List of practice Questions

Find : \( \int \frac{x - \sin x}{1 - \cos x} \, dx \).
Evaluate : \( \int_0^2 \frac{1}{\sqrt{x^2 + 2x + 3}} \, dx \).
Solve the following Linear Programming Problem graphically :
Maximize \( Z = \frac{2x}{5} + \frac{3y}{10} \)
subject to constraints
\( 2x + y \leq 1000 \)
\( x + y \leq 800 \)
\( x, y \geq 0 \).
Three students A, B and C go to a book-store to buy art books, story books and puzzle solving books. A buys one of each type of book for a total of RS 21. B buys 4 art books, 3 story books and 2 puzzle solving books for RS 60. C buys 6 art books, 2 story books and 3 puzzle solving books and pays RS 10 more than B. Use matrix method to find the cost of each type of book.
If \( y = P \cos ux + Q \sin ux \), show that \( \frac{d^2y}{dx^2} + u^2y = 0 \).
Three honey bees were found flying along the vectors \( \vec{a} = 2\hat{i} - 3\hat{j} + \hat{k} \), \( \vec{b} = 4\hat{j} - 2\hat{k} \) and \( \vec{c} = 3\hat{i} + 2\hat{k} \) respectively. Find the value of \( \lambda \) such that the path for \( \vec{a} + \lambda \vec{b} \) is perpendicular to \( \vec{c} \).
If \( A, B \) and \( C \) be three non-collinear points such that \( \vec{AB} = \hat{i} + 2\hat{j} - \hat{k} \) and \( \vec{AC} = 2\hat{i} - 3\hat{j} \), then find the area of \( \triangle ABC \).
The volume of a wooden block in the shape of a cube increases at a constant rate as the air becomes moist during the rainy season. Show that the rate of change of its surface area varies inversely as the length of edge of the cube.
A die is rolled. Consider events : \( A = \{1, 2, 5\} \), \( B = \{3, 5\} \), \( C = \{2, 3, 4, 5\} \). Find \( P(A|C) \) and \( P(C|A) \).
A die is rolled. Consider events : \( A = \{1, 2, 5\} \), \( B = \{3, 5\} \), \( C = \{2, 3, 4, 5\} \). Find \( P(A \cap B|C) \) and \( P(A \cup B|C) \).
A box contains 6 cards numbered 1 to 6. A student is asked to pick up two cards, one by one after replacement and note down the numbers on the cards. Let A be the event of getting sum of the numbers on two cards as 10, and B, the event of a number other than 4 on the first card selected. Find \( P(A \text{ and } B) \) and find whether the events A and B are independent events or not.
Absolute minimum value of \( f(x) = (x - 2)^2 + 5 \) in the interval \( [-3, 2] \) is :
Assertion (A) : A relation \( R \) on the set \( \{1, 2, 3\} \) defined as \( R = \{(1, 1), (1, 2), (2, 1), (2, 2), (3, 3)\ \) is an equivalence relation.
Reason (R) : A relation that is reflexive, symmetric and transitive is an equivalence relation.}
Assertion (A) : Consider a Linear Programming Problem with minimise \( Z = x + 2y \) subject to constraints \( 2x + y \geq 3, x + 2y \geq 6, x, y \geq 0 \) which gives minimum \( Z \) at infinitely many points. The corner points of feasible region are \( (0, 3) \) and \( (6, 0) \).
Reason (R) : If two corner points produce the same minimum value of the objective function, then every point on the line segment joining the points will give the same minimum value.
Evaluate \( \tan \left[ \cos^{-1} \left( \tan \frac{3\pi}{4} \right) \right] \).
Find the angle between the following pair of lines : \( \frac{x-2}{3} = \frac{y+5}{2} = \frac{1-z}{-6} \) and \( \frac{x-7}{1} = \frac{y}{2} = \frac{6-z}{-2} \).
Differentiate \( x^x \) with respect to \( x \log x \).
The corner points of the feasible region determined by the system of linear constraints are \( (0, 0) \), \( (0, 40) \), \( (20, 40) \), \( (60, 20) \) and \( (60, 0) \). If the objective function of an LPP is \( Z = 4x + 3y \), then the maximum value is :
An ant is observed crawling on a sheet of paper along a straight line given by equation \( y = 2x - 4 \). Area of the surface covered by the ant bounded by y-axis, x-axis and \( x = 1 \) is :
The general solution for the differential equation \( \frac{dy}{dx} = e^{3x-y} \) is :
Derivative of \( \cos^{-1} \left( \frac{\sin x + \cos x}{\sqrt{2}} \right) \), \( -\frac{\pi}{4} < x < \frac{\pi}{4} \) with respect to \( x \) is :
If vectors \( \vec{a} = 3\hat{i} + 2\hat{j} + \lambda\hat{k} \) and \( \vec{b} = 2\hat{i} - 4\hat{j} + 5\hat{k} \), represent the two strips of the Red Cross sign placed outside a doctor's clinic, then the value of \( \lambda \) is :
If the area of \( \triangle ABC \) with vertices \( A(3, 1) \), \( B(-2, 1) \) and \( C(0, k) \) is 5 sq. units, then values of \( k \) are :
\( \int \sqrt{\frac{1 + \cos x}{1 - \cos x}} \, dx \) is equal to :
If \( \int_0^1 (6x^2 - 4x + k) \, dx = 0 \), then the value of \( k \) is :