SECTION A
Question 1:Let ∗ be a binary operation on the set of all non-zero real numbers, given by
\[ a ∗ b = \frac{ab}{5} \quad \text{for all } a, b \in \mathbb{R} – \{0\}. \] Find the value of \( x \), given that \( 2 ∗ (x ∗ 5) = 10 \).
Question 2: If
\[
\sin\left(\sin^{-1}\frac{1}{5} + \cos^{-1}x\right) = 1,
\] then find the value of \( x \).
—
Question 3: If
\[
2 \begin{bmatrix} 3 & 4 \\ 5 & x \end{bmatrix} + \begin{bmatrix} 1 & y \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 7 & 0 \\ 10 & 5 \end{bmatrix},
\] find \( x – y \).
—
Question 4: Solve the following matrix equation for \( x \):
\[
[x \, 1] \begin{bmatrix} 1 & 0 \\ -2 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \end{bmatrix}.
\]
Question 5: If
\[
\begin{vmatrix}
2x & 5 \\
8 & x
\end{vmatrix} =
\begin{vmatrix}
6 & -2 \\
7 & 3
\end{vmatrix},
\] write the value of \( x \).
Question 6: Write the antiderivative of
\[
\left(3\sqrt{x} + \frac{1}{\sqrt{x}}\right).
\]
Question 7: Evaluate:
\[
\int_0^\infty \frac{dx}{9 + x^2}.
\]
Question 8: Find the projection of the vector
\[
\mathbf{i} + 3\mathbf{j} + 7\mathbf{k}
\] on the vector
\[
2\mathbf{i} – 3\mathbf{j} + 6\mathbf{k}.
\]
Question 9: If \( \mathbf{a} \) and \( \mathbf{b} \) are two unit vectors such that \( \mathbf{a} + \mathbf{b} \) is also a unit vector, then find the angle between \( \mathbf{a} \) and \( \mathbf{b} \).
Question 10: Write the vector equation of the plane, passing through the point \((a, b, c)\) and parallel to the plane
\[
\mathbf{r} \cdot (\mathbf{i} + \mathbf{j} + \mathbf{k}) = 2.
\]
SECTION B
Question 11: Let \( A = \{1, 2, 3, \ldots, 9\} \) and \( R \) be the relation in \( A \times A \) defined by \((a, b) R (c, d)\) if \( a + d = b + c \) for \((a, b), (c, d) \in A \times A\). Prove that \( R \) is an equivalence relation. Also, obtain the equivalence class \([2, 5]\).
Question 12: Prove that
\[
\cot^{-1} \left( \frac{1 + \sin x + \frac{1 – \sin x}{\sqrt{1 + \sin x} – \sqrt{1 – \sin x}}}{\sqrt{1 + \sin x} + \sqrt{1 – \sin x}} \right) = \frac{x}{2}, \quad x \in \left( 0, \frac{\pi}{4} \right).
\] OR
Prove that
\[
2 \tan^{-1} \left( \frac{1}{5} \right) + \sec^{-1} \left( \frac{5\sqrt{2}}{7} \right) + 2 \tan^{-1} \left( \frac{1}{8} \right) = \frac{\pi}{4}.
\]
Question 13: Using properties of determinants, prove that
\[
\begin{vmatrix}
2y & y – z & x \\
z – x & 2x & y \\
x – y & z & 2z
\end{vmatrix} = (x + y + z)^3.
\]
Question 14:Differentiate
\[
\tan^{-1}\left(\frac{\sqrt{1 – x^2}}{x}\right)
\] with respect to
\[
\cos^{-1}\left(2x\sqrt{1 – x^2}\right), \quad \text{when } x \neq 0.
\]
Question 15: If \( y = x^t \), prove that:
\[
\frac{d^2y}{dx^2} – \frac{1}{y} \left(\frac{dy}{dx}\right)^2 – \frac{y}{x} = 0.
\]
—
Question 16: (a) Find the intervals in which the function:
\[
f(x) = 3x^4 – 4x^3 – 12x^2 + 5
\] is:
1. Strictly increasing.
2. Strictly decreasing.
OR
(b) Find the equations of the tangent and normal to the curve:
\[
x = a\sin^3\theta, \quad y = a\cos^3\theta
\] at \( \theta = \frac{\pi}{4} \).
