8 Questions with detailed solutions.
Consider the function \(f:(0,\infty)\to(-\infty,\infty)\) given by
$$f(x)=\sqrt{x}\,\log_e(x)-x+1.$$
Then which one of the following statements is TRUE?
Step 1: Find the first derivative
To determine local extrema, first find the critical points of \(f(x)\) by computing its first derivative, \(f'(x)\).
\[ f(x)=\sqrt{x}\,\log_e(x) \]
Applying the product rule,
\[ f'(x) =\frac{1}{2\sqrt{x}}\log_e(x)+\sqrt{x}\cdot\frac{1}{x} =\frac{\log_e(x)+2}{2\sqrt{x}} \]
Step 2: Find the second derivative
To study the monotonicity of \(f'(x)\) and test statement (A), compute the second derivative.
\[ f''(x) = -\frac{\log_e(x)}{4x^{3/2}} \]
Step 3: Analyze intervals for the derivatives
Examine the sign of \(f''(x)\) to understand the behavior of \(f'(x)\).
For the interval \((0,1)\):
Since \[ \log_e(x)<0, \] we have \[ -\log_e(x)>0. \] Therefore, \[ f''(x)>0. \] Hence \(f'(x)\) is strictly increasing on \((0,1)\). This shows statement (A) is false.
At \(x=1\):
\[ f''(1)=0. \]
For the interval \((1,\infty)\):
Since \[ \log_e(x)>0, \] we have \[ -\log_e(x)<0. \] Therefore, \[ f''(x)<0. \] Hence \(f'(x)\) is strictly decreasing on \((1,\infty)\).
Step 4: Determine local extrema
Since \(f'(x)\) increases on \((0,1)\) and decreases on \((1,\infty)\), its maximum value occurs at \(x=1\).
\[ f'(1) = \frac{\log_e(1)+2}{2} = 1 \]
Now solve
\[ f'(x)=0 \]
\[ \frac{\log_e(x)+2}{2\sqrt{x}}=0 \]
\[ \log_e(x)=-2 \]
\[ x=e^{-2}. \]
Observe the sign change:
\[ f'(x)<0 \quad\text{for}\quad 0<x<e^{-2} \]
\[ f'(x)>0 \quad\text{for}\quad x>e^{-2} \]
Thus \(f'(x)\) changes from negative to positive at \(x=e^{-2}\), so \(f\) has a local minimum at
\[ x=e^{-2}. \]
Therefore the statement
(D) The function has neither a local maximum nor a local minimum
is false.
Consider the function \(f:(0,\infty)\to(-\infty,\infty)\) given by $$f(x)=\sqrt{x}\log_e(x)-x+1.$$ Then which one of the following statements is TRUE?
Which one of the following matrices can be obtained by performing elementary row transformations on the \(3\times3\) identity matrix?
Considering only the principal values of the inverse trigonometric functions, the value of $$\cot^{-1}(\cot(-11))+10\sin\left(2\cos^{-1}\left(\frac{1}{\sqrt2}\right)\right)+10\sin\left(2\tan^{-1}(2)\right)$$ is
Suppose that Box I contains 6 red balls and 9 green balls, and Box II contains 8 red balls and 12 green balls. All the balls of Box I and Box II are mixed together and a ball is chosen at random from them. Let \(E_1\) be the event that the ball chosen belonged to Box I and let \(E_2\) be the event that the ball chosen belonged to Box II. Let \(F_1\) be the event that the ball chosen is red and let \(F_2\) be the event that the ball chosen is green. Then which of the following statements is (are) TRUE?
Let \(P\) be the plane such that it contains the straight line $$\frac{x-1}{2}=\frac{y-3}{3}=\frac{z+2}{1}$$ and is perpendicular to the plane $$x+2y+3z=4.$$ Let \(P_1\) be the plane which passes through the point \((4,2,2)\) and is parallel to \(P\). Then which of the following statements is (are) TRUE?
Let \(\mathbb{R}\) denote the set of all real numbers. Let \(f:\mathbb{R}\to\mathbb{R}\) be an arbitrary function and let \(g:\mathbb{R}\to\mathbb{R}\) be the function defined by $$g(x)=x f(x),\quad \forall x\in\mathbb{R}.$$ Then which of the following statements is (are) TRUE?
Consider the matrix $$M=\begin{bmatrix}2&-1\\1&0\end{bmatrix}.$$ Let \(p,q,r,s,a,b,c,d\) be integers such that $$M^{26}=\begin{bmatrix}p&q\\r&s\end{bmatrix}$$ and $$\sum_{k=1}^{26}M^k=\begin{bmatrix}a&b\\c&d\end{bmatrix}.$$ Then which of the following statements is (are) TRUE?
The only critical point is x = 1. The derivative remains negative on both sides of x = 1, so there is no sign change and therefore no local maximum or local minimum. Hence option (D) is correct.
