Subject

Mathematics

Class

JEE Class 12

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 Multiple Choice QuestionsMultiple Choice Questions

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11. Statement space minus 1 space colon space sum from straight r space equals space 0 to straight n of space left parenthesis straight r plus 1 right parenthesis space to the power of straight n straight C subscript straight r space equals space left parenthesis straight n plus 2 right parenthesis 2 to the power of straight n minus 1 end exponent
Statement space minus space 2 colon thin space sum from straight r equals 0 to straight n of space left parenthesis straight r plus 1 right parenthesis space to the power of straight n straight C subscript straight r straight x to the power of straight r space equals space left parenthesis 1 plus straight x right parenthesis to the power of straight n space plus space nx space left parenthesis 1 plus straight x right parenthesis to the power of straight n minus 1 end exponent
  • Statement −1 is false, Statement −2 is true

  • Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1

  • Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.

  • Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.


B.

Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1

sum from straight r space equals 0 to straight n of space left parenthesis straight r plus 1 right parenthesis space to the power of straight n straight C subscript straight r space equals space sum from straight r space equals space 0 to straight n of space straight r to the power of straight n straight C subscript straight r space plus space to the power of straight n straight C subscript straight r
space equals space sum from straight r space equals space 0 to straight n of space straight r space straight n over straight r space to the power of straight n minus 1 end exponent straight C subscript straight r minus 1 end subscript space plus space sum from straight r space equals space 0 to straight n of space to the power of straight n straight C subscript straight r space equals space straight n 2 to the power of straight n minus 1 end exponent space plus space 2 to the power of straight n
space equals space 2 to the power of straight n minus 1 end exponent space left parenthesis straight n plus 2 right parenthesis
Statement space minus 1 space true
sum for space of space left parenthesis straight r plus 1 right parenthesis to the power of straight n straight C subscript straight r space end subscript straight x to the power of straight r space equals space sum for space of straight r to the power of straight n space straight C subscript straight r straight x to the power of straight r space plus space sum for space of to the power of straight n straight C subscript straight r straight x to the power of straight r
space equals space straight n space sum from straight r space equals space 0 space to straight n of space to the power of straight n straight C subscript straight r minus 1 end subscript space straight x to the power of straight r space plus space sum from straight r space equals space 0 to straight n of space to the power of straight n straight C subscript straight r straight x to the power of straight r space equals space straight n space straight x space left parenthesis 1 space plus straight x right parenthesis to the power of straight n minus 1 end exponent space plus space left parenthesis 1 plus straight x right parenthesis to the power of straight n
substituting space straight x space equals space 1
sum for space of space left parenthesis straight r plus 1 right parenthesis space to the power of straight n straight C subscript straight r space equals space straight n 2 to the power of straight n minus 1 end exponent space plus space 2 to the power of straight n
Hence Statement −2 is also true and is a correct explanation of Statement −1.
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12.

Let p be the statement “x is an irrational number”, q be the statement “y is a transcendental number”, and r be the statement “x is a rational number iff y is a transcendental number”.
Statement –1: r is equivalent to either q or p
Statement –2: r is equivalent to ∼ (p ↔ ∼ q).

  • Statement −1 is false, Statement −2 is true

  • Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1

  • Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.

  • Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.

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13.

The statement p → (q → p) is equivalent to

  • p → (p → q) 

  • p → (p ∨ q)

  • p → (p ∧ q)

  • p → (p ∧ q)

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14.

The value of cot open parentheses cosec to the power of negative 1 end exponent space 5 over 3 space plus space tan to the power of negative 1 end exponent 2 over 3 close parentheses space is

  • 6/17

  • 5/17

  • 4/17

  • 4/17

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15.

The quadratic equations x2 – 6x + a = 0 and x2 – cx + 6 = 0 have one root in common. The other roots of the first and second equations are integers in the ratio 4 : 3. Then the common root is

  • 1

  • 4

  • 3

  • 3

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16.

How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent?

  • 8 . 6C4 . 7C4

  • 6 . 7 . 8C4

  • 6 . 8 . 7C4

  • 6 . 8 . 7C4

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17.

Let f : N → Y be a function defined as f (x) = 4x + 3, where Y = {y ∈ N : y = 4x + 3 for some x ∈ N}.Show that f is invertible and its inverse is 

  • straight g space left parenthesis straight y right parenthesis space equals space fraction numerator 3 straight y space plus space 4 over denominator 3 end fraction
  • straight g space left parenthesis straight y right parenthesis space equals space 4 plus fraction numerator straight y space plus space 3 over denominator 4 end fraction
  • straight g space left parenthesis straight y right parenthesis space equals fraction numerator straight y space plus space 3 over denominator 4 end fraction
  • straight g space left parenthesis straight y right parenthesis space equals fraction numerator straight y space plus space 3 over denominator 4 end fraction
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18.

Let R be the real line. Consider the following subsets of the plane R × R.
S = {(x, y) : y = x + 1 and 0 < x < 2}, T = {(x, y) : x − y is an integer}. Which one of the following is true?

  • neither S nor T is an equivalence relation on R

  • both S and T are equivalence relations on R

  • S is an equivalence relation on R but T is not 

  • S is an equivalence relation on R but T is not 

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19.

Let A be a 2 × 2 matrix with real entries. Let I be the 2 × 2 identity matrix. Denote by tr (A), the sum of diagonal entries of A. Assume that A2= I.
Statement −1: If A ≠ I and A ≠ − I, then det A = − 1.
Statement −2: If A ≠ I and A ≠ − I, then tr (A) ≠ 0.

  • Statement −1 is false, Statement −2 is true

  • Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1

  • Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.

  • Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.

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20.

Let f(x) = open curly brackets table attributes columnalign left end attributes row cell left parenthesis straight x minus 1 right parenthesis space sin space open parentheses fraction numerator 1 over denominator straight x minus 1 end fraction close parentheses end cell row cell 0 comma space space space space space space space space space space space space space space space space space space space space space space space space space space space if space straight x space equals 1 space space space space space space space space space space end cell end table close comma space if space straight x space not equal to space 1Then which one of the following is true?

  • f is neither differentiable at x = 0 nor at x = 1

  • f is differentiable at x = 0 and at x = 1

  • f is differentiable at x = 0 but not at x = 1 

  • f is differentiable at x = 0 but not at x = 1 

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