Two circles with radius 2R and 2 R \sqrt{2R} 2 R <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"> intersect each other at points A and B. The centers of both the circles are on the same side of AB. O is the center of the bigger circle and ∠AOB is 60°. Find the area of the common region between two circles.

( 13 π 6 + 1 − 3 ) R 2 (\frac{13\pi}{6}+1-\sqrt{3})R^2 ( 6 1 3 π + 1 − 3 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"> ) R 2

( 3 − π − 1 ) R 2 (\sqrt{3}-\pi-1)R^{2} ( 3 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"> − π − 1 ) R 2

( 3 − π ) R 2 (\sqrt{3}-\pi)R^{2} ( 3 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"> − π ) R 2

( 13 π 6 + 3 ) R 2 (\frac{13\pi}{6}+\sqrt{3})R^2 ( 6 1 3 π + 3 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"> ) R 2

XAT Algebra Practice Questions With Solutions

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XAT Quantitative Ability & Data Interpretation Algebra Practice Questions

Verbal and Logical Ability Decision Making General Knowledge

Question 1.

Find the value of
$\frac{\sin^{6}15^{\circ} + \sin^{6}75^{\circ} + 6\sin^{2}15^{\circ}\sin^{2}75^{\circ}}{\sin^{4}15^{\circ} + \sin^{4}75^{\circ} + 5\sin^{2}15^{\circ}\sin^{2}75^{\circ}}$

$\sin 15^{\circ} + \sin 75^{\circ}$

$\frac{6}{5}$

$\sin 15^{\circ} \cos 15^{\circ}$

None of the above

Question 2.

Ramesh and Reena are playing with triangle ABC. Ramesh draws a line that bisects $\angle BAC$ ; this line cuts BC at D. Reena then extends AD to a point P. In response, Ramesh joins B and P. Reena then announces that BD bisects $\angle PBA$ , hat a surprise! Together, Ramesh and Reena find that BD= 6 cm, AC= 9 cm, DC= 5 cm, BP=8 cm, and DP = 5 cm.
How long is AP?

11.5 cm

11.75 cm

10.5 cm

11 cm

10.75 cm

Question 3.

Fatima found that the profit earned by the Bala dosa stall today is a three-digit number. She also noticed that the middle digit is half of the leftmost digit, while the rightmost digit is three times the middle digit. She then randomly interchanged the digits and obtained a different number. This number was more than the original number by 198.
What was the middle digit of the profit amount?

This cannot be solved with only the given information

Question 4.

Rahul has just made a $3 \times 3$ magic square, in which, the sum of the cells along any row, column or diagonal, is the same number N. The entries in the cells are given as expressions in x, y, and Z. Find N?

Question 5.

An encryption system operates as follows:
Step 1. Fix a number k $(k \leq 26)$ .
Step 2. For each word, swap the first k letters from the front with the last k letters from the end in reverse order. If a word contains less than 2k letters, write the entire word in reverse order.
Step 3. Replace each letter by a letter k spaces ahead in the alphabet. If you cross Z in the process to move k steps ahead, start again from A.
Example: k = 2: zebra --> arbez --> ctdgb.
If the word “flight” becomes “znmorl” after encryption, then the value of k:

Cannot be determined uniquely from the given information

Question 6.

Two circles P and Q, each of radius 2 cm, pass through each other’s centres. They intersect at points A and B. A circle R is drawn with diameter AB. What is the area of overlap (in square cm) between the circles R and P?

$\frac{8 \pi}{3} - 2 \sqrt 3$

$\frac{8 \pi}{3}$

$\frac{13 \pi}{3} - \sqrt 3$

$\frac{17 \pi}{6} - 2 \sqrt 3$

$\frac{17 \pi}{6} - \sqrt 3$

Question 7.

An article is marked x% above the cost price. A discount of $\frac{2}{3}$ x% is given on the marked price. If the profit is 4% of the cost price and the value of x lies between 25 and 50, then the value of 50% of x is?

Question 8.

Let ABC be an isosceles triangle. Suppose that the sides AB and AC are equal and let the length of AB be x cm. Let b denote the angle ∠ABC and sin b = 3/5. If the area of the triangle ABC is M square cm, then which of the following is true about M?

$M < \frac{x^2}{4}$

$\frac{3x^2}{4} \leq M < x^2$

$M \geq x^2$

$\frac{x^2}{2} \leq M < \frac{3x^2}{4}$

$\frac{x^2}{4} \leq M < \frac{x^2}{2}$

Question 9.

