PUC Science Class 11 - Karnataka Board PUC Question Bank Solutions for Mathematics

Each set X, contains 5 elements and each set Y, contains 2 elements and \[\cup^{20}_{r = 1} X_r = S = \cup^n_{r = 1} Y_r\] If each element of S belong to exactly 10 of the X_r's and to eactly 4 of Y_r's, then find the value of n.

For any two sets A and B, prove that : 

\[A' - B' = B - A\] 

For any two sets A and B, prove the following: 

\[A \cap \left( A' \cup B \right) = A \cap B\] 

For any two sets A and B, prove the following: 

\[A - \left( A - B \right) = A \cap B\]

For any two sets A and B, prove the following: 

\[A \cap \left( A \cup B \right)' = \phi\] 

For any two sets A and B, prove the following:

\[A - B = A \Delta\left( A \cap B \right)\]

Let A and B be two sets such that : \[n \left( A \right) = 20, n \left( A \cup B \right) = 42 \text{ and } n \left( A \cap B \right) = 4\] \[n \left( A - B \right)\]

Let A and B be two sets such that : \[n \left( A \right) = 20, n \left( A \cup B \right) = 42 \text{ and } n \left( A \cap B \right) = 4\] \[n \left( B - A \right)\]

A survey shows that 76% of the Indians like oranges, whereas 62% like bananas. What percentage of the Indians like both oranges and bananas? 

In a group of 950 persons, 750 can speak Hindi and 460 can speak English. Find: how many can speak Hindi only

In a group of 950 persons, 750 can speak Hindi and 460 can speak English. Find: 

how many can speak English only. 

In a survey it was found that 21 persons liked product P₁, 26 liked product P₂ and 29 liked product P₃. If 14 persons liked products P₁ and P₂; 12 persons liked product P₃ and P₁ ; 14 persons liked products P₂ and P₃ and 8 liked all the three products. Find how many liked product P₃ only.

Let A and B be two sets in the same universal set. Then,\[A - B =\]

Let U be the universal set containing 700 elements. If A, B are sub-sets of U such that \[n \left( A \right) = 200, n \left( B \right) = 300 \text{ and } \left( A \cap B \right) = 100\].Then \[n \left( A' \cap B' \right) =\] 

Let A and B be two sets that \[n \left( A \right) = 16, n \left( B \right) = 14, n \left( A \cup B \right) = 25\] Then, \[n \left( A \cap B \right)\] 

Show that the expansion of \[\left( x^2 + \frac{1}{x} \right)^{12}\]  does not contain any term involving x⁻¹.

Show that the statement
p : "If x is a real number such that x³ + x = 0, then x is 0"
is true by
(i) direct method
(ii) method of contrapositive
(iii) method of contradition.

Show that the following statement is true by the method of contrapositive
p : "If x is an integer and x² is odd, then x is also odd" 

Show that the following statement is true
"The integer n is even if an only if n² is even"

Find the vertex, focus, axis, directrix and latus-rectum of the following parabola:

y² = 8x