List of practice Questions

Find points at which the tangent to the curve y = x³ − 3x² − 9x + 7 is parallel to the x axis.

Find the slope of the normal to the curve \(x = 1 − a\, sin θ,\, y = b\, cos^2 θ\) at θ=\(\frac{π}{2}\).

Find the slope of the normal to the curve x = acos³ θ, y = asin³ θ at θ=π/4.

Find the slope of the tangent to the curve y = x³ − 3x + 2 at the point whose x-coordinate is 3.

Find the slope of the tangent to curve y = x³ − x + 1 at the point whose x-coordinate is 2

Recently Govt. of India has allowed mixing of alcohol in petrol. What is the amount of alcohol permitted for mixing in petrol

Find the slope of the tangent to the curve,y = \(\frac{x-1}{x-2}\), x ≠ 2 at x = 10.

Find the slope of the tangent to the curve y = 3x⁴ − 4x at x = 4.

By using properties of determinants,show that:
(i)\(\begin{vmatrix} a-b-c & 2a & 2a\\ 2b & b-c-a & 2b \\ 2c&2c&c-a-b\end{vmatrix}\)=(a+b+c)³ 
(ii)\(\begin{vmatrix} x+y+2z & x & y\\ z & y+z+2x & y\\ z &x&z+x+2y \end{vmatrix}\)=2(x+y+z)³

By using properties of determinants, show that:
(i)\(\begin{vmatrix} x+4 & 2x & 2x\\ 2x & x+4 & 2x \\2x &2x&x+4\end{vmatrix}\)=(5x+4)(4-x)² 
(II)\(\begin{vmatrix} y+k & y & y\\ y & y+k & y \\y &y&y+k\end{vmatrix}\)=k²(3y+k)

By using properties of determinants,show that:I\(\begin{vmatrix} x & x^2 & yz\\ y & y^2 & zx\\z&z^2&xy \end{vmatrix}\)=(x-y)(y-z)(z-x)(xy+yz+zx)

Using the property of determinants and without expanding,prove that:\(\begin{vmatrix} b+c & q+r & y+z\\ c+a & r+p & z+x\\ a+b&p+q&x+y \end{vmatrix}\)=2\(\begin{vmatrix} a & p & x\\ b & q & y\\c&r&z \end{vmatrix}\)

If A is square matrix such that A²=A,then (I+A)³-7A is equal to

Choose the correct answer in the following questions: If \(\begin{bmatrix} α & β  \\ \gamma & -\alpha \end{bmatrix}\) is such that A²=I then  

If A= \(\begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}\) show that A²-5A+7I=0

Show that the matrix \(B'AB\) is symmetric or skew symmetric according as A is symmetric or skew symmetric.

Find the inverse of each of the matrices, if it exists\(\begin{bmatrix} 2 & 1 \\ 4 & 2 \end{bmatrix}\)

Find the inverse of each of the matrices, if it exists. \(\begin{bmatrix} 2 & -3\\ -1 & 2 \end{bmatrix}\)

During replication of a bacterial chromosomes DNA synthesis starts from a replication origin site and

Find the inverse of each of the matrices, if it exists. \(\begin{bmatrix} 2 & -6 \\ 1 & -2 \end{bmatrix}\)

Find the inverse of each of the matrices, if it exists. \(\begin{bmatrix} 3 & -1 \\ -4 & 2 \end{bmatrix}\)

Find the inverse of each of the matrices, if it exists.
\(\begin{bmatrix} 3 & 10\\ 2 & 7\end{bmatrix}\)

Prove \(tan^{-1}\sqrt{x}=\frac{1}{2}cos^{-1}(\frac{1-x}{1+x}),x∈[0,1]\)

Consult internet and find out how to make orally active protein pharmaceutical. What is the major problem to be encountered?

Which of the following differential equations has y=x as one of its particular solution?