Math Practice MCQs

Mathematics - Algebra

1. If $x^2 - x + 1 = 0$, then what is $(x - \frac{1}{x})^2 + (x - \frac{1}{x})^4 + (x - \frac{1}{x})^8$ equal to?

A.81B.85C.87D.89

Mathematics - Number Theory

2. If $1! + 3! + 5! + 7! + \dots + 199!$ is divided by $24$, what is the remainder?

A.3B.6C.7D.9

Mathematics - Coordinate Geometry

3. If any point on an ellipse is $(3\sin\alpha, 5\cos\alpha)$, then what is the eccentricity of the ellipse?

A.4/3B.4/5C.3/4D.1/2

Mathematics - Matrices and Determinants

4. If $\begin{vmatrix} a & -b & a - b - c \\ -a & b & -a + b - c \\ -a & -b & -a - b + c \end{vmatrix} - kabc = 0$ ($a \ne 0, b \ne 0, c \ne 0$), then what is the value of $k$?

A.-4B.-2C.2D.4

Mathematics - Sequences and Series

5. Let $x > 1, y > 1, z > 1$ be in G.P. Then $\frac{1}{1 + \ln x}, \frac{1}{1 + \ln y}, \frac{1}{1 + \ln z}$ are

A.in A.PB.in G.PC.in H.PD.neither in A.P nor in G.P nor in H.P

Mathematics - Complex Numbers

6. If $x, y$ and $z$ are the cube roots of unity, then what is the value of $xy + yz + xz$?

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

Mathematics - Vectors

7. For any vector $\vec{r}$, what is $(\vec{r} \cdot \hat{i})(\vec{r} \times \hat{i}) + (\vec{r} \cdot \hat{j})(\vec{r} \times \hat{j}) + (\vec{r} \cdot \hat{k})(\vec{r} \times \hat{k})$?

A.$\vec{0}$B.$\vec{r}$C.$2\vec{r}$D.$3\vec{r}$

Mathematics - Trigonometry

8. The minimum value of $f(x) = \sin^2 x + \cos^4 x$ is

A.1B.3/4C.3/8D.0

Mathematics - Calculus

9. Let $(x + y)^{p+q} = x^p y^q$, where $p, q$ are positive integers and $p + q = 10$, then $\frac{dy}{dx}$ is equal to

A.$(\frac{y}{x})^{10}$B.$xy$C.$\frac{y}{x}$D.$x^{10}y^{10}$

Mathematics - Inverse Trigonometry

10. If $k$ is a root of $x^2 - 4x + 1 = 0$, then what is $\tan^{-1}k + \tan^{-1}\frac{1}{k}$ equal to?

A.$-\frac{\pi}{2}$B.0C.$\frac{\pi}{4}$D.$\frac{\pi}{2}$

Mathematics - Calculus

11. Derivative of $y = (\sin x)^{(\sin x)^{(\sin x)^{\dots \infty}}}$ with respect to $x$ is

A.$\frac{y^2 \cot x}{1 + y\log(\sin x)}$B.$\frac{y^2 \tan x}{1 + y\log(\sin x)}$C.$\frac{y^2 \cot x}{1 - y\log(\sin x)}$D.$\frac{y^2 \tan x}{1 - y\log(\sin x)}$

Mathematics - Calculus

12. What is the derivative of $\tan^{-1}\left(\frac{\sqrt{1 + x^2} - 1}{x}\right)$ with respect to $\tan^{-1}x$?

A.0B.1/2C.1D.$x$

Mathematics - Binomial Theorem

13. If the coefficients of the $r^{\text{th}}$, $(r+1)^{\text{th}}$ and $(r+2)^{\text{th}}$ terms in the expansion of $(1+x)^{14}$ are in A.P., then the value of $r$ is:

A.11B.5C.7D.6

Mathematics - Probability

14. Two dice are thrown simultaneously. The probability of obtaining a total score of $5$ is

A.1/9B.1/18C.1/12D.1/36

Mathematics - Calculus

15. $\lim_{x \to 0} \frac{6^x - 3^x - 2^x + 1}{x^2} =$

A.$\log\frac{2}{3}$B.$\log 2 + \log 3$C.$(\log 2)(\log 3)$D.none of these

Mathematics - Calculus

16. If $\sum_{r=1}^n (2r - 1) = x$, then $\lim_{n \to \infty} \left[ \frac{1^3}{x^2} + \frac{2^3}{x^2} + \frac{3^3}{x^2} + \dots + \frac{n^3}{x^2} \right] =$

