If n is a positive integer, then  is from Mathematics JEE Yea

Subject

Mathematics

Class

JEE Class 12

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 Multiple Choice QuestionsMultiple Choice Questions

1.

The equation esinx-e-sinx -4 = 0 has

  • infinite number of real roots

  • No real root

  • exactly one real root

  • exactly one real root

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2.

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 true; statement 2 is not a correct explanation for statement 1

178 Views

3.

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

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

208 Views

4.

Statement I An equation of a common tangent to the parabola straight y squared space equals space 16 space square root of 3 straight x end root and the ellipse 2x2 +y2 =4 is space straight y space equals 2 straight x space plus 2 square root of 3.
Statement II If the line straight Y space equals space mx space plus fraction numerator 4 square root of 3 over denominator straight m end fraction comma space left parenthesis straight m space not equal to 0 right parenthesis is a common tangent to the parabola straight y squared space equals space 16 space square root of 3 straight x and the ellipse 2x2 +y2 =4, then m satisfies m4 +2m2 =24

  • 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

197 Views

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5.

If n is a positive integer, then open parentheses square root of 3 plus 1 close parentheses to the power of 2 straight n end exponent minus space left parenthesis square root of 3 minus 1 right parenthesis to the power of 2 straight n end exponent space is

  • an irrational number

  • an odd positive integer

  • an even positive integer

  • an even positive integer


A.

an irrational number

left parenthesis straight x plus straight a right parenthesis to the power of straight n space equals space to the power of straight n straight C subscript straight o space straight x to the power of straight n space plus to the power of straight n straight C subscript 1 straight x to the power of straight n minus 1 end exponent space straight a space plus to the power of straight n straight C subscript 1 straight x to the power of straight n minus 2 end exponent straight a squared space plus..... plus straight n space to the power of straight n straight C subscript straight n straight a to the power of straight n
and
left parenthesis straight x minus straight a right parenthesis to the power of straight n space equals space to the power of straight n straight C subscript straight o space straight x to the power of straight n space minus to the power of straight n straight C subscript 1 straight x to the power of straight n minus 1 end exponent space straight a space plus to the power of straight n straight C subscript 1 straight x to the power of straight n minus 2 end exponent straight a squared space plus..... plus left parenthesis negative 1 right parenthesis straight n space to the power of straight n straight C subscript straight n straight a to the power of straight n
left parenthesis square root of 3 plus 1 end root right parenthesis to the power of 2 straight n end exponent space equals space to the power of 2 straight n end exponent straight C subscript straight o space left parenthesis square root of 3 right parenthesis to the power of 2 straight n end exponent space plus to the power of 2 straight n end exponent straight C subscript 1 left parenthesis square root of 3 right parenthesis to the power of 2 straight n minus 1 end exponent space plus to the power of 2 straight n end exponent straight C subscript 2 space left parenthesis square root of 3 right parenthesis to the power of 2 straight n minus 2 end exponent space
plus........ plus to the power of 2 straight n end exponent straight C subscript 2 straight n end subscript left parenthesis square root of 3 right parenthesis to the power of 2 straight n minus 2 straight n end exponent

left parenthesis square root of 3 minus 1 end root right parenthesis to the power of 2 straight n end exponent space equals space to the power of 2 straight n end exponent straight C subscript straight o space left parenthesis square root of 3 right parenthesis to the power of 2 straight n end exponent space left parenthesis negative 1 right parenthesis to the power of 0 plus to the power of 2 straight n end exponent straight C subscript 1 left parenthesis square root of 3 right parenthesis to the power of 2 straight n minus 1 end exponent space left parenthesis negative 1 right parenthesis squared space plus
to the power of 2 straight n end exponent straight C subscript 2 space left parenthesis square root of 3 right parenthesis to the power of 2 straight n minus 2 end exponent left parenthesis negative 1 right parenthesis squared space plus........ plus to the power of 2 straight n end exponent straight C subscript 2 straight n end subscript left parenthesis square root of 3 right parenthesis to the power of 2 straight n minus 2 straight n end exponent left parenthesis negative 1 right parenthesis to the power of 2 straight n end exponent
Adding both the binomial expansions above, we get
left parenthesis square root of 3 plus 1 right parenthesis to the power of 2 straight n end exponent space minus space left parenthesis square root of 3 straight n end root minus 1 right parenthesis to the power of 2 straight n end exponent space equals space 2 left square bracket to the power of 2 straight n end exponent straight C subscript 1 left parenthesis square root of 3 right parenthesis to the power of 2 straight n minus 1 end exponent
plus to the power of 2 straight n end exponent straight C subscript 3 space left parenthesis square root of 3 right parenthesis to the power of 2 straight n minus 3 end exponent space plus to the power of 2 straight n end exponent straight C subscript 5 space left parenthesis square root of 3 right parenthesis to the power of 2 straight n minus 5 end exponent space plus....... space plus to the power of 2 straight n end exponent straight C subscript 2 straight n minus 1 end subscript space left parenthesis square root of 3 right parenthesis to the power of 2 straight n minus left parenthesis 2 straight n minus 1 right parenthesis end exponent right square bracket
It is the irrational number because of odd power of square root of 3 appears in each of the terms.


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6.

If 100 times the 100th term of an AP with non zero common difference equals the 50 times its 50th term, then the 150th term of this AP is

  •  –150

  • 150

  • times its 50th term

  • times its 50th term

263 Views

7.

In a ∆PQR, if 3 sin P + 4 cos Q = 6 and 4 sin Q + 3 cos P = 1, then the angle R is equal to

  • 5π/6

  • π/6

  • π/4

  • π/4

163 Views

8.

An equation of a plane parallel to the plane x – 2y + 2z – 5 = 0 and at a unit distance from the origin is

  • x – 2y + 2z – 3 = 0

  • x – 2y + 2z + 1 = 0

  • x – 2y + 2z – 1 = 0

  • x – 2y + 2z – 1 = 0

457 Views

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9.

If the line 2x + y = k passes through the point which divides the line segment joining the points (1, 1) and (2, 4) in the ratio 3 : 2, then k equals

  • 29/5

  • 5

  • 6

  • 6

224 Views

10.

Let x1, x2, ......, xn be n observations, and let top enclose straight x be their arithmetic mean and σ2 be their variance.
Statement 1: Variance of 2x1, 2x2, ......, 2xn is 4 σ2.
Statement 2: Arithmetic mean of 2x1, 2x2, ......, 2xn is 4top enclose straight x.

  • 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

213 Views

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