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Mathematics - 3 — MH-CET Full Length Paper MCQs with Solutions
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Sample questions with solutions
Q1 — easy
Which of the following statements is a contingency (neither a tautology nor a contradiction)?
A. $(p \wedge q) \to p$
B. $(p \vee q) \wedge \sim p$ ✓ Correct
C. $p \vee \sim p$
D. $p \wedge \sim p$
Solution: $(p \vee q) \wedge \sim p$: when $p=T, q=F \to F$; when $p=F, q=T \to T$. Neither all true nor all false, so it is a contingency.
Q2 — easy
If $A$ is a square matrix of order 3 such that $A(\text{adj}\,A) = 6I$, then the value of $|\text{adj}\,A|$ is:
A. $6$
B. $36$ ✓ Correct
C. $216$
D. $1$
Solution: $A(\text{adj}\,A) = |A|I \implies |A| = 6$. For order $n = 3$: $|\text{adj}\,A| = |A|^{n-1} = 6^2 = 36$.
Q3 — easy
If the matrix $A = \begin{bmatrix} 1 & 2 & 1 \\ 0 & 1 & -1 \\ 3 & -1 & 1 \end{bmatrix}$, then the element $c_{23}$ in the cofactor matrix of $A$ is:
A. $7$ ✓ Correct
B. $-7$
C. $5$
D. $-5$
Solution: $M_{23} = \begin{vmatrix} 1 & 2 \\ 3 & -1 \end{vmatrix} = -1 - 6 = -7$. Cofactor $c_{23} = (-1)^{2+3}(-7) = 7$.
Q4 — easy
The value of $\cos\left(2\sin^{-1}\frac{3}{5}\right)$ is equal to:
A. $\frac{7}{25}$ ✓ Correct
B. $\frac{24}{25}$
C. $\frac{16}{25}$
D. $\frac{9}{25}$
Solution: Let $\theta = \sin^{-1}(3/5)$. $\cos 2\theta = 1 - 2\sin^2\theta = 1 - 18/25 = 7/25$.
Q5 — easy
In any $\triangle ABC$, if the sides are $a = 7$, $b = 8$, and $c = 9$, then the value of $\cos A$ is:
A. $\frac{2}{3}$ ✓ Correct
B. $\frac{11}{12}$
C. $\frac{3}{4}$
D. $\frac{5}{6}$
Solution: $\cos A = \frac{b^2 + c^2 - a^2}{2bc} = \frac{64 + 81 - 49}{144} = \frac{96}{144} = \frac{2}{3}$.
Q6 — easy
The acute angle between the pair of straight lines represented by $x^2 - 4xy + y^2 = 0$ is:
A. $\frac{\pi}{6}$
B. $\frac{\pi}{4}$
C. $\frac{\pi}{3}$ ✓ Correct
D. $\frac{\pi}{2}$
Solution: $\tan\theta = \frac{2\sqrt{h^2-ab}}{|a+b|} = \frac{2\sqrt{4-1}}{2} = \sqrt{3} \implies \theta = \pi/3$.
Q7 — easy
The coordinates of the foot of the perpendicular drawn from the origin to the plane $2x - 3y + 4z - 29 = 0$ are:
A. $(2, -3, 4)$ ✓ Correct
B. $(1, -2, 3)$
C. $(4, -6, 8)$
D. $(2, 3, -4)$
Solution: Ratio $= -(-29)/(4+9+16) = 1$. Foot: $(2(1), -3(1), 4(1)) = (2, -3, 4)$.
Q8 — easy
The acute angle between the two planes $2x - y + z = 6$ and $x + y + 2z = 7$ is:
A. $\frac{\pi}{6}$
B. $\frac{\pi}{4}$
C. $\frac{\pi}{3}$ ✓ Correct
D. $\frac{\pi}{2}$
Solution: $\cos\theta = \frac{|2-1+2|}{\sqrt{6}\sqrt{6}} = \frac{3}{6} = \frac{1}{2} \implies \theta = \pi/3$.
Q9 — easy
If $x = a(t + \sin t)$ and $y = a(1 - \cos t)$, then the value of $\frac{dy}{dx}$ at $t = \frac{\pi}{3}$ is:
A. $\sqrt{3}$
B. $\frac{1}{\sqrt{3}}$ ✓ Correct
C. $1$
D. $\frac{\sqrt{3}}{2}$
Solution: $\frac{dy}{dx} = \frac{\sin t}{1 + \cos t} = \tan(t/2)$. At $t = \pi/3$: $\tan(\pi/6) = 1/\sqrt{3}$.
Q10 — easy
The equation of the normal to the curve $y = x^3 - 3x$ at the point where $x = 2$ is:
A. $x + 9y - 20 = 0$ ✓ Correct
B. $9x + y - 20 = 0$
C. $x - 9y + 16 = 0$
D. $9x - y - 16 = 0$
Solution: At $x = 2$: $y = 2$. Tangent slope $= 9$. Normal slope $= -1/9$. Equation: $x + 9y - 20 = 0$.
Q11 — easy
The function $f(x) = 2x^3 - 9x^2 + 12x + 5$ is strictly decreasing in the interval:
A. $(1, 2)$ ✓ Correct
B. $(0, 1)$
C. $(2, 3)$
D. $(-\infty, 1)$
Solution: $f'(x) = 6(x-1)(x-2) < 0$ for $x \in (1, 2)$.
Q12 — easy
The indefinite integral $\int \frac{dx}{\sqrt{x^2 - 4x + 13}}$ is equal to:
A. $\ln|(x-2) + \sqrt{x^2 - 4x + 13}| + C$ ✓ Correct
B. $\sin^{-1}\left(\frac{x-2}{3}\right) + C$
C. $\frac{1}{3}\tan^{-1}\left(\frac{x-2}{3}\right) + C$
D. $\frac{1}{6}\ln\left|\frac{x+1}{x-5}\right| + C$
Solution: $x^2 - 4x + 13 = (x-2)^2 + 9$. Standard form: $\ln|(x-2) + \sqrt{(x-2)^2 + 9}| + C$.
Q13 — easy
The value of the definite integral $\int_0^{\pi/2} \ln(\tan x)\,dx$ is:
A. $0$ ✓ Correct
B. $\frac{\pi}{2}\ln 2$
C. $-\frac{\pi}{2}\ln 2$
D. $\pi$
Solution: Using $x \to \pi/2 - x$: $I = \int_0^{\pi/2}\ln(\cot x)dx = -I \implies I = 0$.
Q14 — easy
The value of the definite integral $\int_0^2 |x^2 - 1|\,dx$ is:
A. $2$ ✓ Correct
B. $\frac{4}{3}$
C. $4$
D. $\frac{8}{3}$
Solution: $\int_0^1(1-x^2)dx + \int_1^2(x^2-1)dx = 2/3 + 4/3 = 2$.
Q15 — easy
The area bounded by the curve $y = \sin x$ and the $x$-axis between $x = 0$ and $x = \pi$ is:
A. $1$ sq. unit
B. $2$ sq. units ✓ Correct
C. $\pi$ sq. units
D. $4$ sq. units
Solution: $\int_0^\pi \sin x\,dx = [-\cos x]_0^\pi = 1 + 1 = 2$ sq. units.
Q16 — easy
The differential equation of the family of all curves $y = Ae^{2x} + Be^{-2x}$ (where $A, B$ are arbitrary constants) is:
A. $\frac{d^2y}{dx^2} - 4y = 0$ ✓ Correct
B. $\frac{d^2y}{dx^2} + 4y = 0$
C. $\frac{d^2y}{dx^2} - 2y = 0$
D. $\frac{d^2y}{dx^2} + 2y = 0$
Solution: $y'' = 4Ae^{2x} + 4Be^{-2x} = 4y \implies y'' - 4y = 0$.
Q17 — easy
The general solution of the differential equation $\frac{dy}{dx} + y\cot x = 2\cos x$ is:
A. $y\sin x = \frac{1}{2}\sin 2x + C$
B. $y\sin x = -\frac{1}{2}\cos 2x + C$ ✓ Correct
C. $y\cos x = \sin 2x + C$
D. $y\sin x = \sin^2 x + C$
Solution: I.F. $= \sin x$. $y\sin x = \int 2\cos x\sin x\,dx = \int \sin 2x\,dx = -\frac{1}{2}\cos 2x + C$.
Q18 — easy
The probability distribution of a discrete random variable $X$ is given by $P(X = x) = \frac{k}{x}$ for $x \in \{1, 2, 3, 4\}$. The value of $P(X \ge 3)$ is:
A. $\frac{7}{25}$ ✓ Correct
B. $\frac{12}{25}$
C. $\frac{1}{2}$
D. $\frac{1}{4}$
Solution: $k(1 + 1/2 + 1/3 + 1/4) = 1 \implies k = 12/25$. $P(X \ge 3) = k(1/3 + 1/4) = (12/25)(7/12) = 7/25$.
Q19 — easy
A merchant makes a profit of Rs. 500 with probability $0.6$ and incurs a loss of Rs. 200 with probability $0.4$. His expected gain is:
A. Rs. 220 ✓ Correct
B. Rs. 300
C. Rs. 180
D. Rs. 250
Solution: $E = 500(0.6) + (-200)(0.4) = 300 - 80 = 220$.
Q20 — easy
A box contains 3 red and 7 black balls. Two balls are drawn successively with replacement. The variance of the number of red balls drawn is:
A. $0.42$ ✓ Correct
B. $0.21$
C. $0.60$
D. $0.36$
Solution: $p = 0.3$, $n = 2$. Variance $= npq = 2(0.3)(0.7) = 0.42$.