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

Mathematics - 3D Geometry

17. The equation of the plane which bisects the line joining $(3, 0, 5)$ and $(1, 2, -1)$ at right angles is

A.$2x + y + 2z = 7$B.$-2x + 2y - 6z = 7$C.$x - y + 2z = 7$D.$x - y + 3z = 7$

Mathematics - Integral Calculus

18. The area enclosed between the parabolas $y^2 = 16x$ and $x^2 = 16y$ is

A.$\frac{64}{3} \text{ sq. units}$B.$\frac{256}{3} \text{ sq. units}$C.$\frac{16}{3} \text{ sq. units}$D.$\frac{8}{3} \text{ sq. units}$

Mathematics - Coordinate Geometry

19. If $C$ is the reflection of $A(2, 4)$ in the x-axis and $B$ is the reflection of $C$ in the y-axis, then $|AB|$ is

A.20B.$2\sqrt{5}$C.$4\sqrt{5}$D.4

Mathematics - Coordinate Geometry

20. The number of points on the line $x + y = 4$ which are unit distance apart from the line $2x + 2y = 5$ is

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

Mathematics - Permutations and Combinations

21. A triangle $PQR$ is such that $3$ points lie on side $PQ$, $4$ points on $QR$ and $5$ points on $RP$ respectively. Triangles are constructed using these points as vertices. What is the number of triangles so formed?

A.205B.206C.215D.220

Mathematics - Circles

22. If two different circles $x^2 + y^2 + 2ax + 2by + 1 = 0$ and $x^2 + y^2 + 2bx + 2ay + 1 = 0$ touch each other, then $(a + b)^2$ is equal to

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

Mathematics - Circles

23. The equation of the chord of the circle $x^2 + y^2 - 4x = 0$ whose midpoint is $(1, 0)$ is

A.$y = 1$B.$y = 2$C.$x = 2$D.$x = 1$

Mathematics - Logarithms

24. If $\log_5 \log_5 \log_2 x = 0$, then the value of $x$ is

A.32B.125C.625D.25

Mathematics - Integral Calculus

25. If $\int 2^{2^x} \cdot 2^x \, dx = A \cdot 2^{2^x} + C$, then $A$ is equal to

A.$\frac{1}{\log 2}$B.$\log 2$C.$(\log 2)^2$D.$\frac{1}{(\log 2)^2}$

Mathematics - Calculus

26. The slope of normal to the curve $y = x^3 + 2x + 6$ which is parallel to line $x + 14y + 4 = 0$ is

A.$-1/3$B.$-1/14$C.-4D.$-1/2$

Mathematics - 3D Geometry

27. The distance between the point $(7, 2, 4)$ and the plane determined by the points $(2, 5, -3)$, $(-2, -3, 5)$ and $(5, 3, -3)$ is

A.$\sqrt{29}$B.29/5C.29D.6

Mathematics - Differential Equations

28. If $x \frac{dy}{dx} = x^2 + y - 2$, $y(1) = 1$, then $y(2)$ is equal to

A.1B.4C.2D.8

Mathematics - Inverse Trigonometry

29. $\cos^{-1}(\cos 12) - \sin^{-1}(\sin 14) =$

A.$\pi - 2$B.$8\pi - 26$C.-2D.$3\pi - 2$

Mathematics - Statistics

30. The mean of a set of 30 observations is 75. If each observation is multiplied by a non-zero number $\lambda$ and then each of them is decreased by 25, their mean remains the same. The $\lambda$ is equal to

A.10/3B.4/3C.1/3D.2/3

1. If $x^2 - x + 1 = 0$, then what is $(x - \frac{1}{x})^2 + (x - \frac{1}{x})^4 + (x - \frac{1}{x})^8$ equal to?

2. If $1! + 3! + 5! + 7! + \dots + 199!$ is divided by $24$, what is the remainder?

3. If any point on an ellipse is $(3\sin\alpha, 5\cos\alpha)$, then what is the eccentricity of the ellipse?

4. If $\begin{vmatrix} a & -b & a - b - c \\ -a & b & -a + b - c \\ -a & -b & -a - b + c \end{vmatrix} - kabc = 0$ ($a \ne 0, b \ne 0, c \ne 0$), then what is the value of $k$?

