Vardaan Institute - Vectors Solution Key

Section 1: Algebra & Binomial Approximation

Solve for the roots of the quadratic equation: $x^2 - 7x + 12 = 0$.

Solution: $x^2 - 3x - 4x + 12 = 0 \Rightarrow x(x-3) - 4(x-3) = 0 \Rightarrow (x-3)(x-4) = 0$.
Roots: $\mathbf{x = 3, 4}$

Find the roots of the equation $2x^2 + 5x - 3 = 0$ using the quadratic formula.

Solution: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-5 \pm \sqrt{25 - 4(2)(-3)}}{4} = \frac{-5 \pm \sqrt{49}}{4} = \frac{-5 \pm 7}{4}$.
Roots: $\mathbf{x = \frac{1}{2}, -3}$

If $\alpha$ and $\beta$ are the roots of the equation $x^2 - 6x + 8 = 0$, find the value of $\alpha^2 + \beta^2$.

Solution: Sum of roots $\alpha + \beta = 6$. Product $\alpha\beta = 8$.
$\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = (6)^2 - 2(8) = 36 - 16 = \mathbf{20}$

Solve the linear equation for $x$: $\frac{3x + 4}{2x - 1} = 5$.

Solution: $3x + 4 = 5(2x - 1) \Rightarrow 3x + 4 = 10x - 5 \Rightarrow 7x = 9$.
$\mathbf{x = \frac{9}{7}}$

Write the binomial expansion for $(1 + x)^n$ up to the first three terms.

Solution: $(1 + x)^n = \mathbf{1 + nx + \frac{n(n-1)}{2!}x^2 + \dots}$

Using binomial approximation, evaluate the value of $(1.02)^4$ up to three decimal places.

Solution: $(1.02)^4 = (1 + 0.02)^4$. Using $(1+x)^n \approx 1 + nx + \frac{n(n-1)}{2}x^2$:
$\approx 1 + 4(0.02) + \frac{4(3)}{2}(0.02)^2 = 1 + 0.08 + 6(0.0004) = 1 + 0.08 + 0.0024 = \mathbf{1.082}$

Evaluate $\sqrt{0.99}$ using binomial theorem approximation, assuming $x \ll 1$.

Solution: $\sqrt{0.99} = (1 - 0.01)^{1/2} \approx 1 + \frac{1}{2}(-0.01) = 1 - 0.005 = \mathbf{0.995}$

Expand the expression $(1 - x)^{-2}$ up to three terms, given that $|x| < 1$.

Solution: $(1 - x)^{-2} = 1 + (-2)(-x) + \frac{(-2)(-3)}{2!}(-x)^2 = \mathbf{1 + 2x + 3x^2}$

Write the formula for the sum of an infinite geometric progression (G.P.).

Solution: $S_{\infty} = \mathbf{\frac{a}{1 - r}}$ (where $a$ is the first term and $|r| < 1$ is the common ratio)

10.

Find the sum of the infinite series: $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots \infty$.

Solution: Here $a = 1$ and $r = \frac{1}{2}$.
$S_{\infty} = \frac{1}{1 - 0.5} = \frac{1}{0.5} = \mathbf{2}$

Section 2: Trigonometry

11.

Convert $150^\circ$ into radians.

Solution: $150^\circ \times \frac{\pi}{180^\circ} = \mathbf{\frac{5\pi}{6} \text{ radians}}$

12.

Convert $\frac{5\pi}{6}$ radians into degrees.

Solution: $\frac{5\pi}{6} \times \frac{180^\circ}{\pi} = 5 \times 30^\circ = \mathbf{150^\circ}$

13.

In a right-angled triangle, what are the standard physics values for $\sin(37^\circ)$ and $\cos(37^\circ)$?

Solution: $\mathbf{\sin(37^\circ) \approx \frac{3}{5}}$ and $\mathbf{\cos(37^\circ) \approx \frac{4}{5}}$

14.

Find the exact value of $\sin(120^\circ)$.

Solution: $\sin(120^\circ) = \sin(180^\circ - 60^\circ) = \sin(60^\circ) = \mathbf{\frac{\sqrt{3}}{2}}$

15.

Evaluate $\cos(210^\circ)$.

Solution: $\cos(210^\circ) = \cos(180^\circ + 30^\circ) = -\cos(30^\circ) = \mathbf{-\frac{\sqrt{3}}{2}}$

16.

Find the value of $\tan(315^\circ)$.

Solution: $\tan(315^\circ) = \tan(360^\circ - 45^\circ) = -\tan(45^\circ) = \mathbf{-1}$

17.

If $\sin \theta = \frac{4}{5}$ and $\theta$ lies in the first quadrant, find $\cos \theta$ and $\tan \theta$.

