Thursday, 21 March 2019

Mathematics 2nd Year Notes Punjab Board Pakistan

12th Class MCQs Chapter Wise Mathematics all punjab board of Pakistan. Many of students of Punajab Board have needed for the solutions of class 12th Mathematics Notes. Here i am sharing important solved Mathematics XII class all Punjab Text Book Board, Lahore Notes. These Mathematics XII FSc Part 2 2nd year important Notes are according to “Punjab Text Book Board, Lahore. All of these important Multiple Choice questions with solved answers of Math Book for Punjab Board Class 12th are helpful for the Board BISE final year exams.

Board of Intermediate and Secondary Education BISE PUNJAB BOARD
12th Class Mathematics Lectures MCQs Notes

Full book solution of 2nd year math book from punjab board pdf are also available here. Solved Mathematics notes for class 12 sindh textbook board pdf, Punjab textbook board, KPK textbook board and Balaochistan Board are also available here for the students. Chemistry Notes,  F.Sc part 2 math notes chapter 1 to 7 are with answers. These notes are also helpful for the teaching jobs, Educators and PPSC jobs tests of Mathematics. English Notes,

12th Class Math MCQs Chapter 1 Functions & Limits Punjab Board


Question 1: Limx→0 tan x⁰/x = ?
  1. 1
  2. 180 π
  3. 180/π
  4. π/180
D
Question 2: x = at², y = 2at represents ------ ?
  1. Hyperbola
  2. Parabola
  3. Ellipse
  4. Circle
B
Question 3: If f(x) + f(- x)/2 = 0, then f(x) is ----- ?
  1. Even
  2. Explicit
  3. Odd
  4. Periodic
C
Question 4: The range of function f(x) = |x|/x is ?
  1. R
  2. [–1, 1]
  3. R – {0}
  4. - 1 and 1
D
Question 5: If f(x) = ex, then f⁻¹(x) = ?
  1. lnx
  2. cosx
  3. ex
  4. Sinx
A
Question 6: Explicit form of y – x² - 2x + 1 = 0 is -------- ?
  1. None
  2. 2x + x² = y + 1
  3. y = x² + 2x – 1
  4. y – x² = 2x – 1
C
Question 7: If f(x) = sinx and g(x) = 1/tan x, then gof(x) = ?
  1. sin(cotx)
  2. cos(tanx)
  3. Cot (sinx)
  4. sin(tanx)
C
Question 8: Equation x = 9 Cost and y = 9Sint represents the equations of ?
  1. Parabola
  2. Hyperbola
  3. Circle
  4. Line
C
Question 9: x² + y² = 16 is ?
  1. Function
  2. Ellipse
  3. Line
  4. Not function
D
Question 10: Limx→∞ 2 – 3x/√3 + 4x² = ------- ?
  1. 1
  2. -3/2
  3. 3/2
B
Question 11: The function f(x) = ax + b is an identity function if ?
  1. a = 1, b = 1
  2. a = 0, b = 1
  3. a = 0, b = 0
  4. a = 1, b = 0
D
Question 12: x →Lim 0 2³x - 1/x = ------ ?
  1. ln3
  2. ln8
  3. ln27
B
Question 13: The value of K for which the function f(x) = {sinx/x k x ≠ 0 x = 0 is continuous is ?
  1. - 1
  2. 1
  3. 0
  4. - 2
B
Question 14: Limx→0 (1 +2x)1/x = ?
  1. e⁻¹
  2. e
B
Question 15: The linear function f(x) = ax + b is constant function if ?
  1. a ≠ 0, b = 1
  2. a = 0 = b
  3. a = 1 = b
  4. a = 1, b = 0
B
Question 16: f(x) = sin x + cos x is -------- function ?
  1. Even
  2. Odd
  3. None of these
  4. Neither even nor odd
D
Question 17: If a function f(x) is not continuous at x = c, then it is is called ------- at c ?
  1. Explicit
  2. Linear
  3. None
  4. Discontinous
D
Question 18: Cosh⁻¹ x = ?
  1. In(x + √x² + 1)
  2. 1/2 ln(x +1/x - 1)
  3. 1/2 ln(1 + x/1 - x)
  4. ln(x + √x² - 1)
D
Question 19: θ→Lim0 1-cos9θ/1-cos3θ = -------- ?
  1. 12
  2. 0
  3. 9
  4. 3
C
Question 20: θ→Lim0 sin 7θ/θ = -------- ?
  1. 1/7
  2. 7
  3. None
  4. 1
B

