ISC Maths 2018 Class-12 Solved Previous Year Question Papers

ISC Maths 2018 Class-12 Solved Previous Year Question Paper for practice. Step by step Solutions with section-A, B and C. Visit official website CISCE for detail information about ISC Board Class-12 Maths.

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ISC Maths 2018 Class-12 Previous Year Question Papers Solved

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Section-A

Section-B

Section-C

ISC Maths 2018 Class-12 Previous Year Question Papers Solved

Time Allowed: 3 Hours
Maximum Marks: 100

(Candidates are allowed additional 15 minutes for only reading the paper. They must NOT start writing during this time.)

The Question Paper consists of three sections A, B and C.
Candidates are required to attempt all questions from Section A and all questions either from Section B or Section C.
Section A: Internal choice has been provided in three questions of four marks each and two questions of six marks each.
Section B: Internal choice has been provided in two questions of four marks each.
Section C: Internal choice has been provided in two questions of four marks each.
All working, including rough work, should be done on the same sheet as, and adjacent to the rest of the answer.
The intended marks for questions or parts of questions are given in brackets [ ].
Mathematical tables and graph papers are provided.

Section – A (80 Marks)

ISC Maths 2018 Class-12 Previous Year Question Papers Solved

Question 1. [10 × 2]
(i) The binary operation * : R × R → R is defined as a * b = 2a + b. Find (2 *3) *4.
(ii) If A = $\begin{pmatrix} 5 & a \\ b & 0 \end{pmatrix}$ and A is symmetric matrix, show that a = b.
(iii) Solve : 3tan^-1x + cot^-1x = π
(iv) Without expanding at any stage, find the value of:

(v) Find the value of constant ‘k’ so that the function f(x) defined as:

(vi) Find the approximate change in the volume ‘V’ of a cube of side x metres caused by decreasing the side by 1%.

(viii) Find the differential equation of the family of concentric circles x² + y² = a².
(ix) If A and B are events such that P(A) = $\frac { 1 }{ 2 }$ , P(B) = $\frac { 1 }{ 3 }$ and P(A∩B) = $\frac { 1 }{ 4 }$ , then find: (a) P(A/B) (b) P(B/A)
(x) In a race, the probabilities of A and B winning the race are $\frac { 1 }{ 3 }$ and $\frac { 1 }{ 6 }$ respectively. Find the probability of neither of them winning the race.
Solution 1.
(i) Given binary operation * : R × R → R is defined as:
a * b = 2a + b
(2 * 3) * 4 = [2(2) +3) * 4 = 7 * 4 = 2(7) + 4 = 18

Question 2. [4]
If the function f(x) = √(2x – 3) is invertible then find its inverse. Hence prove that (fof^-1)(x) = x.
Solution 2.

Question 3. [4]
If tan^-1a + tan^-1b + tan^-1c = π, prove that a + b + c = abc.
Solution 3.

Question 4. [4]
Use properties of determinants to solve for x:

Solution 4.

Question 5.
(a) Show that the function $f(x)=\left\{\begin{array}{ll}{x^{2}} & {, \quad x \leq 1} \\ {\frac{1}{x}} & {, \quad x>1}\end{array}\right.$ is continuous at x = 1 but not differentiable.
Or
(b) Verify Rolle’s theorem for the following function:
f(x) = e^-x sin x on [0, π]
Solution 5.

Question 6. [4]

Solution 6.

Question 7.
Evaluate: ∫ tan-1√x dx [4]
Solution 7.

Question 8. [4]
(a) Find the points on the curves = 4x³ – 3x + 5 at which the equation of the tangent is parallel to the x-axis.
Or
(b) Water is dripping out from a conical funnel of semi-vertical angle $\frac { \pi }{ 4 }$ at the uniform rate of 2 cm²/sec in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is 4 cm, find the rate of decrease of the slant height of the water.
Solution 8.

Question 9.
(a) Solve: sin x $\frac { dy }{ dx }$ – y = sin x. tan $\frac { x }{ 2 }$ .
or
(b) The population of a town grows at the rate of 10% per year. Using differential equation, find how long will it take for the population to grow 4 times.
Solution 9.

Hence, the time required is 6.021 years.

