Verify point and find gradient

A question is this type if and only if it asks to verify that a point lies on a curve and then find the gradient at that point.

6 questions · Moderate -0.2

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CAIE P2 2009 November Q8
9 marks Standard +0.3
8 The equation of a curve is \(y ^ { 2 } + 2 x y - x ^ { 2 } = 2\).
  1. Find the coordinates of the two points on the curve where \(x = 1\).
  2. Show by differentiation that at one of these points the tangent to the curve is parallel to the \(x\)-axis. Find the equation of the tangent to the curve at the other point, giving your answer in the form \(a x + b y + c = 0\).
OCR MEI C3 Q1
6 marks Moderate -0.3
1 A curve has implicit equation \(y ^ { 2 } + 2 x \ln y = x ^ { 2 }\).
Verify that the point \(( 1,1 )\) lies on the curve, and find the gradient of the curve at this point.
OCR MEI C3 Q7
6 marks Moderate -0.3
7 A curve is defined by the equation \(\sin 2 x + \cos y = \sqrt { 3 }\).
  1. Verify that the point \(\mathrm { P } \left( \frac { 1 } { 6 } \pi , \frac { 1 } { 6 } \pi \right)\) lies on the curve.
  2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). Hence find the gradient of the curve at the point P .
OCR MEI C3 2011 June Q6
6 marks Moderate -0.8
6 A curve is defined by the equation \(\sin 2 x + \cos y = \sqrt { 3 }\).
  1. Verify that the point \(\mathrm { P } \left( \frac { 1 } { 6 } \pi , \frac { 1 } { 6 } \pi \right)\) lies on the curve.
  2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). Hence find the gradient of the curve at the point P .
  3. Multiply out \(\left( 3 ^ { n } + 1 \right) \left( 3 ^ { n } - 1 \right)\).
  4. Hence prove that if \(n\) is a positive integer then \(3 ^ { 2 n } - 1\) is divisible by 8 .
OCR MEI C3 2015 June Q5
6 marks Moderate -0.3
5 A curve has implicit equation \(y ^ { 2 } + 2 x \ln y = x ^ { 2 }\).
Verify that the point \(( 1,1 )\) lies on the curve, and find the gradient of the curve at this point.
AQA C4 2011 June Q6
10 marks Standard +0.3
6 A curve is defined by the equation \(2 y + \mathrm { e } ^ { 2 x } y ^ { 2 } = x ^ { 2 } + C\), where \(C\) is a constant. The point \(P \left( 1 , \frac { 1 } { \mathrm { e } } \right)\) lies on the curve.
  1. Find the exact value of \(C\).
  2. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
  3. Verify that \(P \left( 1 , \frac { 1 } { \mathrm { e } } \right)\) is a stationary point on the curve.