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In an experiment to measure focal length (f) of convex lens, the least counts of the measuring scales for the position of object (u) and for the position of image (v) are $$\Delta u$$ and $$\Delta v$$, respectively. The error in the measurement of the focal length of the convex lens will be:
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The equation of stationary wave is: $$y = 2a\sin\left(\frac{2\pi nt}{\lambda}\right)\cos\left(\frac{2\pi x}{\lambda}\right)$$. Which of the following is NOT correct:
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A body travels 102.5 m in the $$n^{th}$$ second and 115.0 m in the $$(n+2)^{th}$$ second. The acceleration is:
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The co-ordinates of a particle moving in x-y plane are given by: $$x = 2 + 4t,\ y = 3t + 8t^2$$. The motion of the particle is:
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A wooden block, initially at rest on the ground, is pushed by a force which increases linearly with time t. Which of the following curve best describes acceleration of the block with time:
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If a rubber ball falls from a height h and rebounds upto the height of h/2. The percentage loss of total energy of the initial system as well as velocity of ball before it strikes the ground, respectively, are:
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A metal wire of uniform mass density having length L and mass M is bent to form a semicircular arc and a particle of mass m is placed at the centre of the arc. The gravitational force on the particle by the wire is:
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Given below are two statements: Statement I: When speed of liquid is zero everywhere, pressure difference at any two points depends on equation $$P_1 - P_2 = \rho g(h_2 - h_1)$$. Statement II: In venturi tube shown, $$2gh = v_1^2 - v_2^2$$. Choose the most appropriate answer from the options given below.
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The resistances of the platinum wire of a platinum resistance thermometer at the ice point and steam point are 8Ω and 10Ω respectively. After inserting in a hot bath of temperature 400°C, the resistance of platinum wire is:
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On Celsius scale the temperature of body increases by 40°C. The increase in temperature on Fahrenheit scale is:
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P-T diagram of an ideal gas having three different densities $$\rho_1, \rho_2, \rho_3$$ (in three different cases) is shown in the figure. Which of the following is correct:
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An infinitely long positively charged straight thread has a linear charge density $$\lambda\ Cm^{-1}$$. An electron revolves along a circular path having axis along the length of the wire. The graph that correctly represents the variation of the kinetic energy of electron as a function of radius of circular path from the wire is:
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To measure the internal resistance of a battery, potentiometer is used. For R = 10Ω, the balance point is observed at l = 500 cm and for R = 1Ω the balance point is observed at l = 400 cm. The internal resistance of the battery is approximately:
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An electron is projected with uniform velocity along the axis inside a current carrying long solenoid. Then:
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In an ac circuit, the instantaneous current is zero, when the instantaneous voltage is maximum. In this case, the source may be connected to: A. pure inductor. B. pure capacitor. C. pure resistor. D. combination of an inductor and capacitor. Choose the correct answer from the options given below :
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The electric field in an electromagnetic wave is given by $$\vec{E} = \hat{i}40\cos\omega(t - z/c)\ NC^{-1}$$. The magnetic field induction of this wave is (in SI unit):
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An effective power of a combination of 5 identical convex lenses which are kept in contact along the principal axis is 25D. Focal length of each of the convex lens is:
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Which figure shows the correct variation of applied potential difference (V) with photoelectric current (I) at two different intensities of light $$(I_1 < I_2)$$ of same wavelengths:
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Which of the following nuclear fragments corresponding to nuclear fission between neutron $$\binom{1}{0}n$$ and uranium isotope $$\left(_{92}^{235}U\right) $$ is correct:
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The value of net resistance of the network as shown in the given figure is:
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Two forces $$\vec{F_1}$$ and $$\vec{F_2}$$ are acting on a body. One force has magnitude thrice that of the other force and the resultant of the two forces is equal to the force of larger magnitude. The angle between $$\vec{F_1}$$ and $$\vec{F_2}$$ is $$\cos^{-1}\left(\frac{1}{n}\right)$$. The value of |n| is _____.