12th Class Math MCQs Chapter 2 Differentiation Punjab Board


Question 1: If f(x) = cos x, then slope of normal at x = 0 is--------- ?
  1. 1
  2. undefined
  3. -1
  4. Zero
B
Question 2: Limx→a f(x) – f(a)/x – a = -------- ?
  1. integral of f(x)
  2. f(a)
  3. f’(a)
  4. f’(x)
C
Question 3: 1/x d/dx (sinx²) = ?
  1. cosx²
  2. 2cosx²
  3. 2xcosx²
  4. 2x cos²x
C
Question 4: d/dx (tan⁻¹x – Cot⁻¹x)= ?
  1. x/1 + x²
  2. 2/1 + x²
  3. 1/1 + x²
  4. 0
B
Question 5: If y = sin x then ?
  1. y₄ = y₂
  2. y₄ = y₁
  3. y₄ ≠ y
  4. y₄ = y
D
Question 6: d/dx(axm + bxn) = ?
  1. (m - 1)xm-2 + (n – 1)xⁿ⁻²
  2. maxm-1 + nbxⁿ⁻¹
  3. a(m – 1) xm-2 + b (n – 1)xⁿ⁻²
  4. mxm-1 + nxⁿ⁻¹
B
Question 7: Notation used by Cauchy for the derivative of f(x) is ?
  1. df/dx
  2. f’(x)
  3. f”(x)
  4. Df(x)
D
Question 8: d/dx sin x - d²/dx² (cos x) = -------- ?
  1. 2 sin x
  2. 2 cos x
  3. 0
  4. - 2 sin x
B
Question 9: If f(x) = 3x² + 1 and f(2) = f(x) – f(2)/x – 2, then f’(2) = ?
  1. 6x
  2. 12
  3. 0
  4. 6
B
Question 10: f(x) = 1/x – 1 ⇒ f’(2) = ?
  1. 1
  2. 0
  3. -2
  4. -1
D
Question 11: d/dx[cot(sec⁻¹ √cot² x + 1/cot² x)] = ?
  1. -cotx
  2. cotx
  3. sec²x
  4. - cosec²x
B
Question 12: If f(x) = 3 + x, then ?
  1. f’(0) < f’(1)
  2. f’(0) ≠ f’(1)
  3. f’(0) > f’(1)
  4. f’(0) = f’(1)
D
Question 13: The function f(x) = ax² + bx + c has maximum value if ?
  1. a > 0
  2. a > 0
  3. a > 1
  4. a > 2
B
Question 14: For a function f(x), if f’ (c) = - & f’’(c) > 0 then f(x) has, at x = c ?
  1. Stationary point
  2. Relative minima
  3. Point of inflection
  4. Relative maxima
B
Question 15: d/dx log₃ x = ?
  1. 1/x1n3
  2. 1/x1na
  3. 1/x
  4. 1n3/x
A
Question 16: if y = e¹⁰x then ys = ?
  1. None of these
  2. 100 e¹⁰x
  3. e¹⁰x
  4. 100000e¹⁰x
D
Question 17: d/dx xⁿ = nxⁿ⁻¹ where ?
  1. n є C
  2. n є R
  3. n є Q
  4. n є Q’
B
Question 18: d/dx sec⁻¹ (x/a)= ?
  1. a/x√x² - a²
  2. a/x√a² - x²
  3. None of these
  4. x/a√x² - a²
A
Question 19: da³/dx = ? where a is a constant ?
  1. 2a
  2. None
  3. 0
  4. a
C
Question 20: Function f(x) = sin x for domain (- π, π) is increasing in interval -------- ?
  1. [- π/2, π/2]
  2. [- π, π/2]
  3. [- π/2, π]
  4. [π/2, π]
A

