Question

For each t ∈R, let [t] be the greatest integer less than or equal to t. Then
$\underset{x \to 0^{+}}{l i m} x ([\frac{1}{x}] + [\frac{2}{x}] + . . . . . . + [\frac{15}{x}])$

does not exist (in R)

is equal to 0

is equal to 15

is equal to 120

Solution

D.

is equal to 120

$\underset{x \to 0^{+}}{l i m} x ([\frac{1}{x}] + [\frac{2}{x}] + . . . . . . . + [\frac{15}{x}]) \Rightarrow N o w \frac{1}{x} - 1 < [\frac{1}{x}] \leq \frac{1}{x} \frac{2}{x} - 1 < [\frac{2}{x}] \leq \frac{2}{x} \frac{15}{x} - 1 < [\frac{15}{x}] \leq \frac{15}{x} \underset{x \to 0^{+}}{l i m} x (\frac{\sum_{}^{} 15}{x} - 15) < \underset{x \to 0^{+}}{l i m} x ([\frac{1}{x}] + [\frac{2}{x}] + . . . . . + [\frac{15}{x}]) \leq \underset{x \to 0^{+}}{l i m} x (\frac{\sum_{} 15}{x}) \underset{x \to 0^{+}}{l i m} x \sum_{} 15 - 5 x < L \leq \underset{x \to 0^{+}}{l i m} \sum_{} 15 L 120$

Question

Wired Faculty App

If the tangent at (1, 7) to the curve x² = y – 6 touches the circle x² + y² + 16x + 12y + c = 0 then the value of c is

95

195

185

85

Solution

A.

95

Equation of tangent at (1,7) to curve x² = y -6 is

$x - 1 = \frac{1}{2} (y + 7) - 6$

2x - y +5 = 0 ... (i)

Centre of circle = (-8,-6)

Radius of circle = $\sqrt{64 + 36 - c} = \sqrt{100 - c}$

$∵ L i n e (i) t o u c h e s t h e c i r c l e ∴ |\frac{2 (- 8) - (- 6) + 5}{\sqrt{4 + 1}}| = \sqrt{100 - C} \sqrt{5} = \sqrt{100 - c} \Rightarrow c = 95$

Question

Wired Faculty App

If α, β ∈ C are the distinct roots, of the equation x² -x + 1 = 0, then α¹⁰¹ + β¹⁰⁷ is equal to

2

-1

0

1

Solution

D.

1

x²-x + 1 = 0

Roots are -ω, -ω²

Let α = -ω, β = -ω²

α¹⁰¹ + β¹⁰⁷ = (-ω)¹⁰¹ + (-ω²)¹⁰⁷

= -( ω¹⁰¹ +ω²¹⁴)
= - (ω² + ω)
= 1

Question

Wired Faculty App

PQR is a triangular park with PQ = PR = 200 m. A T.V. tower stands at the mid-point of QR. If the angles of elevation of the top of the tower at P, Q and R are respectively 45^o, 30^o and 30^o, then the height of the tower (in m) is

$50 \sqrt{2}$

100

50

$100 \sqrt{3}$

Solution

B.

100

Let ST = h (height of tower)

PT = ST = h

$\frac{S T}{Q T} = \tan 30 ° Q T = h \sqrt{3} N o w P T^{2} + Q T^{2} = 200^{2} 4 h^{2} = 200^{2} h = 100$

Question

Wired Faculty App

The sum of the coefficients of all odd degree terms in the expansion of ${(x + \sqrt{x^{3} - 1})}^{5} + {(x - \sqrt{x^{3} - 1})}^{5}, (x > 1) i s$

2

-1

0

1

Solution

A.

