of molecules at temperature T2T_{2}, volume V2V_{2}, and pressure P2P_{2}, are mixed resulting in pressure P and volume V of the mixture. The temperature of the mixture is ____\_\_\_\_ .

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  11. 11
    An ideal gas undergoes a process maintaining relation between pressure ( P ) and volume ( V ) as P=Po(1+(VoV)2)−1P =P_{o}\left( 1 +\left( \frac{V_{o}}{V} \right)^{2} \right)^{- 1}, where PoP_{o} and VoV_{o} are constants. If two samples A and B (two moles each) with initial volume VoV_{o} and 3 Vo3{\text{ }V}_{o} respectively undergo above mentioned process and attain same pressure, then the difference at the temperatures of these samples, TB−TAT_{B}-T_{A} is ____\_\_\_\_ ( R=R = gas constant)

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  12. 12
    A voltmeter with internal resistance of xΩx\Omega can be used to measure upto 20 V . In order to increase its measuring range to 30 V , the required modification is to ____\_\_\_\_ .

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  13. 13
    Two 4 bits binary numbers, A=1101A = 1101 and B=1010B = 1010 are given in the inputs of a logic circuit shown in figure below. The output (Y) will be :

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  14. 14
    A rod of length 10 cm lies along the principle axis of a concave mirror of focal length 10 cm as shown in figure. The length of the image is ____\_\_\_\_ cm.

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  15. 15
    A parallel plate air capacitor is connected to a battery. The plates are pulled apart at uniform speed vv. If x is the separation between the plates at any instant, then the time rate of change of electrostatic energy of the capacitor is proportional to xαx^{\alpha}, where α\alpha is ____\_\_\_\_ .

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  16. 16
    An insulated wire is wound so that it forms a flat coil with N=200N = 200 turns. The radius of the innermost turn is r1=3 cmr_{1}= 3\text{ }cm, and of the outermost turn r2=6 cmr_{2}= 6\text{ }cm. If 20 mA current flow in it then the magnetic moment will be α×10−2 A.m2\alpha \times10^{- 2}\text{ }A.m^{2}. The value of α\alpha is ____\_\_\_\_ .

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  17. 17
    Consider a circuit consisting of a capacitor (20μ F)(20\mu\text{ }F), resistor ( 100Ω100\Omega ) and two identical diodes as shown in figure. The resistance of diode under forward biasing condition is 10Ω10\Omega. The time constant of the circuit is α×10−3 s\alpha \times10^{- 3}\text{ }s. The value of α\alpha is ____\_\_\_\_ .

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  18. 18
    The voltage and the current between A and B points shown in the circuit are ____\_\_\_\_ .

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  19. 19
    A telescope with objective diameter R is used to observe a distant star emitting light of wavelength 500 nm , at a resolution of 5×10−75 \times10^{- 7} radian. The value of RR is ____\_\_\_\_ cm .

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  20. 20
    An unpolarized light is incident on the plane interface of air-dielectric medium shown in figure. If the incident angle is equal to Brewster angle, identify the expression representing reflected wave.

