Number of correct statements among the following is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>. then <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>k</mi></math> is _______ Statement I: Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>7</mn><mo>}</mo><mo>.</mo></math> Define <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><mo>{</mo><mi>T</mi><mo>⊆</mo><mi>A</mi><mo>:</mo><mtext> either </mtext><mn>1</mn><mo>∉</mo><mi>T</mi><mtext> or </mtext><mn>2</mn><mo>∈</mo><mi>T</mi><mo>}</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mo>{</mo><mi>T</mi><mo>⊆</mo><mi>A</mi><mo>:</mo></math> the sum of all the elements of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>'</mo><mi>T</mi><mo>'</mo></math> is a prime number<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>}</mo></math>. Then the number of elements in the set <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>∪</mo><mi>C</mi></math> is 107. Statement II: If sum of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> terms of any sequence is a quadratic in <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>'</mo><mi>n</mi><mo>'</mo></math>, then the sequence is an AP. Statement III: Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msup><mrow><mo>(</mo><mn>8</mn><msqrt><mn>3</mn></msqrt><mo>+</mo><mn>13</mn><mo>)</mo></mrow><mn>13</mn></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mrow><mo>(</mo><mn>7</mn><msqrt><mn>2</mn></msqrt><mo>+</mo><mn>9</mn><mo>)</mo></mrow><mn>9</mn></msup><mo>.</mo></math> If <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mi>t</mi><mo>]</mo></math> denotes the greatest integer <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>≤</mo><mi>t</mi></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mi>x</mi><mo>]</mo><mo>+</mo><mo>[</mo><mi>y</mi><mo>]</mo></math> is even. Statement IV: The expression <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>x</mi><mo>+</mo><msup><mrow><msup><mrow><mo>(</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>1</mn><mo>)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo>)</mo></mrow><mn>5</mn></msup><mo>+</mo><mo>(</mo><mi>x</mi><mo>-</mo><msup><mrow><msup><mrow><mo>(</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>1</mn><mo>)</mo></mrow><mfrac><mstyle displaystyle="true"><mn>1</mn></mstyle><mstyle displaystyle="true"><mn>2</mn></mstyle></mfrac></msup><mo>)</mo></mrow><mn>5</mn></msup></math> is a polynomial of degree 7.

Number of correct statements among the following is k . then 2 k is _ Statement I: Let A = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } . Define B = { T A : either 1 ? T or 2 ? T } and C = { T ? A : the sum of all the elements of ' T ' is a prime number

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Q.
Number of correct statements among the following is $k$ . then $2 k$ is _______

Statement I: Let $A = {1, 2, 3, 4, 5, 6, 7} .$ Define $B = {T \subseteq A : either 1 \notin T or 2 \in T}$ and $C = {T \subseteq A :$ the sum of all the elements of $' T'$ is a prime number $}$ . Then the number of elements in the set $B \cup C$ is 107.

Statement II: If sum of $n$ terms of any sequence is a quadratic in $' n'$ , then the sequence is an AP.

Statement III: Let $x = {(8 \sqrt{3} + 13)}^{13}$ and $y = {(7 \sqrt{2} + 9)}^{9} .$ If $[t]$ denotes the greatest integer $\leq t$ , then $[x] + [y]$ is even.

Statement IV: The expression $(x + {{(x^{3} - 1)}^{\frac{1}{2}})}^{5} + (x - {{(x^{3} - 1)}^{\frac{1}{2}})}^{5}$ is a polynomial of degree 7.

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