An antique store has a collection of eight clocks. At a particular moment, the displayed times on seven of the eight clocks were as follows: 1:55 pm, 2:03 pm, 2:11 pm, 2:24 pm, 2:45 pm, 3:19 pm and 4:14 pm. If the displayed times of all eight clocks form a mathematical series, then what was the displayed time on the remaining clock?

1:53 pm

1:58 pm

2:18 pm

3:08 pm

5:08 pm

Question 10.

A cone of radius 4 cm with a slant height of 12 cm was sliced horizontally, resulting into a smaller cone (upper portion) and a frustum (lower portion). If the ratio of the curved surface area of the upper smaller cone and the lower frustum is 1:2, what will be the slant height of the frustum?

$12-\sqrt{3}$

$12-2\sqrt{3}$

$12-3\sqrt{3}$

$12-4\sqrt{3}$

None of the above

Great Job! continue working on more practice questions?

Question 1.

Two circles with radius 2R and $\sqrt{2R}$ intersect each other at points A and B. The centers of both the circles are on the same side of AB. O is the center of the bigger circle and ∠AOB is 60°. Find the area of the common region between two circles.

$(\sqrt{3}-\pi-1)R^{2}$

$(\sqrt{3}-\pi)R^{2}$

$(\frac{13\pi}{6}+1-\sqrt{3})R^2$

$(\frac{13\pi}{6}+\sqrt{3})R^2$

None of the above

Question 2.

If $N = (11^{p + 7})(7^{q - 2})(5^{r + 1})(3^{s})$ is a perfect cube, where $p, q, r$ and $s$ are positive integers, then the smallest value of $p + q + r + s$ is :

Question 3.

a, b, c are integers, |a| ≠ |b| ≠|c| and -10 ≤ a, b, c ≤ 10. What will be the maximum possible value of [abc - (a + b + c)]?

524

693

731

970

None of the above

Question 4.

If a, b and c are 3 consecutive integers between -10 to +10 (both inclusive), how many integer values are possible for the expression?
$\frac{a^3+b^3+c^3+3abc}{(a+b+c)^2}$ =?

Question 5.

The parallel sides of a trapezoid ABCD are in the ratio of 4 : 5. ABCD is divided into an isosceles triangle ABP and a parallelogram PBCD (as shown below). ABCD has a perimeter equal to 1120 meters and PBCD has a perimeter equal to 1000 meters. Find Sin $\angle$ ABC, given 2 $\angle$ DAB = $\angle$ BCD.

4/5

16/25

5/6

24/25

A single solution is not possible

Question 6.

If the last 6 digits of [(M)! - (N)!] are 999000, which of the following option is not possible for (M) × (M - N)? Both (M) and (N) are positive integers and M > N. (M)! is factorial M.

150

180

200

225

234

Question 7.

There are two circles $C_{1}$ and $C_{2}$ of radii 3 and 8 units respectively. The common internal tangent, T, touches the circles at points $P_{1}$ and $P_{2}$ respectively. The line joining the centers of the circles intersects T at X. The distance of X from the center of the smaller circle is 5 units. What is the length of the line segment $P_{1} P_{2}$ ?

≤ 13

> 13 and ≤ 14

> 14 and ≤ 15

> 15 and ≤ 16

> 16

Question 8.

A polynomial y= $ax^{3} + bx^{2 }+ cx + d$ intersects x-axis at 1 and -1, and y-axis at 2. The value of b is:

- 2

Cannot be determined

Question 9.

Two numbers, $297_{B}$ and $792_{B}$ , belong to base B number system. If the first number is a factor of the second number then the value of B is:

Question 10.

Read the following instruction carefully and answer the question that follows:
Expression $\sum_{n=1}^{13}\frac{1}{n}$ can also be written as $\frac{x}{13!}$ What would be the remainder if x is divided by 11?

None of the above

Great Job! continue working on more practice questions?

XAT Algebra Practice Questions With Solutions

XAT Quantitative Ability & Data Interpretation Algebra Practice Questions

Question 1.

Question 2.

Question 3.

Question 4.

Rahul has just made a 3×33 \times 33×3 magic square, in which, the sum of the cells along any row, column or diagonal, is the same number N. The entries in the cells are given as expressions in x, y, and Z. Find N?

Question 5.

Question 6.