12th Class Math MCQs Chapter 3 Integration Punjab Board


Question 1: ∫ 2x (ln2) dx = ?
  1. 2x /ln 2 + c
  2. 1/2x ln2
  3. ln 2/2x + c
  4. 2x+ c
D
Question 2: ∫ e x[Co sec⁻¹ x – 1/x√x² - 1]dx = ?
  1. e x Cosec⁻¹ x + c
  2. Cosec⁻¹ x + c
  3. Cosec⁻¹/e⁺ x + c
  4. e x/ Cosec⁻¹ x + c
A
Question 3: Order of differential equation x² (2y + 1) dy/dx – 1 = 0 is ?
  1. 3rd
  2. none
  3. 1st
  4. 2nd
C
Question 4: 3∫₀ dx/x² + 9 = ?
  1. π/6
  2. π/16
  3. π/12
  4. π/4
C
Question 5: The solution of differential equations x dy/dx = 1 + y is ?
  1. None
  2. y = cx – 1
  3. y = x – 1
  4. y = cx + 1
B
Question 6: The definite integral b∫af(x) dx represents the --------- above the x-axis and under the curve y = f(x) from x = a to x = b ?
  1. Area
  2. integral
  3. Function
  4. None
A
Question 7: If y = 1nx, then dy is equal to ?
  1. 1/x dx
  2. 1/x . 1/1n a
  3. xdx
  4. 1/x
A
Question 8: ∫ cosec²x/cotx dx = ?
  1. -ln(Cotx) + c
  2. -cotx + c
  3. ln |cotx| + c
  4. None of these
A
Question 9: ∫xex dx = ?
  1. ex + x + c
  2. Xex + ex + c
  3. xex + c
  4. xex – ex + c
D
Question 10: If ²∫₁(3x² + 2x + k) dx = 12 then k = ------- ?
  1. 0
  2. -1
  3. 2
  4. 1
A
Question 11: The conditions through which the orbitrary constants involving in the differential equation can be determined is called ?
  1. Initial value conditions
  2. Final value condition
  3. n-initial value conditions
  4. None
A
Question 12: To evaluate ∫ 1/√a²+x² dx, the best substitution is ?
  1. x = sin θ
  2. X = a cos θ
  3. x = a sec θ
  4. x = a tan θ
D
Question 13: ∫ex (1/x – 1/x²) dx = ----------- ?
  1. none
  2. lnx + c
  3. exl/x + c
  4. -ex l/x² + c
C
Question 14: ∫ x⁻¹ dx = ?
  1. ln|x| + c
  2. ln(lnx) + c
  3. l/x²
  4. x⁰/0
A
Question 15: ∫ dx/√a² - x² dx = -------- ?
  1. sin⁻¹ a/x + c
  2. cos⁻¹ x/c + c
  3. sin⁻¹ x/a + c
  4. sin⁻¹x + c
C
Question 16: ------ is inverse process of differentiation ?
  1. Limit
  2. Linear programming
  3. integration
  4. none
C
Question 17: ∞∫∞ dx/1 + x² = ?
  1. π/3
  2. π/4
  3. π/6
  4. π
D
Question 18: If ¹∫₋₂f(x)dx = 5,³∫₁f(x) dx = 3, then ³∫₋₂f(x)dx = ------- ?
  1. 8
  2. 5
  3. 3
  4. 1
A
Question 19: If a < c < b b∫af (x) dx = c∫af(x) dx + -------- ?
  1. c∫bf(x) dx
  2. b∫af(x)dx
  3. C
  4. b∫c f(x)dx
D
Question 20: ∫ sec5x tan5x dx = ?
  1. tan 5x/5 +c
  2. 5sec5x
  3. 1/5 sec 5x + c
  4. secx/5 +c
C

