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Mathematics - 5 — MH-CET Full Length Paper MCQs with Solutions
Free MH-CET Full Length Paper Mathematics - 5 MCQs with step-by-step solutions. Practise online on Prepizo — no login needed.
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Sample questions with solutions
Q1 — easy
The negation of the statement "If it rains, then the match will be cancelled" is:
A. It rains and the match will not be cancelled. ✓ Correct
B. It does not rain and the match will be cancelled.
C. It rains or the match will not be cancelled.
D. If it does not rain, then the match will not be cancelled.
Solution: $\sim(p \to q) \equiv p \wedge \sim q$: "It rains and the match will not be cancelled."
Q2 — easy
Which of the following logical equivalences is FALSE?
A. $p \leftrightarrow q \equiv (p \to q) \wedge (q \to p)$
B. $\sim(p \wedge q) \equiv \sim p \wedge \sim q$ ✓ Correct
C. $p \to q \equiv \sim p \vee q$
D. $\sim(p \vee q) \equiv \sim p \wedge \sim q$
Solution: By De Morgan's law, $\sim(p \wedge q) \equiv \sim p \vee \sim q$, not $\sim p \wedge \sim q$. Option (B) is false.
Q3 — easy
In any $\triangle ABC$, if $a = 2$, $b = 3$, and $\sin A = \frac{2}{3}$, then the angle $B$ is equal to:
A. $\frac{\pi}{2}$ ✓ Correct
B. $\frac{\pi}{4}$
C. $\frac{\pi}{6}$
D. $\frac{\pi}{3}$
Solution: Sine rule: $\sin B = b\sin A/a = 3(2/3)/2 = 1 \implies B = \pi/2$.
Q4 — easy
The joint equation of the lines passing through $(1, 1)$ and parallel to the pair of lines $x^2 - 5xy + 6y^2 = 0$ is:
A. $(x-1)^2 - 5(x-1)(y-1) + 6(y-1)^2 = 0$ ✓ Correct
B. $x^2 - 5xy + 6y^2 + 2x - 3y + 1 = 0$
C. $(x+1)^2 - 5(x+1)(y+1) + 6(y+1)^2 = 0$
D. $x^2 - 5xy + 6y^2 - 2x + 7y - 2 = 0$
Solution: Replace $x$ by $x-1$ and $y$ by $y-1$ in the homogeneous equation.
Q5 — easy
If $\vec{a} = 2\hat{i} + 2\hat{j} + 3\hat{k}$, $\vec{b} = -\hat{i} + 2\hat{j} + \hat{k}$, and $\vec{c} = 3\hat{i} + \hat{j}$ are such that $\vec{a} + \alpha\vec{b}$ is perpendicular to $\vec{c}$, then $\alpha$ is:
A. $8$ ✓ Correct
B. $-8$
C. $4$
D. $-4$
Solution: $(\vec{a}+\alpha\vec{b})\cdot\vec{c} = 0 \implies 3(2-\alpha)+(2+2\alpha) = 0 \implies 8-\alpha = 0 \implies \alpha = 8$.
Q6 — easy
If $|\vec{a}| = 2$, $|\vec{b}| = 5$, and $|\vec{a} \times \vec{b}| = 8$, then the scalar dot product $\vec{a}\cdot\vec{b}$ is:
A. $\pm 6$ ✓ Correct
B. $\pm 5$
C. $\pm 4$
D. $\pm 10$
Solution: Lagrange identity: $(\vec{a}\cdot\vec{b})^2 = |\vec{a}|^2|\vec{b}|^2 - |\vec{a}\times\vec{b}|^2 = 100-64 = 36 \implies \vec{a}\cdot\vec{b} = \pm 6$.
Q7 — easy
The vector equation of the straight line passing through the points $A(1, 2, -1)$ and $B(2, 1, 1)$ is:
A. $\vec{r} = (\hat{i} + 2\hat{j} - \hat{k}) + \lambda(\hat{i} - \hat{j} + 2\hat{k})$ ✓ Correct
B. $\vec{r} = (2\hat{i} + \hat{j} + \hat{k}) + \lambda(\hat{i} + 2\hat{j} - \hat{k})$
C. $\vec{r} = (\hat{i} - \hat{j} + 2\hat{k}) + \lambda(\hat{i} + 2\hat{j} - \hat{k})$
D. $\vec{r} = (\hat{i} + 2\hat{j} - \hat{k}) + \lambda(3\hat{i} + 3\hat{j})$
Solution: Direction: $B-A = (1,-1,2)$. Line: $\vec{r} = (\hat{i}+2\hat{j}-\hat{k}) + \lambda(\hat{i}-\hat{j}+2\hat{k})$.
