Math Practice MCQs

Mathematics - Logarithms

1. If $\log_3(x + 2) = 2$, then $x =$

A.1B.3C.5D.7

Mathematics - Sequences and Series

2. If $y = \frac{x}{1} + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + \dots$, then $x =$

A.$e^{-y}$B.$1 - \log y$C.$1 - e^{-y}$D.$e^y + 1$

Mathematics - Matrices and Determinants

3. If $A$ is a skew-symmetric matrix, then the trace of $A$ is

A.1B.0C.-1D.2

Mathematics - Complex Numbers

4. The value of $i^{50} + i^{51} + i^{52} + i^{53}$ is

A.-1B.1C.$i$D.0

Mathematics - Trigonometry

5. In $\triangle ABC$, the value of $4\Delta \cot A =$

A.$b^2 + c^2 - a^2$B.$\frac{a+b+c}{2}$C.$a^2 + b^2 - c^2$D.$\frac{4bc}{a+b+c}$

Mathematics - Inverse Trigonometry

6. $\sin^{-1}x - \cos^{-1}(-x) =$

A.$\frac{\pi}{2}$B.$-\frac{\pi}{2}$C.$\frac{\pi}{3}$D.$\pi$

Mathematics - Coordinate Geometry

7. The distance between the parallel lines $y = 2x + 4$ and $6x - 3y = 5$ is

A.$\frac{17}{\sqrt{45}}$B.$\frac{5}{\sqrt{17}}$C.$\frac{17}{\sqrt{5}}$D.$\sqrt{17}$

Mathematics - Coordinate Geometry

8. If the equation $kx^2 + 4xy + 5y^2 = 0$ represents two lines inclined at an angle $\pi$, then $k =$

A.5/4B.4/7C.3/2D.4/5

Mathematics - Conic Sections

9. The eccentricity of the conic $y^2 + 6y - 16x - 71 = 0$ is

A.0B.1C.1/2D.3

Mathematics - Circles

10. The length of the y-intercept made by the circle $x^2 + y^2 - 8x + y - 20 = 0$ is

A.7B.9C.11D.15

Mathematics - Calculus

11. If $\lim_{x \to 0} \frac{\sin 4x}{\tan kx} = 5$, then $k =$

A.5/4B.4/5C.1D.5

Mathematics - Calculus

12. If $y = x^2 + 1$, then the actual change in $y$ when $x = 1$ and $\Delta x = dx = 0.1$ is

A.0.25B.0.24C.0.015D.0.21

Mathematics - Integral Calculus

13. $\int \frac{x}{x^2 - 3} \, dx =$

A.$\ln(x^2 - 3) + c$B.$2\ln(x^2 - 3) + c$C.$\frac{1}{2}\ln(x^2 - 3) + c$D.$\frac{1}{6}\ln(x^2 - 3) + c$

Mathematics - Definite Integrals

14. If $\int_{0}^{a} 3x^2 \, dx = 8$, then the value of $a$ is

A.1B.-1C.2D.-2

Mathematics - Differential Equations

15. The order and degree of differential equation $\left(\frac{dy}{dx}\right)^3 + 3x \frac{d^2y}{dx^2} = 0$ are respectively

A.2, 1B.2, 3C.1, 3D.3, 1

Mathematics - Vectors

16. If the vectors $\vec{a} = \hat{i} - 2\hat{j} + 4\hat{k}$ and $\vec{b} = 2\hat{i} + 7\hat{j} + m\hat{k}$ are perpendicular to each other, then $m =$

A.1B.2C.3D.4

Mathematics - Coordinate Geometry

17. A parallelogram has three consecutive vertices $(-3, 4)$, $(0, -4)$ and $(5, 2)$. The fourth vertex is

A.$(2, 10)$B.$(2, 9)$C.$(3, 9)$D.$(4, 10)$

Mathematics - Trigonometry

18. If $\tan x = -\frac{3}{4}$ and $x$ is in the second quadrant, then what is the value of $\sin x \cdot \cos x$?

A.6/25B.12/25C.-6/25D.-12/25

Mathematics - Statistics

19. For which of the following sets of numbers do the mean, median and mode have the same value?

A.$12, 12, 12, 12, 24$B.$6, 18, 18, 18, 30$C.$6, 6, 12, 30, 36$D.$6, 6, 6, 12, 30$

Mathematics - Probability

20. A coin is tossed twice. If $E$ and $F$ denote occurrence of head on first toss and second toss respectively, then what is $P(E \cup F)$ equal to?

A.1/4B.1/2C.3/4D.1/3

Mathematics - Calculus

21. What is the derivative of $e^x$ with respect to $x^e$?

A.$\frac{xe^x}{ex^e}$B.$\frac{e^x}{x^e}$C.$\frac{xe^x}{x^e}$D.$\frac{e^x}{ex^e}$

Mathematics - Functions

22. What is the domain of the function $f(x) = 3^x$?

A.$(-\infty, \infty)$B.$(0, \infty)$C.$[0, \infty)$D.$(-\infty, \infty) - \{0\}$

Mathematics - Differential Equations

23. A particle starts from origin with a velocity (in m/s) given by the equation $\frac{dx}{dt} = x + 1$. The time (in seconds) taken by the particle to traverse a distance of 24 m is

