SECTION I
Composite Functions
Which of the following statements is incorrect regarding the composite functions?
In the given mapping diagram, how is the function defined from A to C denoted?
If \(f(x) = \{(3,-1), (2,0), (1,1)\}\) and \(g(x) = \{(-1,4), (0,5), (1,6)\}\), what is \(gf(x)\)?
If \(f(x) = 3x - 1\) and \(g(x) = x + 1\), what is \(fg(x)\)?
If \(f(x) = 1 - 3x\) and \(g(x) = 2x + 1\), what is \(gf\!\left(\dfrac{1}{2}\right)\)?
SECTION II
Inverse Functions
Which statement about the inverse function is correct?
If \(f(x) = \{(1, 1), (2, 4), (3, 9), (4, 16)\}\) is a one to one and onto function, what is \(f^{-1}(x)\)?
If \(f(x) = 4x + 5\) is a one to one and onto function, what is the value of \(f^{-1}(x)\)?
If \(g(x) = \dfrac{1 - 3x}{5}\) is a one to one and onto function, what is the value of \(g^{-1}(x)\)?
If \(h(x) = \dfrac{1 + 2x}{3}\) is a one to one and onto function, what is the value of \(h^{-1}(-1)\)?
SECTION III
Matrices: Transpose & Operations
What is meant by the transpose of a matrix?
If \(C = \begin{bmatrix} 5 & -1 & -2 \\ 4 & 6 & 8 \end{bmatrix}\), then, which of the following is the matrix \(C^T\)?
If the symmetric matrix \(A = \begin{bmatrix} 5 & -3 \\ -3 & 4 \end{bmatrix}\), then, which of the following is \(A^T\)?
If \(P = \begin{bmatrix} 0 & 3 \\ 2 & 1 \\ 4 & 7 \end{bmatrix}\) and \(Q = \begin{bmatrix} 3 & -5 \\ -6 & 1 \\ 8 & 2 \end{bmatrix}\), then, which of the following is \((P+Q)^T\)?
SECTION IV
Determinants
If the matrix \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\), then, what is the value of \(|A|\) among the following?
If the matrix \(A = [-5]\), then, what is the value of \(|A|\) among the following?
If the matrix \(A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}\), then, what is the value of \(|A|\) among the following?
If \(\begin{vmatrix} 2 & x \\ 1 & x \end{vmatrix} = 0\), then, what is the value of \(x\)?
SECTION V
Inverse Matrices
If the matrix \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\), which of the following is the inverse matrix \(A^{-1}\)?
In which of the following situations is the inverse matrix \(A^{-1}\) of matrix \(A\) not defined?
Among the following properties, which one is the property of an inverse matrix?
If the matrix \(A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}\), which of the following is the inverse matrix \(A^{-1}\)?
SECTION VI
Trigonometry
Which one is equal to \(\sin(A+B)\)?
Which is equal to \(\cos(A-B)\)?
If \(A = 30^\circ\) and \(B = 60^\circ\), what is the value of \(\sin(A+B)\)?
If \(\sin(A+B) = \dfrac{1}{2}\) and \(B = 30^\circ\), what is the value of \(A\)?
Which of the following relations is correct?
SECTION VII
Geometric Transformations
Which of the following is a translation?
What effect does translation by a vector have on the shape and area of a triangle?
What is the formula to find the image of a point \(P(x,y)\) when it is translated by the vector \(T = \begin{pmatrix} a \\ b \end{pmatrix}\)?
If the point \(Q(-2,4)\) is translated by the translation vector \(T = \begin{pmatrix} 3 \\ 2 \end{pmatrix}\), which of the following is the image \(Q'\)?
If the point \(P(3,-4)\) is translated and its image is \(P'(-2,-1)\), then, what is the value of the translation vector \(T\)?
If the point \(P(-1,4)\) is translated 5 units to the right along the X-axis, what will be the coordinates of its image?
If a point \((x, y)\) is translated by \((-3, 5)\), which of the following is the resulting point?
SECTION VIII
Vectors & Scalar Products
If angle between vectors \(\vec{a}\) and \(\vec{b}\) is \(\theta\), which one is scalar product of \(\vec{a}\) and \(\vec{b}\)?
If \(\vec{a} = (x_1, y_1)\) and \(\vec{b} = (x_2, y_2)\), what is the value of \(\vec{a} \cdot \vec{b}\)?
If \(\vec{a} \cdot \vec{b} = 0\), \(|\vec{a}| \neq 0\) and \(|\vec{b}| \neq 0\), what is the angle between \(\vec{a}\) and \(\vec{b}\)?
If \(\vec{a}\) and \(\vec{b}\) are parallel and in the same direction, what is the value of \(\vec{a} \cdot \vec{b}\)?
If \(\vec{a} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}\) and \(\vec{b} = \begin{pmatrix} -3 \\ 2 \end{pmatrix}\), what is the value of \(\vec{a} \cdot \vec{b}\)?
If \(\vec{a} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}\), what is the value of \(a^2\)?
If \(\vec{OA} = \begin{pmatrix} 5 \\ m \end{pmatrix}\), \(\vec{OB} = \begin{pmatrix} 4 \\ -2 \end{pmatrix}\) and \(\angle AOB = 90^\circ\), which is the value of \(m\)?
SECTION IX
Statistics: Quartile Deviation
What is another term for quartile deviation?
What percentage of the data does quartile deviation include in a distribution?
Which formula is used to find Quartile Deviation (Q.D.)?
If \(Q_1 = 10\) and \(Q_3 = 30\), what is the value of the quartile deviation?
Which values are ignored while calculating quartile deviation?
If quartile deviation is 10 and the upper quartile is 30, what is the value of the lower quartile?
SECTION X
Coordinate Geometry: Straight Lines
Which one is the equation of straight line in two points form?
Which one is the equation of line passing through the points \((2,3)\) and \((-1,0)\) from given below?
Which is the equation of line passing through \((a,0)\) and \((0,b)\)?
Which one of the following represents the equation of line passing through the points \((-4,-2)\) and \((-3,5)\) from given below?
SECTION XI
Matrix Multiplication & Properties
Under which condition is the multiplication of two matrices possible?
If \(A\), \(B\), and \(C\) are three matrices and \((AB)C = A(BC)\), what is the name of this property?
If \(A\), \(B\), and \(C\) are three matrices and \(A(B + C) = AB + AC\), what is the name of this property?
If \(A\) and \(I\) are respectively a square matrix and an identity matrix of the same order and \(AI = IA = A\), what is the name of this property?
If \(P = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\) and \(I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\), then, which of the following is the product \(PI\)?
If \(W = \begin{bmatrix} -3 \\ 4 \\ 5 \end{bmatrix}\) and \(R = \begin{bmatrix} 3 & 2 & 1 \end{bmatrix}\), then, what is the order of matrix \(WR\)?
SECTION XII
Limits & Continuity
If a curve can be drawn without lifting the pen, what is it called?
Which of the following may represent continuity in daily life?
Which parts of the number line are represented by the real numbers from \(-2\) to \(+2\)?
For a function to be continuous at \(x=a\), which of the following conditions is necessary?
Which limit of the function \(f(x)\) is represented by \(\lim_{x \to a^-} f(x)\)?
Which condition must be satisfied for the limit of a function \(f(x)\) to exist at a point \(x=c\)?
What is the value of the function \(f(x) = 3x\) at \(x = 1\)?
Which of the following conditions is necessary for a function \(f(x)\) to be continuous at the point \(x=a\)?
If \(f(x) = x + 1\), then, what is the value of \(\lim_{x \to 1} \dfrac{1}{f(x)}\)?