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A solid sphere and a hollow cylinder roll up without slipping on same inclined plane with same initial speed $$\upsilon$$. The sphere and the cylinder reaches upto maximum heights $$h_1$$ and $$h_2$$, respectively, above the initial level. The ratio $$h_1 : h_2$$ is $$\frac{n}{10}$$. The value of n is _____.
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An elastic spring under tension of 3 N has a length a. Its length is b under tension 2 N. For its length (3a − 2b), the value of tension will be ______ N.
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A soap bubble is blown to a diameter of 7 cm. 36960 erg of work is done in blowing it further. If surface tension of soap solution is 40 dyne/cm then the new radius is ______ cm. (Take $$\pi = \frac{22}{7}$$)
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An infinite plane sheet of charge having uniform surface charge density $$+\sigma_s\ C/m^2$$ is placed on x−y plane. Another infinitely long line charge having uniform linear charge density $$+\lambda_e\ C/m$$ is placed at z = 4 m plane and parallel to y-axis. If the magnitude values $$|\sigma_s| = 2|\lambda_e|$$, then at point (0, 0, 2), the ratio of magnitudes of electric field values due to sheet charge to that of line charge is $$\pi\sqrt{n} : 1$$. The value of n is _____.
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Twelve wires each having resistance 2Ω are joined to form a cube. A battery of emf 6 V is joined across point a and c. The voltage difference between e and f is ______ V.
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The magnetic field existing in a region is given by $$\vec{B} = 0.2(1 + 2x)\hat{k}\ T$$. A square loop of edge 50 cm carrying 0.5 A current is placed in x−y plane with its edges parallel to the x-y axes, as shown in figure. The magnitude of the net magnetic force experienced by the loop is ______ mN.
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A alternating current at any instant is given by $$i = [6 + \sqrt{56}\sin(100\pi t + \pi/3)]\ A$$. The rms value of the current is ______ A.
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Two wavelengths $$\lambda_1$$ and $$\lambda_2$$ are used in Young's double slit experiment. $$\lambda_1 = 450\ nm$$ and $$\lambda_2 = 650\ nm$$. The minimum order of fringe produced by $$\lambda_2$$ which overlaps with the fringe produced by $$\lambda_1$$ is n. The value of n is _____.
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A hydrogen atom changes its state from n = 3 to n = 2. Due to recoil, the percentage change in the wavelength of emitted light is approximately $$1 \times 10^{-n}$$. The value of n is _____. [Given Rhc = 13.6 eV, hc = 1242 eVnm, h = 6.6×10⁻³⁴ Js, mass of hydrogen atom = 1.6×10⁻²⁷ kg]
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Number of elements from the following that CANNOT form compounds with valencies which match with their respective group valencies is ______. B, C, N, S, O, F, P, Al, Si
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The correct order of first ionization enthalpy values of the following elements is: (A) O (B) N (C) Be (D) F (E) B. Choose the correct answer from the options given below :
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Which one of the following molecules has maximum dipole moment?
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Number of molecules/ions from the following in which the central atom is involved in sp³ hybridization is: $$NO_3^-,\ BCl_3,\ ClO_2^-,\ ClO_3$$
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Match List I with List II :
Choose the correct answer from the options given below :
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Which of the following nitrogen containing compound does not give Lassaigne's test?
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Which among the following is incorrect statement?
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Identify (B) and (C) and how are (A) and (C) related ?
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The Molarity (M) of an aqueous solution containing 5.85 g of NaCl in 500 mL water is: (Given: Molar Mass Na: 23 and Cl: 35.5 g mol⁻¹)
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What pressure (bar) of H₂ would be required to make emf of hydrogen electrode zero in pure water at 25°C?
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One of the commonly used electrode is calomel electrode. Under which of the following categories, calomel electrode comes?
