List of practice Questions

If y = y(x) is the solution of the differential equation
$x$ $\frac{dy}{dx}$ $+ 2y =$ $xe^x , y(1) = 0$
then the local maximum value of the function
$z(x) = x²y(x) - e^x , x ∈ R$
is

The acute angle between the planes P₁ and P₂, when P₁ and P₂ are the planes passing through the intersection of the planes 5x + 8y + 13z – 29 = 0 and 8x – 7y + z – 20 = 0 and the points (2, 1, 3) and (0, 1, 2), respectively, is

The shortest distance between the lines $\frac{x−3}{2}=\frac{y−2}{3}=\frac{z−1}{−1}$ and $\frac{x+3}{2}=\frac{y-6}{1}=\frac{z−5}{3}$ is

Let the sum of an infinite G.P., whose first term is a and the common ratio is r, be 5. Let the sum of its first five terms be 98/25. Then the sum of the first 21 terms of an AP, whose first term is 10ar, n^th term is a_n and the common difference is 10ar², is equal to

A circle C₁ passes through the origin O and has diameter 4 on the positive x-axis. The line y = 2x gives a chord OA of circle C₁. Let C₂ be the circle with OA as a diameter. If the tangent to C₂ at the point A meets the x-axis at P and y-axis at Q, then QA :AP is equal to

Let α and β be the roots of the equation x² + (2i – 1) = 0. Then, the value of |α² + β²| is equal to

If the plane 2x + y – 5z = 0 is rotated about its line of intersection with the plane 3x – y + 4z – 7 = 0 by an angle of π/2, then the plane after the rotation passes through the point:

If two distinct points Q, R lie on the line of intersection of the planes –x + 2y – z = 0 and 3x – 5y + 2z = 0 and
$PQ = PR = \sqrt{18}$
where the point P is (1, –2, 3), then the area of the triangle PQR is equal to

Let
$\frac{x-2}{3} = \frac{y+1}{-2} = \frac{z+3}{-1}$
lie on the plane px – qy + z = 5, for some p, q ∈ ℝ. The shortest distance of the plane from the origin is :

The mean and variance of the data $4, 5, 6, 6, 7, 8, x, y$, where $x<y$, are $6$ and $\frac{9}{4}$, respectively. Then $x^4+y^2$ is equal to