#mht-cet pcm
- 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 . 1
- 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 . 1
- 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. 1
- 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 . 1
- 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 . 1
- 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 . 1
- The value of \((1+i)^8\) is Answer: C. 16 (1+i)²=2i⇒(1+i)⁸=(2i)⁴=16 . 1
- 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 . 1
- 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). 1
- 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 . 1
- 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 . 1
- 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 . 1
- 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 . 1
- 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 . 1
- \(\lim_{x\to0}\frac{1-\cos4x}{x^2}=\) Answer: B. 8 1-cos4x=2sin²2x ; lim2sin²2x/x²=2·4=8 . 1
- The value of \(\cos15^\circ-\sin15^\circ\) is Answer: A. 1/sqrt2 cos15^°-sin15^°=sqrt2cos(15^°+45^°)=sqrt2·1/2=frac1sqrt2 . 1
- 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 . 1
- 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. 1
- 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 . 1
- 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. 1
- 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. 1
- The displacement of a particle in SHM is \(x=5\sin\left(\pi t+\frac{\pi}{3}\right)\) cm. Its maximum speed is Answer: B. 5π cm/s v_max=Aω=5×π=5π cm/s. 1
- The polarising (Brewster) angle for a glass of refractive index \(\sqrt3\) is Answer: D. 60° tanθ_p=μ=sqrt3⇒θ_p=60^° . 1
- 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. 1