Question 1 |

An opaque pyramid (shown below), with a square base and isosceles faces, is
suspended in the path of a parallel beam of light, such that its shadow is cast on a
screen oriented perpendicular to the direction of the light beam. The pyramid can
be reoriented in any direction within the light beam. Under these conditions, which
one of the shadows P, Q, R, and S is NOT possible?

P | |

Q | |

R | |

S |

Question 2 |

How many pairs of sets (S,T) are possible among the subsets of {1, 2, 3, 4, 5, 6}
that satisfy the condition that S is a subset of T?

729 | |

728 | |

665 | |

664 |

Question 2 Explanation:

Take one element {1}

T=\phi ,S=\phi \rightarrow (\phi, \phi ) \Rightarrow 1\; pair

T=1,S=\phi \rightarrow (\phi, 1 ),(1,1) \Rightarrow 2\; pair

For 1 element total pair = 3^1

Similarly

For 2 element total pair = 3^2

For 3 element total pair = 3^3

For 4 element total pair = 3^4

For 5 element total pair = 3^5

For 6 element total pair = 3^6=729

T=\phi ,S=\phi \rightarrow (\phi, \phi ) \Rightarrow 1\; pair

T=1,S=\phi \rightarrow (\phi, 1 ),(1,1) \Rightarrow 2\; pair

For 1 element total pair = 3^1

Similarly

For 2 element total pair = 3^2

For 3 element total pair = 3^3

For 4 element total pair = 3^4

For 5 element total pair = 3^5

For 6 element total pair = 3^6=729

Question 3 |

Which one of the sentence sequences in the given options creates a coherent
narrative?

(i) I could not bring myself to knock.

(ii) There was a murmur of unfamiliar voices coming from the big drawing room and the door was firmly shut.

(iii) The passage was dark for a bit, but then it suddenly opened into a bright kitchen.

(iv) I decided I would rather wander down the passage.

(i) I could not bring myself to knock.

(ii) There was a murmur of unfamiliar voices coming from the big drawing room and the door was firmly shut.

(iii) The passage was dark for a bit, but then it suddenly opened into a bright kitchen.

(iv) I decided I would rather wander down the passage.

(iv), (i), (iii), (ii) | |

(iii), (i), (ii), (iv) | |

(ii), (i), (iv), (iii) | |

(i), (iii), (ii), (iv) |

Question 4 |

Consider the following inequalities

p^2-4q \lt 4

3p+2q \lt 6

where p and q are positive integers.

The value of (p+q) is _______.

p^2-4q \lt 4

3p+2q \lt 6

where p and q are positive integers.

The value of (p+q) is _______.

2 | |

1 | |

3 | |

4 |

Question 4 Explanation:

\begin{aligned}
p^2-4q &\lt 4 \\
p^2-4&\ lt 4q \;\;...(i)\\
3p+2Q& \lt 6 \\
6p-12 &\lt -4q\;\;...(ii)\\
&\text{by equation (i) + (ii)} \\
p^2+6p-16 &\lt 0 \\
(p+8)(p-2)& \lt 0 \\
\therefore \; p&\in (-8,2)
\end{aligned}

Given p is positive integer

\therefore \;p=1

Now, from equation (i), 1-4 \lt 4q

q \gt \frac{-3}{4}

from equation (ii),

q \lt 3/2

\therefore \; \frac{-3}{4} \lt q \lt \frac{3}{2}

Given q is positive integer

\therefore \;\; q=1

Thus p + q = 1+1 = 2

Given p is positive integer

\therefore \;p=1

Now, from equation (i), 1-4 \lt 4q

q \gt \frac{-3}{4}

from equation (ii),

q \lt 3/2

\therefore \; \frac{-3}{4} \lt q \lt \frac{3}{2}

Given q is positive integer

\therefore \;\; q=1

Thus p + q = 1+1 = 2

Question 5 |

In a recently held parent-teacher meeting, the teachers had very few complaints
about Ravi. After all, Ravi was a hardworking and kind student. Incidentally, almost
all of Ravi's friends at school were hardworking and kind too. But the teachers drew
attention to Ravi's complete lack of interest in sports. The teachers believed that,
along with some of his friends who showed similar disinterest in sports, Ravi needed
to engage in some sports for his overall development.

Based only on the information provided above, which one of the following statements can be logically inferred with certainty?

Based only on the information provided above, which one of the following statements can be logically inferred with certainty?

All of Ravi's friends are hardworking and kind. | |

No one who is not a friend of Ravi is hardworking and kind. | |

None of Ravi's friends are interested in sports. | |

Some of Ravi's friends are hardworking and kind. |

There are 5 questions to complete.