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GMAT Straight Lines and Conic Sections — practice questions

Explore 52 free Straight Lines and Conic Sections questions in GMAT Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Straight Lines and Conic Sections questions and exercises with answers

Showing questions 1–30 of 52.

$\int x^{2}e^{x^{3}}\,dx$ equalsRead the answer and worked solution →$\int e^{x}\sec x\,(1+\tan x)\,dx$ equalsRead the answer and worked solution →By the Second Fundamental Theorem of Calculus, if $F'=f$ on $[a,b]$, then $\int_a^b f(x)\,dx$ equalsRead the answer and worked solution →$\displaystyle\int_2^3 x^2\,dx$ equalsRead the answer and worked solution →$\displaystyle\int_1^{\sqrt{3}} \dfrac{dx}{1+x^2}$ equalsRead the answer and worked solution →$\displaystyle\int_0^{2/3} \dfrac{dx}{4+9x^2}$ equalsRead the answer and worked solution →$\displaystyle\int_{-1}^{1} 5x^4\sqrt{x^5+1}\,dx$ equalsRead the answer and worked solution →$\displaystyle\int_0^{1} \dfrac{\tan^{-1}x}{1+x^2}\,dx$ equalsRead the answer and worked solution →If $f(x)=\displaystyle\int_0^{x} t\sin t\,dt$, then $f'(x)$ isRead the answer and worked solution →If $f$ is an odd function, then $\displaystyle\int_{-a}^{a} f(x)\,dx$ equalsRead the answer and worked solution →$\displaystyle\int_{-1}^{1} \sin^5 x\cos^4 x\,dx$ equalsRead the answer and worked solution →$\displaystyle\int_{-\pi/4}^{\pi/4} \sin^2 x\,dx$ equalsRead the answer and worked solution →$\displaystyle\int_0^{\pi/2} \dfrac{\sin^4 x}{\sin^4 x+\cos^4 x}\,dx$ equalsRead the answer and worked solution →$\displaystyle\int_{\pi/6}^{\pi/3} \dfrac{dx}{1+\sqrt{\tan x}}$ equalsRead the answer and worked solution →$\displaystyle\int_0^{\pi/2} \log\sin x\,dx$ equalsRead the answer and worked solution →Using property $\mathbf{P_2}$, $\displaystyle\int_{-1}^{2} |x^3-x|\,dx$ equalsRead the answer and worked solution →The value of $\displaystyle\int_0^{\pi/2} \log\!\left(\dfrac{4+3\sin x}{4+3\cos x}\right)dx$ isRead the answer and worked solution →$\displaystyle\int_0^{1} x e^{x^2}\,dx$ equalsRead the answer and worked solution →$\displaystyle\int_0^{\pi/4} \tan x\,dx$ equalsRead the answer and worked solution →$\displaystyle\int \frac{dx}{e^{x}+e^{-x}}$ is equal toRead the answer and worked solution →$\displaystyle\int_0^{2a} f(x)\,dx$ equals $2\displaystyle\int_0^{a} f(x)\,dx$ precisely whenRead the answer and worked solution →$\displaystyle\int_2^3 \dfrac{x\,dx}{x^2+1}$ equalsRead the answer and worked solution →In a right-angled triangle, which side is always directly opposite the right angle?Read the answer and worked solution →Does the tangent ratio have a maximum value?Read the answer and worked solution →A right-angled triangle has shorter sides of 6 cm and 8 cm. How long is the hypotenuse?Read the answer and worked solution →A sailor in a boat looks up at the top of a cliff. What is the angle between the line of sight and the horizontal called?Read the answer and worked solution →An observer stands on top of a cliff and looks down at a boat below. What is the angle between the line of sight and the horizontal called?Read the answer and worked solution →Which result is used alongside the three trigonometric ratios to solve two-dimensional problems?Read the answer and worked solution →What is a mathematical statement that equates one quantity with another called?Read the answer and worked solution →Which rule fits a triangle when two angles and one side are known?Read the answer and worked solution →

Worked practice questions

$\int x^{2}e^{x^{3}}\,dx$ equals

  1. $\dfrac{1}{3}e^{x^{2}}+C$
  2. $\dfrac{1}{2}e^{x^{3}}+C$
  3. $\dfrac{1}{3}e^{x^{3}}+C$
  4. $\dfrac{1}{2}e^{x^{2}}+C$
Show answer and solution

Correct answer: C. $\dfrac{1}{3}e^{x^{3}}+C$

1. Put $t=x^{3}$, so $dt=3x^{2}\,dx$, i.e. $x^{2}\,dx=\dfrac{1}{3}dt$. 2. Integral $=\dfrac{1}{3}\int e^{t}\,dt=\dfrac{1}{3}e^{t}$. 3. Back-substitute: $\dfrac{1}{3}e^{x^{3}}+C$. 4. Options with $e^{x^{2}}$ misread the exponent; option B has wrong constant. _Source: syllabus Mathematics Ch 7 "Integrals", p.329_

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$\int e^{x}\sec x\,(1+\tan x)\,dx$ equals

  1. $e^{x}\cos x+C$
  2. $e^{x}\sec x+C$
  3. $e^{x}\sin x+C$
  4. $e^{x}\tan x+C$
Show answer and solution

Correct answer: B. $e^{x}\sec x+C$

1. Expand: $e^{x}(\sec x+\sec x\tan x)$. 2. Let $f(x)=\sec x$, then $f'(x)=\sec x\tan x$. 3. So the integrand is $e^{x}[f(x)+f'(x)]$, giving $e^{x}f(x)+C$. 4. Result: $e^{x}\sec x+C$. _Source: syllabus Mathematics Ch 7 "Integrals", p.329_

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By the Second Fundamental Theorem of Calculus, if $F'=f$ on $[a,b]$, then $\int_a^b f(x)\,dx$ equals

  1. $F(a)-F(b)$
  2. $F(b)-F(a)$
  3. $F(b)+F(a)$
  4. $F'(b)-F'(a)$
Show answer and solution

Correct answer: B. $F(b)-F(a)$

1. The theorem states $\int_a^b f(x)\,dx=[F(x)]_a^b$. 2. This means upper-limit value minus lower-limit value. 3. Hence $\int_a^b f(x)\,dx=F(b)-F(a)$. _Source: syllabus Mathematics Ch 7 "Integrals", p.334_

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