Question 1

A number when multiplied by 4 and then increased by 7 equals 35. Find the number.

Accepted Answer

Set up the equation 4x + 7 = 35, subtract 7 to get 4x = 28, then divide by 4 to get x = 7.

Question 2

The sum of two consecutive numbers is 49. Find the smaller number.

Accepted Answer

Let the smaller number be x, then x + (x + 1) = 49, so 2x + 1 = 49, giving x = 24.

Question 3

If (x - 3) is a factor of the polynomial q(x) = x² - 5x + 6, what is the value of q(3)?

Accepted Answer

By the Factor Theorem, if (x - 3) is a factor, then q(3) = 0; verify: q(3) = 9 - 15 + 6 = 0.

Question 4

If (a + b)² = 81 and ab = 18, find the value of a² + b².

Accepted Answer

Using the identity (a + b)² = a² + 2ab + b², we get 81 = a² + b² + 2(18), so a² + b² = 81 - 36 = 45.

Question 5

Simplify (x + 5)(x - 5) using the algebraic identity (a + b)(a - b) = a² - b².

Accepted Answer

Applying the difference of squares identity with a = x and b = 5 gives (x + 5)(x - 5) = x² - 25.

Question 6

If 5x + 3 = 28, what is the value of x?

Accepted Answer

Subtract 3 from both sides: 5x = 25, then divide by 5 to get x = 5.

Question 7

What is the degree of the polynomial 4x⁵ + 3x³ - 2x + 7?

Accepted Answer

The degree of a polynomial is the highest power of the variable; here the highest power is x⁵, so degree = 5.

Question 8

If a - b = 7 and ab = 12, find the value of (a - b)² - 4ab.

Accepted Answer

Substituting values into (a - b)² - 4ab = 7² - 4(12) = 49 - 48 = 1.

Question 9

If p(x) = 3x² + 5x - 2, find the value of p(2).

Accepted Answer

Substitute x = 2: p(2) = 3(2)² + 5(2) - 2 = 12 + 10 - 2 = 24.

Question 10

The sum of three consecutive integers is 84. What is the smallest of the three integers?

Accepted Answer

Let the three consecutive integers be x, x+1, x+2; then x + (x+1) + (x+2) = 84 gives 3x + 3 = 84, so x = 27.

Question 11

A shopkeeper sells books at ₹40 each. After selling some books, he earns ₹320. How many books did he sell?

Accepted Answer

Using 40x = 320, where x is the number of books, we get x = 320 ÷ 40 = 8 books.

Question 12

If 3x + 7 = 22, what is the value of x?

Accepted Answer

Subtracting 7 from both sides gives 3x = 15, then dividing by 3 gives x = 5.

Question 13

If (a + b)² = 64 and ab = 12, find the value of a² + b².

Accepted Answer

Using (a + b)² = a² + 2ab + b², we get 64 = a² + b² + 24, so a² + b² = 40.

Question 14

Simplify: (x + 5)² − (x − 5)²

Accepted Answer

Using (a + b)² − (a − b)² = 4ab, with a = x and b = 5, we get 4(x)(5) = 20x.

Question 15

If a² + b² = 25 and a + b = 7, find the value of ab.

Accepted Answer

From (a + b)² = a² + 2ab + b², we have 49 = 25 + 2ab, so ab = 12.

Question 16

Expand (2m − 3n)(2m + 3n) using algebraic identities.

Accepted Answer

Using (a − b)(a + b) = a² − b², with a = 2m and b = 3n, we get (2m)² − (3n)² = 4m² − 9n².

Question 17

If x² + 5x + 6 = 0, what are the roots of this quadratic equation?

Accepted Answer

Factoring: (x + 2)(x + 3) = 0 gives x = -2 and x = -3.

Question 18

A number is 5 more than another number. If the product of the two numbers is 24, what is the smaller number?

Accepted Answer

Let the smaller number be x; then x(x + 5) = 24 gives x² + 5x - 24 = 0, which factors as (x + 8)(x - 3) = 0, so x = 3.

Question 19

What is the value of the discriminant for the quadratic equation 2x² - 4x + 2 = 0?

Accepted Answer

Using Δ = b² - 4ac: Δ = (-4)² - 4(2)(2) = 16 - 16 = 0.

Question 20

A rectangular plot has length 3 m more than its width. If the area is 40 m², what is the width of the plot?

Accepted Answer

Let width = x; then x(x + 3) = 40 gives x² + 3x - 40 = 0, which factors as (x + 8)(x - 5) = 0, so x = 5 m.

Agniveer Army CEE Algebra Questions

Algebra Practice Questions — Graded Difficulty