All questions

• Testnwiw s s kw s skw a am Chapter 1 Relations and Functions no answer
• The general solution of \(\frac{dy}{dx}+\frac yx=x^2\) is Answer: B. xy=x⁴/4+C I.F. =e^(∫ dx/x)=x ; d/dx(xy)=x³⇒ xy=x⁴/4+C . Differential Equations 1 answer
• A bacterial population grows at a rate proportional to its size and doubles in 2 hours. The time in which it becomes 8 times the original is Answer: B. 6 hours N=N₀e^kt , e^2k=2 . 8=2³⇒ t=3×2=6 h. Differential Equations 1 answer
• A random variable \(X\) has \(P(X=x)=kx\) for \(x=1,2,3,4\). Then \(E(X)\) is Answer: C. 3 k(1+2+3+4)=1⇒ k=frac110 ; E(X)=1+4+9+16/10=3 . Probability Distributions 1 answer
• The p.d.f. of \(X\) is \(f(x)=kx^2\) for \(0\le x\le3\) and 0 otherwise. Then \(P(X\le1)\) is Answer: C. frac127 ∫₀³kx²dx=9k=1⇒ k=1/9 ; P(X≤1)=1/9·1/3=frac127 . Probability Distributions 1 answer
• A fair coin is tossed 6 times. The probability of getting exactly 3 heads is Answer: C. 5/16 binom63(1/2)⁶=20/64=frac516 . Binomial Distribution 1 answer
• The mean and variance of a binomial distribution are 4 and 3 respectively. The number of trials n is Answer: D. 16 q=3/4⇒ p=1/4 ; n=4/1/4=16 . Binomial Distribution 1 answer
• The value of \((1+i)^8\) is Answer: C. 16 (1+i)²=2i⇒(1+i)⁸=(2i)⁴=16 . Complex Numbers 1 answer
• If \(\omega\) is a complex cube root of unity, then \((1+\omega-\omega^2)^3=\) Answer: D. −8 1+ω=-ω²⇒(-2ω²)³=-8ω⁶=-8 . Complex Numbers 1 answer
• The number of distinct arrangements of the letters of the word BANANA is Answer: C. 60 6!/3! 2!=60 (A repeated 3 times, N twice). Permutations and combinations 1 answer
• A committee of 5 is to be formed from 6 men and 4 women. The number of ways in which it contains at least 3 women is Answer: B. 66 binom43binom62+binom44binom61=60+6=66 . Permutations and combinations 1 answer
• The distance between the parallel lines \(3x+4y=5\) and \(6x+8y=25\) is Answer: B. 3/2 Write the second as 3x+4y=12.5 : d=|12.5-5|/5=1.5 . Straight Line 1 answer
• The centre and radius of the circle \(x^2+y^2-6x+8y=0\) are Answer: B. (3,-4) and 5 Centre (-g,-f)=(3,-4) , r=√(9+16)=5 . Circle 1 answer
• The eccentricity of the hyperbola \(\frac{x^2}{16}-\frac{y^2}{9}=1\) is Answer: D. 5/4 e=√(1+b²/a²)=sqrt1+frac916=5/4 . Conic Sections 1 answer
• The length of the latus rectum of the ellipse \(\frac{x^2}{25}+\frac{y^2}{16}=1\) is Answer: A. 32/5 2b²/a=2×16/5=32/5 . Conic Sections 1 answer
• \(\lim_{x\to0}\frac{1-\cos4x}{x^2}=\) Answer: B. 8 1-cos4x=2sin²2x ; lim2sin²2x/x²=2·4=8 . Limits 1 answer
• The value of \(\cos15^\circ-\sin15^\circ\) is Answer: A. 1/sqrt2 cos15^°-sin15^°=sqrt2cos(15^°+45^°)=sqrt2·1/2=frac1sqrt2 . Trigonometry II 1 answer
• In Young's double slit experiment, slit separation is 0.5 mm, the screen is 1 m away and \(\lambda=600\) nm. The fringe width is Answer: A. 1.2 mm β=λ D/d=frac600×10⁻⁹×10.5×10⁻³=1.2×10⁻³ m. Wave Optics 1 answer
• The polarising (Brewster) angle for a glass of refractive index \(\sqrt3\) is Answer: D. 60° tanθ_p=μ=sqrt3⇒θ_p=60^° . Wave Optics 1 answer
• Light of wavelength 500 nm falls normally on a single slit of width 0.2 mm. The width of the central maximum on a screen 2 m away is Answer: D. 10 mm W=2λ D/a=frac2×5×10⁻⁷×22×10⁻⁴=10⁻² m =10 mm. Wave Optics 1 answer
• A parallel plate capacitor has capacitance \(C\) with air between the plates. A dielectric slab of dielectric constant 4 and thickness equal to half the plate separation is inserted between the plates. The new capacitance is Answer: B. 1.6 C C'=ε₀A/d-t+t/K=ε₀A/d/2+d/8=8/5·ε₀A/d=1.6C . Electrostatics 1 answer
• The energy stored in a 10 μF capacitor charged to 100 V is Answer: A. 0.05 J U=1/2CV²=1/2×10⁻⁵×10⁴=0.05 J. Electrostatics 1 answer
• Capacitors of 3 μF and 6 μF are connected in series across a 90 V supply. The potential difference across the 3 μF capacitor is Answer: A. 60 V C_s=2 μF, Q=180 μC, V₃=180/3=60 V. Electrostatics 1 answer
• In a metre bridge, a 2 Ω resistance is in the left gap and an unknown resistance \(X\) in the right gap. The null point is at 40 cm from the left end. Then \(X\) is Answer: D. 3 Ω 2/X=40/60⇒ X=3\ Ω . Current Electricity 1 answer