12th Class Math MCQs Chapter 4 Introduction to Analytical Geometry Punjab Board


Question 1: If the lines 3x + ky + 5 = 0 and 2x – y + 5 are parallel then value of k is ?
  1. 2/3
  2. -2/3
  3. 3/2
  4. -3/2
D
Question 2: Slope of y-axis is ?
  1. -1
  2. 1
  3. Undefined
  4. 0
C
Question 3: Equation of line through origin and parallel to x/a + y/b = 1 is ?
  1. x/a + y/b = 0
  2. x/a + y/b = 2
  3. x/b + y/a = 0
  4. x/a + y/b = 1
A
Question 4: Every homogenous equation of 2nd degree ax² + 2hxy + by² = 0 represents two straight lines ------- ?
  1. two parallel lines
  2. none of these
  3. Through origin
  4. not through origin
C
Question 5: If the distance of the point (5, b) from x – axis is 3 then b =---------- ?
  1. b – 3
  2. 5
  3. 3
  4. 0
C
Question 6: Two lines xy + x² + y² = 0 are -------- ?
  1. Real & distinct
  2. none of these
  3. imaginary
  4. real and coincident
C
Question 7: If α is the Inclination of a non-horizontal line than ?
  1. 0 ≤ α ≤ 180°
  2. 0 < α < 180°
  3. - 90° < α < 90°
  4. - 90° ≤ α ≤ 90°
B
Question 8: The point of concurrency of the medians of a triangle is called---------- ?
  1. Centroid
  2. circum centre
  3. orthocenter
  4. in centre
A
Question 9: The symmetric form of equation of line through point (x₁, y₁) and having Inclination α is ?
  1. x - x₁/Cosα = y - y₁/Sinα
  2. x + x₁/Sinα = y - y₁/Cosα
  3. x - x₁/Sinα = y - y₁/Cosα
  4. x + x₁/Cosα = y - y₁/Sinα
A
Question 10: If (2,2) is the mid point of the segment joining points (4,6) and (a,b) then ?
  1. a = 2, b = 1
  2. a = 0, b = -2
  3. A = 0, b = 0
  4. a = -2, b = 0
B
Question 11: In translation of axes, origin is shifted but axes remain --------- to the old axes ?
  1. Co-planer
  2. Perpendicular
  3. Parallel
  4. Collinear
C
Question 12: If slope of l₁ ⊥ to l₂, then slope of l₂ is ?
  1. -1
  2. 0
  3. -2
  4. 2
D
Question 13: The distance of point (1/2, -√3/2) from origin ?
  1. √3/2
  2. √3/4
  3. 1/2
  4. 1
D
Question 14: The point P(x₁, y₁) lies ---- the line ax + by + c = 0 if ax₁ + by₁ + c < 0 ?
  1. below
  2. Above
  3. none of these
  4. on
A
Question 15: If m is slope of line passing through origin then its equation is ---------- ?
  1. y = mx + c
  2. Y = mx
  3. Y=x
  4. y = - x
B
Question 16: General form of equation of line is---------- ?
  1. ax + y = c
  2. none
  3. ax + by + c = 0
  4. X + y + c = 0
C
Question 17: If a ≠ 0, b ≠ 0 then slope of line ax + by + c = 0 is ?
  1. - b/c
  2. c/b
  3. b/a
  4. - a/b
D
Question 18: The two lines a₁x + b₁y + c₁ = 0 & a₂x + b₂y + c₂ = 0 are parallel if ?
  1. a₁/a₂ = b₁/b₂
  2. b₁/c₂ = b₂/c₂
  3. a₁/c₁ = a₂/c₂
  4. a₁/b₁ = -a₂/b₂
A
Question 19: If (3, 1) & (5, 1) are the end points of diameter of a circle then coordinates of its center are -------- ?
  1. (0, 0)
  2. (-2, 1)
  3. (4, 1)
  4. (8, 2)
C
Question 20: The points A (7, 9), B (3, -7) and c (-3, 3) are the vertices of ?
  1. A right angle isosceles triangle
  2. An equilateral triangle
  3. An isosceles triangle
  4. A right triangle
A

