Board Paper of Class 12 2018 Mathematics - Solutions

(a) The binary operation ∗ : R × R → R is defined as a ∗ b = 2a + b. Find (2 ∗ 3) ∗ 4.

(b) If $A=\left(\begin{array}{cc}5& a\\ b& 0\end{array}\right)$and A is symmetric matrix, show that 𝑎 = b

(c) Solve: 3 tan^–1x + cot^–1x = π

(d) Without expanding at any stage, find the value of:

$\left|\begin{array}{ccc}a& b& c\\ a+2x& b+2y& c+2z\\ x& y& z\end{array}\right|$

(e) Find the value of constant ‘k’ so that the function f(x) defined as:

$f\left(x\right)=\left\{\begin{array}{ll}\frac{x^2-2x-3}{x+1},& x\ne -1\\ k,& x=-1\end{array}\right.$ 

is continuous at x = −1.

(f) Find the approximate change in the volume ‘𝑉’ of a cube of side x metres caused by decreasing the side by 1%.

(g) Evaluate : $\int \frac{x^3+5x^2+4x+1}{x^2}dx.$

(h) Find the differential equation of the family of concentric circles x² + y² = a²

(i) If A and B are events such that $P\left(A\right)=\frac{1}{2}, P\left(B\right)=\frac{1}{3}$ and $P\left(A\cap B\right)=\frac{1}{4},$ then find:
(a) P(A⁄B)
(b) P(B⁄A)

(j) In a race, the probabilities of A and B winning the race are $\frac{1}{3}$ and $\frac{1}{6}$ respectively. Find the probability of neither of them winning the race. VIEW SOLUTION

If the function $f\left(x\right)=\sqrt{2x-3}$ is invertible then find its inverse. Hence prove that (𝑓𝑜𝑓^–1) (x) = x. VIEW SOLUTION

If tan^–1 𝑎 + tan^–1 b + tan^–1 𝑐 = π, prove that a + b + c = abc. VIEW SOLUTION

Use properties of determinants to solve for x:

$\left|\begin{array}{ccc}x+a& b& c\\ c& x+b& a\\ a& b& x+c\end{array}\right|=0 \mathrm{and} x\ne 0.$ VIEW SOLUTION

Show that the function $f\left(x\right)=\left\{\begin{array}{ll}{x}^2,& x\le 1\\ \frac{1}{x},& x>1\end{array}\right.$is continuous at x = 1 but not differentiable.

OR

Verify Rolle’s theorem for the following function: f(x) = e^–x sin x on [0, π] VIEW SOLUTION

If $x=\tan \left(\frac{1}{a}\log y\right),$ prove that $\left(1+x^2\right)\frac{d^{2}y}{d{x}^2}+\left(2x-a\right)\frac{dy}{dx}=0$ VIEW SOLUTION

Evaluate: $\int {\tan }^{-1}\sqrt{x} dx$ VIEW SOLUTION

Find the points on the curve y = 4x³ – 3x + 5 at which the equation of the tangent is parallel to the x-axis.

OR

Water is dripping out from a conical funnel of semi-vertical angle $\frac{\pi }{4}$ at the uniform rate of 2 cm²/sec in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is 4 cm, find the rate of decrease of the slant height of the water. VIEW SOLUTION

Solve: $\sin x\frac{dy}{dx}-y=\sin x·\tan \frac{x}{2}$

OR

The population of a town grows at the rate of 10% per year. Using differential equation, find how long will it take for the population to grow 4 times.
  VIEW SOLUTION

Using matrices, solve the following system of equations:
2x – 3y + 5z = 11
3x + 2y – 4z = −5
x + y – 2z = – 3

OR

Using elementary transformation, find the inverse of the matrix:

$\left[\begin{array}{ccc}0& 1& 2\\ 1& 2& 3\\ 3& 1& 1\end{array}\right]$ VIEW SOLUTION

A speaks truth in 60% of the cases, while B in 40% of the cases. In what percent of cases are they likely to contradict each other in stating the same fact? VIEW SOLUTION

A cone is inscribed in a sphere of radius 12 cm. If the volume of the cone is maximum, find its height. VIEW SOLUTION

Evaluate: $\int \frac{x-1}{\sqrt{x^2-x}}dx$

OR

Evaluate:${\int }_0^{\pi /2}\frac{{\cos }^{2}x}{1+\sin x \cos x}dx$ VIEW SOLUTION

From a lot of 6 items containing 2 defective items, a sample of 4 items are drawn at random. Let the random variable X denote the number of defective items in the sample. If the sample is drawn without replacement, find:
(a) The probability distribution of X
(b) Mean of X
(c) Variance of X VIEW SOLUTION