Solution: Using $P=4, H=5 \Rightarrow B=3$.
$\mathbf{\cos \theta = \frac{3}{5}}$ and $\mathbf{\tan \theta = \frac{4}{3}}$

18.

Evaluate $\sin(15^\circ)$ using the compound angle formula for $\sin(A - B)$.

Solution: $\sin(45^\circ - 30^\circ) = \sin 45^\circ \cos 30^\circ - \cos 45^\circ \sin 30^\circ = \left(\frac{1}{\sqrt{2}}\right)\left(\frac{\sqrt{3}}{2}\right) - \left(\frac{1}{\sqrt{2}}\right)\left(\frac{1}{2}\right) = \mathbf{\frac{\sqrt{3} - 1}{2\sqrt{2}}}$

19.

State the formula for $\sin(2\theta)$ in terms of $\sin \theta$ and $\cos \theta$.

Solution: $\sin(2\theta) = \mathbf{2\sin\theta\cos\theta}$

20.

Write the three standard formulas for $\cos(2\theta)$.

Solution: $\mathbf{\cos^2\theta - \sin^2\theta}$, $\mathbf{2\cos^2\theta - 1}$, and $\mathbf{1 - 2\sin^2\theta}$

21.

What is the approximate value of $\sin \theta$ and $\tan \theta$ if $\theta$ is very small (in radians)?

Solution: For small angles, $\mathbf{\sin \theta \approx \theta}$ and $\mathbf{\tan \theta \approx \theta}$

22.

What is the approximate value of $\cos \theta$ for very small angles?

Solution: For small angles, $\mathbf{\cos \theta \approx 1}$

23.

Simplify the expression: $\sin(180^\circ + \theta)$.

Solution: (Third quadrant where sine is negative) $\mathbf{-\sin\theta}$

24.

Simplify the expression: $\cos(90^\circ + \theta)$.

Solution: (Second quadrant where cosine is negative, and $90^\circ$ flips ratio) $\mathbf{-\sin\theta}$

25.

Solve for $\theta$ if $2\cos\theta - \sqrt{3} = 0$, given $0 \le \theta \le 90^\circ$.

Solution: $2\cos\theta = \sqrt{3} \Rightarrow \cos\theta = \frac{\sqrt{3}}{2}$. Therefore, $\mathbf{\theta = 30^\circ}$ (or $\frac{\pi}{6}$ rad)

Section 3: Logarithms & Exponents

26.

Evaluate the base-2 logarithm: $\log_2 64$.

Solution: Since $2^6 = 64$, $\log_2 64 = \mathbf{6}$

27.

Find the value of $x$ if $\log_5 x = 3$.

Solution: $x = 5^3 = \mathbf{125}$

28.

Simplify the expression: $\log_{10} 20 + \log_{10} 5$.

Solution: $\log_{10}(20 \times 5) = \log_{10}(100) = \mathbf{2}$

29.

Write the expression $3\ln(x) - 2\ln(y)$ as a single natural logarithm.

Solution: $\ln(x^3) - \ln(y^2) = \mathbf{\ln\left(\frac{x^3}{y^2}\right)}$

30.

Solve the exponential equation for $x$: $e^{3x} = 15$.

Solution: Take $\ln$ of both sides: $3x = \ln(15) \Rightarrow \mathbf{x = \frac{\ln(15)}{3}}$

31.

What is the value of $\ln(e^4)$?

Solution: $4 \ln(e) = 4(1) = \mathbf{4}$

32.

If $\log_x 81 = 4$, determine the base $x$.

Solution: $x^4 = 81 \Rightarrow \mathbf{x = 3}$ (Base must be positive)

33.

What is the numerical value of $\ln(1)$ and $\log_{10}(1)$?

Solution: Logarithm of $1$ for any valid base is $\mathbf{0}$.

34.

Simplify the general logarithmic expression: $\log_a (a^x)$.

Solution: $x\log_a a = \mathbf{x}$

35.

State the relationship formula used to convert a natural logarithm ($\ln x$) to a common logarithm ($\log_{10} x$).

Solution: $\mathbf{\ln x \approx 2.303 \log_{10} x}$

Section 4: Coordinate Geometry & Graphs

36.

Find the slope ($m$) and y-intercept ($c$) of the straight line represented by $4x + 3y = 24$.

Solution: $3y = -4x + 24 \Rightarrow y = -\frac{4}{3}x + 8$.
Slope $\mathbf{m = -\frac{4}{3}}$, Intercept $\mathbf{c = 8}$

37.

Write the equation of a straight line passing through the origin with a slope of $-3$.

Solution: $y = mx + c \Rightarrow y = -3x + 0 \Rightarrow \mathbf{y = -3x}$

38.

Calculate the slope of the line passing through the points $(2, 3)$ and $(5, 9)$.