A.$\ln 24$B.$\ln 5$C.$2\ln 5$D.$2\ln 4$

Mathematics - Integral Calculus

24. What is the area bounded by $y = \sqrt{16 - x^2}$, $y \ge 0$ and the x-axis?

A.$16\pi\text{ sq. units}$B.$8\pi\text{ sq. units}$C.$4\pi\text{ sq. units}$D.$2\pi\text{ sq. units}$

Mathematics - Calculus

25. What is the derivative of $\sin(\ln x) + \cos(\ln x)$ with respect to $x$ at $x = e$?

A.$\frac{\cos 1 - \sin 1}{e}$B.$\frac{\sin 1 - \cos 1}{e}$C.$\frac{\cos 1 + \sin 1}{e}$D.1

Mathematics - 3D Geometry

26. What is the angle between the two lines having direction ratios $(6, 3, 6)$ and $(3, 3, 0)$?

A.$\frac{\pi}{6}$B.$\frac{\pi}{4}$C.$\frac{\pi}{3}$D.$\frac{\pi}{2}$

Mathematics - Binomial Theorem

27. How many terms are there in the expansion of $\left(\frac{a^2}{b^2} + \frac{b^2}{a^2} + 2\right)^{21}$ where $a \ne 0, b \ne 0$?

A.21B.22C.42D.43

Mathematics - Set Theory

28. If a set $A$ contains 3 elements and another set $B$ contains 6 elements, then what is the minimum number of elements that $(A \cup B)$ can have?

A.3B.6C.8D.9

Mathematics - Quadratic Equations

29. The roots of the equation $(q - r)x^2 + (r - p)x + (p - q) = 0$ are

A.$\frac{(r - p)}{(q - r)}, \frac{1}{2}$B.$\frac{(p - q)}{(q - r)}, 1$C.$\frac{(q - r)}{(p - q)}, 1$D.$\frac{(r - p)}{(p - q)}, \frac{1}{2}$

Mathematics - Conic Sections

30. If $5x^2 + \lambda y^2 = 20$ represents a rectangular hyperbola, then $\lambda$ equals

A.5B.4C.-5D.20

1. If $\log_3(x + 2) = 2$, then $x =$

2. If $y = \frac{x}{1} + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + \dots$, then $x =$

3. If $A$ is a skew-symmetric matrix, then the trace of $A$ is

4. The value of $i^{50} + i^{51} + i^{52} + i^{53}$ is

5. In $\triangle ABC$, the value of $4\Delta \cot A =$

6. $\sin^{-1}x - \cos^{-1}(-x) =$

7. The distance between the parallel lines $y = 2x + 4$ and $6x - 3y = 5$ is

8. If the equation $kx^2 + 4xy + 5y^2 = 0$ represents two lines inclined at an angle $\pi$, then $k =$

9. The eccentricity of the conic $y^2 + 6y - 16x - 71 = 0$ is

10. The length of the y-intercept made by the circle $x^2 + y^2 - 8x + y - 20 = 0$ is

11. If $\lim_{x \to 0} \frac{\sin 4x}{\tan kx} = 5$, then $k =$

12. If $y = x^2 + 1$, then the actual change in $y$ when $x = 1$ and $\Delta x = dx = 0.1$ is

13. $\int \frac{x}{x^2 - 3} \, dx =$

14. If $\int_{0}^{a} 3x^2 \, dx = 8$, then the value of $a$ is

15. The order and degree of differential equation $\left(\frac{dy}{dx}\right)^3 + 3x \frac{d^2y}{dx^2} = 0$ are respectively

16. If the vectors $\vec{a} = \hat{i} - 2\hat{j} + 4\hat{k}$ and $\vec{b} = 2\hat{i} + 7\hat{j} + m\hat{k}$ are perpendicular to each other, then $m =$

17. A parallelogram has three consecutive vertices $(-3, 4)$, $(0, -4)$ and $(5, 2)$. The fourth vertex is

18. If $\tan x = -\frac{3}{4}$ and $x$ is in the second quadrant, then what is the value of $\sin x \cdot \cos x$?

19. For which of the following sets of numbers do the mean, median and mode have the same value?

20. A coin is tossed twice. If $E$ and $F$ denote occurrence of head on first toss and second toss respectively, then what is $P(E \cup F)$ equal to?

21. What is the derivative of $e^x$ with respect to $x^e$?

22. What is the domain of the function $f(x) = 3^x$?

23. A particle starts from origin with a velocity (in m/s) given by the equation $\frac{dx}{dt} = x + 1$. The time (in seconds) taken by the particle to traverse a distance of 24 m is

24. What is the area bounded by $y = \sqrt{16 - x^2}$, $y \ge 0$ and the x-axis?

25. What is the derivative of $\sin(\ln x) + \cos(\ln x)$ with respect to $x$ at $x = e$?

26. What is the angle between the two lines having direction ratios $(6, 3, 6)$ and $(3, 3, 0)$?

27. How many terms are there in the expansion of $\left(\frac{a^2}{b^2} + \frac{b^2}{a^2} + 2\right)^{21}$ where $a \ne 0, b \ne 0$?

28. If a set $A$ contains 3 elements and another set $B$ contains 6 elements, then what is the minimum number of elements that $(A \cup B)$ can have?

29. The roots of the equation $(q - r)x^2 + (r - p)x + (p - q) = 0$ are

30. If $5x^2 + \lambda y^2 = 20$ represents a rectangular hyperbola, then $\lambda$ equals

Test Result