All questions

• If \(x^2+y^2=25\), then \(\frac{d^2y}{dx^2}\) at the point \((3,4)\) is Answer: D. -25/64 y'=-x/y=-3/4 ; y''=-y-xy'/y²=-4+9/4/16=-25/64 . Differentiation 1 answer
• The absolute minimum value of \(f(x)=x^3-3x\) on \([0,3]\) is Answer: C. −2 f'(x)=3x²-3=0⇒ x=1 . f(0)=0,\ f(1)=-2,\ f(3)=18 . Minimum =-2 . Applications of Derivatives 1 answer
• The radius of a circle increases at 0.5 cm/s. The rate of increase of its area when the radius is 4 cm is Answer: A. 4π cm2/s dA/dt=2π rdr/dt=2π(4)(0.5)=4π . Applications of Derivatives 1 answer
• The equation of the tangent to \(y=x^2-2x+3\) at \(x=2\) is Answer: B. y=2x-1 Point (2,3); slope =2x-2=2 . Tangent: y-3=2(x-2)⇒ y=2x-1 . Applications of Derivatives 1 answer
• The value of \(c\) in Lagrange's mean value theorem for \(f(x)=x^3\) on \([0,3]\) is Answer: D. sqrt3 3c²=27-0/3=9⇒ c=sqrt3in(0,3) . Applications of Derivatives 1 answer
• \(\int\frac{dx}{x^2+4x+13}=\) Answer: A. 1/3tan⁻¹(x+2/3)+C x²+4x+13=(x+2)²+3² ; use ∫dx/u²+a²=1/atan⁻¹u/a . Indefinite Integration 1 answer
• \(\int e^x\left(\frac1x-\frac1{x^2}\right)dx=\) Answer: A. e^x/x+C Form ∫ e^x(f+f')dx=e^xf+C with f=1/x . Indefinite Integration 1 answer
• \(\int\frac{dx}{1+e^x}=\) Answer: C. x-ln(1+e^x)+C 1/1+e^x=1-e^x/1+e^x ; integrate term by term. Indefinite Integration 1 answer
• \(\int_0^{\pi/2}\frac{\sin x}{\sin x+\cos x}\,dx=\) Answer: C. π/4 Using x→fracπ2-x , I=∫cos x/sin x+cos xdx ; adding, 2I=fracπ2 . Definite Integration 1 answer
• \(\int_0^1 x(1-x)^5\,dx=\) Answer: B. 1/42 Using x→1-x : ∫₀¹(1-x)x⁵dx=1/6-1/7=frac142 . Definite Integration 1 answer
• The area enclosed between the curves \(y=x^2\) and \(y=x\) is Answer: B. 1/6 sq. unit ∫₀¹(x-x²)dx=1/2-1/3=1/6 . Application of Definite Integration 1 answer
• The area of the region bounded by the ellipse \(\frac{x^2}{16}+\frac{y^2}{9}=1\) is Answer: C. 12π sq. units Area =π ab=π·4·3=12π . Application of Definite Integration 1 answer
• The order and degree of the differential equation \(\left(\frac{d^2y}{dx^2}\right)^3+\left(\frac{dy}{dx}\right)^4+y=0\) are respectively Answer: B. 2 and 3 Highest derivative d²y/dx² (order 2), raised to power 3 (degree 3). Differential Equations 1 answer
• A solid sphere rolls without slipping on a horizontal surface. The fraction of its total kinetic energy that is rotational is Answer: B. 2/7 KE_rot=1/2·2/5 mR²ω²=1/5 mv² , KE_total=1/2mv²+1/5mv²=7/10mv² . Fraction =1/5/7/10=2/7 . Rotational Dynamics 1 answer
• A flywheel of moment of inertia 2 kg m2 is speeded up from 10 rad/s to 20 rad/s. The work done on it is Answer: D. 300 J W=Δ KE=1/2 I(ω₂²-ω₁²)=1/2(2)(400-100)=300 J. Rotational Dynamics 1 answer
• A curved road of radius 50 m is to be banked so that a car moving at 10 m/s needs no friction. The angle of banking is (g = 10 m/s2) Answer: C. tan⁻¹(0.2) tanθ=v²/rg=100/50×10=0.2 . Rotational Dynamics 1 answer
• A small stone tied to a string of length 0.9 m is whirled in a vertical circle. The minimum speed it must have at the lowest point to just complete the circle is (g = 10 m/s2) Answer: A. 6.7 m/s v_min=√(5gr)=√(5×10×0.9)=√(45)≈6.7 m/s. Rotational Dynamics 1 answer
• The excess pressure inside a soap bubble of radius 2 cm is (surface tension of soap solution = 0.03 N/m) Answer: C. 6 Pa A soap bubble has two surfaces: Δ P=4T/r=4×0.03/0.02=6 Pa. Mechanical Properties of Fluids 1 answer
• Eight identical rain drops, each falling with terminal velocity \(v\), coalesce to form a single drop. The terminal velocity of the big drop is Answer: A. 4v Volume conserved: R³=8r³⇒ R=2r . Since v_t∝ r² , new terminal velocity =4v . Mechanical Properties of Fluids 1 answer
• Water flows through a horizontal pipe whose cross-sectional area at a wide section is twice that at a narrow section. If the speed at the wide section is 2 m/s, the pressure difference between the wide and narrow sections is (density of water = 1000… Answer: C. 6000 Pa Continuity: v₂=2×2=4 m/s. Bernoulli: Δ P=1/2ρ(v₂²-v₁²)=500(16-4)=6000 Pa. Mechanical Properties of Fluids 1 answer
• The r.m.s. speed of the molecules of a gas at 27 °C is \(v\). The temperature at which the r.m.s. speed becomes \(2v\) is Answer: D. 927 °C v_rms∝√(T) : T₂=4×300=1200 K =927 °C. Kinetic Theory of Gases and Radiation 1 answer
• If the absolute temperature of a perfect black body is doubled, the energy radiated by it per second becomes Answer: B. 16 times Stefan's law: E∝ T⁴ , so 2⁴=16 times. Kinetic Theory of Gases and Radiation 1 answer
• The wavelength of maximum emission from the sun is 480 nm. Taking Wien's constant \(b=2.88\times10^{-3}\) m K, the surface temperature of the sun is Answer: C. 6000 K T=fracbλ_max=frac2.88×10⁻³480×10⁻⁹=6000 K. Kinetic Theory of Gases and Radiation 1 answer
• An ideal gas has density 1.2 kg/m3 at a pressure of \(1\times10^5\) Pa. The r.m.s. speed of its molecules is Answer: B. 500 m/s P=1/3ρ v_rms²⇒ v_rms=√(3P/ρ)=√(3×10⁵/1.2)=500 m/s. Kinetic Theory of Gases and Radiation 1 answer