Solution: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9 - 3}{5 - 2} = \frac{6}{3} = \mathbf{2}$

39.

Identify the geometric shape of the graph for the equation $y = 5x^2$. Where does its vertex lie?

Solution: It is an upward-opening $\mathbf{Parabola}$. Vertex is at the origin $\mathbf{(0, 0)}$.

40.

Identify the geometric shape of the graph for the equation $x^2 + y^2 = 36$. What is its radius?

Solution: Shape is a $\mathbf{Circle}$. Radius $r = \sqrt{36} = \mathbf{6 \text{ units}}$.

41.

Write the standard equation of an ellipse centered at the origin with semi-major axis $a$ and semi-minor axis $b$.

Solution: $\mathbf{\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1}$

42.

If a physical quantity $y$ is inversely proportional to $x$ ($y \propto \frac{1}{x}$), what is the shape of the graph plotted between $y$ and $x$?

Solution: $\mathbf{Rectangular\ Hyperbola}$

43.

Find the straight-line distance between the points $(4, 3)$ and the origin $(0, 0)$.

Solution: Distance $d = \sqrt{(4-0)^2 + (3-0)^2} = \sqrt{16 + 9} = \sqrt{25} = \mathbf{5 \text{ units}}$

44.

Describe the shape of the exponential decay curve given by the equation $y = e^{-x}$. Does it ever touch the x-axis?

Solution: It falls rapidly from $y=1$ (at $x=0$) towards zero as $x$ increases. $\mathbf{No}$, it asymptotically approaches the x-axis but theoretically never touches it.

45.

What is the equation of a parabola that opens leftwards and has its vertex at the origin?

Solution: $\mathbf{y^2 = -4ax}$ (where $a > 0$)

Section 5: Differentiation

46.

Find the derivative $\frac{dy}{dx}$ if $y = x^6$.

Solution: $\frac{dy}{dx} = \mathbf{6x^5}$

47.

Differentiate $y = 4x^3 - 5x^2 + 7x - 2$ with respect to $x$.

Solution: $\frac{dy}{dx} = 4(3x^2) - 5(2x) + 7 - 0 = \mathbf{12x^2 - 10x + 7}$

48.

Find the derivative $\frac{d}{dx}(\sqrt{x})$.

Solution: $\frac{d}{dx}(x^{1/2}) = \frac{1}{2}x^{-1/2} = \mathbf{\frac{1}{2\sqrt{x}}}$

49.

Differentiate $y = \frac{1}{x^3}$ with respect to $x$.

Solution: $y = x^{-3} \Rightarrow \frac{dy}{dx} = -3x^{-4} = \mathbf{-\frac{3}{x^4}}$

50.

Find the derivative of the trigonometric function $y = \sin x - \cos x$.

Solution: $\frac{dy}{dx} = \cos x - (-\sin x) = \mathbf{\cos x + \sin x}$

51.

Using the Product Rule, differentiate $y = x^2 \sin x$.

Solution: $(u \cdot v)' = uv' + u'v \Rightarrow \frac{dy}{dx} = x^2(\cos x) + (\sin x)(2x) = \mathbf{x^2\cos x + 2x\sin x}$

52.

Using the Quotient Rule, differentiate $y = \frac{\ln x}{x^2}$.

Solution: $\frac{v u' - u v'}{v^2} = \frac{x^2(\frac{1}{x}) - (\ln x)(2x)}{(x^2)^2} = \frac{x - 2x\ln x}{x^4} = \mathbf{\frac{1 - 2\ln x}{x^3}}$

53.

Using the Chain Rule, find $\frac{dy}{dx}$ if $y = (3x + 4)^5$.

Solution: $y' = 5(3x + 4)^4 \cdot \frac{d}{dx}(3x+4) = 5(3x + 4)^4 \cdot 3 = \mathbf{15(3x + 4)^4}$

54.

Differentiate the composite function $y = \cos(x^3)$.

Solution: $y' = -\sin(x^3) \cdot \frac{d}{dx}(x^3) = \mathbf{-3x^2\sin(x^3)}$

55.

Find the derivative of the exponential function $y = e^{4x}$.

Solution: $y' = e^{4x} \cdot \frac{d}{dx}(4x) = \mathbf{4e^{4x}}$

56.

Calculate the double derivative $\frac{d^2y}{dx^2}$ for the function $y = x^4 - 2x^3$.

Solution: $\frac{dy}{dx} = 4x^3 - 6x^2 \Rightarrow \frac{d^2y}{dx^2} = \frac{d}{dx}(4x^3 - 6x^2) = \mathbf{12x^2 - 12x}$

57.

The position of a moving particle is given by $x(t) = 3t^2 - 6t + 2$. Find its velocity equation $v(t) = \frac{dx}{dt}$.