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  21. 21
    A 1 kg block subjected to two simultaneous forces (2i^+3j^+4k^)N(2\widehat{i} + 3\widehat{j} + 4\widehat{k})N and (3i^−j^−2k^)N(3\widehat{i} - \widehat{j} - 2\widehat{k})N is moved a distance of 25 m along (3i^−4j^)(3\widehat{i} - 4\widehat{j}) direction. The work done in this process is ____\_\_\_\_ J.
  22. 22
    The surface tension of a soap solution is 3.5×10−2 N/m3.5 \times 10^{- 2}\text{ }N/m. The work required to increase the radius of a soap bubble from 1 cm to 2 cm is α×10−6 J\alpha \times 10^{- 6}\text{ }J. The value of α\alpha is ____\_\_\_\_ . (π=22/7)(\pi = 22/7)
  23. 23
    The velocity of a particle executing simple harmonic motion along x -axis is described as v2=50−x2v^{2} = 50 - x^{2}, where x represents displacement. If the time period of motion is x7s\frac{x}{7}s, the value of xx is ____\_\_\_\_ .
  24. 24
    A body of mass 2 kg begins to move under the influence of time dependent force F→=\overrightarrow{F} = (2ti^+6t2j^)N\left( 2t\widehat{i} + 6t^{2}\widehat{j} \right)N, where i^\widehat{i} and j^\widehat{j} are unit vectors along xx and y -axis respectively. The power produced by the force at t=2 st = 2\text{ }s is ____\_\_\_\_ W.
  25. 25
    An inductor of 10 mH , capacitor of 0.1μ F0.1\mu\text{ }F and a resistor of 100Ω100\Omega are connected in series aross an a.c power supply 220 V,70 Hz220\text{ }V,70\text{ }Hz. The power factor of the given circuit is 0.5 . The difference in the inductive reactance and capacitance reactance is 3αΩ\sqrt{3}\alpha\Omega. The value of α\alpha is ____\_\_\_\_ .

Chemistry

25 questions

  1. 26
    Number of moles and number of molecules in 1.4187 L of SO2SO_{2} at STP respectively are

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  2. 27
    What is the ratio of wave number of first line (lowest energy line) of Balmer series of H atomic spectrum to first line of its Brackett series?

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  3. 28
    Which of the following is correct set of 4 quantum number of 19th19^{\text{th}} electron in Chromium (Atomic number =24= 24 ) in accordance with Aufbau principle?

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  4. 29
    Given below are two statements: Statement I: For an ideal gas, heat capacity at constant volume is always greater than the heat capacity at constant pressure. Statement II : In a constant volume process, no work is produced, and all the heat withdrawn goes into the chaotic motion and is reflected by a temperature increase of the ideal gas. In the light of the above statements, choose the correct answer from the options given below

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  5. 30
    At T(K)T(K), the equilibrium constant of A2( g)+B2( g)⇌C(g)A_{2}(\text{ }g) +B_{2}(\text{ }g) \rightleftharpoons C(g) is 2.7×10−52.7 \times10^{- 5}. What is the equilibrium constant for 13 A2( g)+13 B2( g)⇌13C(g)\frac{1}{3}{\text{ }A}_{2}(\text{ }g) +\frac{1}{3}{\text{ }B}_{2}(\text{ }g) \rightleftharpoons\frac{1}{3}C(g) at the same temperature?

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  6. 31
    In order to oxidise a mixture of 1 mole each of FeC2O4,Fe2(C2O4)3,FeSO4FeC_{2}O_{4},Fe_{2}\left( C_{2}O_{4} \right)_{3},FeSO_{4} and Fe2(SO4)3Fe_{2}\left( SO_{4} \right)_{3} in acidic medium, the number of moles of KMnO4KMnO_{4} required is :

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  7. 32
    Consider the first order reaction R→PR \rightarrow P.The fraction of molecules decomposed in the given first order reaction can be expressed as

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  8. 33
    A monoatomic anion (A−)\left( A^{-} \right)has 45 neutrons and 36 electrons. Atomic mass, group in the periodic table and physical state at room temperature of the element (A) respectively are :

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  9. 34
    Given below are two statements : Statement I : The covalency of oxygen is generally two but it can exceed upto four. The oxidation state of oxygen in SO2{SO}_{2} is -2 and in OF2{OF}_{2} it is +2 . Statement II: The anomalous behavior of oxygen when compared to the other elements of group 16 is due to its small size and high electronegativity . In the light of the above statements, choose the correct answer from the options given below:

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  10. 35
    The correct statements among the following are, (A) Mo(VI)Mo(VI) and W(VI)W(VI) are less stable than Cr(VI)Cr(VI). (B) Ce4+{Ce}^{4 +} and Tb4+{Tb}^{4 +} are oxidant while Eu2+{Eu}^{2 +} and Yb2+{Yb}^{2 +} are reductant. (C) Cm and Am have seven unpaired electrons. (D) Actinoid contraction is greater from element to element than lanthanoid contraction. Choose the correct answer from the options given below :

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  11. 36
    Correct statements from the following are (A) Potassium dichromate is an oxidising agent and it oxidises FeSO4{FeSO}_{4} to Fe2(SO4)3{Fe}_{2}\left( {SO}_{4} \right)_{3} in acidic medium. (B) Sodium dichromate can be used as primary standard in volumetric estimation. (C) CrO42−{CrO}_{4}^{2 -} and Cr2O72−{Cr}_{2}O_{7}^{2 -} are interconvertible in aqueous solution by varying the pH of the solution. (D) Cr−O−CrCr - O - Cr bond angle in Cr2O72−{Cr}_{2}O_{7}^{2 -} is 126∘126^{\circ}. Choose the correct answer from the options given below :

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  12. 37
    Match List-I with List-II
    List-I Complex ionList-II Calculated spin only magnetic moment (BM)
    (A)
    [Cr(H2O)6]2+\left\lbrack Cr\left( H_{2}O \right)_{6} \right\rbrack^{2 +}
    		(I)3.87
    (B)
    [Co(H2O)6]2+\left\lbrack Co\left( H_{2}O \right)_{6} \right\rbrack^{2 +}
    		(II)5.92
    (C)
    [Cu(H2O)6]2+\left\lbrack Cu\left( H_{2}O \right)_{6} \right\rbrack^{2 +}
    		(III)4.90
    (D)
    [Mn(H2O)6]2+\left\lbrack Mn\left( H_{2}O \right)_{6} \right\rbrack^{2 +}
    		(IV)1.73
    Choose the correct answer from the options given below:

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  13. 38
    Increasing order of electron withdrawing power of following functional groups is : (A) -CN (B) -COOH (C) −NO2- {NO}_{2} (D) -I

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  14. 39
    An alkene (X) on ozonolytic followed by reduction gives following products. The alkene ( X ) is :

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  15. 40
    Match the List-I with List-II
    List-I Name of reactionList-II Reagent or catalyst used
    (A)Finkelstein reaction(I)SbF3{SbF}_{3}
    (B)Swarts reaction(II)Na, dry ether
    (C)Sandmeyer's reaction(III)NaI
    (D)Fittig reaction(IV)Cu2Cl2{Cu}_{2}{Cl}_{2}
    Choose the correct answer from the options given below :

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  16. 41
    Amongst the following, the total number of compounds soluble in aqueous NaOH at room temperature is: (I) (II) (III) (IV) (V) (VI) (VII) (VIII) (IX)

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  17. 42
    Product C of the following reaction sequence will be

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  18. 43
    Statement-I : The structure of maltose is given below, with the caption "Maltose is a non reducing sugar". Statement-II : The structure of lactose is given below, with the caption "Lactose is a reducing sugar". In the light of the above statements, choose the correct answer from the options given below

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  19. 44
    Match the LIST-I with LIST-II :
    List-I Name of amino acidList-II One letter symbol/type
    (A)Arginine(I)D / Non-essential
    (B)Aspartic acid(II)R / Essential
    (C)Lysine(III)E / Non-essential
    (D)Glutamic acid(IV)K/Essential
    Choose the correct answer from the options given below:

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  20. 45
    Identify the colour of compound ' X ' in the sequence of the reaction.

