CUET UG Mathematics · Paper 2

The general solution of the differential equation $\frac{dy}{dx} = \frac{\sqrt{y}}{\sqrt{x}}$ is

If the sum of all the elements of a $3 \times 3$ scalar matrix is 9, then the product of all its elements is

The value of the definite integral $\int_0^{\pi/4}\tan^2 x\,dx$ is

If $A = \begin{bmatrix} 1 & -1 \\ 2 & 3 \end{bmatrix}$ and $A^{-1} = \frac{1}{5}(kI - A)$, then the value of $k$ is

Minimise $Z = 3x + 5y$ subject to the constraints $x + 3y \ge 3$, $x + y \ge 2$, $x \ge 0$, $y \ge 0$. The minimum value of $Z$ is

The domain of $f(x) = \cos^{-1}(2x - 5)$ is

If $e^{-x} + e^{-y} = 2$, then $\frac{dy}{dx}$ is

The volume of a spherical balloon is increasing at the rate of 25 $\text{cm}^3$/s. The rate at which its radius is increasing when the radius is 5 cm is

The area (in square units) of the region bounded by the parabola $y^2 = 4x$ and the line $x = 3$ is

Consider the linear programming problem: maximise $Z = 3x + 2y$ subject to $x + 2y \le 10$, $3x + y \le 15$, $x \ge 0$, $y \ge 0$.

Which of the following statements are true?

I. $(4, 3)$ is a corner point of the feasible region.

II. The maximum value of $Z$ is 18.

III. $Z$ is maximum at $(5, 0)$.

IV. The feasible region is unbounded.

If $\vec{a} + \vec{b} + \vec{c} = \vec{0}$, $|\vec{a}| = 3$, $|\vec{b}| = 5$ and $|\vec{c}| = 7$, then the angle between $\vec{a}$ and $\vec{b}$ is

In a 1 km race, player P beats player Q by 18 metres or 9 seconds. What is P's time to complete the race?

Which of the following is a parameter?

If for a Poisson variate X, $P(X = k) = P(X = k + 1)$, then the variance of X is

Pipes P and Q can fill a tank in 12 hours and 15 hours respectively, and pipe R can empty the full tank in 20 hours. All three pipes are opened together. After 4 hours, pipe R is closed. In how many more hours will the tank be full?

The total cost function of a firm is $C(x) = x^3 - 12x^2 + 60x + 100$, where $x$ is the output. The minimum value of the marginal cost is

If $P = \begin{bmatrix} 1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{bmatrix}$ is the adjoint of a $3 \times 3$ matrix $A$ and $|A| = 4$, then $\alpha$ is equal to

A bond of face value ₹1,000 pays an annual coupon at 8% and is redeemable at par after 3 years. If the investor's required yield is 10% per annum, the price of the bond is (use $(1.1)^{-3} = 0.7513$)