Solution: $v(t) = \frac{d}{dt}(3t^2 - 6t + 2) = \mathbf{6t - 6}$

58.

For the particle in the previous question, calculate its acceleration $a(t) = \frac{dv}{dt}$. Is the acceleration constant?

Solution: $a(t) = \frac{d}{dt}(6t - 6) = \mathbf{6}$. $\mathbf{Yes}$, the acceleration is constant ($6 \text{ units/s}^2$).

59.

Find the value of $x$ for which the function $y = x^2 - 6x + 9$ is at a minimum. What is the minimum value?

Solution: For extrema, $y' = 2x - 6 = 0 \Rightarrow \mathbf{x = 3}$.
Minimum value = $(3)^2 - 6(3) + 9 = 9 - 18 + 9 = \mathbf{0}$.

60.

For what value(s) of $x$ does the function $y = x^3 - 12x$ have a local maximum or minimum?

Solution: $y' = 3x^2 - 12 = 0 \Rightarrow x^2 = 4 \Rightarrow \mathbf{x = 2, -2}$

Section 6: Integration

61.

Evaluate the indefinite integral: $\int x^4 \, dx$.

Solution: $\mathbf{\frac{x^5}{5} + C}$

62.

Find $\int (3x^2 - 4x + 2) \, dx$.

Solution: $\frac{3x^3}{3} - \frac{4x^2}{2} + 2x + C = \mathbf{x^3 - 2x^2 + 2x + C}$

63.

Evaluate the standard integral: $\int \frac{1}{x} \, dx$.

Solution: $\mathbf{\ln|x| + C}$

64.

Find the integral $\int \cos x \, dx$.

Solution: $\mathbf{\sin x + C}$

65.

Evaluate the integral with substitution: $\int \sin(3x) \, dx$.

Solution: $\mathbf{-\frac{\cos(3x)}{3} + C}$

66.

Find the integral of the exponential function: $\int e^{5x} \, dx$.

Solution: $\mathbf{\frac{e^{5x}}{5} + C}$

67.

Evaluate the definite integral: $\int_{1}^{2} x^3 \, dx$.

Solution: $[\frac{x^4}{4}]_1^2 = \frac{2^4}{4} - \frac{1^4}{4} = \frac{16}{4} - \frac{1}{4} = \mathbf{\frac{15}{4}}$ (or 3.75)

68.

Evaluate the definite trigonometric integral: $\int_{0}^{\pi} \sin x \, dx$.

Solution: $[-\cos x]_0^\pi = (-\cos\pi) - (-\cos 0) = -(-1) - (-1) = 1 + 1 = \mathbf{2}$

69.

Calculate the area under the curve $y = 3x^2$ from $x = 0$ to $x = 2$.

Solution: $\text{Area} = \int_0^2 3x^2 dx = [\frac{3x^3}{3}]_0^2 = [x^3]_0^2 = 2^3 - 0^3 = \mathbf{8}$

70.

Evaluate $\int \left(x + \frac{1}{x}\right)^2 \, dx$ by first expanding the algebraic expression.

Solution: $\int (x^2 + 2(x)(\frac{1}{x}) + \frac{1}{x^2}) dx = \int (x^2 + 2 + x^{-2}) dx = \mathbf{\frac{x^3}{3} + 2x - \frac{1}{x} + C}$

71.

The velocity of a particle is given by $v(t) = 4t^3$. Find its displacement equation $x(t)$ given that $x(0) = 0$.

Solution: $x(t) = \int v(t) dt = \int 4t^3 dt = t^4 + C$. Since $x(0) = 0 \Rightarrow C = 0$.
Therefore, $\mathbf{x(t) = t^4}$

72.

Evaluate the infinite definite integral: $\int_{0}^{\infty} e^{-x} \, dx$.

Solution: $[-e^{-x}]_0^\infty = (-e^{-\infty}) - (-e^0) = 0 - (-1) = \mathbf{1}$

73.

Integrate the polynomial: $\int (6x^5 - 2x) \, dx$.

Solution: $\frac{6x^6}{6} - \frac{2x^2}{2} + C = \mathbf{x^6 - x^2 + C}$

74.

Evaluate the integral involving roots: $\int \sqrt{x} \, dx$.

Solution: $\int x^{1/2} dx = \frac{x^{3/2}}{3/2} + C = \mathbf{\frac{2}{3}x^{3/2} + C}$ (or $\frac{2}{3}x\sqrt{x} + C$)

75.

Find the integral $\int \frac{1}{\sqrt{x}} \, dx$.

Solution: $\int x^{-1/2} dx = \frac{x^{1/2}}{1/2} + C = \mathbf{2\sqrt{x} + C}$

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