12th Class Math MCQs Chapter 5 Math Online Test Linear Inequalities and Linear Programming Punjab Board


Question 1: Most general form of 2nd degree ax² + 2hxy + by² + 2gx + 2fy + c = 0 form a circle if ------ ?
  1. h² - ab < 0
  2. h² - ab = 0
  3. none of these
  4. h² - ab > 0
A
Question 2: The equation of circle whose center is g & f radius is ---- ?
  1. √g² - f² - c
  2. √g² + f² + c
  3. √g² + f² - c
  4. √g² - f² + c
C
Question 3: The equation of conjugate axis of x²/16 -y²/9 = 1 is ?
  1. Y = 0
  2. y = 9
  3. x = 16
  4. x = 0
D
Question 4: If a plane passes through the vertex of cone then the intersection is ?
  1. A point circle
  2. Parabola
  3. Ellipse
  4. Hyperbola
A
Question 5: The points where the hyperbola meets the focal axis are called its ?
  1. co-vertices
  2. Foci
  3. vertices
  4. Directrices
C
Question 6: The eccentricity of ellipse x²/16 + y²/9 = 1 is ?
  1. 16
  2. 9
  3. 7/4
  4. √7/4
D
Question 7: The equation of asymptotes of x²/16 - y²/25 = 1 are ?
  1. y = ± 5/4 x
  2. y = ± x
  3. Y = ± 4/5 x
  4. y = ± 3/5 x
A
Question 8: The focus of parabola y² = -8x is ?
  1. (0, 2)
  2. (2, 0)
  3. (0, 0)
  4. (-2, 0)
D
Question 9: If e = o, then conic is called -------- ?
  1. Circle
  2. Hyperbola
  3. Parabola
  4. Ellipse
A
Question 10: The line through focus and perpendicular to directrix of parabola is called ?
  1. Vertex Parabola
  2. Latusrectum Parabola
  3. Axis of Parabola
  4. None of these
C
Question 11: If r = 0 then x² + y² = r² is called ----------- circle ?
  1. large
  2. small
  3. Point
  4. Zero
C
Question 12: The end points of major axis of ellipse are called ?
  1. Co-vertices
  2. Foci
  3. Vertices
  4. Directrices
C
Question 13: The vertex of parabola x² - 4x – 8y + 4 = 0 is ?
  1. (-2, 1)
  2. (2, 0)
  3. (1, -2)
  4. (-1, 2)
B
Question 14: Ax² + By² + Gx + Fy + C = 0 form ellipse if ?
  1. None of these
  2. A ≠ B, both have same signs
  3. A ≠ B, both have opposite signs
  4. A = B
B
Question 15: Ax ² + By² + Gx + Fy + C = 0 form hyperbola if ------ ?
  1. None of these
  2. A ≠ B, both have opposite signs
  3. A ≠ B, both have same signs
  4. A = B
B
Question 16: The condition that line y = mx + c may touch the circle x² + y² = a² is ?
  1. c = ±m√1 - a²
  2. c = ±a√1 + m²
  3. c = ±m√1 + a²
  4. c = ±a√1 - m²
B
Question 17: The centre of the circle (x - 5)² + (y - 3)² = 8 is ?
  1. (-5, 3)
  2. (5, 3)
  3. (-5, -3)
  4. (5, -3)
B
Question 18: The parabola y² = 4ax passes through the point (2, 3) then the value of “a” is ?
  1. -1/3
  2. 9/8
  3. 8/9
  4. 1/3
B
Question 19: Then length of latusrectum of ellipse x²/9 + y²/16 is ?
  1. 9/2
  2. 2/9
  3. 1/9
  4. 9
A
Question 20: The length of latusrectum of ellipse x²/a² + y²/b² = 1 is ?
  1. 2ab
  2. 2b²/a
  3. a²/2b
  4. a/2b²
B