2

${(x + \sqrt{x^{3} - 1})}^{5} + {(x - \sqrt{x^{3} - 1})}^{5} = 2 [{}^{2}C_{0} x^{5} + {}^{5}C_{2} x^{3} (x^{3} - 1) + {}^{5}C_{4} x (x^{3} - 1)^{2}] = 2 [x^{5} + 10 (x^{6} - x^{3}) + 5 x (x^{6} - 2 x^{3} + 1)] = 2 [x^{5} + 10 x^{6} - 10 x^{3} + 5 x^{7} - 10 x^{4} + 5 x] = 2 [5 x^{7} + 10 x^{6} + x^{5} - 10 x^{4} - 10 x^{3} + 5 x]$

Sum of odd degree terms coefficients
= 2(5 + 1 – 10 + 5)
= 2

For Daily Free Study Material Join wiredfaculty

WhatsApp Group

Download Android App

Ncert Book Download

JEE mathematics

For each t ∈R, let [t] be the greatest integer less than or equal to t. Then
$\underset{x \to 0^{+}}{l i m} x ([\frac{1}{x}] + [\frac{2}{x}] + . . . . . . + [\frac{15}{x}])$

does not exist (in R)

is equal to 0

is equal to 15

is equal to 120

If the tangent at (1, 7) to the curve x² = y – 6 touches the circle x² + y² + 16x + 12y + c = 0 then the value of c is

95

195

185

85

If α, β ∈ C are the distinct roots, of the equation x² -x + 1 = 0, then α¹⁰¹ + β¹⁰⁷ is equal to

2

-1

0

1

PQR is a triangular park with PQ = PR = 200 m. A T.V. tower stands at the mid-point of QR. If the angles of elevation of the top of the tower at P, Q and R are respectively 45^o, 30^o and 30^o, then the height of the tower (in m) is

$50 \sqrt{2}$

100

50

$100 \sqrt{3}$

The sum of the coefficients of all odd degree terms in the expansion of ${(x + \sqrt{x^{3} - 1})}^{5} + {(x - \sqrt{x^{3} - 1})}^{5}, (x > 1) i s$

2

-1

0

1

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Loaction

For Daily Free Study Material Join wiredfaculty

WhatsApp Group

Download Android App

Ncert Book Download

JEE mathematics

For each t ∈R, let [t] be the greatest integer less than or equal to t. Thenlimx→0+ x1x+2x+......+15x does not exist (in R) is equal to 0 is equal to 15 is equal to 120

If the tangent at (1, 7) to the curve x2 = y – 6 touches the circle x2 + y2 + 16x + 12y + c = 0 then the value of c is 95 195 185 85

If α, β ∈ C are the distinct roots, of the equation x2 -x + 1 = 0, then α101 + β107 is equal to 2 -1 0 1

PQR is a triangular park with PQ = PR = 200 m. A T.V. tower stands at the mid-point of QR. If the angles of elevation of the top of the tower at P, Q and R are respectively 45o, 30o and 30o, then the height of the tower (in m) is 502 100 50 1003

The sum of the coefficients of all odd degree terms in the expansion of x + x3 -15 + x - x3-15, (x>1) is 2 -1 0 1

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

For each t ∈R, let [t] be the greatest integer less than or equal to t. Then
$\underset{x \to 0^{+}}{l i m} x ([\frac{1}{x}] + [\frac{2}{x}] + . . . . . . + [\frac{15}{x}])$

does not exist (in R)

is equal to 0

is equal to 15

is equal to 120

If the tangent at (1, 7) to the curve x² = y – 6 touches the circle x² + y² + 16x + 12y + c = 0 then the value of c is

95

195

185

85

If α, β ∈ C are the distinct roots, of the equation x² -x + 1 = 0, then α¹⁰¹ + β¹⁰⁷ is equal to

2

-1

0

1

PQR is a triangular park with PQ = PR = 200 m. A T.V. tower stands at the mid-point of QR. If the angles of elevation of the top of the tower at P, Q and R are respectively 45^o, 30^o and 30^o, then the height of the tower (in m) is

$50 \sqrt{2}$

100

50

$100 \sqrt{3}$

The sum of the coefficients of all odd degree terms in the expansion of ${(x + \sqrt{x^{3} - 1})}^{5} + {(x - \sqrt{x^{3} - 1})}^{5}, (x > 1) i s$

2

-1

0

1