Mathematics & Statistics Part 2
- 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 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 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
- 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
- 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
- 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 . 1
- 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 . 1
- 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 . 1
- 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 . 1
- 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 . 1
- 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) . 1
- \(\int_0^{\pi/2}\frac{\sin x}{\sin x+\cos x}\,dx=\) Answer: C. π/4 Using x→fracπ2-x , I=∫cos x/sin x+cos xdx ; adding, 2I=fracπ2 . 1
- \(\int_0^1 x(1-x)^5\,dx=\) Answer: B. 1/42 Using x→1-x : ∫₀¹(1-x)x⁵dx=1/6-1/7=frac142 . 1
- 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). 1
- If \(y=x^x\), then \(\frac{dy}{dx}\) at \(x=e\) is Answer: B. 2e^e dy/dx=x^x(1+ln x) ; at x=e : e^e(1+1)=2e^e . 1
- If \(x=a\cos^3t\) and \(y=a\sin^3t\), then \(\frac{dy}{dx}=\) Answer: A. -tan t dy/dx=3asin²tcos t/-3acos²tsin t=-tan t . 1
- 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 . 1
- \(\int\frac{dx}{x^2+4x+13}=\) Answer: A. 1/3tan⁻¹(x+2/3)+C x²+4x+13=(x+2)²+3² ; use ∫dx/u²+a²=1/atan⁻¹u/a . 1
- \(\int e^x\left(\frac1x-\frac1{x^2}\right)dx=\) Answer: A. e^x/x+C Form ∫ e^x(f+f')dx=e^xf+C with f=1/x . 1
- \(\int\frac{dx}{1+e^x}=\) Answer: C. x-ln(1+e^x)+C 1/1+e^x=1-e^x/1+e^x ; integrate term by term. 1