Question 10.
(a) Using matrices, solve the following system of equations:
2x – 3y + 5z = 11
3x + 2y – 4z = -5
x + y – 2z = -3
Or
(b) Using elementary transformation, find the inverse of the matrix:
$\left[ \begin{matrix} 0 & 1 & 3 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{matrix} \right]$
Solution 10.
(a) Given system of equations are:
2x – 3y + 5z = 11
3x + 2y – 4z = -5
x + y – 2z = -3
Corresponding matrix equation is:
AX = B

Question 11. [4]
A speaks the truth in 60% of the cases, while B in 40% of the cases. In what per cent of cases are they likely to contradict each other in stating the same fact?
Solution 11.
Let E be the event of A speaking truth and F be the event of B speaking truth.
P(E) = $\frac { 60 }{ 100 }$ = $\frac { 3 }{ 5 }$ , P(F) = $\frac { 40 }{ 100 }$ = $\frac { 2 }{ 5 }$
Probability of A and B likely to contradict each other in stating the same fact

Question 12.
A cone is inscribed in a sphere of radius 12 cm. If the volume of the cone is maximum, find its height. [6]

Solution 12.

Question 13.

Solution 13.

Question 14. [6]
From a lot of 6 items containing 2 defective items, a sample of 4 items are drawn at random. Let the random variable X denote the number of defective items in the sample. If the sample is drawn without replacement, find:
(a) The probability distribution of X
(b) Mean of X
(c) Variance of X
Solution 14.
In a lot of 6 items, 2 items are defective. A sample of 4 items is drawn at random.
Let the random variable X denote the number of defective items in the sample.
X may have value 0, 1, 2

Section – B (20 Marks)

ISC Maths 2018 Class-12 Previous Year Question Papers Solved

Question 15. [3 × 2]
(a) Find λ if the scalar projection of $\vec{a}=\lambda \hat{i}+\hat{j}+4 \hat{k} \text { on } \vec{b}=2 \hat{i}+6 \hat{j}+3 \hat{k}$ is 4 units.
(b) The Cartesian equation of a line is: 2x – 3 = 3y + 1 = 5 – 6z. Find the vector equation of a line passing through (7, -5, 0) and parallel to the given line.
(c) Find the equation of the plane through the intersection of the planes $\vec{r} \cdot(\hat{i}+3 \hat{j}-\hat{k})=9 \text { and } \vec{r} \cdot(2 \hat{i}-\hat{j}+\hat{k})=3$ and passing through the origin.
Solution 15.

Question 16. [4]
(a) If A, B, C are three non-collinear points with position vectors $\vec{a}, \vec{b}, \vec{c}$ respectively, then show that the length of the perpendicular from C on AB is $\frac{|(\vec{a} \times \vec{b})+(\vec{b} \times \vec{c})+(\vec{c} \times \vec{a})|}{|\vec{b}-\vec{a}|}$
Or
(b) Show that the four points A, B, C and D with position vectors $4 \hat{i}+5 \hat{j}+\hat{k},-\hat{j}-\hat{k}, \hat{3} \hat{i}+9 \hat{j}+4 \hat{k} \text { and } 4(-\hat{i}+\hat{j}+\hat{k})$ respectively, are coplanar.
Solution 16.
(a) Let h be the perpendicular distance from point C to the line segment AB.

Question 17. [4]
(a) Draw a rough sketch of the curve and find the area of the region bounded by curve y² = 8x and the line x = 2.
Or
(b) Sketch the graph of y = |x + 4|. Using integration, find the area of the region bounded by the curve y = |x + 4| and x = -6 and x = 0.
Solution 17.
(a) Given curves are:
y² = 8x …(i)
and x = 2 …(ii)
Putting x = 2 in eqn. (i),
we have y² = 16
⇒ y = ±4
when x = 2, y = 4
when x = 2, y = -4
Points of intersections are (2, 4) and (2, -4)

Question 18.
Find the image of a point having position vector: $3 \hat{i}-2 \hat{j}+\hat{k}$ in the Plane $\vec{r} \cdot(3 \hat{i}-\hat{j}+4 \hat{k})=2$ .
Solution 18.
Given point is P(3, -2, 1) and plane is 3x – y + 4z = 2.
D.R’s of normal to the plane are < 3, -1, 4 >
D.R’s of line PQ are <3, -1, 4 >
Equation of line PQ is, where Q is the foot of a perpendicular

Section – C (20 Marks)

