Hard Level Worksheet

Very Short Answer Questions (1 Mark Each)

(1) Write an algebraic expression for: "Four more than thrice a number."(Let the variable be x)

Correct! "Thrice a number" = 3x, "four more than" means + 4, so 3x + 4.

(2) If a = 5, b = 2, find the value of 2a – b.

Perfect! 2(5) - 2 = 10 - 2 = 8.

(3) What is the coefficient of x in –7x + 9?

Excellent! The coefficient includes the sign: -7.

(4) Write the constant term in the expression 3m2 + 5m – 12.

Great! The constant term has no variable: -12.

(5) If x = 4, find the value of x2 – 2x.

Correct! x2 - 2x = 42 - 2(4) = 16 - 8 = 8.

Short Answer Questions (2 Marks Each)

Answer each question clearly

(1) Form expressions for:

The product of a number and 6, decreased by 4: (Let the variable be x)

The quotient of a number and 3, increased by 2: (Let the variable be x) x3+23x+2

Excellent! "Product of a number and 6, decreased by 4" = 6x - 4, "Quotient of a number and 3, increased by 2" = x3 + 2.

(2) Simplify: 3p + 5q – 2p + 7 – 4q.

Perfect! Combining like terms: (3p - 2p) + (5q - 4q) + 7 = p + q + 7.

(3) Write an expression for the perimeter of a rectangle whose length = 3x + 2, breadth = 2x – 1.

Perimeter:

Correct! Perimeter = 2((3x + 2) + (2x - 1)) = 2(5x + 1) = 10x + 2.

(4) If a = 2, b = 3, evaluate ab + a2 + b2.

Great! ab + a2 + b2 = (2)(3) + 22 + 32 = 6 + 4 + 9 = 19.

(5) Subtract 4x2−3x+5 from 7x2+2x−8.

Result: + +

Perfect! (7x2 + 2x - 8) - (4x2 - 3x + 5) = 7x2 + 2x - 8 - 4x2 + 3x - 5 = 3x2 + 5x - 13.

Long Answer Questions (4 Marks Each)

(1) A square has side 2x + 3. Write an expression for its area. Evaluate it when x = 5.

Expression: 2x+3242x+32 Value when x = 5:

Correct! Area = 2x+32. When x = 5: 25+32 = 132 = 169.

(2) A rectangle has length x + 5 and breadth x – 2. Write an expression for its area. Find its value when x = 10.

Expression: x+5x−2x+52x−22 Value when x = 10:

Perfect! Area = (x + 5)(x - 2). When x = 10: (10 + 5)(10 - 2) = 15 × 8 = 120.

(3) A man has x pens. His friend has 3 times as many pens plus 4 more. Write an expression for the total number of pens. Evaluate it when x = 6.

Expression: Value when x = 6:

Excellent! Friend has 3x + 4 pens. Total = x + (3x + 4) = 4x + 4. When x = 6: 4(6) + 4 = 28.

(4) A farmer has a rectangular field of length l and breadth b. He increases both length and breadth by 5 units. Write an expression for the new area. Find its value when l = 20, b = 15.

Expression: l+5b+5l−5b−5 Value:

Great! New area = (l + 5)(b + 5). When l = 20, b = 15: (20 + 5)(15 + 5) = 25 × 20 = 500.

(5) The sum of three consecutive even numbers can be expressed algebraically. Form the expression and find their sum. (Let the variable be x)

Expression:

Correct! Let first even number = x. Consecutive evens: x, x+2, x+4. Sum = x + (x+2) + (x+4) = 3x + 6.

Part B: Objective Questions (1 Mark Each)

Choose the correct answer and write the option (a/b/c/d)

(1) The coefficient of x2 in 5x2 – 3x + 2 is:

(a) 5 (b) –3 (c) 2 (d) x

Correct! The coefficient of x2 is 5.

(2) Which of the following is a polynomial in one variable?

(a) 2x + 5 (b) 7 + y – z (c) 3pq (d) 12

Correct! A polynomial in one variable has only one variable. 2x + 5 has only variable x.

(3) If x = 3, the value of x2 + 2x is:

(a) 9 (b) 12 (c) 15 (d) 21

Correct! x2 + 2x = 32 + 2(3) = 9 + 6 = 15.

(4) The perimeter of a triangle with sides a, b, c is:

(a) a + b (b) a + b + c (c) ab + c (d) a × b × c

Correct! Perimeter = sum of all sides = a + b + c.

(5) The constant term in 2x2 – 7x + 11 is:

(a) –7 (b) 2 (c) 11 (d) x2

Correct! The constant term has no variable: 11.

(6) The expression for "five less than the square of a number" is:

(a) x2 + 5 (b) x2 – 5 (c) 5 – x2 (d) 5x2

Correct! "Five less than x2" means x2 - 5.

(7) If y = 2, the value of 3y2 – 4y + 1 is:

(a) 3 (b) 5 (c) 7 (d) 9

Correct! 3y2 - 4y + 1 = 3(2)² - 4(2) + 1 = 12 - 8 + 1 = 5.

(8) Which of these represents the product of two consecutive numbers?

(a) x + (x + 1) (b) x(x + 1) (c) x – (x + 1) (d) x+12

Correct! Product means multiplication: x × (x + 1) = x(x + 1).

(9) The value of 2a2 – 3a + 4 when a = 2 is:

(a) 6 (b) 8 (c) 10 (d) 16

Correct! 2a2 - 3a + 4 = 222 - 3(2) + 4 = 8 - 6 + 4 = 6.

(10) The area of a square of side x is expressed as:

(a) 4x (b) 2x (c) x2 (d) x + x

Correct! Area of square = side × side = x × x = x2.

Expert Algebra Challenge

Determine whether these statements are True or False:

Algebra Quiz

🎉 Congratulations! What You've Mastered:

You have successfully completed the "Expert Introduction to Algebra" worksheet and learned:

(1) Advanced Expression Formation: Creating complex algebraic expressions with multiple operations and variables

(2) Quadratic Expressions: Working with expressions containing x² terms and higher powers

(3) Polynomial Operations: Adding, subtracting, and simplifying multi-term expressions

(4) Complex Evaluation: Substituting values in expressions with squares and products

(5) Product Expressions: Understanding and working with expressions like (x+a)(x+b)

(6) Advanced Geometric Applications: Using algebra for areas involving variable dimensions

(7) Consecutive Number Mastery: Expressing complex number patterns algebraically

(8) Multi-variable Complexity: Handling expressions with multiple variables and operations

(9) Advanced Simplification: Reducing complex polynomial expressions systematically

(10) Foundation for Advanced Algebra: Building skills for factoring, expanding, and equation solving

Exceptional achievement! You have mastered advanced algebraic concepts and are ready for high school algebra including factoring, expanding, and solving equations!