CUET UG Mathematics · Paper 5

$\int e^x(\sin x + \cos x)\,dx$ is equal to

The region lying above the x-axis and under $y = 3x^2$, between the vertical lines $x = 1$ and $x = 2$, has an area (in square units) of

If $x = t^2$ and $y = t^3$, then $\frac{d^2y}{dx^2}$ is equal to

The corner points of the bounded feasible region determined by a system of linear constraints are $(0, 3)$, $(1, 1)$ and $(3, 0)$. Let $Z = px + qy$, where $p, q > 0$. The condition on $p$ and $q$ so that the minimum of $Z$ occurs at $(3, 0)$ and $(1, 1)$ is

Assertion: One of the particular solutions of the differential equation $\frac{dy}{dx} = e^{x+y}$ can be $e^x + e^{-y} = -2$.

Reason: $e^x + e^{-y} = C$ is the general solution of the differential equation $\frac{dy}{dx} = e^{x+y}$.

Let $R$ be the relation on the set $\mathbb{Z}$ of integers given by $R = \{(a, b) : a - b \text{ is divisible by } 5\}$. The equivalence class $[2]$ is

The value of $m$ for which the points with position vectors $-\hat{i} - \hat{j} + 2\hat{k}$, $2\hat{i} + m\hat{j} + 5\hat{k}$ and $3\hat{i} + 11\hat{j} + 6\hat{k}$ are collinear, is

The number of points of discontinuity of $f(x) = \begin{cases} x + 3, & \text{if } x \le -3 \\ -2x, & \text{if } -3 < x < 3 \\ 6x + 2, & \text{if } x \ge 3 \end{cases}$ is

If $A^2 = 4A + 3I$ and $A^{-1} = xA + yI$, then the value of $(x + y)$ is

If $f(a + b - x) = f(x)$, then $\int_a^b x f(x)\,dx$ is equal to

For $x > 0$, which function satisfies $\frac{dy}{dx} + \frac{y}{x} = x^2$ together with $y = 1$ at $x = 1$?

Meena and Kabir throw a fair die alternately, Meena first. The first person to throw a six wins. The probability that Meena wins is

Using flat rate method, the EMI to repay a loan of ₹20,000 in $2\frac{1}{2}$ years at an interest rate of 8% p.a. is

Match List I (component of a time series) with List II (example).

Choose the correct answer.

A vessel contains 80 litres of milk. 8 litres of milk is taken out and replaced by water. This process is repeated once more. The quantity of milk now in the vessel is

Pipes X and Y can fill a tank in 20 minutes and 30 minutes respectively. Both pipes are opened together. After how many minutes should pipe Y be closed so that the tank is full in 15 minutes in all?

The demand function of a monopolist is $p = 60 - 2x$ and the total cost function is $C(x) = x^2 + 12x + 50$, where $x$ is the output. The output that maximises the profit is

In a binomial distribution, $n = 200$ and $p = 0.04$. Taking Poisson distribution as an approximation to the binomial distribution:

Assertion: Mean of Poisson distribution = 8.

Reason: $P(X = 4) = \frac{512}{3e^8}$.

A ₹10,000 bond carries interest at 10% per annum, paid every six months, and will be repaid at its face value after 2 years. An investor wants a return of 8% per annum compounded half-yearly. What should the investor pay for it? (use $(1.04)^{-4} = 0.8548$)