All questions

• The vectors \(\hat i+2\hat j+3\hat k\), \(2\hat i+\hat j+\lambda\hat k\) and \(3\hat i+\hat j+2\hat k\) are coplanar if \(\lambda=\) Answer: C. 9/5 beginvmatrix1&2&3\\2&1&λ\\3&1&2endvmatrix=(2-λ)-2(4-3λ)+3(2-3)=5λ-9=0 . Vectors 1 answer
• If \(|\vec a|=2\), \(|\vec b|=3\) and \(|\vec a+\vec b|=\sqrt{19}\), the angle between \(\vec a\) and \(\vec b\) is Answer: B. π/3 19=4+9+2vec a·vec b⇒vec a·vec b=3=6cosθ⇒cosθ=1/2 . Vectors 1 answer
• The shortest distance between the lines \(\vec r=(\hat i+2\hat j+\hat k)+\lambda(\hat i-\hat j+\hat k)\) and \(\vec r=(2\hat i-\hat j-\hat k)+\mu(2\hat i+\hat j+2\hat k)\) is Answer: D. 3/sqrt2 vec b₁×vec b₂=-3hat i+3hat k (magnitude 3sqrt2 ); vec a₂-vec a₁=hat i-3hat j-2hat k ; d=|-3-6|/3sqrt2=3/sqrt2 . Line and Plane 1 answer
• The angle between the line \(\frac{x-1}{2}=\frac{y+2}{1}=\frac{z}{2}\) and the plane \(2x-y+2z=5\) is Answer: C. sin⁻¹(7/9) sinθ=|vec b·vec n|/|vec b||vec n|=|4-1+4|/3·3=7/9 . Line and Plane 1 answer
• The distance of the point \((2,1,-1)\) from the plane \(x-2y+4z=9\) is Answer: C. 13/√(21) d=|2-2-4-9|/√(1+4+16)=13/√(21) . Line and Plane 1 answer
• The maximum value of \(z=3x+2y\) subject to \(x+y\le4,\ x+3y\le6,\ x\ge0,\ y\ge0\) is Answer: C. 12 Corner points: (0,0)→0, (4,0)→12, (3,1)→11, (0,2)→4. Maximum =12 . Linear Programming 1 answer
• The minimum value of \(z=2x+5y\) subject to \(x+y\ge4,\ x+3y\ge6,\ x,y\ge0\) is Answer: C. 11 Corner points of the (unbounded) region: (6,0)→12, (3,1)→11, (0,4)→20. Since both coefficients are positive, the minimum is 11... Linear Programming 1 answer
• 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 . Differentiation 1 answer
• 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 . Differentiation 1 answer
• 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