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Question 1:

Let \(L_{1}\) be the line of intersection of the planes given by the equations \(2x+3y+z=4\) and \(x+2y+z=5\). Let \(L_{2}\) be the line passing through the point \(P(2,-1,3)\) and parallel to \(L_{1}\). Let \(M\) denote the plane given by the equation \(2x+y-2z=6\).
Suppose that the line \(L_{2}\) meets the plane \(M\) at the point \(Q\). Let \(R\) be the foot of the perpendicular drawn from \(P\) to the plane \(M\).
Then which of the following statements is (are) TRUE?

The length of the line segment \(PQ\) is \(9 \sqrt{3}\)

The length of the line segment \(QR\) is 15

The area of \(\triangle PQR\) is \(\tfrac{3}{2}\sqrt{234}\)

The acute angle between the line segments \(PQ\) and \(PR\) is \(\cos^{-1}(\tfrac{1}{2\sqrt{3}})\)

Question 2:

Three students \( S_{1}, S_{2} \), and \( S_{3} \) are given a problem to solve. Consider the following events:
U : At least one of \( S_{1}, S_{2} \), and \( S_{3} \) can solve the problem,
V: \( S_{1} \) can solve the problem, given that neither \( S_{2} \) nor \( S_{3} \) can solve the problem,
W: \( S_{2} \) can solve the problem and \( S_{3} \) cannot solve the problem,
T: \( S_{3} \) can solve the problem.
For any event \( E \), let \( P(E) \) denote the probability of \( E \). If
\( P(U)=\frac{1}{2}, P(V)=\frac{1}{10} \), and \( P(W)=\frac{1}{12} \),
then \( P(T) \) is equal to

13/36

1/3

19/60

1/4

Question 3:

Let \( \mathbb{R} \) denote the set of all real numbers. Let \( a_{i}, b_{i} \in \mathbb{R} \) for \( i \in\{1,2,3\} \).
Define the functions \( f: \mathbb{R} \rightarrow \mathbb{R}, g: \mathbb{R} \rightarrow \mathbb{R} \), and \( h: \mathbb{R} \rightarrow \mathbb{R} \) by
\[
\begin{array}{l}
f(x)=a_{1}+10 x+a_{2} x^{2}+a_{3} x^{3}+x^{4} \\
g(x)=b_{1}+3 x+b_{2} x^{2}+b_{3} x^{3}+x^{4} \\
h(x)=f(x+1)-g(x+2)
\end{array}
\]
If \( f(x) \neq g(x) \) for every \( x \in \mathbb{R} \), then the coefficient of \( x^{3} \) in \( h(x) \) is

8

2

-4

-6

Question 4:

Let \( \mathbb{R} \) denote the set of all real numbers. Define the function \( f: \mathbb{R} \rightarrow \mathbb{R} \) by
\[
f(x)=\left\{\begin{array}{cl}
2-2 x^{2}-x^{2} \sin \frac{1}{x} & \text { if } x \neq 0 \\
2 & \text { if } x=0
\end{array}\right.
\]

Then which one of the following statements is TRUE?

The function \( f \) is NOT differentiable at \( x=0 \)

There is a positive real number \( \delta \), such that \( f \) is a decreasing function on the interval \( (0, \delta) \)

For any positive real number \( \delta \), the function \( f \) is NOT an increasing function on the interval \( (-\delta, 0) \)

\( x=0 \) is a point of local minima of \( f \)

Question 5:

Consider the matrix
\[
P=\begin{pmatrix}
2 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 3
\end{pmatrix}.
\]
Let the transpose of a matrix \(X\) be denoted by \(X^{T}\). Then the number of \(3\times3\) invertible matrices \(Q\) with integer entries, such that \(Q^{-1}=Q^{T}\) and \(PQ=QP\), is

32

8

16

24

Kusum Ma'am JEE

00:00:00

Question 1:

Question 2:

Question 3:

Question 4:

Question 5: