The equation e^sinx-e^-sinx-4 = 0 has

infinite number of real roots

No real root

exactly one real root

exactly four real roots

Solution

No real root

straight e to the power of sinx space minus straight e to the power of negative sin space straight x end exponent space equals space 4 rightwards double arrow space straight e to the power of sin space straight x end exponent space equals space straight t straight t minus 1 over straight t space equals space 4 straight t squared minus 4 straight t minus 1 space equals space 0 rightwards double arrow space straight t space equals space fraction numerator 4 space plus-or-minus square root of 16 plus 4 end root over denominator 2 end fraction rightwards double arrow straight t space equals space fraction numerator 4 space plus-or-minus 2 square root of 5 over denominator 2 end fraction rightwards double arrow space straight t space equals space 2 plus-or-minus square root of 5 straight e to the power of sin space straight x end exponent space equals space space 2 plus-or-minus square root of 5 space minus space less or equal than space sin space straight x less or equal than 1 straight e to the power of sin space straight x end exponent space equals space 2 space plus square root of 5 space not space possible straight e to the power of sin space straight x end exponent space equals space 2 space minus square root of 5 space not space possible Hence comma space no space solution

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Question

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Statement 1: The sum of the series 1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) + ...... + (361 + 380 +400) is 8000.
Statement 2: $sum from straight k space equals 1 to straight n equals space 1 of left square bracket straight k cubed minus left parenthesis straight k minus 1 cubed right parenthesis right square bracket space equals space straight n cubed$ , for any natural number n.

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 false

Solution

Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1

Statement 1 has 20 terms whose sum is 8000 And statement 2 is true and supporting statement 1.
k^th bracket is (k – 1)² + k(k – 1) + k² = 3k² – 3k + 1.

Question

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The negation of the statement “If I become a teacher, then I will open a school” is

I will become a teacher and I will not open a school

Either I will not become a teacher or I will not open a school

Neither I will become a teacher nor I will open a school

I will not become a teacher or I will open a school

Solution

I will become a teacher and I will not open a school

Let us assume that
p: I become a teacher' and
q: I will open a school
Then, we can easily as certain that
Negation of (p →q)
~(~p ∨ q) = p ∧ ~q
Which means that ' l' will become a teacher and I will not open a school.