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  21. 46
    According to Lewis theory, the total number of σ\sigma bond-pairs and lone pair of electrons around the central atom of XeO64−{XeO}_{6}^{4 -} ion is ____\_\_\_\_ .
  22. 47
    Consider the following sequence of reactions to give the major product (X). P g of the major product ( X ) formed is reacted with NaHCO3{NaHCO}_{3} solution to liberate a gas which occupied 11.2dm311.2{dm}^{3} at STP. P=P = ____\_\_\_\_ . g. (Given molar mass in gmol−1H:1,C:12g{mol}^{- 1}H:1,C:12, O:16,Cl:35.5)O:16,Cl:35.5)
  23. 48
    2.0 g of a bromo hydrocarbon ( X ) was subjected to Carius analysis, gave 3.36 g of AgBr . The percentage of carbon in the compound ( X ) is 26.7%26.7\%. Total number of carbon atoms in the empirical formula for compound ( X ) is ____\_\_\_\_ . (Given molar mass in gmol−1H:1,C:12g{mol}^{- 1}H:1,C:12, Br : 80, Ag : 108)
  24. 49
    The pH a solution obtained by mixing 5 mL of 0.1MNH4OH0.1M{NH}_{4}OH solution with 250 mL of 0.1 M NH4Cl{NH}_{4}Cl solution is ____\_\_\_\_ ×10−2\times 10^{- 2}. (Nearest integer) Given : pKb(NH4OH)=4.74{pK}_{b}\left( {NH}_{4}OH \right) = 4.74
    log⁡2=0.30\log2 = 0.30
    log⁡3=0.48\log3 = 0.48
    log⁡5=0.70\log5 = 0.70
  25. 50
    A non-volatile, non-electrolyte solid solute when dissolved in 40 g of a solvent, the vapor pressure of the solvent decreased from 760 mm Hg to 750 mm Hg . If the same solution boils at 320 K , then the number of moles of the solvent present in the solution is ____\_\_\_\_ . (Nearest integer) [Given : boiling point of the pure solvent =319.5 K, Kb= 319.5\text{ }K,{\text{ }K}_{b} of the solvent =0.3 K kg mol−1= 0.3\text{ }K\text{ }kg{\text{ }mol}^{- 1} ]

Mathematics

25 questions

  1. 51
    Let [ • ] denote the greatest integer function. If the domain of the function f(x)=cos⁡−1(4x+2[x]3)f(x) =\cos^{- 1}\left( \frac{4x+ 2\lbrack x\rbrack}{3} \right) is [α,β]\lbrack\alpha,\beta\rbrack, then 12(α+β)12(\alpha+\beta) is equal to :

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  2. 52
    If the set of all solutions of ∣x2+x−9∣=∣x∣+∣x2−9∣\left| x^{2}+x- 9 \right|= |x| +\left| x^{2}- 9 \right| is [α,β]∪[γ,∞)\lbrack\alpha,\beta\rbrack \cup \lbrack\gamma,\infty), then (α2+β2+γ2)\left( \alpha^{2}+\beta^{2}+\gamma^{2} \right) is equal to :

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  3. 53
    Let z be a complex number such that ∣z+2∣=∣z−2∣|z + 2| = |z - 2| and arg(z+3z−i)=π4arg\left( \frac{z+ 3}{z-i} \right)=\frac{\pi}{4}. Then ∣z∣2|z|^{2} is equal to :

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  4. 54
    The number of functions f:{1,2,3,4}→{a,b,c}\mathbf{f:\{ 1,2,3,4\} \rightarrow \{ a,b,c\}}, which are not onto, is :

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  5. 55
    Let S={A=[abcd]:a,b,c,d∈{0,1,2,3,4}S =\left\{ A =\begin{bmatrix} a & b \\ c & d \end{bmatrix}\mathbf{:a,b,c,d \in \{ 0,1,2,3,4\}} \right. and  A2−4 A+3I=0}\left. \ A^{2}- 4\text{ }A + 3I = 0 \right\} be a set of 2×22 \times 2 matrices. Then the number of matrices in S, for which the sum of the diagonal elements is equal to 4 , is:

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  6. 56
    Let A=[112−201135]A =\begin{bmatrix} 1 & 1 & 2 \\ - 2 & 0 & 1 \\ 1 & 3 & 5 \end{bmatrix}. Then the sum of all elements of the matrix adj(adj(2(adj A)−1))adj\left( adj\left( 2(adj\text{ }A)^{- 1} \right) \right) is equal to :

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  7. 57
    The first term of an A.P. of 30 non-negative terms is 103\frac{10}{3}. If the sum of this A.P. is the cube of its last term, then its common difference is :

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  8. 58
    The number of ways, of forming a queue of 4 boys and 3 girls such that all the girls are not together, is:

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  9. 59
    Let the smallest value of k∈Nk \in\mathbb{N}, for which the coefficient of x3x^{3} in (1+x)3+(1+x)4+(1+x)5+…(1 + x)^{3}+ (1 + x)^{4}+ (1 + x)^{5}+ \ldots +(1+x)99+(1+kx)100,x≠0+ (1 +x)^{99}+ (1 +kx)^{100},x\neq 0, is (43n+1014)\left( 43n+\frac{101}{4} \right) (  100C3\ ^{100}C_{3} ) for some n∈Nn \in\mathbb{N}, be pp. Then the value of pp +n is:

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  10. 60
    Suppose that the mean and median of the nonnegative numbers 21, 8, 17, a, 51,103,b,13,6751,103,b,13,67, (a>b)(a > b), are 40 and 21, respectively. If the mean deviation about the median is 26 , then 2a2a is equal to :

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  11. 61
    Let the line L1:x+3=0L_{1}:x + 3 = 0 intersect the lines L2:x−y=0L_{2}:x - y = 0 and L3:3x+y=0L_{3}:3x + y = 0 at the points A and B , respectively. Let the bisector of the obtuse angle between the lines L2L_{2} and L3L_{3} intersect the line L1L_{1} at the point C . Then BC2:AC2BC^{2}:AC^{2} is equal to :

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  12. 62
    Let the vertex A of a triangle ABC be (1, 2), and the mid-point of the side AB be (5,−1)(5, - 1). If the centroid of this triangle is (3,4)(3,4) and its circumcenter is (α,β)(\alpha,\beta), then 21(α+β)21(\alpha+\beta) is equal to:

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  13. 63
    Suppose the two chords, drawn from the point (1, 2) on the circle x2+y2+x−3y=0x^{2}+y^{2}+x- 3y= 0 are bisected by the y -axis. If the other ends of these chords are R and S , and the midpoint of the line segment RS is (α,β)(\alpha,\beta), then 6(α+β)6(\alpha+\beta) is equal to :

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  14. 64
    A line with direction ratios 1,−1,21, - 1,2 intersects the lines x2=y3=z+13\frac{x}{2}=\frac{y}{3}=\frac{z + 1}{3} and x+1−1=y−21=z4\frac{x + 1}{- 1}=\frac{y - 2}{1}=\frac{z}{4} at the points P and Q , respectively. If the length of the line segment PQ is α\alpha, then 225α2225\alpha^{2} is equal to:

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  15. 65
    The square of the distance of the point ( −2,−8,6- 2, - 8,6 ) from the line x−11=y−12=z−1\frac{x- 1}{1}=\frac{y- 1}{2}=\frac{z}{- 1} along the line x+51=y+5−1=z2\frac{x+ 5}{1}=\frac{y+ 5}{- 1}=\frac{z}{2} is equal to:

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  16. 66
    If y=tan⁡−1(3cos⁡x−4sin⁡x4cos⁡x+3sin⁡x)+2tan⁡−1(x1+1−x2)y=\tan^{- 1}\left( \frac{3\cos x- 4\sin x}{4\cos x+ 3\sin x} \right)+ 2\tan^{- 1}\left( \frac{x}{1 +\sqrt{1 -x^{2}}} \right), then dydx\frac{dy}{dx} at x=32x =\frac{\sqrt{3}}{2} is equal to :