A matrix obtained from the identity matrix using elementary row operations must be invertible. Option (A) has rank 1 and is singular. Option (C) has determinant 0. Option (D) has determinant 0. Option (B) has determinant 1 and is invertible, hence it can be obtained from the identity matrix by a sequence of elementary row operations. Therefore, option (B) is correct.
Using principal values, \(\cot^{-1}(\cot(-11))=4\pi-11\). Also, \(\cos^{-1}(1/\sqrt2)=\pi/4\), so \(10\sin(2\cdot\pi/4)=10\sin(\pi/2)=10\). Further, if \(\theta=\tan^{-1}(2)\), then \(\sin(2\theta)=\frac{2\tan\theta}{1+\tan^2\theta}=\frac{4}{5}\). Hence \(10\sin(2\tan^{-1}(2))=8\). Therefore the value is \((4\pi-11)+10+8=4\pi+7\). Thus option (C) is correct.
Total balls = 15 + 20 = 35. Red balls = 6 + 8 = 14 and green balls = 9 + 12 = 21. Therefore P(F_1)=14/35=2/5 and P(F_2)=21/35=3/5. Also P(E_1)=15/35=3/7 and P(E_2)=20/35=4/7. Since P(F_1|E_1)=6/15=2/5=P(F_1), the events E_1 and F_1 are independent, so (A) is true. Similarly, P(F_2|E_2)=12/20=3/5=P(F_2), hence E_2 and F_2 are also independent, making (B) false. Further, P(F_1|E_1)=6/15=2/5 and P(F_1|E_2)=8/20=2/5, so (C) is true. Finally, P(F_1|E_1)=2/5 while P(F_2|E_2)=3/5, therefore (D) is false. Hence the correct options are (A) and (C).
The line has direction vector \((2,3,1)\). The given plane has normal vector \((1,2,3)\). Since plane P contains the line and is perpendicular to the given plane, a normal vector to P is \((2,3,1)\times(1,2,3)=(7,-5,1)\). Using point \((1,3,-2)\) on the line, the equation of P is \(7x-5y+z=-10\). Hence (A) is true. The distance between P and the parallel plane through \((4,2,2)\) is \(\frac{|7(4)-5(2)+2+10|}{\sqrt{75}}=\frac{30}{5\sqrt3}=2\sqrt3\), so (B) is false. The distance of P from the origin is \(\frac{10}{\sqrt{75}}=\frac{2}{\sqrt3}\), so (C) is false. The angle between planes equals the angle between their normals. Normals are \((7,-5,1)\) and \((2,2,1)\). Therefore \(\cos\theta=\frac{|14-10+1|}{\sqrt{75}\cdot3}=\frac{5}{15\sqrt3}=\frac1{3\sqrt3}\), so (D) is true.
Statement (A) is false because g(0)=0·f(0)=0 only if g is defined that way, and continuity is not guaranteed for arbitrary f. For (B), if f is continuous at 0, then f(0) exists and is finite. Hence \(g'(0)=\lim_{h\to0}\frac{hf(h)-0}{h}=\lim_{h\to0}f(h)=f(0)\), so g is differentiable at 0. Thus (B) is true. For (C), consider a function f that is not continuous at 0 but has a finite limit as x→0; then g may still be differentiable. Hence (C) is false. For (D), if g is differentiable at 0, then \(\lim_{h\to0}\frac{g(h)-g(0)}{h}=\lim_{h\to0}f(h)\) exists, implying that \(\lim_{x\to0}f(x)\) exists. Therefore (D) is true.
The characteristic polynomial of M is $$\lambda^2-2\lambda+1=(\lambda-1)^2.$$ Hence M has the single eigenvalue 1 with algebraic multiplicity 2 and is similar to the Jordan block $$\begin{bmatrix}1&1\\0&1\end{bmatrix}.$$ Therefore statement (A) is true. Since M=I+N where $$N=\begin{bmatrix}1&-1\\1&-1\end{bmatrix}$$ and N²=0, we have $$M^k=I+kN.$$ Thus $$M^{26}=I+26N=\begin{bmatrix}27&-26\\26&-25\end{bmatrix},$$ so p=27, q=-26, r=26 and s=-25. The determinant is ps-qr=1, therefore the matrix is unimodular and for every pair of integers (m,n), there exists a unique integer solution (x,y). Hence (C) is true. Also, $$\sum_{k=1}^{26}M^k=26I+\left(\sum_{k=1}^{26}k\right)N=26I+351N,$$ giving $$\begin{bmatrix}377&-351\\351&-325\end{bmatrix}.$$ Therefore a=377, not 378, so (B) is false. Let A denote this matrix. Then $$A+tI=\begin{bmatrix}377+t&-351\\351&-325+t\end{bmatrix}.$$ Its determinant is $$\det(A+tI)=(377+t)(-325+t)+351^2=t^2+52t=t(t+52).$$ For every positive real t, this determinant is nonzero, so the system has a unique solution. Hence (D) is true. Therefore the correct options are (A), (C) and (D).