12th Class Math MCQs Chapter 6 Math Online Test Conic Sections Punjab Board


Question 1: If θ is angle between two vectors u & v then u . v ≥ 0 only for ?
  1. π/2 ≤ θ ≤ π
  2. O ≤ θ ≤ π
  3. O ≤ θ ≤ π/2
  4. O < θ < π/2
C
Question 2: The work done by a force 2i – i – k in moving on object through a displacement 3i + 2i – 5k is ------- ?
  1. 5
  2. 9
  3. 7
  4. 3
B
Question 3: If ⊻ = I – 3 j + 4k and w = ai + 9 j – 12k are parallel then the value of a is ?
  1. 1
  2. 3
  3. -1
  4. -3
D
Question 4: If the vectors 2i + α j + 5k and 3i + j + α k are perpendicular then value of α is ?
  1. None of these
  2. 0
  3. -1
  4. 1
C
Question 5: Explicit form of y - x² - 2x + 1 = 0 is -------- ?
  1. Y = x² + 2x – 1
  2. 2x + x² = y + 1
  3. y – x² = 2x – 1
  4. None
A
Question 6: If 4i + 0i – 3k be a vector them cos γ = ----------- ?
  1. 1/5
  2. 2/5
  3. 4/5
  4. -3/5
D
Question 7: If 2 and 2 are x and y-components of a vector then its angle with x-axis ?
  1. 45°
  2. 60°
  3. 30°
  4. 90°
D
Question 8: The norm of non zero vector u is --------- ?
  1. |u| < 0
  2. |u| ≥ 0
  3. |u|= 0
  4. |u| > 0
D
Question 9: The vectors u 2i + 3 j + k and w = - 6i – 9 j – 3 k are ?
  1. Parallel but have opposite direction
  2. Perpendicular
  3. Neither parallel nor perpendicular
  4. Parallel and have same direction
A
Question 10: The projection of – 2i + 3 j + 7k on 2 j + k ?
  1. 13/5
  2. 13
  3. 13/4
  4. 13/√5
A
Question 11: If u . i = -2, u . i = 3 then u = ---------- ?
  1. 3i – 2i
  2. -2i + 3i
  3. 2i – 3i
  4. 3i + 2i
B
Question 12: A vector parallel and perpendicular to every vector is -------- ?
  1. negative vector
  2. zero vector
  3. unit
  4. Position vector
B
Question 13: Two vectors u & v, written as pu + qv = 0 are collinear if ?
  1. p = 0 , q ≠ 0
  2. p = 0 = q
  3. p ≠ 0 , q = 0
  4. p ≠ 0 , q ≠ 0
D
Question 14: A vector whose magnitude is 2 and is in the direction of 5i + 3i + 2k is ?
  1. 10i/√38 + 6j/√38+4k/√38
  2. 5/√38 i + 3/√38 i 2/√38 k
  3. 10i + 6i + 4k
  4. -10i – 6i -4k
A
Question 15: A unit vector in the direction of ⊻ = [3, -4] is ?
  1. [3/5, +4/5]
  2. [3/5, -4/5]
  3. none of these
  4. [-3/5, 4/5]
B
Question 16: If the magnitude of dot product and cross product of two vectors are 6√3 and 6 respectively then the angle between vectors is ?
  1. 60°
  2. 90°
  3. 30°
  4. 45°
A
Question 17: If P = (2, 3) and Q = (6 , -2), then PQ = ?
  1. -4i + 5j
  2. 4i – 5 j
  3. 4i + 5j
  4. 5i - 4j
B
Question 18: (i - i) . (i - k) x (k - i) = ------- ?
  1. -1
  2. 2
  3. 1
  4. 0
D
Question 19: Angle between √3 i + j with x-axis is ?
  1. 90°
  2. 45°
  3. 30°
  4. 60°
C
Question 20: If A, B, C, D are vertices of parallelepiped, its volume = ------- ?
  1. [AB→BC→ CD→]
  • [AB→AC→ BD→]
  • [BC→BD→ AD→]
  • [AB→AC→ AD→]