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What will be the decreasing order of basic strength of the following conjugate bases? $$^-OH,\ R\bar{O},\ CH_3CO\bar{O},\ C\bar{l}$$
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The element which shows only one oxidation state other than its elemental form is:
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Number of complexes from the following with even number of unpaired "d" electrons is: $$[V(H_2O)_6]^{3+}$$, $$\ [Cr(H_2O)_6]^{2+}$$, $$\ [Fe(H_2O)_6]^{3+}$$, $$\ [Ni(H_2O)_6]^{3+}$$, $$\ [Cu(H_2O)_6]^{2+}$$. [Given atomic numbers: V=23, Cr=24, Fe=26, Ni=28, Cu=29]
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The correct sequence of ligands in the order of decreasing field strength is:
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Identify the correct set of reagents or reaction conditions 'X' and 'Y' in the following set of transformation:
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Given below are two statements: Statement I: Acidity of α-hydrogens of aldehydes and ketones is responsible for Aldol reaction. Statement II: Reaction between benzaldehyde and ethanal will NOT give Cross-Aldol product. Choose the most appropriate answer from the options given below :
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Identify the product in the following reaction :
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In the precipitation of the iron group (III) in qualitative analysis, ammonium chloride is added before adding ammonium hydroxide to:
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Which of the following is the correct structure of L-Glucose? (Four Fischer projection structures shown)
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The de-Broglie's wavelength of an electron in the 4th orbit is ______ $$\pi a_0$$. ($$a_0$$ = Bohr's radius)
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Number of molecules/species from the following having one unpaired electron is ______. $$O_2,\ O_2^{-1},\ NO,\ CN^{-1},\ O_2^{2-}$$
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The enthalpy of formation of ethane (C₂H₆) from ethylene by addition of hydrogen where the bond-energies of C−H, C−C, C=C, H−H are 414 kJ, 347 kJ, 615 kJ and 435 kJ respectively is ______ kJ.
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Only 2 mL of KMnO₄ solution of unknown molarity is required to reach the end point of a titration of 20 mL of oxalic acid (2M) in acidic medium. The molarity of KMnO₄ solution should be ______ M.
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The number of different chain isomers for C₇H₁₆ is ______.
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2.5 g of a non-volatile, non-electrolyte is dissolved in 100 g of water at 25°C. The solution showed a boiling point elevation by 2°C. Assuming the solute concentration is negligible with respect to the solvent concentration, the vapor pressure of the resulting aqueous solution is ______ mm of Hg (nearest integer). [Given: Kb = 0.52 K·kgmol⁻¹, 1 atm = 760 mm Hg, molar mass of water = 18 g mol⁻¹]
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Consider the following transformation involving first order elementary reaction in each step at constant
temperature as shown below.
Some details of the above reactions are listed below.
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Consider the following reaction: $$MnO_2 + KOH + O_2 \to A + H_2O$$. Product A in neutral or acidic medium disproportionates to give products B and C along with water. The sum of spin-only magnetic moment values of B and C is ______ BM. (nearest integer) [Given atomic number of Mn is 25]
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The number of the correct reaction(s) among the following is ______
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X g of ethylamine is subjected to reaction with NaNO₂/HCl followed by water; evolved dinitrogen gas which occupied 2.24 L volume at STP. X is ______ × 10⁻¹ g.
