Mathematics 2023 Question Paper Online Test (12th Standard)

Q1

A square matrix A of order n has inverse if and only if :

A.\(\rho(A) > n\) B.\(\rho(A) = n\) C.\(\rho(A) \neq n\) D.\(\rho(A) < n\)

Q2

Distance from the origin to the plane \(3x-6y+2z+7=0\) is :

A.2 B.0 C.3 D.1

Q3

If \(3\cos^{-1}x = \cos^{-1}(4x^3-3x)\),

A.\(x \in \left(\frac{1}{2}, 1\right)\) B.\(x \in \left[\frac{1}{2}, 1\right]\) C.\(x \in (-\infty, 1]\) D.\(x \in \left[\frac{1}{2}, \infty\right)\)

Q4

The general solution of the differential equation \(\frac{dy}{dx} = \frac{y}{x}\) is :

A.\(y = kx\) B.\(xy = k\) C.\(\log y = kx\) D.\(y = k\log x\)

Q5

The number of normals that can be drawn from a point to the parabola \(y^2 = 4ax\) is :

A.3 B.2 C.0 D.1

Q6

If \(\vec{a}\) and \(\vec{b}\) are parallel vectors then \([\vec{a}, \vec{c}, \vec{b}]\) is equal to :

A.1 B.2 C.0 D.-1

Q7

The number of real numbers in \([0, 2\pi]\) satisfying \(\sin^4 x - 2\sin^2 x + 1\) is :

A.1 B.2 C.\(\infty\) D.4

Q8

Suppose that X takes on one of the values 0, 1, 2. If for some constant k, \(P(X=i) = kP(X=i-1)\) for \(i = 1, 2\) and \(P(X=0) = \frac{1}{7}\), then the value of k is :

A.3 B.1 C.4 D.2

Q9

The maximum value of the function \(x^2 e^{-2x}, x > 0\) is :

A.\(\frac{1}{e^2}\) B.\(\frac{1}{e}\) C.\(\frac{4}{e^4}\) D.\(\frac{1}{2e}\)

Q10

The operation * defined by \(a * b = \frac{ab}{7}\) is not a binary operation on :

A.R B.\(Q^+\) C.C D.Z

Q11

The area between \(y^2 = 4x\) and its latus rectum is :

A.\(\frac{8}{3}\) B.\(\frac{2}{3}\) C.\(\frac{5}{3}\) D.\(\frac{4}{3}\)

Q12

Angle between the curves \(y^2 = x\) and \(x^2 = y\) at the origin is :

A.\(\frac{\pi}{2}\) B.\(\tan^{-1}\left(\frac{3}{4}\right)\) C.\(\frac{\pi}{4}\) D.\(\tan^{-1}\left(\frac{4}{3}\right)\)

Q13

\(|\text{adj}(\text{adj}A)| = |A|^{16}\), then the order of the square matrix A is :

A.2 B.3 C.5 D.4

Q14

The value of \(\left(\frac{1+i}{\sqrt{2}}\right)^8 + \left(\frac{1-i}{\sqrt{2}}\right)^8\) is :

A.8 B.4 C.2 D.6

Q15

If \(|z| = 1\), then the value of \(\frac{1+z}{1+\bar{z}}\) is :

A.\(\frac{1}{z}\) B.\(z\) C.1 D.\(\bar{z}\)

Q16

The abscissa of the point on the curve \(f(x) = \sqrt{8-2x}\) at which the slope of the tangent is \(-0.25\) ?

A.-2 B.-8 C.0 D.-4

Q17

The value of \(\int_0^{\pi/3} \tan x\,dx\) is :

A.\(-\log 2\) B.\(\log 2\) C.\(-\log 3\) D.\(\log 3\)

Q18

The number of positive zeros of the polynomial \(\sum_{r=0}^{n} {}^nC_r (-1)^r x^r\) is :

A.\(< n\) B.0 C.r D.n

Q19

The Principal value of \(\sin^{-1}\left(\frac{-1}{2}\right)\) is :

A.\(\frac{-\pi}{6}\) B.0 C.\(\frac{-\pi}{2}\) D.\(\frac{\pi}{2}\)

Q20

Area of the greatest rectangle inscribed in the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) is :

A.\(\sqrt{ab}\) B.2ab C.\(\frac{a}{b}\) D.ab

12th Standard Mathematics March 2023: Online Practice Test

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