12th Science Maharashtra Board
In this category students supposed to ask questions which belong to std 12th maharashtra board Science
- 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
- 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