Ctet Algebra Questions with Detailed Solutions

Given:
​$$P = 5x^2 - 3x + 2\\Q = -3x^2 + 4x - 7\\R = x^2 - x + 5\\P - Q + R = ax^2 + bx + c$$​​
Formula Used:
Addition and subtraction of algebraic expressions.
Solution:
Substitute the expressions into the given equation:
​$$(5x^2 - 3x + 2) - (-3x^2 + 4x - 7) + (x^2 - x + 5)$$​​
Distribute the negative sign:
​$$5x^2 - 3x + 2 + 3x^2 - 4x + 7 + x^2 - x + 5$$​​
Group the like terms together:
​$$(5x^2 + 3x^2 + x^2) + (-3x - 4x - x) + (2 + 7 + 5)\\9x^2 - 8x + 14$$​​
By comparing with $$ax^2 + bx + c,$$​ we get:
a = 9
b = -8
c = 14
Calculate the value of a + b - c:
9 + (-8) - 14 = 1 - 14 = -13
Final Answer
So the correct answer is (a)

Given:
Algebraic expression:
​$$6a^2 - 11ab + 4b^2 + 10ac - 5bc$$​​
Formula Used:
Factorization by grouping.
Solution:
Split the middle term of the quadratic part:
-11ab = -8ab - 3ab
Factorize the quadratic expression:
​$$6a^2 - 8ab - 3ab + 4b^2 = 2a(3a - 4b) - b(3a - 4b) = (2a - b)(3a - 4b)$$​​
Factorize the remaining part of the expression:
10ac - 5bc = 5c(2a - b)
Combine the two parts:
(2a - b)(3a - 4b) + 5c(2a - b)
Factor out the common term:
(2a - b)(3a - 4b + 5c)
The factors are:
(2a - b)
and
(3a - 4b + 5c)
No errors found in Question and Solution
Final Answer
So the correct answer is (a)

Given:
Three consecutive integers are multiplied by 2, 4, and 3 respectively.
Their sum is 82.
Formula Used:
Consecutive integers can be represented as x, x+1, x+2.
Solution:
Let the three consecutive integers be x, x+1, and x+2.
According to the given condition:
2x + 4(x+1) + 3(x+2) = 82
2x + 4x + 4 + 3x + 6 = 82
9x + 10 = 82
9x = 72
x = 8
The integers are 8, 9, and 10.
The sum of the first and third integers = 8 + 10 = 18.
Final Answer
So the correct answer is (b)

Given:
Equation: $$\frac{x - 4}{7} - 1 = \frac{5 - x}{3} + x$$​​
The solution to the equation is x = k.

Formula Used:
Algebraic simplification and solving linear equations.

Solution:
Simplify the left side of the equation:
​$$\frac{x - 4 - 7}{7} = \frac{x - 11}{7}$$​​
Simplify the right side of the equation:
​$$\frac{5 - x + 3x}{3} = \frac{5 + 2x}{3}$$​​
Equate both sides:
​$$\frac{x - 11}{7} = \frac{5 + 2x}{3}$$​​
Cross-multiply to remove fractions:
​$$3(x - 11) = 7(5 + 2x)\\3x - 33 = 35 + 14x\\3x - 14x = 35 + 33\\-11x = 68\\x = \frac{-68}{11}$$​​
Since $$x = k, k = \frac{-68}{11}$$​. This means  $$11k = -68$$​.
Substitute 11k into the given expression $$\frac{11k + 18}{11k - 12}:$$​​
​$$\frac{-68 + 18}{-68 - 12}= \frac{-50}{-80} = \frac{5}{8}$$​​

Final Answer
So the correct answer is (c)

Given:

Find the coefficient of ab in

(4a − 5b)(3a + 2b) − (2a − 3b)(5a − 4b)

Formula Used:
Expand each expression and combine like terms.

Solution:

First expression:

(4a − 5b)(3a + 2b)

= 12a² + 8ab − 15ab − 10b²

= 12a² − 7ab − 10b²

Second expression:

(2a − 3b)(5a − 4b)

= 10a² − 8ab − 15ab + 12b²

= 10a² − 23ab + 12b²

Now subtract the second expression from the first:

(12a² − 7ab − 10b²) − (10a² − 23ab + 12b²)

= 12a² − 7ab − 10b² − 10a² + 23ab − 12b²

= 2a² + 16ab − 22b²

Therefore, the coefficient of ab is 16.

The correct option is (B) 16.

