K-12 Math Questions and Answers

$\dfrac{ 1 } { 1 + \sqrt{2} } + \dfrac{ 1 } { \sqrt{2}+\sqrt{3} } + \dfrac{ 1 } { \sqrt{3}+\sqrt{4 } } + \ldots + \dfrac{ 1 } { \sqrt{ 15 } + \sqrt{ 16 } } = ? \space\space\space\space\space$

Question 271:

$ABCD$ is a parallelogram where

$P$ and

$R$ are the midpoints of sides

$BC$ and

$DC$ respectively. If the line

$PR$ intersects the diagonal

$AC$ at

$Q$ , prove that

$AC = 4CQ$ .

Question 272: Kenneth is brother of Robert. Lisa is sister of Kenneth. Mark is brother of Laura. Laura is daughter of Robert. Richard is father of Lisa. Who is the uncle of Mark?

Question 273: In the image given below, if the radius of all the large circles is r, then what is the radius of the inner smaller circle?

Question 274: The two sides of a rectangle are x and x + 1. If the length of the diagonal of the rectangle is 5 cm, then what is the area of the rectangle?

Question 275: We draw a square inside a rectangle. The ratio of the rectangle's width to the square's side is

$2:1$ , and the ratio of the rectangle's length to its width is

$4:1$ . What percentage of the rectangle's area is inside the square?

Question 276: n is a positive integer. Let's say that {n} denotes the sum of all the positive divisors of n except for n itself. For example, {4} = 1 + 2 = 3 and {8} = 1 + 2 + 4 = 7. What is {{{496}}}?

Question 277: Consider a square ABCD of area 25 cm². L is the midpoint of AB, M the midpoint of BC, N the midpoint of CD, and O the midpoint of DA. These points are used to construct a new square LMNO. The same process is repeated on LMNO to construct a smaller square QRST (where Q is the midpoint of LM and so on). What is the perimeter of square QRST?

Question 278: In a triangle ABC, AD is the median. F is the point on AC such that line BF bisect AD at E. Prove that AC = 3AF.

Question 279: In a triangle

$ABC$ ,

$AD$ is a median.

$F$ is a point on

$AC$ such that the line

$BF$ bisects

$AD$ at

$E$ . If

$AD = 9 \space cm$ and

$AF = 3 \space cm$ , find the measure of

$AC$ .

Question 280: ABCD is a parallelogram with E and F as the mid-points of side AB and CD. Prove that the line segment AF and CE trisect the diagonal BD.

Question 281: Prove that the line segment joining the midpoints of the diagonals of a trapezium is parallel to each of the parallel sides and is equal to half of the difference of these sides.

Question 282: In a triangle, ABC, D is a point on AB such that AB = 4AD and E is a point on AC such that AC = 4AE. Prove that BC = 4ED.

Question 283: A Trapezoid

$PQRS$ has two parallel bases

$PQ$ and

$RS.$ The diagonals of the Trapezoid intersect at a point

$O.$ Suppose that

$PQ = 10, SR = 15,$ and the area of the triangle

$POS$ is

$30.$ What is the area of the trapezoid

$PQRS?$

Question 284: Leap day, 29th February, in 2004 fell on a Sunday. On what day of the week will the leap day in 2012 fall?

Question 285: There is a sequence of numbers a₁, a₂, ... where a₁ = 2, a₂ = 3, and a_n =

a_n-1

a_n-2

for n ≥ 3. What is the value of a₇₅₆ ?

Question 286: How many 3-digit numbers satisfy the property that the digit in the middle is the average of the first and the third digit?

Question 287: The sum of the digits of a 2-digit number is subtracted from the number. The resulting number has the units digit of 6. Which numbers have this property?

Question 288: Coin A is flipped 4 times and coin B is flipped 5 times. What is the probability that the number of heads obtained from flipping the two coins is the same?

Question 289: Area of parallelogram

$ABCD$ is

$x \space cm^2.$ If

$E, F, G,$ and

$H$ are mid-points of the sides, find the area of

$EFGH.$

Question 290: ABCD is a square. Two arcs are drawn with A and B as centers, and radius equal to the side of square. If arcs intersects at point E, find the angle ∠ACE.

Question 291:

$ABCD$ is a rectangle and point

$P$ is such that

$PA = 5 \text { cm }, PB = 4 \sqrt { 2 } \text { cm, } \text { and } PD = 3 \text { cm. }$ Find the length of

$PC$ .

Question 292: If there are 9 diagonals in a convex polygon, find the number of sides of the polygon.

Question 293: How many times do a clock's hour, minute and second hands overlap in a day?

Question 294: Assume that the symbol

$\ddagger x \ddagger$ denotes the largest integer not exceeding

$x$ . For example,

$\ddagger 3 \ddagger = 3$ , and

$\ddagger 4.9 \ddagger = 4$ . What is the value of

$\ddagger \sqrt{1}\ddagger + \ddagger\sqrt{2}\ddagger + \ddagger\sqrt{3}\ddagger \space + ...... + \ddagger\sqrt { 16 }\ddagger.$

Question 295: You are given a set of tiles which are numbered from 1 to 121. You do the following operations repeatedly: you remove all those tiles that are numbered with a perfect square, and renumber the remaining tiles consecutively starting with 1. How many times must you perform the operation before you are left with 1 tile?

Question 296: {a_m} is a sequence of integers such that a₁ = 1, and a_m+n = a_m + a_n + mn, for all positive integers m and n. What is the value of a₁₁?

Question 297: If

$cot \theta = \dfrac{ a } { b }$ and

$90^\circ > \theta > 0^\circ,$ find value of

$cosec \theta.$

Question 298: If

$sec \theta = \dfrac{ a } { b }$ and

$90^\circ > \theta > 0^\circ,$ find value of

$tan \theta.$

Question 299: If

$\triangle ABC$ is right angled at

$C$ , find the value of

$cos(A+B)$ .