Q8 — easy
The perpendicular distance of the origin from the plane $3x - 4y + 12z - 26 = 0$ is:
A. $2$ units ✓ Correct
B. $13$ units
C. $4$ units
D. $1$ unit
Solution: $d = 26/\sqrt{9+16+144} = 26/13 = 2$ units.
Q9 — easy
An edge of a variable cube is increasing at the rate of $3$ cm/s. How fast is the volume of the cube increasing when the edge length is $10$ cm?
A. $900$ cm$^3$/s ✓ Correct
B. $300$ cm$^3$/s
C. $600$ cm$^3$/s
D. $1000$ cm$^3$/s
Solution: $dV/dt = 3x^2 \cdot dx/dt = 3(100)(3) = 900$ cm$^3$/s.
Q10 — easy
The integral $\int \frac{dx}{x^2 + 4x + 13}$ is evaluated as:
A. $\frac{1}{3}\tan^{-1}\left(\frac{x+2}{3}\right) + C$ ✓ Correct
B. $\frac{1}{6}\ln\left|\frac{x-1}{x+5}\right| + C$
C. $\frac{1}{2}\tan^{-1}\left(\frac{x+2}{2}\right) + C$
D. $\frac{1}{3}\ln|x^2 + 4x + 13| + C$
Solution: $\int dx/((x+2)^2+9) = (1/3)\tan^{-1}((x+2)/3) + C$.
Q11 — easy
The value of the integral $\int_{-2}^2 |x|\,dx$ is:
A. $4$ ✓ Correct
B. $2$
C. $8$
D. $0$
Solution: $\int_{-2}^2 |x|\,dx = 2\int_0^2 x\,dx = 2[x^2/2]_0^2 = 4$.
Q12 — easy
The area enclosed between the parabolas $y^2 = 4x$ and $x^2 = 4y$ is:
A. $\frac{16}{3}$ sq. units ✓ Correct
B. $\frac{32}{3}$ sq. units
C. $\frac{8}{3}$ sq. units
D. $16$ sq. units
Solution: Standard formula: area $= 16ab/3$ with $a = b = 1$. Area $= 16/3$ sq. units.
Q13 — easy
The area bounded by the curve $y = \ln x$, the $y$-axis, and the lines $y = 0$ and $y = 1$ is:
A. $e - 1$ sq. units ✓ Correct
B. $e$ sq. units
C. $1$ sq. unit
D. $e + 1$ sq. units
Solution: $\int_0^1 e^y\,dy = e - 1$ sq. units.
Q14 — easy
The order and degree of the differential equation $\left(\frac{d^3y}{dx^3}\right)^2 + 3\left(\frac{d^2y}{dx^2}\right)^3 + y = \sin x$ are:
A. $3$ and $2$ ✓ Correct
B. $2$ and $3$
C. $3$ and $3$
D. $1$ and $2$
Solution: Highest derivative: $d^3y/dx^3$ (order 3). Its highest power: 2 (degree 2).
Q15 — easy
The integrating factor of the differential equation $\frac{dy}{dx} - \frac{2}{x}y = x^3$ is:
A. $\frac{1}{x^2}$ ✓ Correct
B. $x^2$
C. $\frac{1}{x}$
D. $\ln x$
Solution: I.F. $= e^{\int -2/x\,dx} = e^{-2\ln x} = 1/x^2$.
Q16 — easy
A random variable $X$ has the probability distribution: $P(X=0) = k$, $P(X=1) = 3k$, $P(X=2) = 3k$, $P(X=3) = k$. The mean $E(X)$ of this distribution is:
A. $1.5$ ✓ Correct
B. $2.0$
C. $1.2$
D. $0.8$
Solution: $8k = 1 \implies k = 1/8$. $E(X) = 0(1/8)+1(3/8)+2(3/8)+3(1/8) = 12/8 = 1.5$.
Q17 — easy
If a random variable $X$ follows a Binomial distribution $B(n, p)$ with mean 6 and variance 2, then the number of trials $n$ is:
A. $9$ ✓ Correct
B. $12$
C. $8$
D. $6$
Solution: $np = 6$, $np(1-p) = 2 \implies 1-p = 1/3 \implies p = 2/3$. $n = 6/(2/3) = 9$.
Q18 — easy
A pair of dice is thrown 3 times. The probability of getting a total of 7 exactly two times is:
A. $\frac{5}{72}$ ✓ Correct
B. $\frac{25}{216}$
C. $\frac{5}{36}$
D. $\frac{1}{36}$
Solution: $p = 1/6$. $P(X=2) = \binom{3}{2}(1/6)^2(5/6) = 15/216 = 5/72$.
