CUET UG 2025 — Mathematics Algebra

When two matrices are equal, each corresponding position has the same value.

$$\begin{bmatrix} 2a + b & a - 2b \\ 5c - d & 4c + 3d \end{bmatrix} = \begin{bmatrix} 4 & -3 \\ 11 & 24 \end{bmatrix}$$[2a+b5c−d​a−2b4c+3d​]=[411​−324​]

Matching each position:

$2a + b = 4$2a+b=4 ... equation (1)

$a - 2b = -3$a−2b=−3 ... equation (2)

$5c - d = 11$5c−d=11 ... equation (3)

$4c + 3d = 24$4c+3d=24 ... equation (4)

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From equation (2):

$$a - 2b = -3$$a−2b=−3

$a = 2b - 3$a=2b−3 ... (5)

Substituting into equation (1):

$$2a + b = 4$$2a+b=4

$$2(2b - 3) + b = 4$$2(2b−3)+b=4

$$4b - 6 + b = 4$$4b−6+b=4

$$5b = 10$$5b=10

$$b = 2$$b=2

Using equation (5):

$$a = 2(2) - 3$$a=2(2)−3

$$a = 4 - 3$$a=4−3

$$a = 1$$a=1

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From equation (3):

$$5c - d = 11$$5c−d=11

$d = 5c - 11$d=5c−11 ... (6)

Substituting into equation (4):

$$4c + 3d = 24$$4c+3d=24

$$4c + 3(5c - 11) = 24$$4c+3(5c−11)=24

$$4c + 15c - 33 = 24$$4c+15c−33=24

$$19c = 57$$19c=57

$$c = 3$$c=3

Using equation (6):

$$d = 5(3) - 11$$d=5(3)−11

$$d = 15 - 11$$d=15−11

$$d = 4$$d=4

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The expression $a + 2b - 3c + 4d$a+2b−3c+4d becomes:

$$a + 2b - 3c + 4d = 1 + 2(2) - 3(3) + 4(4)$$a+2b−3c+4d=1+2(2)−3(3)+4(4)

$$= 1 + 4 - 9 + 16$$=1+4−9+16

$$= 12$$=12

Therefore, the value of $a + 2b - 3c + 4d = 12$a+2b−3c+4d=12