Maths-
General
Easy
Question
If a tangent to ellipse
+
= 1 makes an angle
with x- axis, then square of length of intercept of tangent cut between axes is-
- None of these
The correct answer is: ![a to the power of 2 end exponent s e c to the power of 2 end exponent invisible function application alpha plus b to the power of 2 end exponent c o s e c to the power of 2 end exponent invisible function application alpha](data:image/png;base64,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)
Let tangent is
+
= 1
Slope is tan
= –
S… (1)
square of length of intercept is
= ![left parenthesis a s e c invisible function application theta right parenthesis to the power of 2 end exponent plus left parenthesis b c o s e c invisible function application theta right parenthesis to the power of 2 end exponent](data:image/png;base64,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)
Length is
… (2)
Now use value of tan from (1)
Related Questions to study
physics-
Vibrating tuning fork of frequency
is placed near the open end of a long cylindrical tube. The tube has a side opening and is fitted with a movable reflecting piston. As the piston is moved through
the intensity of sound changes from a maximum to minimum. If the speed of sound is
then
is
![](data:image/png;base64,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)
Vibrating tuning fork of frequency
is placed near the open end of a long cylindrical tube. The tube has a side opening and is fitted with a movable reflecting piston. As the piston is moved through
the intensity of sound changes from a maximum to minimum. If the speed of sound is
then
is
![](data:image/png;base64,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)
physics-General
maths-
PQ and QR are two focal chords of an ellipse and the eccentric angles of P,Q,R and
,
respectively then tan
tan
is equal to -
PQ and QR are two focal chords of an ellipse and the eccentric angles of P,Q,R and
,
respectively then tan
tan
is equal to -
maths-General
maths-
The radius of the circle passing through the points of intersection of ellipse
= 1 and x2 – y2 = 0 is -
The radius of the circle passing through the points of intersection of ellipse
= 1 and x2 – y2 = 0 is -
maths-General
maths-
If
are the eccentric angles of the extremities of a focal chord of an ellipse, then the eccentricity of the ellipse is -
If
are the eccentric angles of the extremities of a focal chord of an ellipse, then the eccentricity of the ellipse is -
maths-General
maths-
If
is the angle between the diameter through any point on a standard ellipse and the normal at the point, then the greatest value of tan
is–
If
is the angle between the diameter through any point on a standard ellipse and the normal at the point, then the greatest value of tan
is–
maths-General
physics-
A radar operates at wavelength 50.0 cm. If the beat freqency between the transmitted singal and the singal reflected from aircraft (
) is 1 kHz, then velocity of the aircraft will be
A radar operates at wavelength 50.0 cm. If the beat freqency between the transmitted singal and the singal reflected from aircraft (
) is 1 kHz, then velocity of the aircraft will be
physics-General
Maths-
The locus of P such that PA2 + PB2 = 10 where A = (2, 0) and B = (4, 0) is
The locus of P such that PA2 + PB2 = 10 where A = (2, 0) and B = (4, 0) is
Maths-General
maths-
Let L = 0 is a tangent to ellipse
+
= 1 and S,
be its foci. If length of perpendicular from S on L = 0 is 2 then length of perpendicular from
on L = 0 is
Let L = 0 is a tangent to ellipse
+
= 1 and S,
be its foci. If length of perpendicular from S on L = 0 is 2 then length of perpendicular from
on L = 0 is
maths-General
maths-
The condition that the line
x + my = n may be a normal to the ellipse
+
= 1 is
The condition that the line
x + my = n may be a normal to the ellipse
+
= 1 is
maths-General
Physics-
Two blocks A and B of equal masses m kg each are connected by a light thread, which passes over a massless pulley as shown in the figure. Both the blocks lie on wedge of mass m kg. Assume friction to be absent everywhere and both the blocks to be always in contact with the wedge. The wedge lying over smooth horizontal surface is pulled towards right with constant acceleration a (
). (g is acceleration due to gravity). Normal reaction (in N) acting on block A.
![](https://mycourses.turito.com/pluginfile.php/1/question/questiontext/472067/1/944861/Picture150.png)
Two blocks A and B of equal masses m kg each are connected by a light thread, which passes over a massless pulley as shown in the figure. Both the blocks lie on wedge of mass m kg. Assume friction to be absent everywhere and both the blocks to be always in contact with the wedge. The wedge lying over smooth horizontal surface is pulled towards right with constant acceleration a (
). (g is acceleration due to gravity). Normal reaction (in N) acting on block A.
![](https://mycourses.turito.com/pluginfile.php/1/question/questiontext/472067/1/944861/Picture150.png)
Physics-General
physics-
Two blocks A and B of equal masses m kg each are connected by a light thread, which passes over a massless pulley as shown in the figure. Both the blocks lie on wedge of mass m kg. Assume friction to be absent everywhere and both the blocks to be always in contact with the wedge. The wedge lying over smooth horizontal surface is pulled towards right with constant acceleration a (
). (g is acceleration due to gravity) Normal reaction (in N) acting on block B is
![](https://mycourses.turito.com/pluginfile.php/1/question/questiontext/472059/1/944859/Picture121.png)
Two blocks A and B of equal masses m kg each are connected by a light thread, which passes over a massless pulley as shown in the figure. Both the blocks lie on wedge of mass m kg. Assume friction to be absent everywhere and both the blocks to be always in contact with the wedge. The wedge lying over smooth horizontal surface is pulled towards right with constant acceleration a (
). (g is acceleration due to gravity) Normal reaction (in N) acting on block B is
![](https://mycourses.turito.com/pluginfile.php/1/question/questiontext/472059/1/944859/Picture121.png)
physics-General
physics-
A light inextensible string connects a block of mass m and top of wedge of mass M. The string is parallel to inclined surface and the inclined surface makes an angle
with horizontal as shown in the figure. All surfaces are smooth. Now a constant horizontal force of minimum magnitude F is applied to wedge towards right such that the normal reaction on block exerted by wedge just becomes zero. The magnitude of acceleration of wedge is
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)
A light inextensible string connects a block of mass m and top of wedge of mass M. The string is parallel to inclined surface and the inclined surface makes an angle
with horizontal as shown in the figure. All surfaces are smooth. Now a constant horizontal force of minimum magnitude F is applied to wedge towards right such that the normal reaction on block exerted by wedge just becomes zero. The magnitude of acceleration of wedge is
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)
physics-General
maths-
The vector
which is perpendicular to (2,-3,1) and (1,-2,3) and which satisfies the condition ![stack x with minus on top times open parentheses stack i with minus on top plus 2 stack j with minus on top minus 7 k close parentheses equals 10](data:image/png;base64,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)
The vector
which is perpendicular to (2,-3,1) and (1,-2,3) and which satisfies the condition ![stack x with minus on top times open parentheses stack i with minus on top plus 2 stack j with minus on top minus 7 k close parentheses equals 10](data:image/png;base64,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)
maths-General
Maths-
Let f : R → R be a differentiable function satisfying f (x) = f (x – y) f (y) " x, y Î R and f ¢ (0) = a, f ¢ (2) = b then f ¢ (-2) is
Let f : R → R be a differentiable function satisfying f (x) = f (x – y) f (y) " x, y Î R and f ¢ (0) = a, f ¢ (2) = b then f ¢ (-2) is
Maths-General
physics-
Four identical metal butterflies are hanging from a light string of length
at equally placed points as shown in the figure . The ends of the string are attached to a horizontal fixed support. The middle section of the string is horizontal. The relation between the angle
and
is given by
![](data:image/png;base64,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)
Four identical metal butterflies are hanging from a light string of length
at equally placed points as shown in the figure . The ends of the string are attached to a horizontal fixed support. The middle section of the string is horizontal. The relation between the angle
and
is given by
![](data:image/png;base64,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)
physics-General