Class 12 Mathematics 2019 Board Exam

Solve : ${\tan }^{-1}4x+{\tan }^{-1}6x=\frac{\pi }{4}$.

Find : $\int \frac{{\sec }^{2}x}{{\tan }^{2}x+4}dx$.

Examine whether the operation $\ast$ defined on $\mathbb{R}$ by $a\ast b=ab+1$ is (i) a binary operation or not.
(ii) if a binary operation, is it associative or not?

Using integration, find the area of triangle ABC, whose vertices are $A(2,5)$, $B(4,7)$ and $C(6,2)$.

OR

Find the area of the region lying above $x$-axis and included between the circle $x^2+y^2=8x$ and inside of the parabola $y^2=4x$.

Find the order and the degree of the differential equation $x^2\frac{d^{2}y}{d{x}^2}={\left\{1+{\left(\frac{dy}{dx}\right)}^2\right\}}^4$.

If the sum of two unit vectors is a unit vector, prove that the magnitude of their difference is $\sqrt{3}$.

OR

If $\vec{a}=2\hat{i}+3\hat{j}+\hat{k}$, $\vec{b}=\hat{i}-2\hat{j}+\hat{k}$ and $\vec{c}=-3\hat{i}+\hat{j}+2\hat{k}$, find $[\vec{a}\vec{b}\vec{c}]$.

Find the direction cosines of a line which makes equal angles with the coordinate axes.

OR

A line passes through the point with position vector $2\hat{i}-\hat{j}+4\hat{k}$ and is in the direction of the vector $\hat{i}+\hat{j}-2\hat{k}$. Find the equation of the line in cartesian form.

A manufacturer has employed 5 skilled men and 10 semi-skilled men and makes two models A and B of an article. The making of one item of model A requires 2 hours work by a skilled man and 2 hours work by a semi-skilled man. One item of model B requires 1 hour by a skilled man and 3 hours by a semi-skilled man. No man is expected to work more than 8 hours per day. The manufacturer's profit on an item of model A is ₹ 15 and on an item of model B is ₹ 10. How many of items of each model should be made per day in order to maximize daily profit? Formulate the above LPP and solve it graphically and find the maximum profit.

Find the equation of tangent to the curve $y=\sqrt{3x-2}$ which is parallel to the line $4x-2y+5=0$. Also, write the equation of normal to the curve at the point of contact.

If $f(x)=x+1$, find $\frac{d}{dx}(f\circ f)(x)$.