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If 2 and 6 are the roots of the equation $$ax^2 + bx + 1 = 0$$, then the quadratic equation whose roots are $$\frac{1}{2a+b}$$ and $$\frac{1}{6a+b}$$ is:
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Let α and β be the sum and the product of all the non-zero solutions of the equation $$(\bar{z})^2 + |z| = 0,\ z \in \mathbb{C}$$. Then $$4(\alpha^2 + \beta^2)$$ is equal to:
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There are 5 points $$P_1, P_2, P_3, P_4, P_5$$ on the side AB, excluding A and B, of a triangle ABC. Similarly there are 6 points $$P_6, P_7,\ldots, P_{11}$$ on the side BC and 7 points $$P_{12}, P_{13},\ldots, P_{18}$$ on the side CA of the triangle. The number of triangles, that can be formed using the points $$P_1, P_2,\ldots, P_{18}$$ as vertices, is:
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Let the first three terms 2, p and q, with q ≠ 2, of a G.P. be respectively the 7th, 8th and 13th terms of an A.P. If the $$5^{th}$$ term of the G.P. is the $$n^{th}$$ term of the A.P., then n is equal to:
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The sum of all rational terms in the expansion of $$\left(2^{\frac{1}{5}} + 5^{\frac{1}{3}}\right)^{15}$$ is equal to:
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The vertices of a triangle are A(−1, 3), B(−2, 2) and C(3, −1). A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is:
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A square is inscribed in the circle $$x^2 + y^2 - 10x - 6y + 30 = 0$$. One side of this square is parallel to y = x + 3. If $$(x_i, y_i)$$ are the vertices of the square, then $$\sum(x_i^2 + y_i^2)$$ is equal to:
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Let $$\alpha, \beta \in \mathbb{R}$$. Let the mean and the variance of 6 observations $$−3, 4, 7, −6, \alpha, \beta$$ be $$2$$ and $$23$$, respectively. The mean deviation about the mean of these 6 observations is:
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Let $$\alpha \in (0,\infty)$$ and $$A = \begin{bmatrix}1 & 2 & \alpha\\ 1 & 0 & 1\\ 0 & 1 & 2\end{bmatrix}$$. If $$\det(\text{adj}(2A-A^T)\cdot\text{adj}(A-2A^T)) = 2^8$$, then $$(\det(A))^2$$ is equal to:
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If the system of equations $$x + (\sqrt{2}\sin\alpha)y + (\sqrt{2}\cos\alpha)z = 0$$, $$x + (\cos\alpha)y + (\sin\alpha)z = 0$$, $$x + (\sin\alpha)y - (\cos\alpha)z = 0$$ has a non-trivial solution, then $$\alpha \in \left(0,\frac{\pi}{2}\right)$$ is equal to:
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If the domain of the function $$\sin^{-1}\left(\frac{3x-22}{2x-19}\right) + \log_e\left(\frac{3x^2-8x+5}{x^2-3x-10}\right)$$ is $$(\alpha, \beta]$$, then $$3\alpha + 10\beta$$ is equal to:
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Let the sum of the maximum and the minimum values of the function $$f(x) = \frac{2x^2-3x+8}{2x^2+3x+8}$$ be $$\frac{m}{n}$$, where gcd(m, n) = 1. Then m + n is equal to:
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Let $$f : R to R$$ be a function given by $$f(x) = \begin{cases}\frac{1-\cos 2x}{x^2}, & x < 0\\ \alpha, & x = 0\\ \frac{\beta\sqrt{1-\cos x}}{x}, & x > 0\end{cases}$$, where $$\alpha, \beta \in R$$. If f is continuous at x = 0, then $$\alpha^2 + \beta^2$$ is equal to:
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Let $$f(x) = x^5 + 2e^{x/4}$$ for all $$x \in R$$. Consider a function g(x) such that $$(g \circ f)(x) = x$$ for all $$x \in R$$. Then the value of $$8g'(2)$$ is:
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Let $$f(x) = \begin{cases}-2, & -2 \le x \le 0\\ x-2, & 0 < x \le 2\end{cases}$$ and $$h(x) = f(|x|) + |f(x)|$$. Then $$\int_{-2}^{2}h(x)dx$$ is equal to:
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One of the points of intersection of the curves $$y = 1 + 3x - 2x^2$$ and $$y = \frac{1}{x}$$ is $$\left(\frac{1}{2}, 2\right)$$. Let the area of the region enclosed by these curves be $$\frac{1}{24}(l\sqrt{5}+m) - n\log_e(1+\sqrt{5})$$, where $$l, m, n \in N$$. Then $$l + m + n$$ is equal to:
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If the solution $$y = y(x)$$ of the differential equation $$(x^4 + 2x^3 + 3x^2 + 2x + 2)dy - (2x^2 + 2x + 3)dx = 0$$ satisfies $$y(-1) = -\frac{\pi}{4}$$, then y(0) is equal to:
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Let a unit vector which makes an angle of 60° with $$2\hat{i} + 2\hat{j} - \hat{k}$$ and angle 45° with $$\hat{i} - \hat{k}$$ be $$\overrightarrow{C}$$. Then $$\overrightarrow{C} + \left(-\frac{1}{2}\hat{i} + \frac{1}{3\sqrt{2}}\hat{j} - \frac{\sqrt{2}}{3}\hat{k}\right)$$ is:
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Let the point, on the line passing through the points P(1, −2, 3) and Q(5, −4, 7), farther from the origin and at distance of 9 units from the point P, be $$(\alpha, \beta, \gamma)$$. Then $$\alpha^2 + \beta^2 + \gamma^2$$ is equal to:
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Three urns A, B and C contain 7 red, 5 black; 5 red, 7 black and 6 red, 6 black balls, respectively. One of the urn is selected at random and a ball is drawn from it. If the ball drawn is black, then the probability that it is drawn from urn A is:
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Let $$a = 1 + \frac{^2C_2}{3!} + \frac{^3C_2}{4!} + \frac{^4C_2}{5!} + \ldots$$, $$b = 1 + \frac{^1C_0 + ^1C_1}{1!} + \frac{^2C_0+^2C_1+^2C_2}{2!} + \frac{^3C_0+^3C_1+^3C_2+^3C_3}{3!} + \ldots$$. Then $$\frac{2b}{a^2}$$ is equal to ______.
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Let the length of the focal chord PQ of the parabola $$y^2 = 12x$$ be 15 units. If the distance of PQ from the origin is p, then $$10p^2$$ is equal to ______.
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Let A be a square matrix of order 2 such that |A| = 2 and the sum of its diagonal elements is −3. If the points (x, y) satisfying $$A^2 + xA + yI = O$$ lie on a hyperbola whose length of semi major axis is x and semi minor axis is y, eccentricity is e and the length of the latus rectum is l, then $$81(e^4 + l^2)$$ is equal to ______.
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If $$\lim_{x\to 1}\frac{(5x+1)^{1/3}-(x+5)^{1/3}}{(2x+3)^{1/2}-(x+4)^{1/2}} = \frac{m\sqrt{5}}{n(2n)^{2/3}}$$, where gcd(m, n) = 1, then $$8m + 12n$$ is equal to ______.
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In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let m and n respectively be the least and the most number of students who studied all the three subjects. Then $$m + n$$ is equal to ______.
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Let A be a 3×3 matrix of non-negative real elements such that $$A\begin{bmatrix}1\\1\\1\end{bmatrix} = 3\begin{bmatrix}1\\1\\1\end{bmatrix}$$. Then the maximum value of det(A) is ______.
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If $$\int_0^{\pi/4}\frac{\sin^2 x}{1+\sin x\cos x}dx = \frac{1}{a}\log_e\left(\frac{a}{3}\right) + \frac{\pi}{b\sqrt{3}}$$, where $$a, b \in N$$, then $$a + b$$ is equal to ______.
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Let the solution $$y = y(x)$$ of the differential equation $$\frac{dy}{dx} - y = 1 + 4\sin x$$ satisfy $$y(\pi) = 1$$. Then $$y\left(\frac{\pi}{2}\right) + 10$$ is equal to ______.
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Let ABC be a triangle of area $$15\sqrt{2}$$ and the vectors $$\overrightarrow{AB} = \hat{i} + 2\hat{j} - 7\hat{k}$$, $$\overrightarrow{BC} = a\hat{i} + b\hat{j} + c\hat{k}$$ and $$\overrightarrow{AC} = 6\hat{i} + d\hat{j} - 2\hat{k}$$, d > 0. Then the square of the length of the largest side of the triangle ABC is ______.
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If the shortest distance between the lines $$\frac{x+2}{2} = \frac{y+3}{3} = \frac{z-5}{4}$$ and $$\frac{x-3}{1} = \frac{y-2}{-3} = \frac{z+4}{2}$$ is $$\frac{38}{3\sqrt{5}}k$$, and $$\int_0^k [x^2]dx = \alpha - \sqrt{\alpha}$$, where [x] denotes the greatest integer function, then $$6\alpha^3$$ is equal to ______.
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