  • D

    12th Class Math MCQs Chapter 7 Math Online Conic Vectors Punjab Board


    Question 1: Explicit form of y - x² - 2x + 1 = 0 is -------- ?
    1. 2x + x² = y + 1
    2. Y = x² + 2x – 1
    3. y – x² = 2x – 1
    4. None
    B
    Question 2: A vector whose magnitude is 2 and is in the direction of 5i + 3i + 2k is ?
    1. 5/√38 i + 3/√38 i 2/√38 k
    2. -10i – 6i -4k
    3. 10i + 6i + 4k
    4. 10i/√38 + 6j/√38+4k/√38
    D
    Question 3: Angle between √3 i + j with x-axis is ?
    1. 60°
    2. 30°
    3. 90°
    4. 45°
    B
    Question 4: If 2 and 2 are x and y-components of a vector then its angle with x-axis ?
    1. 45°
    2. 60°
    3. 90°
    4. 30°
    C
    Question 5: The projection of – 2i + 3 j + 7k on 2 j + k ?
    1. 13/5
    2. 13/4
    3. 13
    4. 13/√5
    A
    Question 6: The work done by a force 2i – i – k in moving on object through a displacement 3i + 2i – 5k is -------- ?
    1. 9
    2. 3
    3. 7
    4. 5
    A
    Question 7: If A, B, C, D are vertices of parallelepiped, its volume = ------- ?
    1. [AB→AC→ AD→]
    2. [AB→BC→ CD→]
    3. [AB→AC→ BD→]
    4. [BC→BD→ AD→]
    A
    Question 8: If θ is angle between two vectors u & v then u . v ≥ 0 only for ?
    1. O < θ < π/2
    2. O ≤ θ ≤ π/2
    3. π/2 ≤ θ ≤ π
    4. O ≤ θ ≤ π
    B
    Question 9: Two vectors u & v, written as pu + qv = 0 are collinear if ?
    1. p = 0 = q
    2. p ≠ 0 , q = 0
    3. p ≠ 0 , q ≠ 0
    4. p = 0 , q ≠ 0
    C
    Question 10: If ⊻ = I – 3 j + 4k and w = ai + 9 j – 12k are parallel then the value of a is ?
    1. 1
    2. -1
    3. -3
    4. 3
    C
    Question 11: If u . i = -2, u . i = 3 then u = ---------- ?
    1. 3i + 2i
    2. -2i + 3i
    3. 2i – 3i
    4. 3i – 2i
    B
    Question 12: (i - i) . (i - k) x (k - i) = ------- ?
    1. 0
    2. -1
    3. 1
    4. 2
    A
    Question 13: A vector parallel and perpendicular to every vector is -------- ?
    1. zero vector
    2. unit
    3. Position vector
    4. negative vector
    A
    Question 14: If the vectors 2i + α j + 5k and 3i + j + α k are perpendicular then value of α is ?
    1. 0
    2. -1
    3. 1
    4. None of these
    B
    Question 15: The norm of non zero vector u is --------- ?
    1. |u|= 0
    2. |u| > 0
    3. |u| ≥ 0
    4. |u| < 0
    B
    Question 16: A unit vector in the direction of ⊻ = [3, -4] is ?
    1. [3/5, +4/5]
    2. [-3/5, 4/5]
    3. none of these
    4. [3/5, -4/5]
    D
    Question 17: The vectors u 2i + 3 j + k and w = - 6i – 9 j – 3 k are ?
    1. Perpendicular
    2. Neither parallel nor perpendicular
    3. Parallel and have same direction
    4. Parallel but have opposite direction
    D
    Question 18: If 4i + 0i – 3k be a vector them cos γ = ---------- ?
    1. 4/5
    2. 1/5
    3. 2/5
    4. -3/5
    D
    Question 19: If the magnitude of dot product and cross product of two vectors are 6√3 and 6 respectively then the angle between vectors is ?
    1. 45°
    2. 90°
    3. 30°
    4. 60°
    D
    Question 20: If P = (2, 3) and Q = (6 , -2), then PQ = ?
    1. 4i + 5j
    2. 4i – 5 j
    3. -4i + 5j
    4. 5i - 4j
    B