—
Question 17: Evaluate:
\[
\int \frac{\sin^6x + \cos^6x}{\sin^2x – \cos^2x} dx
\] OR
\[
\int (x – 3)\sqrt{x^2 + 3x – 18} dx.
\]
—
Question 18: Find the particular solution of the differential equation:
\[
e^y\sqrt{1-y^2} dx + \frac{y}{x} dy = 0,
\] given \( y = 1 \) when \( x = 0 \).
Question 19: Solve the following differential equation:
\[
(x^2 – 1) \frac{dy}{dx} + 2xy = \frac{2}{x^2 – 1}.
\]
Question 20: (a) Prove that, for any three vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \):
\[
[\mathbf{a} + \mathbf{b}, \mathbf{b} + \mathbf{c}, \mathbf{c} + \mathbf{a}] = 2[\mathbf{a}, \mathbf{b}, \mathbf{c}].
\] OR
(b) Vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are such that \( \mathbf{a} + \mathbf{b} + \mathbf{c} = 0 \) and \( |\mathbf{a}| = 3, |\mathbf{b}| = 5, |\mathbf{c}| = 7 \). Find the angle between \( \mathbf{a} \) and \( \mathbf{b} \).
Question 21: Show that the lines:
\[
\frac{x + 1}{3} = \frac{y + 3}{5} = \frac{z + 5}{7},
\quad \text{and} \quad \frac{x – 2}{1} = \frac{y – 4}{3} = \frac{z – 6}{5}
\] intersect. Also find their point of intersection.
Question 22: Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls, given that:
1. The youngest is a girl.
2. At least one is a girl.
SECTION C
Question 23: Two schools P and Q want to award their selected students on the values of Discipline, Politeness, and Punctuality. The school P wants to award \( x \) each, \( y \) each, and \( z \) each for the three respective values to its 3, 2, and 1 students with a total award money of ₹1,000. School Q wants to spend ₹1,500 to award its 4, 1, and 3 students on the respective values (by giving the same award money for the three values as before). If the total amount of awards for one prize on each value is ₹600, using matrices, find the award money for each value. Apart from the above three values, suggest one more value for awards.
—
Question 24: Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is:
\[
\cos^{-1}\left(\frac{1}{\sqrt{3}}\right).
\]
—
Question 25: Evaluate:
\[
\int_{\pi/6}^{\pi/3} \frac{dx}{1 + \sqrt{\cot x}}.
\]
—
Question 26: Find the area of the region in the first quadrant enclosed by the \( x \)-axis, the line \( y = x \), and the circle \( x^2 + y^2 = 32 \).
Question 27: (a) Find the distance between the point \((7, 2, 4)\) and the plane determined by the points \( A(2, 5, -3) \), \( B(-2, -3, 5) \), and \( C(5, 3, -3) \).
OR
(b) Find the distance of the point \((-1, -5, -10)\) from the point of intersection of the line:
\[
\mathbf{r} = 2\mathbf{i} – \mathbf{j} + 2\mathbf{k} + \lambda (3\mathbf{i} + 4\mathbf{j} + 2\mathbf{k})
\] and the plane:
\[
\mathbf{r} \cdot (\mathbf{i} – \mathbf{j} + \mathbf{k}) = 5.
\]
—
Question 28: A dealer in rural areas wishes to purchase a number of sewing machines. He has only ₹5,760 to invest and has space for at most 20 items for storage. An electronic sewing machine costs ₹360 and a manually operated sewing machine ₹240. He can sell an electronic sewing machine at a profit of ₹22 and a manually operated sewing machine at a profit of ₹18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit? Make it as a LPP and solve it graphically.
—
Question 29: A card from a pack of 52 playing cards is lost. From the remaining cards of the pack, three cards are drawn at random (without replacement) and are found to be all spades. Find the probability of the lost card being a spade.
OR
From a lot of 15 bulbs which include 5 defectives, a sample of 4 bulbs is drawn one by one with replacement. Find the probability distribution of number of defective bulbs. Hence find the mean of the distribution.