ISC Maths 2018 Class-12 Previous Year Question Papers Solved

Question 19. [3 × 2]
(a) Given the total cost function for x units of a commodity as:
C(x) = $\frac { 1 }{ 3 }$ x³ + 3x² – 16x + 2.
Find:
(i) Marginal cost function
(ii) Average cost function
(b) Find the coefficient of correlation from the regression lines: x – 2y + 3 = 0 and 4x – 5y + 1 = 0.
(c) The average cost function associated with producing and marketing x units of an item is given by AC = 2x – 11 + $\frac { 50 }{ x }$ . Find the range of values of the outputx, for which AC is increasing.
Solution 19.
(a) Given the total cost function for x units of a commodity is:
C(x) = $\frac { 1 }{ 3 }$ x³ + 3x² – 16x + 2
(i) C'(x) = x² + 6x – 16
Which is the required marginal cost function
(ii) Average cost function = $\frac { C(x) }{ x }$
$\frac { 1 }{ 3 }$ x + 3x – 16 + $\frac { 2 }{ x }$
(b) Given regression lines are:
x – 2y + 3 = 0 …..(i)
and 4x – 5y + 1 = 0 …..(ii)
From eqn. (i), we have
x = 2y – 3
Reg. of x on y = 2
From eqn. (ii), we have

(c) The average cost function associated with producing and marketing x units of an item is given as:
AC = 2x – 11 + $\frac { 50 }{ x }$
Output’for which AC increases is:
$\frac { d }{ dx }$ (AC) > 0
⇒ $\frac { d }{ dx }$ (2x – 11 + $\frac { 50 }{ x }$ ) > 0
⇒ $2-\frac{50}{x^{2}}>0$
⇒ x² – 25 > 0
⇒ (x – 5)(x + 5) > 0
⇒ x > 5 [∵ x > 0]
Clearly, the average cost increases, if the output x > 5.

Question 20.
(a) Find the line of regression of y on x from the following table. [4]

Hence, estimate the value of y when x = 6.
Or
(b) From the given data:

and correlation coefficient: $\frac { 2 }{ 3 }$ . Find:
(i) Regression coefficients b_yx and b_xy
(ii) Regression line x on y
(iii) Most likely value of x when y = 14
Solution 20.
(a) Given that:

Question 21.
(a) A product can be manufactured at a total cost $C(x)=\frac{x^{2}}{100}+100 x+40$ , where x is the number of units produced. The price at which each unit can be sold is given by P = (200 – $\frac { x }{ 400 }$ )
Determine the production level x at which the profit is maximum. What are the price per unit and total profit at the level of production?
Or
(b) A manufacturer’s marginal cost function is $\frac{500}{\sqrt{2 x+25}}$ . Find the cost involved to increase production from 100 units to 300 units.
Solution 21.

Question 22.
A manufacturing company makes two types of teaching aids A and B of Mathematics for Class X. Each type of A requires 9 labour hours for fabricating and 1 labour hour for finishing. Each type of B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available per week are 180 and 30 respectively. The company makes a profit of ₹ 80 on each piece of type A and ₹ 120 on each piece of type B. How many pieces of type A and type B should be manufactured per week to get a maximum profit? Formulate this as Linear Programming Problem and solve it. Identify the feasible region from the rough sketch. [6]
Solution 22.
Let x and y be the number of teaching aids of type A and type B be produced by the company.
Objective Function (Z) = 80x + 120y
Subject to constraints
9x + 12y ≤ 180
or 3x + 4y ≤ 60,
x + 3y ≤ 30
and x, y ≥ 0
Table of solutions of 3x + 4y = 60

Table of solutions of x + 3y = 30

Plot the points A(0, 15), B(12, 6), C (20, 0), D(0, 10) and E(30, 0) to get the required graph as shown in the figure. Shaded region is the required feasible region and comer points of bounded feasible region are:

O(0, 0), B(12, 6), C(20, 0) and D(0, 10)
At O(0, 0), Z = 0 + 0 = 0
At C(20, 0), Z = 20 × 80 + 0 = 1600
At B(12, 6), Z = 12 × 80 + 120 × 6 = 1680 → Maximum
At D(0, 10), Z = 0 + 120 × 10 = 1200
Hence, maximum profit can be made by manufacturing 12 teaching aids of type A and 6 teaching aids of type B.