Given:
​$$x - \frac{1}{x} = 4$$​​
Formula Used:
​$$(a - b)^2 = a^2 - 2ab + b^2$$​​
Solution:
Square both sides of the given equation:
​$$(x - \frac{1}{x})^2 = 4^2$$​​
Expand the left side using the algebraic identity:
​$$x^2 - 2(x)(\frac{1}{x}) + \frac{1}{x^2} = 16$$​​
Simplify the middle term:
​$$x^2 - 2 + \frac{1}{x^2} = 16$$​​
Add 2 to both sides to isolate the desired expression:
​$$x^2 + \frac{1}{x^2} = 16 + 2\\x^2 + \frac{1}{x^2} = 18$$​​
Final Answer
So the correct answer is (c)

Given:
Linear equation: 3(x - 2) = 2(x + 4)

Formula Used:
Distributive property: a(b + c) = ab + ac

Solution:
Expand both sides of the equation:
3x - 6 = 2x + 8
3x - 2x = 8 + 6
x = 14

Final Answer
So the correct answer is (c)

Given:
Quadratic equation: $$x^2 - 11x + 28 = (x - p)(x - q)$$​​
p and q are positive integers.

Formula Used:
​$$(x - p)(x - q) = x^2 - (p + q)x + pq$$​​
Solution:

By expanding the right side and comparing the coefficients with the original polynomial:

​$$x^2 - 11x + 28 = (x - p)(x - q)\\ \ \\ = x^2 - 11x + 28 = x^2 - (p + q)x + pq$$​​
The sum of the roots is p + q = 11.
The product of the roots is pq = 28.
We need to find the value of $$(p + q)^2.$$​​
Substitute the value of the sum:
​$$(p + q)^2 = (11)^2 = 121$$​​
Final Answer
So the correct answer is (c)

Given:
Expression =$$3x(4x - 5) + 3$$​​
Value of $$x = 3$$​​

Formula Used:
Algebraic substitution.

Solution:
Substitute x = 3 into the expression:
3(3)(4(3) - 5) + 3
= 9(12 - 5) + 3
= 9(7) + 3
= 63 + 3

= 66
Final Answer
So the correct answer is (b)

Given:
Equation: $$\frac{3-4x}{5+2x} = -\frac{2}{3}$$​​

Formula Used:
Cross-multiplication

Solution:

​$$\frac{3-4x}{5+2x} = -\frac{2}{3}$$​​

​$$3(3 - 4x) = -2(5 + 2x)\\9 - 12x = -10 - 4x\\19 = 8x \implies x = \frac{19}{8}$$​​
The value of $$\frac{3x+2}{x-1} =$$

$$\frac{3(\frac{19}{8})+2}{\frac{19}{8}-1}\\ \ \\= \frac{\frac{57}{8}+2}{\frac{11}{8}}\\ \ \\ =\frac{\frac{73}{8}}{\frac{11}{8}}\\ \ \\ = \frac{73}{11}$$​​​
Final Answer
So the correct answer is (b)

Given:
The equation to solve is:
​$$\frac{3x - 2}{4} - \frac{2x + 3}{3} = \frac{1}{6}$$​​
Formula Used:
Algebraic simplification by finding a common denominator.
Solution:
Find the least common multiple of the denominators 4, 3, and 6, which is 12.
Multiply the entire equation by 12 to eliminate fractions:
​$$12 \times \frac{3x - 2}{4} - 12 \times \frac{2x + 3}{3} = 12 \times \frac{1}{6}\\ \ \\3(3x - 2) - 4(2x + 3) = 2$$​​
Expand the brackets:
9x - 6 - 8x - 12 = 2
Combine the like terms:
x - 18 = 2
x = 20
The solution is m = 20.
Calculate the value of 12m:
​$$12 \times 20 = 240$$​​
Final Answer
So the correct answer is (a)

Given:
Charge for 1st km = 50
Charge for additional km = 15
Total distance = 8 km
Formula Used:
Total Fare = First km charge + (Total distance - 1) $$\times$$​ Additional km charge
Solution:
Distance remaining after 1st km = 8 - 1 = 7
Charge for remaining 7 km = $$7 \times 15 = 105$$​​
Total Fare = 50 + 105 = 155
Final Answer
So the correct answer is (a)

Given:
A number is doubled and 4 is added.
The result is 20.
Formula Used:
Linear equation formulation: 2x + 4 = 20
Solution:
Let the original number be x.
According to the condition:
2x + 4 = 20
2x = 16
x = 8
Final Answer
So the correct answer is (a)

Given:
A number increased by 36 equals five times the number.

Formula Used:
Linear equation in one variable.

Solution:
Let the unknown number be x.
According to the question:
x + 36 = 5x
Transposing x to the right side:
36 = 5x - x
36 = 4x
​$$x = \frac{36}{4}$$​​
x = 9

Final Answer
So the correct answer is (b)