Two circles P and Q, each of radius 2 cm, pass through each other’s centres. They intersect at points A and B. A circle R is drawn with diameter AB. What is the area of overlap (in square cm) between the circles R and P?

Question 7.

An article is marked x% above the cost price. A discount of 23\frac{2}{3}32​x% is given on the marked price. If the profit is 4% of the cost price and the value of x lies between 25 and 50, then the value of 50% of x is?

Question 8.

Let ABC be an isosceles triangle. Suppose that the sides AB and AC are equal and let the length of AB be x cm. Let b denote the angle ∠ABC and sin b = 3/5. If the area of the triangle ABC is M square cm, then which of the following is true about M?

Question 9.

Question 10.

A cone of radius 4 cm with a slant height of 12 cm was sliced horizontally, resulting into a smaller cone (upper portion) and a frustum (lower portion). If the ratio of the curved surface area of the upper smaller cone and the lower frustum is 1:2, what will be the slant height of the frustum?

Question 1.

Two circles with radius 2R and 2R\sqrt{2R}2R​ intersect each other at points A and B. The centers of both the circles are on the same side of AB. O is the center of the bigger circle and ∠AOB is 60°. Find the area of the common region between two circles.

Question 2.

If N=(11p+7)(7q−2)(5r+1)(3s)N = (11^{p + 7})(7^{q - 2})(5^{r + 1})(3^{s})N=(11p+7)(7q−2)(5r+1)(3s) is a perfect cube, where p,q,rp, q, rp,q,r and sss are positive integers, then the smallest value of p+q+r+sp + q + r + sp+q+r+s is :

Question 3.

a, b, c are integers, |a| ≠ |b| ≠|c| and -10 ≤ a, b, c ≤ 10. What will be the maximum possible value of [abc - (a + b + c)]?

Question 4.

If a, b and c are 3 consecutive integers between -10 to +10 (both inclusive), how many integer values are possible for the expression?a3+b3+c3+3abc(a+b+c)2\frac{a^3+b^3+c^3+3abc}{(a+b+c)^2}(a+b+c)2a3+b3+c3+3abc​=?

Question 5.

Question 6.

If the last 6 digits of [(M)! - (N)!] are 999000, which of the following option is not possible for (M) × (M - N)? Both (M) and (N) are positive integers and M > N. (M)! is factorial M.

Question 7.

Question 8.

A polynomial y=ax3+bx2+cx+dax^{3} + bx^{2 }+ cx + dax3+bx2+cx+d intersects x-axis at 1 and -1, and y-axis at 2. The value of b is:

Question 9.

Two numbers, 297B297_{B}297B​ and 792B792_{B}792B​ , belong to base B number system. If the first number is a factor of the second number then the value of B is:

Question 10.

Read the following instruction carefully and answer the question that follows:Expression ∑n=1131n\sum_{n=1}^{13}\frac{1}{n}∑n=113​n1​ can also be written as x13!\frac{x}{13!}13!x​ What would be the remainder if x is divided by 11?

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Rahul has just made a $3 \times 3$ magic square, in which, the sum of the cells along any row, column or diagonal, is the same number N. The entries in the cells are given as expressions in x, y, and Z. Find N?

An article is marked x% above the cost price. A discount of $\frac{2}{3}$ x% is given on the marked price. If the profit is 4% of the cost price and the value of x lies between 25 and 50, then the value of 50% of x is?

Two circles with radius 2R and $\sqrt{2R}$ intersect each other at points A and B. The centers of both the circles are on the same side of AB. O is the center of the bigger circle and ∠AOB is 60°. Find the area of the common region between two circles.

If $N = (11^{p + 7})(7^{q - 2})(5^{r + 1})(3^{s})$ is a perfect cube, where $p, q, r$ and $s$ are positive integers, then the smallest value of $p + q + r + s$ is :

If a, b and c are 3 consecutive integers between -10 to +10 (both inclusive), how many integer values are possible for the expression?
$\frac{a^3+b^3+c^3+3abc}{(a+b+c)^2}$ =?

A polynomial y= $ax^{3} + bx^{2 }+ cx + d$ intersects x-axis at 1 and -1, and y-axis at 2. The value of b is:

Two numbers, $297_{B}$ and $792_{B}$ , belong to base B number system. If the first number is a factor of the second number then the value of B is:

Read the following instruction carefully and answer the question that follows:
Expression $\sum_{n=1}^{13}\frac{1}{n}$ can also be written as $\frac{x}{13!}$ What would be the remainder if x is divided by 11?