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Q1. Multiple choice questions (Attempt all · 20 × 1 = 20)

Which of the following quantities is a vector?
1. Charge
2. Mass
3. Momentum
4. Time
What does the Bisection method do?
1. Finds the maximum of a function
2. Estimates the area under the curve
3. Finds the root of the function
4. Differentiates a function
The point of intersection of right bisectors of a triangle is called:
1. incenter
2. circumcenter
3. centroid
4. orthocenter
The focus of the parabola x²= − 16y is:
1. (4 , 0)
2. (– 4 , 0)
3. (0 , 4)
4. (0 , – 4)
If $f$ is differentiable for every x, then $f$ is
1. Discontinuous
2. Continuous
3. inite
4. Infinite
The logarithmic function $f(x) = \log_b x$
1. all real numbers
2. x < 0
3. x > 0
4. x ≥ 0
$\text{What results } \sec^{-1}(1) - \sec^{-1}(\sqrt{2})$
1. $\frac{\pi}{6}$
2. $\frac{\pi}{2}$
3. $\frac{\pi}{3}$
4. $\frac{\pi}{4}$
$\text{If } f \text{ is integrable, then it is .}$
1. discontinous
2. unbounded
3. continuous
4. linear
The solution of the differential equation dy = dx is:
1. y = x + c
2. y = x² + c
3. y² = x² + c
4. y² = x + c
$\text{For what value of } x \in (0, 2\pi) \text{, } \sin x = -\frac{1}{2}?$
1. $\frac{7\pi}{6}, \frac{\pi}{6}$
2. $\frac{7\pi}{6}, \frac{11\pi}{6}$
3. $\frac{5\pi}{6}, \frac{11\pi}{6}$
4. $\frac{\pi}{6}, \frac{5\pi}{6}$
Which of following can be used to determine the magnitude of velocity?
1. Area under acceleration–time graph
2. Area under velocity-time graph
3. Gradient of an acceleration–time graph
4. Gradient of a velocity–time graph
Numerical techniques more commonly involve:
1. Elimination
2. Reduction
3. Iterative method
4. Direct methods
If the determinant of three points is zero then the points are:
1. collinear
2. non-collinear
3. imaginary
4. concurrent
Equation of conjugate axis of the hyperbola $\frac{(x-1)^2}{4} - \frac{(y+3)^2}{12} = 1 \text{ is:}$
1. x = 1
2. x = - 1
3. y = 3
4. y = - 3
If $f(2) = 2$ , $g(x) = x^2$ , then $\frac{d}{dx} \left[ \frac{3g(x)}{f(2)} \right]$
1. 2x
2. 3x
3. $\frac{3}{2}x$
4. $\frac{3}{2}x^2$
Scaling refers to:
1. increasing the size of an object.
2. decreasing the size of an object.
3. maintaining the properties while resizing an object.
4. changing the shape of an object.
For what value of x ∈ (0 , 2π) , Sin x = $-\frac{1}{2}$
1. $\frac{7\pi}{6}, \frac{\pi}{6}$
2. $\frac{7\pi}{6}, \frac{11\pi}{6}$
3. $\frac{5\pi}{6}, \frac{11\pi}{6}$
4. $\frac{\pi}{6}, \frac{5\pi}{6}$
$\int \sqrt{x} \, dx$ is:
1. $x^{3/2} + C$
2. $\frac{2}{3} x^{3/2} + C$
3. $\frac{3}{2} x^{3/2} + C$
4. $x^{1/2} + C$
y = 8 is a solution of the differential equation:
1. $\frac{dy}{dx}+ 8y = 32$
2. $\frac{dy}{dx}+ 6y = 32$
3. $\frac{dy}{dx}+ 5y = 32$
4. $\frac{dy}{dx}+ 4y = 32$
$\text{If } \cos^2 x = \frac{1}{2} \text{ with } x \in [0, \pi] \text{ then value of } x \text{ is :}$
1. $\frac{\pi}{4}, \frac{7\pi}{4}$
2. $\frac{\pi}{4}, \frac{3\pi}{4}$
3. $\frac{3\pi}{4}, \frac{5\pi}{4}$
4. $\frac{5\pi}{4}, \frac{7\pi}{4}$

Q2. Short questions (Attempt any 10 of 13 · 10 × 4 = 40)

Plot the graph of the following functions. $f(x) = 2x^3$
Solve the initial value problems. (x²+ 2y²)dx = xy dy ; y (1) = 1
Prove that: ∫ tanx dx = - ln(cos) + c = ln (secx) + c
Find the derivative of the vector function at $f(t) = 2t^2\hat{i} + 8j - \frac{1}{t^3}\hat{k} \quad \text{at } t = 5.$
Evaluate. $\lim_{x \to 5} 10x$
Solve graphically the trigonometric equation : $\sin \theta = \frac{\theta}{2}$ where $\theta \in \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$
Check whether the point P(2 , 4) lies outside, on or inside the circle: 2x² + 2y² - 6x + 8x + 1 = 0
Find the lines represented by each of the following joint equation: 7x²− 2xy − 9y²= 0
Draw the graph of each of the following functions against the intervals mentioned: y = sin⁻¹ (2x) for x ∈ $\begin{bmatrix} -\frac{1}{2}, \frac{1}{2} \end{bmatrix}.$ .
Determine whether the given function is one-to-one by examining its graph. If the function is one-to-one, find its inverse. Also draw the graphs of the inverse function. $f(x) = \frac{1}{3}x + 3$
Solve the differential equation by separating the variable. $x\frac{dy}{dx} =4y$
Find the area of the region bounded by the graphs of y = x , y = -2x and x = 3.
Find the domain of $r(t) = \ln|t - 1| \, \hat{t} + e^t \, \hat{j} + \sqrt{t} \, \hat{k}$

Q3. Long questions (Attempt any 5 of 7 · 5 × 8 = 40)

Using Newton–Raphson Method, find a root correct to three decimal places of the following. Initial guess is x₀ = 2 , 3x − 1 = cos⁡(x)
Find the velocity and acceleration of the function at t = 0. $r(t) = (3t + 1)i + \sqrt{3}tj + t^2k$
Evaluate the definite integral. $\int_{2}^{5} \frac{1}{x(x+1)} \, dx$
Find the general solution, in radians, of the following equations: cos4x + cos6x = cos5x
Find circumcenter of a triangle ABC with A(0 , 0) , B(6 , 0) and C(0 , 6). Also prove that circumcenter, centroid and orthocenter are collinear.
Verify that the indicated function is a solution of the given differential equation: 2y′ + y = 0 ; $y= e^-\frac{x}{2}$
The side of a cube increases at a rate of 5 cm/hr. At what rate does the diagonal of the cube increase?