ISC Maths 2019 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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Previous Year Question Paper ISC Maths 2019 Class-12 Solved

Section-A

Section-B

Section-C

ISCMaths 2019 Class-12 Solved Previous Year Question Paper

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 2019 Class-12 Solved Previous Year Question Paper )

Question 1. [10 × 2]

(i) If f: R → R, f(x) = x3 and g: R → R, g (x) = 2x2 + 1, and R is the set of real numbers, then find fog(x) and gof(x).
(ii) Solve: sin (2 tan-1x) = 1.
(iii) Using determinants, find the values of k, if the area of triangle with vertices (-2, 0), (0, 4) and (0, k) is 4 square units.
(iv) Show that (A + A’) is symmetric matnx. if A = $\begin{pmatrix} 2 & 4 \\ 3 & 5 \end{pmatrix}$
(v) f(x) = $\frac { { x }^{ 2 }-9 }{ x-3 }$ is not defined at x = 3. What value should be assigned to f(3) for continuity of f(x) at x = 3?
(vi) Prove that the function f(x) = x3 – 6x2 + 12x + 5 is increasing on R.
(vii) Evaluate:

(viii) UsingL Hospital’s Rule, evaluate:

(ix) Two balls are drawn from an urn containing 3 white, 5 red and 2 black balls, one by one without replacement. What is the probability that at least one ball is red?
(x) If events A and B are independent, such that P( A) = $\frac { 3 }{ 5 }$, P (B) = $\frac { 2 }{ 3 }$, find P(A∪B).
Solution:

Question 2.

If f: A → A and A = R – {$\frac { 8 }{ 5 }$}, show that the function f(x) = $\frac { 8x+3 }{ 5x-8 }$ is one-one onto. Hence, find f-1.
Solution:
Given function is:

(a) Solve for x:

Solution:

Question 4.

Use properties of determinants prove that:

Solution:

Question 5. [4]

(a) Show that the function f(x) = |x – 4|, x ∈ R is continuous, but not differentiable at x = 4.
Or
(b) Verify the Lagrange’s mean value theorem for the function:
f(x) = x + $\frac { 1 }{ x }$ in the interval [1, 3] Solution:

Thus, f(x) is continuous at x = 4.
For differentiability at x = 4.
When x < 4, f(x) = -(x – 4), which being polynomial function is differentiable for all x < 4.
When x > 4, f(x) = x – 4, which being polynomial function is differentiable for all x > 4.
When x = 4

Hence, f(x) is not differentiable at x = 4.
Or
(b) Let f(x) = x + $\frac { 1 }{ x }$ in the inverval [1, 3] Since, f(x) is a polynomial function, therefore, it is continuous and derivable in (1, 3).
⇒ f satisfies conditions of Mean Value theorem in [1, 3],
Thus, there exists atleast one real c ∈ (1, 3) such that

Hence, Mean Value theorem for the given function is verified in the given interval

Solution:

Question 7.

A 13 m long ladder is leaning against a wall, touching the wall at a certain height from the ground level. The bottom of the ladder is pulled away from the wall, along the ground, at the rate of 2 m/s. How fast is the height on the wall decreasing when the foot of the ladder is 5 m away from the wall?
Solution:
Let at any instant of time t, the height of the top of the ladder be y and its foot be at distance x from the wall, then

Solution:

Question 9.

Solve the differential equation:

Solution:
Given the differential equation is

Question 10. [4]
Bag A contains 4 white balls and 3 black balls, while Bag B contains 3 white balls and 5 black balls. Two balls are drawn from Bag A and placed in Bag B. Then, what is the probability of drawing a white ball from Bag B?
Solution:
Here, we have three cases
Case 1: Two balls drawn from Bag A are White.
Case 2: Two balls drawn from Bag A are Black.
Case 3: Two balls drawn from Bag A are one White and other Black.

Question 11. [6]

Solve the following system of linear equations using matrix method:

Solution:

Question 12. [6]

(a) The volume of a closed rectangular metal box with a square base is 4096 cm3. The cost of polishing the outer surface of the box is ₹ 4 per cm2. Find the dimensions of the box for the minimum cost of polishing it.
Or
(b) Find the point on the straight line 2x + 3y = 6, which is closest to the origin.
Solution:

(a) Let x be the side of the square base andy be its height.

Question 13. [6]

Solution:

Question 14.

(a) Given three identical Boxes A, B and C, Box A contains 2 gold and 1 silver coin, Box B contains 1 gold and 2 silver coins and Box C contains 3 silver coins. A person chooses a Box at random and takes out a coin. If the coin drawn is of silver, find the probability that it has been drawn from the Box which has the remaining two coins also of silver.
Or
(b) Determine the binomial distribution where mean is 9 and standard deviation is $\frac { 3 }{ 2 }$. Also, find the probability of obtaining at most one success.
Solution:
(a) Let E1, E2, E3 be the three events of the selection of three bags A, B and C respectively.
Let S be the event that the coin drawn is of silver, we have

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