5. Let $x > 1, y > 1, z > 1$ be in G.P. Then $\frac{1}{1 + \ln x}, \frac{1}{1 + \ln y}, \frac{1}{1 + \ln z}$ are

6. If $x, y$ and $z$ are the cube roots of unity, then what is the value of $xy + yz + xz$?

7. For any vector $\vec{r}$, what is $(\vec{r} \cdot \hat{i})(\vec{r} \times \hat{i}) + (\vec{r} \cdot \hat{j})(\vec{r} \times \hat{j}) + (\vec{r} \cdot \hat{k})(\vec{r} \times \hat{k})$?

8. The minimum value of $f(x) = \sin^2 x + \cos^4 x$ is

9. Let $(x + y)^{p+q} = x^p y^q$, where $p, q$ are positive integers and $p + q = 10$, then $\frac{dy}{dx}$ is equal to

10. If $k$ is a root of $x^2 - 4x + 1 = 0$, then what is $\tan^{-1}k + \tan^{-1}\frac{1}{k}$ equal to?

11. Derivative of $y = (\sin x)^{(\sin x)^{(\sin x)^{\dots \infty}}}$ with respect to $x$ is

12. What is the derivative of $\tan^{-1}\left(\frac{\sqrt{1 + x^2} - 1}{x}\right)$ with respect to $\tan^{-1}x$?

13. If the coefficients of the $r^{\text{th}}$, $(r+1)^{\text{th}}$ and $(r+2)^{\text{th}}$ terms in the expansion of $(1+x)^{14}$ are in A.P., then the value of $r$ is:

14. Two dice are thrown simultaneously. The probability of obtaining a total score of $5$ is

15. $\lim_{x \to 0} \frac{6^x - 3^x - 2^x + 1}{x^2} =$

16. If $\sum_{r=1}^n (2r - 1) = x$, then $\lim_{n \to \infty} \left[ \frac{1^3}{x^2} + \frac{2^3}{x^2} + \frac{3^3}{x^2} + \dots + \frac{n^3}{x^2} \right] =$

17. The equation of the plane which bisects the line joining $(3, 0, 5)$ and $(1, 2, -1)$ at right angles is

18. The area enclosed between the parabolas $y^2 = 16x$ and $x^2 = 16y$ is

19. If $C$ is the reflection of $A(2, 4)$ in the x-axis and $B$ is the reflection of $C$ in the y-axis, then $|AB|$ is

20. The number of points on the line $x + y = 4$ which are unit distance apart from the line $2x + 2y = 5$ is

21. A triangle $PQR$ is such that $3$ points lie on side $PQ$, $4$ points on $QR$ and $5$ points on $RP$ respectively. Triangles are constructed using these points as vertices. What is the number of triangles so formed?

22. If two different circles $x^2 + y^2 + 2ax + 2by + 1 = 0$ and $x^2 + y^2 + 2bx + 2ay + 1 = 0$ touch each other, then $(a + b)^2$ is equal to

23. The equation of the chord of the circle $x^2 + y^2 - 4x = 0$ whose midpoint is $(1, 0)$ is

24. If $\log_5 \log_5 \log_2 x = 0$, then the value of $x$ is

25. If $\int 2^{2^x} \cdot 2^x \, dx = A \cdot 2^{2^x} + C$, then $A$ is equal to

26. The slope of normal to the curve $y = x^3 + 2x + 6$ which is parallel to line $x + 14y + 4 = 0$ is

27. The distance between the point $(7, 2, 4)$ and the plane determined by the points $(2, 5, -3)$, $(-2, -3, 5)$ and $(5, 3, -3)$ is

28. If $x \frac{dy}{dx} = x^2 + y - 2$, $y(1) = 1$, then $y(2)$ is equal to

29. $\cos^{-1}(\cos 12) - \sin^{-1}(\sin 14) =$

30. The mean of a set of 30 observations is 75. If each observation is multiplied by a non-zero number $\lambda$ and then each of them is decreased by 25, their mean remains the same. The $\lambda$ is equal to

Test Result