Mathematics & Statistics Part 1
- 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 . 1
- 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 . 1
- 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) . 1
- 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 . 1
- 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... 1
- The negation of \(p\rightarrow q\) is Answer: B. pwedgesim q p→ qequivsim pvee q ; its negation is pwedgesim q . 1
- Which of the following statements is a tautology? Answer: A. (pwedge q)→ p If pwedge q is true then p is true, so the implication is always true. The others fail for (T,F), (F,T) and always,... 1
- If \(A=\begin{bmatrix}1&2\\3&4\end{bmatrix}\) satisfies \(A^2-5A-2I=O\), then \(A^{-1}\) equals Answer: A. 1/2(A-5I) A(A-5I)=2I⇒ A⁻¹=1/2(A-5I) . 1
- If \(A\) is a square matrix of order 3 with \(|A|=4\), then \(|\operatorname{adj}(2A)|\) is Answer: B. 1024 |2A|=2³·4=32 ; |adjB|=|B|² for order 3, so 32²=1024 . 1
- The system \(x+y+z=1,\ x+2y+3z=2,\ x+3y+kz=3\) has infinitely many solutions when \(k=\) Answer: B. 5 Δ=k-5=0⇒ k=5 . Then E₂-E₁ and E₃-E₂ both give y+2z=1 , so the system is consistent. 1
- The acute angle between the lines represented by \(x^2-4xy+y^2=0\) is Answer: B. 60° tanθ=|2√(h²-ab)/a+b|=2√(4-1)/2=sqrt3⇒θ=60^° . 1
- If the lines represented by \(kx^2+6xy-7y^2=0\) are perpendicular, then \(k=\) Answer: A. 7 Perpendicular when a+b=0 : k-7=0⇒ k=7 . 1
- The joint equation of the lines through the origin perpendicular to the lines \(x^2-5xy+4y^2=0\) is Answer: A. 4x²+5xy+y²=0 Given lines (x-y)(x-4y)=0 have slopes 1,1/4 . Perpendicular slopes -1,-4 : (x+y)(4x+y)=0 . 1
- The general solution of \(2\cos^2\theta+3\sin\theta=0\) is Answer: D. θ=nπ+(-1)ⁿ(-π/6),\ ninmathbb Z 2-2sin²θ+3sinθ=0⇒(2sinθ+1)(sinθ-2)=0⇒sinθ=-1/2 . 1
- The value of \(\tan^{-1}2+\tan^{-1}3\) is Answer: D. 3π/4 Since 2·3>1 and both are positive, the sum =π+tan⁻¹2+3/1-6=π-fracπ4=3π/4 . 1
- In \(\triangle ABC\), \(a=3,\ b=5,\ c=7\). Then angle \(C\) is Answer: A. 2π/3 cos C=9+25-49/2·3·5=-1/2⇒ C=2π/3 . 1
- The area of the parallelogram whose diagonals are \(3\hat i+\hat j-2\hat k\) and \(\hat i-3\hat j+4\hat k\) is Answer: B. 5sqrt3 vec d₁×vec d₂=-2hat i-14hat j-10hat k , magnitude √(300)=10sqrt3 . Area =1/2|vec d₁×vec d₂|=5sqrt3 . 1
- 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 . 1
- 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 . 1