Q21 — easy
In 8 independent trials of a Binomial experiment, the probability of getting exactly 3 successes is equal to the probability of getting exactly 5 successes. The parameter $p$ is:
A. $\frac{1}{4}$
B. $\frac{1}{2}$ ✓ Correct
C. $\frac{3}{4}$
D. $\frac{1}{3}$
Solution: $\binom{8}{3}p^3 q^5 = \binom{8}{5}p^5 q^3 \implies q^2 = p^2 \implies p = q = 1/2$.
Q22 — easy
The distance of the plane $3x - 6y + 2z + 11 = 0$ from the origin is:
A. $\frac{11}{7}$ units ✓ Correct
B. $11$ units
C. $\frac{11}{49}$ units
D. $\frac{7}{11}$ units
Solution: $d = |11|/\sqrt{9+36+4} = 11/7$.
Q23 — easy
If a straight line makes equal angles $\alpha$ with the three coordinate axes, then the value of $\sin^2\alpha$ is:
A. $\frac{1}{3}$
B. $\frac{2}{3}$ ✓ Correct
C. $\frac{3}{4}$
D. $1$
Solution: $3\cos^2\alpha = 1 \implies \cos^2\alpha = 1/3 \implies \sin^2\alpha = 2/3$.
Q24 — easy
The vector equation of a line passing through $(2, -1, 4)$ and parallel to the vector $\hat{i} + \hat{j} - 2\hat{k}$ is:
A. $\vec{r} = (2\hat{i} - \hat{j} + 4\hat{k}) + \lambda(\hat{i} + \hat{j} - 2\hat{k})$ ✓ Correct
B. $\vec{r} = (\hat{i} + \hat{j} - 2\hat{k}) + \lambda(2\hat{i} - \hat{j} + 4\hat{k})$
C. $\vec{r} = (2\hat{i} + \hat{j} - 2\hat{k}) + \lambda(\hat{i} - \hat{j} + 4\hat{k})$
D. $\vec{r} = (2\hat{i} - \hat{j} + 4\hat{k}) + \lambda(3\hat{i} + 2\hat{k})$
Solution: $\vec{r} = \vec{a} + \lambda\vec{b}$ where $\vec{a} = 2\hat{i} - \hat{j} + 4\hat{k}$ and $\vec{b} = \hat{i} + \hat{j} - 2\hat{k}$.
Q25 — easy
If $A + B + C = 180°$, then the value of $\tan A + \tan B + \tan C$ is identically equal to:
A. $\tan A \tan B \tan C$ ✓ Correct
B. $0$
C. $1$
D. $\cot A \cot B \cot C$
Solution: Standard identity: $\tan A + \tan B + \tan C = \tan A \tan B \tan C$ when $A + B + C = \pi$.
Q26 — easy
The coordinates of the foot of the perpendicular drawn from the point $(2, 3)$ to the straight line $x - y - 1 = 0$ are:
A. $(3, 2)$ ✓ Correct
B. $(2, 1)$
C. $(1, 0)$
D. $(2.5, 1.5)$
Solution: Foot formula: ratio $= -(2-3-1)/2 = 1$. Foot: $(2+1, 3-1) = (3, 2)$.
Q27 — easy
The equation of the director circle of the circle $x^2 + y^2 = 36$ is:
A. $x^2 + y^2 = 72$ ✓ Correct
B. $x^2 + y^2 = 18$
C. $x^2 + y^2 = 144$
D. $x^2 + y^2 = 54$
Solution: Director circle radius $= \sqrt{2}r$. Equation: $x^2 + y^2 = 2(36) = 72$.
Q28 — easy
If the standard deviation of the numbers $2, 4, 6, 8, 10$ is $\sigma$, then the standard deviation of the numbers $12, 14, 16, 18, 20$ is:
A. $\sigma + 10$
B. $10\sigma$
C. $\sigma$ ✓ Correct
D. $\sigma^2$
Solution: Adding a constant to each observation does not change the standard deviation.
Q29 — easy
A bag contains 5 white and 3 red balls. If two balls are drawn at random without replacement, the probability that both balls are red is:
A. $\frac{3}{28}$ ✓ Correct
B. $\frac{9}{64}$
C. $\frac{3}{8}$
D. $\frac{5}{28}$
Solution: $P = \binom{3}{2}/\binom{8}{2} = 3/28$.
Q30 — easy
If $(1 + i)^8 + (1 - i)^8 = K$, then the value of the integer $K$ is:
A. $16$
B. $32$ ✓ Correct
C. $64$
D. $128$
Solution: $(1+i)^8 = (\sqrt{2})^8 e^{i2\pi} = 16$. Similarly $(1-i)^8 = 16$. $K = 32$.