Commerce (English Medium) Class 12 - CBSE Question Bank Solutions

The slope of the tangent to the curve x = a sin t, y = a{cot t + log(tan `"t"/2`)} at the point ‘t’ is ____________.

The tangent to the parabola x² = 2y at the point (1, `1/2`) makes with the x-axis an angle of ____________.

The two curves x³ - 3xy² + 5 = 0 and 3x²y - y³ - 7 = 0

The distance between the point (1, 1) and the tangent to the curve y = e^2x + x² drawn at the point x = 0

The tangent to the curve y = 2x² - x + 1 is parallel to the line y = 3x + 9 at the point ____________.

The tangent to the curve y = x² + 3x will pass through the point (0, -9) if it is drawn at the point ____________.

Find a point on the curve y = (x – 2)². at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).

Tangents to the curve x² + y² = 2 at the points (1, 1) and (-1, 1) are ____________.

The line y = x + 1 is a tangent to the curve y² = 4x at the point

Find points on the curve `x^2/9 + "y"^2/16` = 1 at which the tangent is parallel to y-axis. 

Find the equation of the tangent line to the curve y = x² − 2x + 7 which is parallel to the line 2x − y + 9 = 0.

If x = a sin t and `y = a (cost+logtan(t/2))` ,find `((d^2y)/(dx^2))`

Evaluate : `int_0^4(|x|+|x-2|+|x-4|)dx`

If y=2 cos(logx)+3 sin(logx), prove that `x^2(d^2y)/(dx2)+x dy/dx+y=0`

Evaluate `∫_0^(3/2)|x cosπx|dx`

Show that four points A, B, C and D whose position vectors are 

`4hati+5hatj+hatk,-hatj-hatk-hatk, 3hati+9hatj+4hatk and 4(-hati+hatj+hatk)` respectively are coplanar.

If x cos(a+y)= cosy then prove that `dy/dx=(cos^2(a+y)/sina)`

Hence show that `sina(d^2y)/(dx^2)+sin2(a+y)(dy)/dx=0`

Evaluate `int_(-1)^2|x^3-x|dx`

Find the coordinate of the point P where the line through A(3, –4, –5) and B(2, –3, 1) crosses the plane passing through three points L(2, 2, 1), M(3, 0, 1) and N(4, –1, 0).
Also, find the ratio in which P divides the line segment AB.

Evaluate :

`∫_(-pi)^pi (cos ax−sin bx)^2 dx`