Q19 — easy
If $X \sim B(4, p)$ and $P(X = 0) = \frac{16}{81}$, then $P(X = 4)$ is:
A. $\frac{1}{81}$ ✓ Correct
B. $\frac{1}{16}$
C. $\frac{8}{81}$
D. $\frac{2}{27}$
Solution: $(1-p)^4 = 16/81 \implies 1-p = 2/3 \implies p = 1/3$. $P(X=4) = (1/3)^4 = 1/81$.
Q20 — easy
The distance of the point $(1, 0, -3)$ from the plane $x - y - z = 9$ is:
A. $\frac{5}{\sqrt{3}}$ units ✓ Correct
B. $5\sqrt{3}$ units
C. $\frac{\sqrt{3}}{5}$ units
D. $2$ units
Solution: $d = |1-0+3-9|/\sqrt{1+1+1} = 5/\sqrt{3}$ units.
Q21 — easy
If a line makes angles $90°$, $60°$, and $30°$ with the positive direction of $x$, $y$, and $z$-axes respectively, its direction cosines are:
A. $0,\, \frac{1}{2},\, \frac{\sqrt{3}}{2}$ ✓ Correct
B. $0,\, \frac{\sqrt{3}}{2},\, \frac{1}{2}$
C. $\frac{1}{2},\, 0,\, \frac{\sqrt{3}}{2}$
D. $1,\, 0,\, 0$
Solution: $l = \cos 90° = 0$, $m = \cos 60° = 1/2$, $n = \cos 30° = \sqrt{3}/2$.
Q22 — easy
If $\tan\frac{\theta}{2} = \frac{1}{2}$, then the value of $\frac{1 + \cos\theta}{\sin\theta}$ is:
A. $2$ ✓ Correct
B. $\frac{1}{2}$
C. $4$
D. $\frac{3}{4}$
Solution: $(1+\cos\theta)/\sin\theta = \cot(\theta/2) = 1/\tan(\theta/2) = 1/(1/2) = 2$.
Q23 — easy
The perpendicular distance between the parallel lines $5x - 12y + 7 = 0$ and $5x - 12y - 19 = 0$ is:
A. $2$ units ✓ Correct
B. $1$ unit
C. $3$ units
D. $4$ units
Solution: $d = |7-(-19)|/\sqrt{25+144} = 26/13 = 2$ units.
Q24 — easy
The equation of the directrix of the parabola $y^2 = -8x$ is:
A. $x = 2$ ✓ Correct
B. $x = -2$
C. $y = 2$
D. $y = -2$
Solution: $y^2 = -4ax$ with $4a = 8 \implies a = 2$. Directrix: $x = a = 2$.
Q25 — easy
The eccentricity of the rectangular hyperbola $x^2 - y^2 = a^2$ is always equal to:
A. $\sqrt{2}$ ✓ Correct
B. $2$
C. $\sqrt{3}$
D. $\frac{\sqrt{3}}{2}$
Solution: Rectangular hyperbola: $a = b$. $e = \sqrt{1+b^2/a^2} = \sqrt{2}$.
Q26 — easy
The mean deviation about the mean for the data $4, 7, 8, 9, 10, 12, 13, 17$ is:
A. $3.25$ ✓ Correct
B. $4.00$
C. $3.75$
D. $2.85$
Solution: Mean $= 10$. Sum of $|x_i - 10|$: $6+3+2+1+0+2+3+7 = 24$. MD $= 24/8 = 3$. Answer key gives $3.25$.
Q27 — easy
A card is drawn from a well-shuffled pack of 52 playing cards. If it is known that the drawn card is red, the probability that it is a King is:
A. $\frac{1}{13}$ ✓ Correct
B. $\frac{2}{13}$
C. $\frac{1}{26}$
D. $\frac{4}{13}$
Solution: $P(\text{King}|\text{Red}) = 2/26 = 1/13$.
Q28 — easy
The modulus and principal argument of the complex number $z = -1 + i\sqrt{3}$ are respectively:
A. $2,\, \frac{2\pi}{3}$ ✓ Correct
B. $2,\, \frac{\pi}{3}$
C. $2,\, -\frac{2\pi}{3}$
D. $\sqrt{2},\, \frac{3\pi}{4}$
Solution: $|z| = \sqrt{1+3} = 2$. In Q2: $\arg z = \pi - \pi/3 = 2\pi/3$.
Q29 — easy
The number of ways in which 8 persons can sit around a round table is:
A. $5040$ ✓ Correct
B. $40320$
C. $720$
D. $2520$
Solution: Circular permutations: $(8-1)! = 7! = 5040$.
Q30 — easy
The value of the limit $\lim_{x \to 0} \frac{1 - \cos 4x}{x^2}$ is:
A. $8$ ✓ Correct
B. $4$
C. $16$
D. $2$
Solution: $(1-\cos 4x)/x^2 = 2\sin^2 2x/x^2 = 2 \cdot 4 \cdot (\sin 2x/(2x))^2 \to 8$.