List of practice Questions

Assertion (A) : One of the particular solutions of the differential equation \(\frac{dy}{dx} = e^{x+y}\) can be \(e^x + e^{-y} = -2\).
Reason (R) : \(e^x + e^{-y} = C\) is the general solution of the differential equation \(\frac{dy}{dx} = e^{x+y}\).

Assertion (A) : The vectors \(\vec{a}\) and \((-2\vec{a})\), where \(\vec{a} \neq \vec{0}\) are collinear vectors.
Reason (R) : \(\vec{a} \cdot (-2\vec{a}) = 0\).

If \(x = t + \frac{1}{t}\) and \(y = t - \frac{1}{t}\), find \(\frac{dy}{dx}\) at \(t = 2\).

Find the sub-interval(s) of \((0, \frac{\pi}{2})\) in which \(f(x) = \tan x - 4x\) is increasing.

Find the value of \(\sin [\cot^{-1} \sqrt{2} (\cos (\tan^{-1} 1))]\).

A relation \(R\) on \(A = \{1, 2, 3\}\) is defined as \(R = \{(1, 1), (3, 3), (1, 2)\\). Is \(R\) a symmetric relation? Justify. Write the smallest relation set \(R_1\) such that \(R \cup R_1\) becomes an equivalence relation on the set \(\{1, 2, 3\}\).}

If for two unit vectors \(\vec{a}\) and \(\vec{b}\), \(|\vec{a} + 2\vec{b}| = |2\vec{a} - \vec{b}|\), then find the angle between \(\vec{a}\) and \(\vec{b}\).

For any two vectors \( \vec{a} \) and \( \vec{b} \), which of the following statements is always true ?

If \( (\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b}) = 198 \) and \( |\vec{a}| = 10|\vec{b}| \), then :

If \(l_1, m_1, n_1\) and \(l_2, m_2, n_2\) are direction cosines of lines \(L_1\) and \(L_2\) respectively and \(\theta\) is the acute angle between them, then :

Direction ratios of lines \(l_1\) and \(l_2\) are \(\) and \(\) respectively. The values of \(p\) and \(q\) for which \(l_1\) and \(l_2\) are parallel are respectively :

If \(E\) and \(F\) are two independent events such that \(P(E) = \frac{3{10}\), \(P(E \cup F) = \frac{1}{2}\), then \(P(E|F) - P(F|E)\) is equal to :}

The rate of change of volume of a sphere with respect to its diameter, when its radius is 5 cm, is :

\(\int \frac{dx}{2^x + 2^{-x}}\) is equal to :

\( \int_{-1}^{1} (1 - |x|) \, dx \) is equal to :

The area of the shaded region of the circle given below is equal to :

\( \frac{dy}{dx} = F(x, y) \) will be a homogeneous differential equation for which of the following functions ?
(i) \( F(x, y) = 3x + 2y \)
(ii) \( F(x, y) = \sin \frac{y{x} + \log y - \log x \)
(iii) \( F(x, y) = e^{y/x} + 1 \)
(iv) \( F(x, y) = \sqrt{x^2 + y^2} - y \)}

If \(A\) and \(B\) are skew-symmetric matrices of same order, then \(AB' + BA'\) is a/an :

If \(\begin{bmatrix} 4 1 3 \end{bmatrix} A = \begin{bmatrix} -4 & 8 & 4 \\ -1 & 2 & 1 \\ -3 & 6 & 3 \end{bmatrix}\), then order of \(A\) must be :

If a square matrix \(A\) is such that \(A^2 = A\) and \((I - A)^3 = xA + I\), then value of \(x\) must be :

If \(B(\text{adj } B) = \begin{bmatrix \frac{1}{3} & 0 & 0 \\ 0 & \frac{1}{3} & 0 \\ 0 & 0 & \frac{1}{3} \end{bmatrix}\), then the value of \(\det (B^{-1}) =\)}

The value of \(k\) for which the function \(f(x) = \begin{cases x^2 \sin \frac{1}{x}, & x \neq 0 \\ k(x + 1), & x = 0 \end{cases}\) is a continuous function, is :}

If \(\sin^{-1} x = y\), then \(\frac{dy}{dx}\) is :

The domain of \(f(x) = \cos^{-1 (2x - 5)\) is :}

If \(A^2 = 4A + 3I\) and \(A^{-1} = xA + yI\), then the value of \((x + y)\) is :