Mathematics - 4 — MH-CET Full Length Paper MCQs with Solutions

If $p$ and $q$ are two propositions, then the logical statement $\sim(p \leftrightarrow q)$ is equivalent to:

The inverse of the statement "If a number is divisible by 6, then it is divisible by 3" is:

If the pair of straight lines given by $ax^2 + 2hxy + by^2 = 0$ makes angles $\alpha$ and $\beta$ with the $x$-axis such that $\alpha + \beta = \frac{\pi}{2}$, then:

The area of the triangle formed by the pair of straight lines $x^2 - 4y^2 = 0$ and the line $x = 4$ is:

If $|\vec{a}| = 3$, $|\vec{b}| = 4$, and $|\vec{a} - \vec{b}| = 5$, then the value of $|\vec{a} + \vec{b}|$ is:

The angle of intersection between the two curves $y^2 = 4x$ and $x^2 = 4y$ at the origin $(0, 0)$ is:

A stone dropped into a quiet lake produces concentric circular ripples. The radius of the outer ripple increases at a uniform rate of $4$ cm/s. When the radius is $10$ cm, the rate at which the enclosed area increases is:

The absolute minimum value of $f(x) = 2x^3 - 3x^2 - 12x + 1$ on the closed interval $[-2, 3]$ is:

The value of $\int_0^3 [x]\,dx$, where $[x]$ denotes the greatest integer function less than or equal to $x$, is:

The area enclosed between the curve $y = 4 - x^2$ and the $x$-axis is:

The differential equation corresponding to the family of parabolas having their vertex at the origin and axis along the positive $y$-axis is:

If the population of a town increases at a rate proportional to the population at any time $t$, and if it doubles in 40 years, the population will triple in:

The probability mass function of a discrete random variable $X$ is given by $P(X = x) = k \cdot \binom{4}{x}$ for $x = 0, 1, 2, 3, 4$. The value of the constant $k$ is:

A random variable $X$ has mean $\mu = 5$ and standard deviation $\sigma = 2$. If $Y = 3X - 4$, then the variance of $Y$ is:

If a fair coin is tossed 10 times, the probability of getting at most 2 heads is:

The distance between the two parallel planes $2x - 2y + z + 3 = 0$ and $4x - 4y + 2z + 12 = 0$ is:

If the direction cosines of a directed line segment are $\frac{1}{c}, \frac{1}{c}, \frac{1}{c}$, then the value of $c$ can be:

The coordinates of the point where the straight line $\frac{x-2}{3} = \frac{y+1}{4} = \frac{z-2}{12}$ intersects the plane $x - y + z = 16$ are:

If $\sin\theta + \cos\theta = 1$, then the value of $\sin 2\theta$ is:

The radius of the circle passing through the points $(0, 0)$, $(a, 0)$, and $(0, b)$ is:

If a focal chord of the parabola $y^2 = 4ax$ has one end at $(at_1^2, 2at_1)$, then its other end is located at:

The eccentricity of the hyperbola conjugate to $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is $e'$. If the eccentricity of the original hyperbola is $e$, then $\frac{1}{e^2} + \frac{1}{e'^2}$ is:

The standard deviation of the first $n$ natural numbers $1, 2, 3, \dots, n$ is given by:

A speaks truth in $75\%$ cases and B in $80\%$ cases. The probability that they contradict each other in narrating the same incident is:

If $z = \cos\theta + i\sin\theta$, then the value of $z^n + \frac{1}{z^n}$ is:

The number of different necklaces that can be formed using 7 distinct colored beads is:

The value of the limit $\lim_{x \to 1} \frac{x^{15} - 1}{x^{10} - 1}$ is:

The indefinite integral $\int \frac{dx}{(x-1)\sqrt{x+2}}$ is equal to:

The integral $\int \frac{x + \sin x}{1 + \cos x}\,dx$ is equal to:

The value of the definite integral $\int_0^{\pi/2} \frac{\sin^2 x}{\sin x + \cos x}\,dx$ is: