🏫 CBSE FINAL PAPER
All India
Session Ending Examination 2022-23
📘 Class XI · Mathematics (041)
⏱️ 3 Hours · Max Marks: 80 · 5 Sections (A-E)
📋
This question paper contains five sections A, B, C, D and E. Each section is compulsory. Internal choices in some questions.
Section A: 18 MCQs + 2 Assertion-Reason (1 mark each)
Section B: 5 VSA (2 marks each)
Section C: 6 SA (3 marks each)
Section D: 4 LA (5 marks each)
Section E: 3 Case-based (4 marks each)
📝 SECTION A
20 × 1 = 20 marks
Sets
The set of all even natural numbers divisible by 5 is:
(A) Empty set
(B) Infinite set
(C) Singleton set
(D) Not a set
🔍 VIEW ANSWER
✅ ANSWER: B (Infinite set)
📐 EXPLANATION: Even natural numbers divisible by 5 are multiples of 10: {10, 20, 30, …}. This set has infinitely many elements.
Let \(A\) and \(B\) be two sets in the same universal set. Then \(A – B =\)
(A) \(A \cap B\)
(B) \(A’ \cap B\)
(C) \(A \cap B’\)
(D) None of these
🔍 VIEW ANSWER
✅ ANSWER: C (\(A \cap B’\))
📐 EXPLANATION: \(A – B\) represents elements in A but not in B, which is intersection of A with complement of B.
If a set contains 6 elements, then total number of proper subsets are:
(A) 65
(B) 64
(C) 63
(D) 32
🔍 VIEW ANSWER
✅ ANSWER: C (63)
📐 EXPLANATION: Total subsets = \(2^6 = 64\). Proper subsets = total – 1 (excluding the set itself) = 63.
If \((x + 1,1) = (3, y – 2)\) then \(x^{2} + y^{2}\) is:
(A) 5
(B) 13
(C) 9
(D) 4
🔍 VIEW ANSWER
✅ ANSWER: B (13)
📐 EXPLANATION: Equate coordinates: \(x+1=3\) \(\Rightarrow x=2; 1=y-2\) \(\Rightarrow y=3\). Then \(x^2+y^2=4+9=13\).
The range of the Signum function is:
(A) R
(B) N
(C) \(R^{+}\)
(D) \(\{-1,0,1\}\)
🔍 VIEW ANSWER
✅ ANSWER: D ({-1,0,1})
Signum function returns -1 for negative, 0 for zero, 1 for positive.
Length of arc of circle radius 5 cm subtending central angle 15°:
(A) \(\frac{\pi}{12}\) cm
(B) \(\frac{5\pi}{12}\) cm
(C) \(\frac{3\pi}{12}\) cm
(D) None
🔍 VIEW ANSWER
✅ ANSWER: B (\(\frac{5\pi}{12}\) cm)
Arc length = \(\frac{\theta}{360°} \times 2\pi r \)\(= \frac{15}{360}\times 2\pi \times5 \)\(= \frac{1}{24}\times10\pi \)\(= \frac{5\pi}{12}\).
If \(x\) in 3rd quadrant, \(\cos x = -\frac{12}{13}\), then \(\tan x\) =
(A) \(\frac{2}{13}\)
(B) \(\frac{5}{12}\)
(C) \(\frac{3}{4}\)
(D) -1
🔍 VIEW ANSWER
✅ ANSWER: B (\(\frac{5}{12}\))
\(\sin^2 x = 1 – \frac{144}{169}\)\( = \frac{25}{169}\) ⇒ \(\sin x = -\frac{5}{13}\) (3rd quadrant negative). Then \(\tan x = \frac{\sin x}{\cos x}\)\(= \frac{-5/13}{-12/13}\)\(= \frac{5}{12}\).
Standard form of \((5-3i)^3\) is:
(A) 10+198i
(B) 10-198i
(C) -10-198i
(D) None
🔍 VIEW ANSWER
✅ ANSWER: C (-10-198i)
\((5-3i)^3 \)\(= 125 – 3\cdot25\cdot3i + 3\cdot5\cdot9i^2 – 27i^3\)\(= 125 -225i -135 +27i\)\(= -10 -198i\).
Solution set of \(\frac{1}{x-2}<0\) is:
(A) \((-\infty,2)\)
(B) \((-\infty,-2)\)
(C) \((-\infty,5)\)
(D) None
🔍 VIEW ANSWER
✅ ANSWER: A (\((-\infty,2)\))
Fraction negative \(⇒ denominator negative\)\( ⇒ x-2<0\) \(⇒x<2\).
A gentleman has 6 friends to invite. In how many ways can he send invitation cards to them, if he has three servants to carry the cards?
(A)18
(B)216
(C)729
(D)1
🔍 VIEW ANSWER
✅ ANSWER: C (729)
Each card can be given to any of 3 servants ⇒ \(3^6 = 729\).
📝 SECTION A (Q11 – Q20)
10 × 1 = 10 marks
🧮 Combinations
From a class of 32 students, 4 are to be chosen for a competition. In how many ways can this be done?
(A) 30560
(B) 35960
(C) 39560
(D) 35690
🔍 VIEW ANSWER
✅ ANSWER: B (35960)
📐 EXPLANATION: Number of ways = \(C(32,4) = \frac{32 \times 31 \times 30 \times 29}{4 \times 3 \times 2 \times 1} \)\(= \frac{863040}{24}\)\( = 35960\).
🔢 Binomial Theorem
In the expansion of \((1 + x)^{20}\), total number of terms are:
(A) 20
(B) 0
(C) 21
(D) 19
🔍 VIEW ANSWER
✅ ANSWER: C (21)
📐 EXPLANATION: In binomial expansion of \((1+x)^n\), number of terms = \(n+1\). Here n=20 ⇒ 21 terms.
📈 AP
Sum of 20 terms of the A.P.: 1,4,7,… is:
(A) 650
(B) 590
(C) 200
(D) 59
🔍 VIEW ANSWER
✅ ANSWER: B (590)
📐 EXPLANATION: First term a=1, common difference d=3. Sum \(S_n = \frac{n}{2}[2a + (n-1)d]\)\( = \frac{20}{2}[2(1) + 19\times3]\)\( = 10[2 + 57]\)\( = 10 \times 59\)\( = 590\).
📍 3D Geometry
The ratio in which the line joining (2,4,5) and (3,5,-9) is divided by the yz-plane is:
(A) 2:3
(B) 3:2
(C) -2:3
(D) 4:-3
🔍 VIEW ANSWER
✅ ANSWER: C (-2:3)
📐 EXPLANATION: In yz-plane, x-coordinate = 0. Using section formula: \(0 = \frac{2 + 3k}{1+k}\) \(⇒ 2+3k=0\)\( ⇒ k = -2/3 \)\(⇒ ratio -2:3 (externally).\)
🌐 Octants
The point \((-5,3,2)\) lies in the:
(A) 2nd octant
(B) 7th octant
(C) X-Axis
(D) 5th octant
🔍 VIEW ANSWER
✅ ANSWER: A (2nd octant)
📐 EXPLANATION: Signs: x negative, y positive, z positive → 2nd octant (as per standard octant numbering).
Which of the following is not a measure of dispersion?
(A) Range
(B) Quartile deviation
(C) Standard Deviation
(D) Median
🔍 VIEW ANSWER
✅ ANSWER: D (Median)
📐 EXPLANATION: Median is a measure of central tendency, not dispersion. Range, quartile deviation, and standard deviation measure spread.
If three coins are tossed together, probability of getting exactly 3 heads is:
(A) \(\frac{3}{8}\)
(B) \(\frac{1}{8}\)
(C) \(\frac{7}{8}\)
(D) \(\frac{1}{2}\)
🔍 VIEW ANSWER
✅ ANSWER: B (\(\frac{1}{8}\))
📐 EXPLANATION: Total outcomes = \(2^3 = 8\). Favorable (HHH) = 1. Probability = \(\frac{1}{8}\).
If A and B are mutually exclusive events then \(P(A \cap B)\) is:
(A) 1
(B) \(\frac{1}{2}\)
(C) 0
(D) Not defined
🔍 VIEW ANSWER
✅ ANSWER: C (0)
📐 EXPLANATION: Mutually exclusive events cannot occur simultaneously, so intersection is empty, probability = 0.
⚖️ Assertion-Reason
Assertion (A): The domain of the function \(f:A \rightarrow B\) is the set A.
Reason (R): The modulus function is defined for all real values. So the domain is the set of real numbers.
Choices:
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
🔍 VIEW ANSWER
✅ ANSWER: (a) Both A and R are true and R is the correct explanation of A.
📐 EXPLANATION: Assertion is true by definition of function domain. Reason is also true (modulus domain is all reals) and it correctly illustrates that domain can be set of real numbers, supporting the concept that domain is the set of inputs.
⚖️ Assertion-Reason
Assertion (A): The function which satisfies the condition \(f(-x) = f(x)\), is known as even function.
Reason (R): \(f(x) = \sin x\) is an even function.
Choices:
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
🔍 VIEW ANSWER
✅ ANSWER: (c) A is true but R is false.
📐 EXPLANATION: Assertion is correct definition of even function. Reason is false because \(\sin(-x) = -\sin x\), so sin x is odd, not even.
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🏆 CBSE FINAL PAPER · Class 11 Mathematics 2022-23
✅ Complete Section A (Q1-Q20) solutions · Answer key embedded · All steps explained