MHT CET 2025 19 April Shift 1 Question Paper (Available): Download Question Paper (PCM) with Answers PDF

MHT CET 2025 April 19 Shift 1 Questions with Solutions

Question 1:

The ratio of areas bounded by curves \( y = \cos x \) and \( y = 0 \) between \( x = 0 \) to \( x = \frac{\pi}{3} \) and \( x = \frac{\pi}{3} \) to \( x = \frac{2\pi}{3} \), with the x-axis is:

• (1) \( 2 : 1 \)
• (2) \( \sqrt{2} : 1 \)
• (3) \( 1 : 1 \)
• (4) \( 1 : 3 \)

If \[ A = \begin{bmatrix} \cos\theta & \sin\theta & 0
-\sin\theta & \cos\theta & 0
0 & 0 & 1 \end{bmatrix} \]
and \( A_{11}, A_{12}, A_{13} \) are the cofactors of \( a_{11}, a_{12}, a_{13} \) respectively, then the value of \( a_{11} A_{11} + a_{12} A_{12} + a_{13} A_{13} \) is:

• (1) \( -1 \)
• (2) \( 1 \)
• (3) \( 0 \)
• (4) \( 2 \)

If \[ f(x) = 2 \left( \cos x + i \sin x \right) \left( \cos 3x + i \sin 3x \right) \cdots \left( \cos (2n-1)x + i \sin (2n-1)x \right) \]
where \( n \in \mathbb{N} \), then what is the value of \( f''(x) \) ?

• (1) \( -n^2 f(x) \)
• (2) \( n^2 f(x) \)
• (3) \( -n^4 f(x) \)
• (4) \( n^4 f(x) \)

Smallest angle of a triangle whose sides are \( 6 + \sqrt{12}, \sqrt{48}, \sqrt{54} \) is:

• (1) \( \frac{\pi}{4} \)
• (2) \( \frac{\pi}{2} \)
• (3) \( \frac{\pi}{6} \)
• (4) \( \frac{\pi}{3} \)

A box contains 9 tickets numbered from 1 to 9 inclusive. 3 tickets are drawn from the box one at a time. What is the probability that they are alternatively either (odd, even, odd) or (even, odd, even)?

• (1) \( \frac{5}{16} \)
• (2) \( \frac{5}{17} \)
• (3) \( \frac{4}{17} \)
• (4) \( \frac{5}{16} \)

A plane passes through the point \( (1, -2, 1) \) and is perpendicular to both the planes \[ 2x - 2y - 2z = 5 \quad and \quad x - y + 2z = 24 \]
Then, the distance of the point \( (1, 2, 2) \) from this plane is:

• (1) \( 2\sqrt{2} \)
• (2) \( 1 \)
• (3) \( \sqrt{2} \)
• (4) \( 2 \)

Solve the equation: \[ x + \log_{15}(5 + 3x) = x \log_{15} 5 + \log_{15} 24 \]

• (1) \( 2 \)
• (2) \( 1 \)
• (3) \( 5 \)
• (4) \( 8 \)

An ellipse has \( OB \) as the semi-minor axis, and \( S \), \( S' \) as the foci. If \( \angle SBS' \) is a right angle, then the eccentricity \( e \) of the ellipse is:

• (1) \( \sqrt{2} \)
• (2) \( \frac{1}{2} \)
• (3) \( \frac{1}{\sqrt{2}} \)
• (4) \( \frac{1}{3} \)

The value of \[ \int_1^4 \log(\lfloor x \rfloor) \, dx \]
where \( \lfloor x \rfloor \) is the greatest integer less than or equal to \( x \), is:

• (1) \( \log 2 \)
• (2) \( \log 5 \)
• (3) \( \log 6 \)
• (4) \( \log 3 \)

In a triangle \( ABC \), with usual notation, if \[ \frac{b + c}{11} = \frac{c + a}{12} = \frac{a + b}{13} \]
then the ratio \( \cos A : \cos B : \cos C \) is:

• (1) \( 19 : 7 : 25 \)
• (2) \( 7 : 19 : 25 \)
• (3) \( 12 : 14 : 20 \)
• (4) \( 19 : 25 : 7 \)

The value of \[ \int_1^4 \log(\lfloor x \rfloor) \, dx \]
where \( \lfloor x \rfloor \) is the greatest integer less than or equal to \( x \), is equal to:

• (1) \( \log 6 \)
• (2) \( \log 5 \)
• (3) \( \log 2 \)
• (4) \( \log 3 \)

A population \( P(t) \) of 1000 bacteria introduced to a nutrient medium grows according to the relation \[ P(t) = \frac{1000t + 1000t}{100 + t^2} \]
The maximum size of this bacterial population is:

• (1) \( 1250 \)
• (2) \( 1100 \)
• (3) \( 1050 \)
• (4) \( 950 \)

If the angle \( \theta \) between the line \[ \frac{2t + 1}{1} = \frac{y - 1}{2} = \frac{z}{2} \]
and the plane \( 2x - y\sqrt{7} + z + 4 = 0 \) is such that \( \sin \theta = \frac{8}{\sqrt{3}} \), then the value of the expression is:

• (1) \( -\frac{5}{\sqrt{3}} \)
• (2) \( \frac{5}{\sqrt{3}} \)
• (3) \( \frac{8}{\sqrt{3}} \)
• (4) \( -\frac{8}{\sqrt{3}} \)

The distance of the point \( (-3, 2, 3) \) from the line passing through \( (4, 6, -2) \) and having direction ratios \( -1, 2, 3 \) is:

• (1) \( 4\sqrt{17} \)
• (2) \( 2\sqrt{17} \)
• (3) \( 2\sqrt{19} \)
• (4) \( 4\sqrt{19} \)

If \( y = y(x) \) satisfies \[ \left( \frac{2 + \sin x}{1 + y} \right) \frac{dy}{dx} = -\cos x, \]
such that \( y(0) = 2 \), then the value of \( y\left( \frac{\pi}{2} \right) \) is:

• (1) \( 3 \)
• (2) \( 4 \)
• (3) \( 2 \)
• (4) \( 1 \)

Let \[ f(x) = (\cos x + \sin x) \cdot \cos(3x + i \sin x) \cdot \left[ (2n - 1)x + i \sin((2n - 1)x) \right], \]
where \( n \in \mathbb{N} \), and \( i = \sqrt{-1} \). Then: \[ f''(x) = \, ? \]

• (1) \( -n^4 f(x) \)
• (2) \( n^2 f(x) \)
• (3) \( -n^2 f(x) \)
• (4) \( n^4 f(x) \)

If \[ [2\vec{p} - 3\vec{q} \ \vec{q} \ \vec{s}] + [3\vec{p} + 2\vec{q} \ \vec{r} \ \vec{s}] = m[\vec{p} \ \vec{r} \ \vec{s}] + n[\vec{q} \ \vec{r} \ \vec{s}] + l[\vec{p} \ \vec{q} \ \vec{s}], \]
then the values of \( m, n, l \) respectively are:

• (1) \( 3, 4, 5 \)
• (2) \( 2, 3, 3 \)
• (3) \( 1, 2, 3 \)
• (4) \( 3, 5, 2 \)

Given: \[ \vec{a} = \hat{j} - \hat{k}, \quad \vec{c} = \hat{i} - \hat{j} - \hat{k} \]
The vector \( \vec{b} \) satisfies: \[ \vec{a} \times \vec{b} + \vec{c} = \vec{0} \quad and \quad \vec{a} \cdot \vec{b} = 3 \]
Find the vector \( \vec{b} \).

• (1) \( \vec{b} = -\hat{i} + \hat{j} - 2\hat{k} \)
• (2) \( \vec{b} = \hat{i} + \hat{j} + 2\hat{k} \)
• (3) \( \vec{b} = \hat{i} - \hat{j} + 2\hat{k} \)
• (4) \( \vec{b} = -\hat{i} + \hat{j} + \hat{k} \)

Evaluate the integral: \[ \int_{-1}^{1} \log\left(\frac{2 - x}{2 + x}\right) dx \]

• (1) \( 2 \log\left(\frac{1}{2}\right) \)
• (2) \( \log\left(\frac{3}{4}\right) \)
• (3) \( 0 \)
• (4) \( \log 2 \)

Find the area bounded between the parabola \( y^2 = 4x \) and the line \( y = 2x - 3 \).

• (1) \( \displaystyle \int_{1 - \sqrt{7}}^{1 + \sqrt{7}} \left( \frac{y + 3}{2} - \frac{y^2}{4} \right) dy \)
• (2) \( \displaystyle \int_{1 - \sqrt{7}}^{1 + \sqrt{7}} \left( \frac{y^2}{4} - \frac{y + 3}{2} \right) dy \)
• (3) \( \displaystyle \int_{-2}^{2} \left( \frac{y^2}{4} - y \right) dy \)
• (4) \( \displaystyle \int_{1 - \sqrt{7}}^{1 + \sqrt{7}} \left( \frac{y^2 + y + 3}{2} \right) dy \)

The magnetic moment of a sample of mass \( 2 \, \text{g} \) is \( 8 \times 10^{-7} \, \text{A} \cdot \text{m}^2 \).
If the density \( \rho = 4 \, \text{g/cm}^3 \), then the magnetisation \( M \) of the sample is:

• (1) \( 0.4 \, A/m \)
• (2) \( 1.6 \, A/m \)
• (3) \( 4.0 \, A/m \)
• (4) \( 6.4 \, A/m \)

Which of the following statements is \emph{correct} regarding the coordination compound \([Fe(CN)_6]^{4-}\) and fructose structure?

• (1) The EAN of iron in \([Fe(CN)_6]^{4-}\) is 36 and fructose forms a pyranose ring.
• (2) The EAN of iron in \([Fe(CN)_6]^{4-}\) is 36 and fructose forms a furanose ring.
• (3) The compound exhibits ionisation isomerism and fructose is a non-reducing sugar.
• (4) The EAN of Fe is 30 and fructose forms a pyranose ring.

The EAN of cobalt in the complex \([Co(NH_3)_6]^{3+}\) is:

• (1) 27
• (2) 30
• (3) 33
• (4) 36

A cube of edge 4 cm has mass 256 g. The density of the material in SI unit is:

• (1) \( 4 \, kg/m^3 \)
• (2) \( 1600 \, kg/m^3 \)
• (3) \( 4000 \, kg/m^3 \)
• (4) \( 1000 \, kg/m^3 \)

A force \( F = 5x \, N \) acts on a body and displaces it from \( x = 0 \) to \( x = 2 \, m \). The work done by the force is:

• (1) \( 10 \, J \)
• (2) \( 20 \, J \)
• (3) \( 5 \, J \)
• (4) \( 15 \, J \)

The van’t Hoff factor for a solution of \( K_2SO_4 \) in water is:

• (1) \( 1 \)
• (2) \( 2 \)
• (3) \( 3 \)
• (4) \( 4 \)

Rosenmund reduction is used to convert acyl chlorides into:

• (1) Alcohols
• (2) Carboxylic acids
• (3) Aldehydes
• (4) Ketones

In the electrolysis of molten NaCl, the product obtained at the cathode is:

• (1) \( Cl_2 \) gas
• (2) \( Na \) metal
• (3) \( NaOH \)
• (4) \( H_2 \) gas

Which of the following is an example of physisorption?

• (1) Adsorption of \( NH_3 \) on charcoal
• (2) Adsorption of \( H_2 \) on Ni
• (3) Adsorption of noble gases on solid surface
• (4) Adsorption of \( O_2 \) on heated metal

For a first-order reaction, the time required to reduce the concentration of the reactant to half its initial value is:

• (1) \( \frac{0.3010}{k} \)
• (2) \( \frac{1}{k} \)
• (3) \( \frac{0.693}{k} \)
• (4) \( \frac{2.303}{k} \)