Question 300: If cos θ + cos²θ = 1, then find the value of expression (sin²θ + sin⁴θ).

Question 301: What is the sum of all two-digit odd positive numbers?

Question 302: What is the sum of all three-digit numbers that are divisible by 13?

Question 303: Write the condition for the given system of equations to be inconsistent.
a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0

Question 304: Simplify sin⁶β + cos⁶β + 3sin²β cos²β.

Question 305: If

$\alpha + \beta = 90^ \circ$ , simplify

$\sqrt{\cos \alpha \mathrm{cosec} \beta - \cos \alpha \sin \beta}.$

Question 306: If

$sin \theta + cos \theta = \sqrt{ 3 },$ simplify

$tan \theta + cot \theta.$

Question 307: Simplify

$\dfrac{\cos \theta } { 1 + \sin \theta } + \dfrac{ 1 + \sin \theta } { \cos \theta }$ .

Question 308: If sinθ + 2cosθ = 1, find the value of 2sinθ - cosθ.

Question 309: A spherical balloon of radius

$15$ feet subtends an angle

$60^\circ$ at the eye of an observer. If the angle of elevation of its center is

$45^\circ,$ find the height of the center of the balloon.

Question 310: sin(45° + θ) - cos(45° - θ) = ?

Question 311: Simplify: 1 +

tan²α

1 + secα

Question 312: Prove that n²-n is even for every positive integer n.

Question 313: Prove that √2 is an irrational number.

Question 314: Prove that

√11

is an irrational number.

Question 315: Jason, Patricia, John, Daniel, Ronald, and Steven are playing game by forming two teams of three players each. In how many different ways can they be put into two teams of three players?

Question 316: Simplify

$\dfrac{ 1 }{ 1 + sec \theta } + \dfrac {1} { 1 - sec \theta }$ .

Question 317: Simplify

$\sqrt { \dfrac { 1 + cos \theta } { 1 - cos \theta } }$ .

Question 318: Simplify

cosθ - 2 cos³θ

2 sin³θ - sinθ

Question 319: Simplify (1 + tanθ + secθ) (1 + cotθ - cosecθ)

Question 320: Simplify

Question 321: Simplify

Question 322: Simplify:

$\dfrac{ \cos \theta } { 1 - \tan \theta } + \dfrac{ \sin^2 \theta } { \sin \theta - \cos \theta }$

Question 323: Simplify:

$1 - \dfrac { sin^3 \theta + \cos^3 \theta } { sin \theta + cos \theta }$

Question 324: Simplify:

$\dfrac{ \cos^2 \theta } { 1 - \tan \theta } + \dfrac{ \sin^3 \theta } { \sin \theta - \cos \theta }$

Question 325: Simplify:

$\dfrac{ \sin x - \sin y } { \cos x + \cos y } + \dfrac{ \cos x - \cos y } { \sin x + \ sin y }$

Question 326: Simplify cos⁶θ + sin⁶θ + 3 cos²θ sin²θ

Question 327: Simplify 3(sin⁴θ + cos⁴θ) - 2(sin⁶θ + cos⁶θ)

Question 328: Simplify

$\dfrac{ ( 1 + \cot \theta + \tan \theta) (\sin \theta - \cos \theta) } { \sec^3 \theta - \mathrm{cosec}^3 \theta }$

Question 329: If

$sec\theta + tan\theta = y,$ simplify

$\dfrac{ y^2 -1 } { y^2 + 1 }$ in terms of

$\theta.$

Question 330: If

$sec\theta + tan\theta = p,$ find value of

$sec\theta$ in terms of

$p.$

Question 331: Simplify cos³θ sin³(90°-θ) + sin³θ cos³(90°-θ) + 3 cos²θ sin²θ

Question 332: Simplify

$\dfrac{ \cos^2 10 ^\circ + \cos^2 80 ^\circ } { \sin^2 10 ^\circ + \sin^2 80 ^\circ } + \dfrac{ \sin (90 ^\circ - \theta ) \sin \theta } { \tan \theta } + \dfrac{ \cos (90 ^\circ - \theta ) \cos \theta } { \cot \theta }$

Question 333: Simplify:

$\dfrac{ \cos^2 20 ^\circ + \cos^2 70 ^\circ } { \sin^2 20 ^\circ + \sin^2 70 ^\circ } + \cos (20 ^\circ + \theta ) - \sin (70 ^\circ - \theta )$

Question 334: Simplify:

$\cot^2 \theta \left( \dfrac{ \sec \theta - 1 } { 1 + \sin \theta } \right) + \sec^2 \theta \left( \dfrac{ \sin\theta - 1 } { 1 + \sec \theta } \right)$

Question 335: Simplify:

$\mathrm{cosec}^2 \theta \left( \dfrac{ \cos\theta - 1 } { 1 + \mathrm{cosec} \theta } \right) - \tan^2 \theta \left( \dfrac{ \mathrm{cosec} \theta - 1 } { 1 + \cos \theta } \right)$

Question 336: Simplify

$\sqrt { \dfrac{ 1 + \cos \theta } { 1 - \cos \theta } } + \sqrt { \dfrac{ 1 - \cos \theta } { 1 + \cos \theta } }$

Question 337: Simplify

$\sqrt { \dfrac{ \mathrm{cosec} \theta - 1 } { \mathrm{cosec} \theta + 1 } } + \sqrt { \dfrac{ \mathrm{cosec} \theta + 1 } { \mathrm{cosec} \theta - 1 } }$

Question 338: Solve quadratic equation a²b²x² + b²x - a²x - 1 = 0 using factorization.

Question 339: Solve quadratic equation