Linear growth and decay

Correct answer: A. \(P(n)=72-5n\)

In \(P(n)=72-5n\), the coefficient of n is −5, so increasing n by 1 changes P by −5. \(72-5n^2\) is not linear because it contains n². Exam tip: a linear expression has the variable to power 1.

Correct answer: A. The quantity decreases by 5 units per unit of time.

The coefficient of \(t\) is \(-5\), so when time increases by 1 unit, \(R\) falls by 5 units. Here 90 is the initial amount, not the rate of growth. Exam tip: a negative coefficient indicates decay.

Correct answer: C. 6

The savings model is \(S(m)=400+60m\). Put \(S(m)=760\): \(400+60m=760\). Thus, \(60m=360\), so \(m=6\). Therefore, the amount will be ₹760 after 6 months. After 5 months, the amount would be ₹700, so that option is incorrect. Exam tip: To find the time for a given value in a linear model, equate the model to the target value first.

Correct answer: C. \(h=7\)

Set \(B(h)=50\) because the battery level is required to be 50%. Thus, \(120-10h=50\), so \(10h=70\) and \(h=7\). At \(h=6\), the battery level would be 60%, so that option is not correct. Exam tip: In linear decay questions, equate the model to the required value first and then solve for the variable.

Correct answer: C. \(y=42+5x\)

A linear rule has the form \(y=a+bx\), where \(a\) is the initial value and \(b\) is the change per step. Here, \(a=42\) and \(b=+5\), so the rule is \(y=42+5x\). In option A, \(-5\) represents decay rather than growth. Exam tip: identify the constant term for the initial value and the coefficient of \(x\) for the rate of change.

Correct answer: D. \(y=210-14x\)

A linear rule has the form \(y=a+bx\), where \(a\) is the initial value and \(b\) is the change per step. Here, the initial value is \(210\), and the value decreases by \(14\) each step, so \(b=-14\). Therefore, the rule is \(y=210-14x\). Option A represents an increase, not a decrease. Exam tip: Words such as “decrease” or “decay” indicate a negative coefficient of \(x\).

Correct answer: A. \(y=40-3x\)

In \(y=40-3x\), the coefficient of x is \(-3\), so y falls by 3 for every increase of 1 in x. \(40+3x\) represents growth, not decay. Exam tip: linear decay has a constant negative coefficient of x.

Correct answer: A. \(y=42-5x\)

In \(y=42-5x\), the coefficient of x is the constant \(-5\), so y falls by an equal amount each time x increases. A relation containing \(x^2\) is not linear. Exam tip: for linear decay, look for x to power 1 with a negative coefficient.

Correct answer: C. 30

Given \(A(t)=a+9t\), substitute \(t=3\): \(A(3)=a+9\times3=a+27\). Since \(A(3)=57\), we get \(a+27=57\), so \(a=30\). Option 27 is the amount added over 3 time units, not the initial value \(a\). Exam tip: substitute the specified value of \(t\) first, then solve the resulting equation.

Correct answer: D. 120

Given R(t)=b-11t and R(4)=76. Substituting t=4 gives b-11(4)=76, so b-44=76. Adding 44 to both sides gives b=120. Hence, 120 is the correct option. If b were 110, then R(4)=110-44=66, not 76. Exam tip: In a linear expression, first substitute the given value of the variable and then isolate the unknown term.

Correct answer: B. \(b<0\)

The coefficient \(b\) is the rate of change, or slope, per unit increase in \(x\). If \(b<0\), \(Q(x)\) decreases as \(x\) rises, so it represents decay. In contrast, \(b=0\) represents a constant quantity. Exam tip: use the sign of the slope to identify growth or decay.

Correct answer: A. हर घंटे टंकी में पानी की मात्रा 5 लीटर कम होती है।

In linear decay, a quantity decreases by a fixed amount in equal time intervals. Losing 5 litres every hour follows this pattern. Becoming half each hour is exponential decay. Exam tip: look for a fixed difference, not a fixed ratio.

Correct answer: C. (y=24+7x)

The direct answer is option C, \(y=24+7x\). A linear rule has the form \(y=mx+c\), where \(m\) is the constant rate of change and \(c\) is the value when \(x=0\). Growth means \(m>0\), and the initial value 24 means \(y=24\) at \(x=0\). In C, the intercept is 24 and the slope is +7, so both conditions hold. Option A has initial value 24 but slope −7, so it describes linear decay. Option B has positive growth but starts at 7, not 24. Option D starts at 24 but contains \(x^2\), so its rate is not constant and it is not linear. Memory cue: positive coefficient of \(x\) means growth; the constant term is the initial value.