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  17. 67
    Let f be a real polynomial of degree n such that f(x)=f′(x)f′′(x)f(x) =f^{'}(x)f^{''}(x), for all x∈Rx \in\mathbb{R}. If f(0)=0f(0) = 0, then 36 (f′(2)+f′′(2)+∫02f(x)dx)\left( f^{'}(2) +f^{''}(2) +\int_{0}^{2} f(x)dx \right) is equal to :

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  18. 68
    The area of the region {(x,y):y≤π−∣x∣, y≤∣xsin⁡x∣, y≥0}\{(x,y):y \leq\pi- |x|,\ y \leq |x\sin x|,\ y \geq 0\} is :

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  19. 69
    Let ∫−22(∣sin⁡x∣+[xsin⁡x])dx=2(3−cos⁡2)+β\int_{- 2}^{2} (|\sin x| + \lbrack x\sin x\rbrack)dx= 2(3 - \cos2) +\beta, where [⋅]\lbrack \cdot \rbrack is the greatest integer function . Then βsin⁡(β2)\beta\sin\left( \frac{\beta}{2} \right) equals.

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  20. 70
    Let y=y(x)y=y(x) be the solution of the differential equation dydx=(1+x+x2)(1−y+y2),y(0)=12\frac{dy}{dx}=\left( 1 + x +x^{2} \right)\left( 1 - y +y^{2} \right),y(0) =\frac{1}{2}. Then ( 2y(1)−12y(1) - 1 ) is equal to

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  21. 71
    A coin is tossed 8 times. If the probability that exactly 4 heads appear in the first six tosses and exactly 3 heads appear in the last five tosses is p , then 96p is equal to ____\_\_\_\_ .
  22. 72
    Consider the parabola P:y2=4kxP:y^{2} = 4kx and the ellipse E:x2a2+y2 b2=1E:\frac{x^{2}}{a^{2}} + \frac{y^{2}}{{\text{ }b}^{2}} = 1. Let the line segment joining the points of intersection of P and E be their latus rectums. If the eccentricity of E is e , then e2+22e^{2} + 2\sqrt{2} is equal to ____\_\_\_\_ .
  23. 73
    If A=sin⁡3∘cos⁡9∘+sin⁡9∘cos⁡27∘+sin⁡27∘cos⁡81∘A = \frac{\sin 3^{\circ}}{\cos 9^{\circ}} + \frac{\sin 9^{\circ}}{\cos 27^{\circ}} + \frac{\sin 27^{\circ}}{\cos 81^{\circ}} and B=tan⁡81∘−tan⁡3∘B = \tan81^{\circ} - \tan3^{\circ}, then BA\frac{B}{A} is equal to ____\_\_\_\_ .
  24. 74
    Let a→k=(tan⁡θk)i^+j^{\overrightarrow{a}}_{k} = \left( \tan\theta_{k} \right)\widehat{i} + \widehat{j} and b→k=i^−(cot⁡θk)j^{\overrightarrow{b}}_{k} = \widehat{i} - \left( \cot\theta_{k} \right)\widehat{j}, where θk=2k−1π2n+1\theta_{k} = \frac{2^{k - 1}\pi}{2^{n} + 1}, for some n∈N,n>5n \in \mathbb{N},n > 5. Then the value of ∑k=1n∣a→k∣2∑k=1n∣b→k∣2\frac{\sum_{k = 1}^{n} \left| {\overrightarrow{a}}_{k} \right|^{2}}{\sum_{k = 1}^{n} \left| {\overrightarrow{b}}_{k} \right|^{2}} is ____\_\_\_\_ .
  25. 75
    The number of points, at which the function f(x)=max{6x,2+3x2}+∣x−1∣cos⁡∣x2−14∣,x∈(−π,π)f(x) = max\left\{ 6x,2 + 3x^{2} \right\} + |x - 1|\cos\left| x^{2} - \frac{1}{4} \right|,x \in ( - \pi,\pi), is not differentiable, is ____\_\_\_\_ .

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