Correct answer: A. \(y=150-12x\)

A linear rule has the form \(y=a+bx\), where the initial value is \(a\), found by putting \(x=0\). In \(y=150-12x\), the initial value is 150 and the coefficient of \(x\) is \(-12\), so the value decreases linearly. \(y=12-150x\) also represents decay, but its initial value is 12. Exam tip: put \(x=0\) to check the initial value, and look for a negative slope to identify decay.

Correct answer: C. 84

Given \(M(t)=m+8t\). Substituting \(t=2\), \(M(2)=m+16=44\), so \(m=28\). Now, at \(t=7\), \(M(7)=28+8\times7=28+56=84\). Therefore, 84 is the correct answer. The value 76 can result from incorrectly counting the increase. Exam tip: first use the given condition to find the unknown constant \(m\), then substitute the required value of \(t\).

Correct answer: A. 28

Given \(N(t)=n-6t\). At \(t=5\), \(N(5)=n-30=70\), so \(n=100\). Therefore, \(N(12)=100-6\times12=100-72=28\). Hence, 28 is correct. A value such as 34 can result from an error in subtraction. Exam tip: first find the unknown constant \(n\) from the given value, then substitute the required value of \(t\).

Correct answer: B. 6

Substituting the given values into y=45+kx gives 81=45+6k. Hence, 6k=36 and k=6. Option 36 is only the difference 81-45; it must be divided by x=6 to find k. Exam tip: In a linear equation, substitute the given x and y values before solving for the unknown coefficient.

Correct answer: B. 8

At \(x=8\), the value of \(y\) is 136. Substituting in the equation gives \(136=200-8k\). Hence, \(8k=64\), so \(k=8\). Note that 64 is the value of \(8k\), not of \(k\). Exam tip: Substitute the given values first, then isolate the unknown term carefully.

Correct answer: B. The value of B(t) is greater

Substituting t=7 gives A(7)=20+5(7)=55 and B(7)=50+2(7)=64. Since 64 is greater than 55, the value of B(t) is greater. They are not equal because the two values are different. Exam tip: To compare linear expressions at a given value, substitute that value into both expressions first.

Correct answer: C. \(C(6)=D(6)\)

Substituting \(t=6\), \(C(6)=140-8\times6=92\) and \(D(6)=110-3\times6=92\). Hence, the two model values are equal, so \(C(6)=D(6)\). Options A and B are incorrect because neither value is greater than the other, and both values are not zero. Exam tip: evaluate each expression separately at the given value of \(t\) before comparing them.

Correct answer: C. \(t=8\)

Putting \(T(t)=102\) gives \(54+6t=102\). Hence, \(6t=48\), so \(t=8\). Therefore, option C is correct. For example, at \(t=7\), \(T(t)=54+42=96\), not 102. Exam tip: equate the function to the target value first, then isolate and solve for the variable.

Correct answer: C. \(t=7\)

Given \(U(t)=150-15t\), put \(U(t)=45\). This gives \(150-15t=45\), so \(-15t=-105\) and hence \(t=7\). If \(t=9\), the value is \(15\), not 45. Exam tip: In linear decay questions, substitute the target value into the function and solve the resulting equation for \(t\).

Correct answer: C. 24

First, f(x+4)=25+6(x+4)=25+6x+24. Therefore, f(x+4)-f(x)=(25+6x+24)-(25+6x)=24. Hence, 24 is the correct option. 18 may seem tempting, but an increase of 4 units at a rate of 6 gives 6×4=24. Exam tip: For a linear function ax+b, increasing x by h always changes the function by ah.

Correct answer: A. \(P(x)=80-6x\)

Linear decay has the form \(a-bx\), where \(b>0\). In \(P(x)=80-6x\), the value decreases by 6 for every increase of 1 in \(x\). \(80+6x\) shows linear growth. In exams, check the sign of the \(x\)-coefficient.

Correct answer: B. \(y=18+7x\)

In a linear expression \(y=a+bx\), \(a=y(0)\), while \(b\) gives the change in \(y\) for every increase of 1 in \(x\). Here, \(y(0)=18\) and the increase is 7, so \(y=18+7x\). In \(y=7+18x\), the initial value is 7 and the rate of increase is 18, so it is not correct. Exam tip: Substitute \(x=0\) to check the constant term quickly.